Abstract

Modeling of radiation characteristics of semitransparent media containing particles or bubbles in the independent scattering limit is examined. The existing radiative properties models of a single particle in an absorbing medium using the approaches based on (1) the classical Mie theory neglecting absorption by the matrix, (2) the far field approximation, and (3) the near field approximation are reviewed. Comparison between models and experimental measurements are carried out not only for the radiation characteristics but also for hemispherical transmittance and reflectance of porous fused quartz. Large differences are found among the three models predicting the bubble radiative properties when the matrix is strongly absorbing and/or the bubbles are optically large. However, these disagreements are masked by the matrix absorption during calculation of radiation characteristics of the participating medium. It is shown that all three approaches can be used for radiative transfer calculations in an absorbing matrix containing bubbles.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  6. D. Baillis and J. F. Sacadura, 'Thermal radiation properties of dispersed media: theoretical prediction and experimental characterization,' J. Quant. Spectrosc. Radiat. Transf. 67, 327-363 (2000).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  18. Q. Fu and W. Sun, 'Mie theory for light scattering by a spherical particle in an absorbing medium,' Appl. Opt. 40, 1354-1361 (2001).
    [CrossRef]
  19. W. Sun, G. N. Loeb, and Q. Fu, 'Light scattering by coated sphere immersed in an absorbing medium: a comparison between the FDTD and analytic solutions,' J. Quant. Spectrosc. Radiat. Transf. 83, 483-492 (2004).
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    [CrossRef]
  37. M. A. Khashan and A. Y. Nassif, 'Dispersion of the optical constants of quartz and polymethyl methacrylate glasses in a wide spectral range: 0.2-3μm,' Opt. Commun. 188, 129-139 (2001).
    [CrossRef]
  38. D. Baillis and J. F. Sacadura, 'Identification of polyurethane foam radiative properties--influence of transmittance measurements number,' J. Thermophys. Heat Transfer 16, 200-206 (2002).
    [CrossRef]
  39. V. P. Nicolau, M. Raynaud, and J. F. Sacadura, 'Spectral radiative properties identification of fiber insulating materials,' Int. J. Heat Mass Transfer 37, 311-324 (1994).
    [CrossRef]

2005 (1)

2004 (4)

R. Coquard and D. Baillis, 'Radiative characteristics of opaque spherical particles beds: a new method of prediction,' J. Thermophys. Heat Transfer 18, 178-186 (2004).
[CrossRef]

D. Baillis, L. Pilon, H. Randrianalisoa, R. Gomez, and R. Viskanta, 'Measurements of radiation characteristics of fused-quartz containing bubbles,' J. Opt. Soc. Am. A 21, 149-159 (2004).
[CrossRef]

L. A. Dombrovsky, 'The propagation of infrared radiation in a semitransparent liquid containing gas bubbles,' High Temp. 42, 133-139 (2004).

W. Sun, G. N. Loeb, and Q. Fu, 'Light scattering by coated sphere immersed in an absorbing medium: a comparison between the FDTD and analytic solutions,' J. Quant. Spectrosc. Radiat. Transf. 83, 483-492 (2004).
[CrossRef]

2003 (1)

S. K. Sharma and A. R. Jones, 'Absorption and scattering of electromagnetic radiation by a large absorbing sphere with highly absorbing spherical inclusions,' J. Quant. Spectrosc. Radiat. Transf. 79-80, 1051-1060 (2003).
[CrossRef]

2002 (4)

2001 (4)

I. W. Sudiarta and P. Chylek, 'Mie-scattering formalism for spherical particle embedded in an absorbing medium,' J. Opt. Soc. Am. A 18, 1275-1278 (2001).
[CrossRef]

I. S. Sudiarta and P. Chylek, 'Mie-scattering efficiency of a large spherical particle embedded in an absorbing medium,' J. Quant. Spectrosc. Radiat. Transf. 70, 709-714 (2001).
[CrossRef]

Q. Fu and W. Sun, 'Mie theory for light scattering by a spherical particle in an absorbing medium,' Appl. Opt. 40, 1354-1361 (2001).
[CrossRef]

M. A. Khashan and A. Y. Nassif, 'Dispersion of the optical constants of quartz and polymethyl methacrylate glasses in a wide spectral range: 0.2-3μm,' Opt. Commun. 188, 129-139 (2001).
[CrossRef]

2000 (2)

D. Baillis and J. F. Sacadura, 'Thermal radiation properties of dispersed media: theoretical prediction and experimental characterization,' J. Quant. Spectrosc. Radiat. Transf. 67, 327-363 (2000).
[CrossRef]

A. G. Fedorov and R. Viskanta, 'Radiation characteristics of glass foams,' J. Am. Ceram. Soc. 83, 2769-2776 (2000).
[CrossRef]

1999 (2)

N. Lebedev, M. Gartz, U. Kreibig, and O. Stenzel, 'Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films,' Eur. Phys. J. D 6, 365-373 (1999).

N. Lebedev and O. Stenzel, 'Optical extinction of an assembly of spherical particles in an absorbing medium: application to silver clusters in absorbing organic materials,' Eur. Phys. J. D 7, 83-88 (1999).
[CrossRef]

1998 (2)

X. Zhang, M. Lewis, and B. Johnson, 'Influence of bubbles on scattering of light in the ocean,' Appl. Opt. 37, 6525-6536 (1998).
[CrossRef]

C. Z. Tan, 'Determination of refractive index of silica glass for infrared wavelengths by IR spectroscopy,' J. Non-Cryst. Solids 223, 158-163 (1998).
[CrossRef]

1994 (1)

V. P. Nicolau, M. Raynaud, and J. F. Sacadura, 'Spectral radiative properties identification of fiber insulating materials,' Int. J. Heat Mass Transfer 37, 311-324 (1994).
[CrossRef]

1991 (1)

B. P. Singh and M. Kaviany, 'Independent theory versus direct simulation of radiative transfer in packed beds,' Int. J. Heat Mass Transfer 34, 2869-2882 (1991).
[CrossRef]

1989 (1)

R. Viskanta and P. Mengüç, 'Radiative transfer in dispersed media,' Appl. Mech. Rev. 42, 241-259 (1989).
[CrossRef]

1987 (1)

A. Ungan and R. Viskanta, 'Three-dimensional numerical modeling of circulation and heat transfer in a glass melting tank: Part 1. Mathematical formulation,' Glastech. Ber. 60, 71-78 (1987).

1977 (1)

1974 (1)

1971 (1)

1965 (1)

Abramowitz, M.

M. Abramowitz, Handbook of Mathematical Functions (Dover, 1970).

Baillis, D.

L. A. Dombrovsky, J. Randrianalisoa, and D. Baillis, 'The use of Mie theory for analyzing experimental data to identify infrared properties of fused quartz containing bubbles,' Appl. Opt. 44, 7021-7031 (2005).
[CrossRef] [PubMed]

D. Baillis, L. Pilon, H. Randrianalisoa, R. Gomez, and R. Viskanta, 'Measurements of radiation characteristics of fused-quartz containing bubbles,' J. Opt. Soc. Am. A 21, 149-159 (2004).
[CrossRef]

R. Coquard and D. Baillis, 'Radiative characteristics of opaque spherical particles beds: a new method of prediction,' J. Thermophys. Heat Transfer 18, 178-186 (2004).
[CrossRef]

D. Baillis and J. F. Sacadura, 'Identification of polyurethane foam radiative properties--influence of transmittance measurements number,' J. Thermophys. Heat Transfer 16, 200-206 (2002).
[CrossRef]

D. Baillis and J. F. Sacadura, 'Thermal radiation properties of dispersed media: theoretical prediction and experimental characterization,' J. Quant. Spectrosc. Radiat. Transf. 67, 327-363 (2000).
[CrossRef]

J. Randrianalisoa, D. Baillis, and L. Pilon, 'Improved inverse method for radiative characteristics of closed-cell absorbing porous media,' J. Thermophys. Heat Transfer (to be published).

Bass, C. D.

Baum, B. A.

Beder, E. C.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Brewster, M. Q.

M. Q. Brewster, Thermal Radiative Transfer and Properties (Wiley, 1992).

Chylek, P.

Coquard, R.

R. Coquard and D. Baillis, 'Radiative characteristics of opaque spherical particles beds: a new method of prediction,' J. Thermophys. Heat Transfer 18, 178-186 (2004).
[CrossRef]

DeWitt, D. P.

Y. S. Touloukian and D. P. DeWitt, Thermal Radiative Properties: Nonmetallic Solids, Vol. 8 of Thermophysical Properties of Matter (Plenum, 1972).

Dombrovsky, L. A.

L. A. Dombrovsky, J. Randrianalisoa, and D. Baillis, 'The use of Mie theory for analyzing experimental data to identify infrared properties of fused quartz containing bubbles,' Appl. Opt. 44, 7021-7031 (2005).
[CrossRef] [PubMed]

L. A. Dombrovsky, 'The propagation of infrared radiation in a semitransparent liquid containing gas bubbles,' High Temp. 42, 133-139 (2004).

L. A. Dombrovsky, Radiation Heat Transfer in Disperse Systems (Begell, 1996).

Fedorov, A. G.

A. G. Fedorov and L. Pilon, 'Glass foam: formation, transport properties, and heat, mass, and radiation transfer,' J. Non-Cryst. Solids 311, 154-173 (2002).

A. G. Fedorov and R. Viskanta, 'Radiation characteristics of glass foams,' J. Am. Ceram. Soc. 83, 2769-2776 (2000).
[CrossRef]

Fu, Q.

Gao, B.-C.

Gartz, M.

N. Lebedev, M. Gartz, U. Kreibig, and O. Stenzel, 'Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films,' Eur. Phys. J. D 6, 365-373 (1999).

Gomez, R.

Hottel, H. C.

H. C. Hottel and A. F. Sarofim, Radiative Transfer (McGraw-Hill, 1967).

Hu, Y. X.

Huang, H.-L.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Johnson, B.

Jones, A. R.

S. K. Sharma and A. R. Jones, 'Absorption and scattering of electromagnetic radiation by a large absorbing sphere with highly absorbing spherical inclusions,' J. Quant. Spectrosc. Radiat. Transf. 79-80, 1051-1060 (2003).
[CrossRef]

Kaviany, M.

B. P. Singh and M. Kaviany, 'Independent theory versus direct simulation of radiative transfer in packed beds,' Int. J. Heat Mass Transfer 34, 2869-2882 (1991).
[CrossRef]

Khashan, M. A.

M. A. Khashan and A. Y. Nassif, 'Dispersion of the optical constants of quartz and polymethyl methacrylate glasses in a wide spectral range: 0.2-3μm,' Opt. Commun. 188, 129-139 (2001).
[CrossRef]

Kreibig, U.

N. Lebedev, M. Gartz, U. Kreibig, and O. Stenzel, 'Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films,' Eur. Phys. J. D 6, 365-373 (1999).

Lebedev, N.

N. Lebedev, M. Gartz, U. Kreibig, and O. Stenzel, 'Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films,' Eur. Phys. J. D 6, 365-373 (1999).

N. Lebedev and O. Stenzel, 'Optical extinction of an assembly of spherical particles in an absorbing medium: application to silver clusters in absorbing organic materials,' Eur. Phys. J. D 7, 83-88 (1999).
[CrossRef]

Lewis, M.

Loeb, G. N.

W. Sun, G. N. Loeb, and Q. Fu, 'Light scattering by coated sphere immersed in an absorbing medium: a comparison between the FDTD and analytic solutions,' J. Quant. Spectrosc. Radiat. Transf. 83, 483-492 (2004).
[CrossRef]

Loeb, N. G.

Malitson, I. H.

Marston, P. L.

P. L. Marston, 'Light scattering from bubbles in water,' in Procceedings of IEEE Oceans '89 (IEEE, 1989), pp. 1186-1193.
[CrossRef]

Mengüç, P.

R. Viskanta and P. Mengüç, 'Radiative transfer in dispersed media,' Appl. Mech. Rev. 42, 241-259 (1989).
[CrossRef]

Mishchenko, M. I.

Modest, M. F.

M. F. Modest, Radiative Heat Transfer (McGraw-Hill, 1993).

Mundy, W. C.

Nassif, A. Y.

M. A. Khashan and A. Y. Nassif, 'Dispersion of the optical constants of quartz and polymethyl methacrylate glasses in a wide spectral range: 0.2-3μm,' Opt. Commun. 188, 129-139 (2001).
[CrossRef]

Nicolau, V. P.

V. P. Nicolau, M. Raynaud, and J. F. Sacadura, 'Spectral radiative properties identification of fiber insulating materials,' Int. J. Heat Mass Transfer 37, 311-324 (1994).
[CrossRef]

Park, S. K.

Pilon, L.

D. Baillis, L. Pilon, H. Randrianalisoa, R. Gomez, and R. Viskanta, 'Measurements of radiation characteristics of fused-quartz containing bubbles,' J. Opt. Soc. Am. A 21, 149-159 (2004).
[CrossRef]

A. G. Fedorov and L. Pilon, 'Glass foam: formation, transport properties, and heat, mass, and radiation transfer,' J. Non-Cryst. Solids 311, 154-173 (2002).

L. Pilon and R. Viskanta, 'Apparent radiation characteristics of semitransparent media containing gas bubbles,' in Proceedings of the Twelfth International Heat Transfer Conference, France (Elsevier, 2002), pp. 645-650.

J. Randrianalisoa, D. Baillis, and L. Pilon, 'Improved inverse method for radiative characteristics of closed-cell absorbing porous media,' J. Thermophys. Heat Transfer (to be published).

Platnick, S. E.

Randrianalisoa, H.

Randrianalisoa, J.

L. A. Dombrovsky, J. Randrianalisoa, and D. Baillis, 'The use of Mie theory for analyzing experimental data to identify infrared properties of fused quartz containing bubbles,' Appl. Opt. 44, 7021-7031 (2005).
[CrossRef] [PubMed]

J. Randrianalisoa, D. Baillis, and L. Pilon, 'Improved inverse method for radiative characteristics of closed-cell absorbing porous media,' J. Thermophys. Heat Transfer (to be published).

Raynaud, M.

V. P. Nicolau, M. Raynaud, and J. F. Sacadura, 'Spectral radiative properties identification of fiber insulating materials,' Int. J. Heat Mass Transfer 37, 311-324 (1994).
[CrossRef]

Roux, J. A.

Sacadura, J. F.

D. Baillis and J. F. Sacadura, 'Identification of polyurethane foam radiative properties--influence of transmittance measurements number,' J. Thermophys. Heat Transfer 16, 200-206 (2002).
[CrossRef]

D. Baillis and J. F. Sacadura, 'Thermal radiation properties of dispersed media: theoretical prediction and experimental characterization,' J. Quant. Spectrosc. Radiat. Transf. 67, 327-363 (2000).
[CrossRef]

V. P. Nicolau, M. Raynaud, and J. F. Sacadura, 'Spectral radiative properties identification of fiber insulating materials,' Int. J. Heat Mass Transfer 37, 311-324 (1994).
[CrossRef]

Sarofim, A. F.

H. C. Hottel and A. F. Sarofim, Radiative Transfer (McGraw-Hill, 1967).

Shackleford, W. L.

Sharma, S. K.

S. K. Sharma and A. R. Jones, 'Absorption and scattering of electromagnetic radiation by a large absorbing sphere with highly absorbing spherical inclusions,' J. Quant. Spectrosc. Radiat. Transf. 79-80, 1051-1060 (2003).
[CrossRef]

Singh, B. P.

B. P. Singh and M. Kaviany, 'Independent theory versus direct simulation of radiative transfer in packed beds,' Int. J. Heat Mass Transfer 34, 2869-2882 (1991).
[CrossRef]

Smith, A. M.

Stenzel, O.

N. Lebedev, M. Gartz, U. Kreibig, and O. Stenzel, 'Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films,' Eur. Phys. J. D 6, 365-373 (1999).

N. Lebedev and O. Stenzel, 'Optical extinction of an assembly of spherical particles in an absorbing medium: application to silver clusters in absorbing organic materials,' Eur. Phys. J. D 7, 83-88 (1999).
[CrossRef]

Sudiarta, I. S.

I. S. Sudiarta and P. Chylek, 'Mie-scattering efficiency of a large spherical particle embedded in an absorbing medium,' J. Quant. Spectrosc. Radiat. Transf. 70, 709-714 (2001).
[CrossRef]

Sudiarta, I. W.

Sun, W.

Tan, C. Z.

C. Z. Tan, 'Determination of refractive index of silica glass for infrared wavelengths by IR spectroscopy,' J. Non-Cryst. Solids 223, 158-163 (1998).
[CrossRef]

Touloukian, Y. S.

Y. S. Touloukian and D. P. DeWitt, Thermal Radiative Properties: Nonmetallic Solids, Vol. 8 of Thermophysical Properties of Matter (Plenum, 1972).

Tsay, S.-C.

Ungan, A.

A. Ungan and R. Viskanta, 'Three-dimensional numerical modeling of circulation and heat transfer in a glass melting tank: Part 1. Mathematical formulation,' Glastech. Ber. 60, 71-78 (1987).

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

Viskanta, R.

D. Baillis, L. Pilon, H. Randrianalisoa, R. Gomez, and R. Viskanta, 'Measurements of radiation characteristics of fused-quartz containing bubbles,' J. Opt. Soc. Am. A 21, 149-159 (2004).
[CrossRef]

A. G. Fedorov and R. Viskanta, 'Radiation characteristics of glass foams,' J. Am. Ceram. Soc. 83, 2769-2776 (2000).
[CrossRef]

R. Viskanta and P. Mengüç, 'Radiative transfer in dispersed media,' Appl. Mech. Rev. 42, 241-259 (1989).
[CrossRef]

A. Ungan and R. Viskanta, 'Three-dimensional numerical modeling of circulation and heat transfer in a glass melting tank: Part 1. Mathematical formulation,' Glastech. Ber. 60, 71-78 (1987).

L. Pilon and R. Viskanta, 'Apparent radiation characteristics of semitransparent media containing gas bubbles,' in Proceedings of the Twelfth International Heat Transfer Conference, France (Elsevier, 2002), pp. 645-650.

Winker, D. M.

Wiscombe, W. J.

Yang, P.

Zhang, X.

Appl. Mech. Rev. (1)

R. Viskanta and P. Mengüç, 'Radiative transfer in dispersed media,' Appl. Mech. Rev. 42, 241-259 (1989).
[CrossRef]

Appl. Opt. (6)

Eur. Phys. J. D (2)

N. Lebedev, M. Gartz, U. Kreibig, and O. Stenzel, 'Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films,' Eur. Phys. J. D 6, 365-373 (1999).

N. Lebedev and O. Stenzel, 'Optical extinction of an assembly of spherical particles in an absorbing medium: application to silver clusters in absorbing organic materials,' Eur. Phys. J. D 7, 83-88 (1999).
[CrossRef]

Glastech. Ber. (1)

A. Ungan and R. Viskanta, 'Three-dimensional numerical modeling of circulation and heat transfer in a glass melting tank: Part 1. Mathematical formulation,' Glastech. Ber. 60, 71-78 (1987).

High Temp. (1)

L. A. Dombrovsky, 'The propagation of infrared radiation in a semitransparent liquid containing gas bubbles,' High Temp. 42, 133-139 (2004).

Int. J. Heat Mass Transfer (2)

B. P. Singh and M. Kaviany, 'Independent theory versus direct simulation of radiative transfer in packed beds,' Int. J. Heat Mass Transfer 34, 2869-2882 (1991).
[CrossRef]

V. P. Nicolau, M. Raynaud, and J. F. Sacadura, 'Spectral radiative properties identification of fiber insulating materials,' Int. J. Heat Mass Transfer 37, 311-324 (1994).
[CrossRef]

J. Am. Ceram. Soc. (1)

A. G. Fedorov and R. Viskanta, 'Radiation characteristics of glass foams,' J. Am. Ceram. Soc. 83, 2769-2776 (2000).
[CrossRef]

J. Non-Cryst. Solids (2)

A. G. Fedorov and L. Pilon, 'Glass foam: formation, transport properties, and heat, mass, and radiation transfer,' J. Non-Cryst. Solids 311, 154-173 (2002).

C. Z. Tan, 'Determination of refractive index of silica glass for infrared wavelengths by IR spectroscopy,' J. Non-Cryst. Solids 223, 158-163 (1998).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

J. Quant. Spectrosc. Radiat. Transf. (4)

I. S. Sudiarta and P. Chylek, 'Mie-scattering efficiency of a large spherical particle embedded in an absorbing medium,' J. Quant. Spectrosc. Radiat. Transf. 70, 709-714 (2001).
[CrossRef]

W. Sun, G. N. Loeb, and Q. Fu, 'Light scattering by coated sphere immersed in an absorbing medium: a comparison between the FDTD and analytic solutions,' J. Quant. Spectrosc. Radiat. Transf. 83, 483-492 (2004).
[CrossRef]

D. Baillis and J. F. Sacadura, 'Thermal radiation properties of dispersed media: theoretical prediction and experimental characterization,' J. Quant. Spectrosc. Radiat. Transf. 67, 327-363 (2000).
[CrossRef]

S. K. Sharma and A. R. Jones, 'Absorption and scattering of electromagnetic radiation by a large absorbing sphere with highly absorbing spherical inclusions,' J. Quant. Spectrosc. Radiat. Transf. 79-80, 1051-1060 (2003).
[CrossRef]

J. Thermophys. Heat Transfer (2)

R. Coquard and D. Baillis, 'Radiative characteristics of opaque spherical particles beds: a new method of prediction,' J. Thermophys. Heat Transfer 18, 178-186 (2004).
[CrossRef]

D. Baillis and J. F. Sacadura, 'Identification of polyurethane foam radiative properties--influence of transmittance measurements number,' J. Thermophys. Heat Transfer 16, 200-206 (2002).
[CrossRef]

Opt. Commun. (1)

M. A. Khashan and A. Y. Nassif, 'Dispersion of the optical constants of quartz and polymethyl methacrylate glasses in a wide spectral range: 0.2-3μm,' Opt. Commun. 188, 129-139 (2001).
[CrossRef]

Other (11)

Y. S. Touloukian and D. P. DeWitt, Thermal Radiative Properties: Nonmetallic Solids, Vol. 8 of Thermophysical Properties of Matter (Plenum, 1972).

J. Randrianalisoa, D. Baillis, and L. Pilon, 'Improved inverse method for radiative characteristics of closed-cell absorbing porous media,' J. Thermophys. Heat Transfer (to be published).

M. F. Modest, Radiative Heat Transfer (McGraw-Hill, 1993).

H. C. Hottel and A. F. Sarofim, Radiative Transfer (McGraw-Hill, 1967).

M. Q. Brewster, Thermal Radiative Transfer and Properties (Wiley, 1992).

L. A. Dombrovsky, Radiation Heat Transfer in Disperse Systems (Begell, 1996).

H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

M. Abramowitz, Handbook of Mathematical Functions (Dover, 1970).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

L. Pilon and R. Viskanta, 'Apparent radiation characteristics of semitransparent media containing gas bubbles,' in Proceedings of the Twelfth International Heat Transfer Conference, France (Elsevier, 2002), pp. 645-650.

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Figures (25)

Fig. 1
Fig. 1

Ratio between the intensities I i and I 0 as a function of the optical radius of a matrix particle, a 0 .

Fig. 2
Fig. 2

Variation of the absorption ( α ) and scattering ( σ ) coefficients of a porous medium with λ = π μ m , x = 200 , and f v = 5 % as a function of κ 0 : solid curve, α 0 ; open symbols, results from the definition I i = γ I 0 ; solid symbols, results from the definition I i = I 0 .

Fig. 3
Fig. 3

Bubble scattering efficiency factor Q s for n 0 = 1.4 as a function of the bubble size parameter x.

Fig. 4
Fig. 4

Bubble scattering efficiency factor Q s for n 0 = 1.7 as a function of the bubble size parameter x.

Fig. 5
Fig. 5

Comparison among the FFA, NFA, and CMT scattering efficiency factors Q s for n 0 = 1.4 versus optical radius of a matrix particle, a 0 . Deviations for κ 0 = 10 3 and κ 0 = 10 5 are overlapping.

Fig. 6
Fig. 6

Comparison among the FFA, NFA, and CMT scattering efficiency factors Q s for n 0 = 1.7 versus optical radius of a matrix particle, a 0 . Deviations for κ 0 = 10 3 and κ 0 = 10 5 are overlapping.

Fig. 7
Fig. 7

Comparison among the FFA, NFA, and CMT scattering efficiency factors Q s versus optical radius of a matrix particle, a 0 . Deviations for κ 0 = 10 3 and κ 0 = 10 5 are overlapping.

Fig. 8
Fig. 8

Bubble asymmetry factor g for n 0 = 1.4 . The predictions of g by the FFA and NFA are identical for the values of κ 0 considered. For κ 0 = 0 , predictions by the FFA and NFA and the CMT are overlapping.

Fig. 9
Fig. 9

Bubble asymmetry factor g for n 0 = 1.7 . The predictions of g by the FFA and NFA are identical for the values of κ 0 considered. For κ 0 = 0 , predictions by the FFA and NFA and the CMT are overlapping.

Fig. 10
Fig. 10

Comparison among the CMT, NFA, and FFA transport extinctions β tr for n 0 = 1.4 and f v = 5 % .

Fig. 11
Fig. 11

Comparison among the CMT, NFA, and FFA transport albedos ω tr for n 0 = 1.4 and f v = 5 % .

Fig. 12
Fig. 12

Bubble normalized size distribution n ( a ) N t for N t = 212 measured bubbles.

Fig. 13
Fig. 13

Refraction index of fused quartz, n 0 , calculated from Eq. (28).

Fig. 14
Fig. 14

Absorption index of fused quartz, κ 0 , and the corresponding optical radius a 0 , with x = 2 π a 32 λ and a 32 = 0.64 mm .

Fig. 15
Fig. 15

Extinction coefficient β of porous fused quartz.

Fig. 16
Fig. 16

Single scattering albedo ω of porous fused quartz.

Fig. 17
Fig. 17

Porous fused quartz asymmetry factor g. The asymmetry factors predicted by the FFA and NFA are identical.

Fig. 18
Fig. 18

Effect of the uncertainty in the porosity measurements on the predictions of the extinction coefficient β using the FFA.

Fig. 19
Fig. 19

Effect of the uncertainty in the porosity measurements on the predictions of the single scattering albedo ω using the FFA.

Fig. 20
Fig. 20

Hemispherical transmittance T + of the 5 mm thick sample. The results from the FFA and NFA are overlapping.

Fig. 21
Fig. 21

Hemispherical reflectance T of the 5 mm thick sample. The results from the FFA and NFA are overlapping.

Fig. 22
Fig. 22

Hemispherical transmittance T + of the 9.9 mm thick sample. The results from the FFA and NFA are overlapping.

Fig. 23
Fig. 23

Hemispherical reflectance T of the 9.9 mm thick sample. The results from the FFA and NFA are overlapping.

Fig. 24
Fig. 24

Effect of the uncertainty in the porosity measurements on the predictions of the hemispherical transmittance T + using the FFA for the 9.9 mm thick sample.

Fig. 25
Fig. 25

Effect of the uncertainty in the porosity measurements on the predictions of the hemispherical reflectance T using the FFA for the 9.9 mm thick sample.

Equations (31)

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α λ = α 0 π 0 Q m a 2 n ( a ) d a = α 0 0.75 f v a 32 Q m ¯ ,
σ λ = π 0 Q s a 2 n ( a ) d a = 0.75 f v a 32 Q s ¯ ,
a 32 = 0 a 3 n ( a ) d a 0 a 2 n ( a ) d a .
Q ¯ j = 0 Q j a 2 n ( a ) d a 0 a 2 n ( a ) d a with j = s or m ,
Φ λ ( Θ ) = π σ λ 0 Q s ϕ λ ( Θ ) a 2 n ( a ) d a = 3 f v 4 σ λ a 32 0 Q s ϕ λ ( Θ ) a 2 n ( a ) d a 0 a 2 n ( a ) d a ,
g = 1 2 0 π Φ ( Θ ) cos Θ sin Θ d Θ .
Q s M ( a ) = 2 ( n 0 x ) 2 p = 1 ( 2 p + 1 ) ( a p 2 + b p 2 ) ,
Q e M ( a ) = 2 ( n 0 x ) 2 Re p = 1 ( 2 p + 1 ) ( a p + b p ) ,
Q a M = Q e M Q s M ,
ϕ M ( Θ ) = S 1 ( Θ ) 2 + S 2 ( Θ ) 2 p = 1 ( 2 p + 1 ) ( a p 2 + b p 2 ) ,
g M = 4 p = 1 { Re [ ( p 2 1 ) ( a p 1 a p * + b p 1 b p * ) p ] + ( 2 p 1 ) ( 1 1 p ) Re ( a p 1 b p 1 * ) } ( n 0 x ) 2 Q s M .
Q m M = 8 κ 0 x 3 = 4 a 0 3 ,
Q s M = 2 ,
g M = 1 0.45 ( n 0 1 ) ,
η = I 0 I i exp ( α 0 r ) ,
Q s F ( r ) = 2 η x 2 m 0 2 p = 1 ( 2 p + 1 ) ( a p 2 + b p 2 ) ,
Q e F ( r ) = 2 η x 2 m 0 2 Re p = 1 ( 2 p + 1 ) ( a p + b p ) ,
Q s F ( a ) = 2 η x 2 m 0 2 p = 1 ( 2 p + 1 ) ( a p 2 + b p 2 ) ,
Q e F ( a ) = 2 η x 2 m 0 2 Re p = 1 ( 2 p + 1 ) ( a p + b p ) ,
η = I 0 I i exp ( a 0 ) .
Q s N ( a ) = 2 I 0 Re { ρ } ρ 2 I i Re { ρ * p = 1 ( 2 p + 1 ) i [ a p 2 ξ p ( ρ ) ξ p * ( ρ ) + b p 2 ξ p ( ρ ) ξ p * ( ρ ) ] } ,
Q a N ( a ) = 2 I 0 Re { ρ } ρ 2 I i Re { ρ * p = 1 ( 2 p + 1 ) i [ ψ p ( ρ ) ψ p * ( ρ ) ψ p ( ρ ) ψ p * ( ρ ) + b p ψ p * ( ρ ) ξ p ( ρ ) + b p * ψ p ( ρ ) ξ p * ( ρ ) a p ψ p * ( ρ ) ξ p ( ρ ) a p * ψ p ( ρ ) ξ p * ( ρ ) + a p 2 ξ p ( ρ ) ξ p * ( ρ ) b p 2 ξ p ( ρ ) ξ p * ( ρ ) ] } ,
Q m N ( a ) = 4 I 0 ρ 2 I i p = 1 ( 2 p + 1 ) Im [ ψ p ( ρ ) ψ p * ( ρ ) ] ,
γ = 1 + ( a 0 1 ) exp ( a 0 ) a 0 2 .
β tr = α + σ ( 1 g ) ,
ω tr = σ ( 1 g ) β tr .
f v = 1 d p d b ,
n 0 ( λ ) = 1 + 0.696 λ 2 λ 2 0.068 2 + 0.407 λ 2 λ 2 0.114 2 + 0.897 λ 2 λ 2 9.891 2 .
κ 0 ( λ ) = 1 4 π e λ ln { [ 1 ε ( λ ) ] 4 + 4 T ( λ ) 2 [ ε ( λ ) ] 2 [ 1 ε ( λ ) ] 2 2 T ( λ ) [ ε ( λ ) ] 2 } ,
ε ( λ ) = [ 1 n 0 ( λ ) ] 2 + [ n 0 ( λ ) ] 2 [ κ 0 ( λ ) ] 2 [ 1 + n 0 ( λ ) ] 2 + [ n 0 ( λ ) ] 2 [ κ 0 ( λ ) ] 2 .
ε ( λ ) = [ 1 n 0 ( λ ) ] 2 [ 1 + n 0 ( λ ) ] 2 .

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