Abstract

The radiative transfer equation in nonemitting media is solved using a four-flux model in the case of Lorenz-Mie scatter centers embedded in a slab. The various coefficients of absorption and scattering appearing in the theory are nonphenomenological but expressed in terms of quantities available from the Lorenz-Mie framework. Formulas for the various transmittances and reflectances are established. Special cases are then discussed, and (potential or actual) applications reported.

© 1984 Optical Society of America

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References

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  1. G. Gouesbet, “Optical Sizing, with Emphasis on Simultaneous Measurement of Velocities and Sizes of Particles Embedded in Flows: a Plenary Lecture,” in Proceedings, Fifteenth International Symposium on Heat and Mass Transfer., 5–9 September 1983, Dubrovnik, Yugoslavia.
  2. G. Gouesbet, G. Gréhan, B. Maheu, “Single Scattering Characteristics of Volume Elements in Coal Clouds,” Appl. Opt. 22, 2038 (1983).
    [CrossRef] [PubMed]
  3. S. Chandrasekhar, Radiative Transfer (Oxford U.P., London, 1950).
  4. H. C. van de Hulst, Multiple Light Scattering, Tables, Formulas and Applications, Vols. 1 and 2 (Academic, New York, 1980).
  5. S. Chandrasekhar, “On the Diffuse Reflection of a Pencil of Radiation by a Plane-Parallel Atmosphere,” Proc. Natl. Acad. Sci. U.S.A. 44, 933 (1958).
    [CrossRef] [PubMed]
  6. G. B. Rybicki, “The Searchlight Problem with Isotropic Scattering,” J. Quant. Spectrosc. Radiat. Transfer 11, 827 (1971).
    [CrossRef]
  7. P. Beckett, P. J. Foster, V. Hutson, R. L. Moss, “Radiative Transfer for a Cylindrical Beam Scattered Isotropically,” J. Quant. Spectrosc. Radiat. Transfer 14, 1115 (1974).
    [CrossRef]
  8. A. L. Crosbie, R. L. Dougherty, “Two-Dimensional Isotropic Scattering in a Finite Thick Cylindrical Medium Exposed to a Laser Beam,” J. Quant. Spectrosc. Radiat. Transfer, 27, 149 (1982).
    [CrossRef]
  9. D. C. Look, H. F. Nelson, A. L. Crosbie, J. Heat Transfer 103, 127 (1981).
    [CrossRef]
  10. G. W. Kattawar, G. N. Plass, “Degree and Direction of Polarization of Multiple Scattered Light. 1: Homogeneous Cloud Layers,” Appl. Opt. 11, 2851 (1972).
    [CrossRef] [PubMed]
  11. G. N. Plass, G. W. Kattawar, “Degree and Direction of Polarization of Multiple Scattered Light. 2: Earth’s Atmosphere with Aerosols,” Appl. Opt. 11, 2866 (1972).
    [CrossRef] [PubMed]
  12. L. L. Carter, H. G. Horak, M. T. Sandford, “An Adjoint Monte Carlo Treatment of the Equations of Radiative Transfer for Polarized Light,” J. Comp. Phys. 26, 119 (1978).
    [CrossRef]
  13. R. R. Meier, J. S. Lee, D. E. Anderson, “Atmospheric Scattering of Middle UV Radiation from an Internal Source,” Appl. Opt. 17, 3216 (1978).
    [CrossRef] [PubMed]
  14. R. B. Lyons, J. Wormhoudt, J. Gruninger, “Scattering of Radiation by Particles in Low-Altitude Plumes,” in Proceedings AIAA Sixteenth Thermophysics Conference, 23–25 June, Palo Alto, Calif. (1981), paper AIAA-81-1053.
  15. P. S. Mudgett, L. W. Richards, “Multiple Scattering Calculations for Technology,” Appl. Opt. 10, 1485 (1971).
    [CrossRef] [PubMed]
  16. C. Sagan, J. B. Pollack, “Anisotropic Nonconservative Scattering and the Clouds of Venus,” J. Geophys. Res. 72, 469 (1967).
    [CrossRef]
  17. J. K. Beasley, J. T. Atkins, F. W. Billmeyer, “Scattering and Absorption of Light in Turbid Media,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, (Gordon and Breach, New York, 1967).
  18. M. Q. Brewster, C. L. Tien, “Examination of the Two-Flux Model for Radiative Transfer in Particular Systems,” Int. J. Heat Mass Transfer, 25, 1905 (1982).
    [CrossRef]
  19. J. W. Ryde, “The Scattering of Light by Turbid Media. Part I,” Proc. R. Soc. A 131, 451 (1931).
    [CrossRef]
  20. J. W. Ryde, B. S. Cooper, “The Scattering of Light by Turbid Media. Part II,” Proc. R. Soc. A 131, 464 (1931).
    [CrossRef]
  21. L. Silberstein, “The Transparency of Turbid Media,” Phil. Mag. 4, 1291 (1927).
  22. A. Schuster, “Radiation through a Foggy Atmosphere,” Astrophys. J. 21, 1 (1905).
    [CrossRef]
  23. P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeits. Tech. Phys. 11a, 593 (1931), and P. Kubelka, “New Contributions to the Optics of Intensely Light-Scattering Materials. Part I,” J. Opt. Soc. Am. 38, 448 (1948).
    [CrossRef] [PubMed]
  24. B. Maheu, G. Gouesbet, “Four-Flux Models to Solve the Scattering Transfer Equation in Terms of Lorenz-Mie Parameters: the Formalisms,” Internal Report MADO/MG/5/83/II.
  25. B. Maheu, G. Gouesbet, “Four-flux Models to Solve the Scattering Transfer Equation in Terms of Lorenz-Mie Parameters,” Internal Report MADO/MG/1/84/I.
  26. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  27. G. Kortüm, Reflectance Spectroscopy (Springer, New York, 1969).
    [CrossRef]
  28. G. Gréhan, G. Gouesbet, “Mie Theory Calculations: New Progress, with Emphasis on Particle Sizing,” Appl. Opt. 18, 3489 (1979).
    [CrossRef] [PubMed]
  29. W. Ishihama, H. Enamoto, “New Experimental Method for Studies of Dust Explosions,” Combust. Flame 21, 177 (1973).
    [CrossRef]
  30. H. C. van de Hulst, “The Spherical Albedo of a Planet Covered with a Homogeneous Cloud Layer,” Astron. Astrophys. 35, 209 (1974).
  31. M. D. King, “A Method for Determining the Single Scattering Albedo of Clouds through Observation of the Internal Scattered Radiation Field,” J. Atmos. Sci. 38, 2031 (1981).
    [CrossRef]

1983

1982

M. Q. Brewster, C. L. Tien, “Examination of the Two-Flux Model for Radiative Transfer in Particular Systems,” Int. J. Heat Mass Transfer, 25, 1905 (1982).
[CrossRef]

A. L. Crosbie, R. L. Dougherty, “Two-Dimensional Isotropic Scattering in a Finite Thick Cylindrical Medium Exposed to a Laser Beam,” J. Quant. Spectrosc. Radiat. Transfer, 27, 149 (1982).
[CrossRef]

1981

D. C. Look, H. F. Nelson, A. L. Crosbie, J. Heat Transfer 103, 127 (1981).
[CrossRef]

M. D. King, “A Method for Determining the Single Scattering Albedo of Clouds through Observation of the Internal Scattered Radiation Field,” J. Atmos. Sci. 38, 2031 (1981).
[CrossRef]

1979

1978

R. R. Meier, J. S. Lee, D. E. Anderson, “Atmospheric Scattering of Middle UV Radiation from an Internal Source,” Appl. Opt. 17, 3216 (1978).
[CrossRef] [PubMed]

L. L. Carter, H. G. Horak, M. T. Sandford, “An Adjoint Monte Carlo Treatment of the Equations of Radiative Transfer for Polarized Light,” J. Comp. Phys. 26, 119 (1978).
[CrossRef]

1974

P. Beckett, P. J. Foster, V. Hutson, R. L. Moss, “Radiative Transfer for a Cylindrical Beam Scattered Isotropically,” J. Quant. Spectrosc. Radiat. Transfer 14, 1115 (1974).
[CrossRef]

H. C. van de Hulst, “The Spherical Albedo of a Planet Covered with a Homogeneous Cloud Layer,” Astron. Astrophys. 35, 209 (1974).

1973

W. Ishihama, H. Enamoto, “New Experimental Method for Studies of Dust Explosions,” Combust. Flame 21, 177 (1973).
[CrossRef]

1972

1971

G. B. Rybicki, “The Searchlight Problem with Isotropic Scattering,” J. Quant. Spectrosc. Radiat. Transfer 11, 827 (1971).
[CrossRef]

P. S. Mudgett, L. W. Richards, “Multiple Scattering Calculations for Technology,” Appl. Opt. 10, 1485 (1971).
[CrossRef] [PubMed]

1967

C. Sagan, J. B. Pollack, “Anisotropic Nonconservative Scattering and the Clouds of Venus,” J. Geophys. Res. 72, 469 (1967).
[CrossRef]

1958

S. Chandrasekhar, “On the Diffuse Reflection of a Pencil of Radiation by a Plane-Parallel Atmosphere,” Proc. Natl. Acad. Sci. U.S.A. 44, 933 (1958).
[CrossRef] [PubMed]

1931

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeits. Tech. Phys. 11a, 593 (1931), and P. Kubelka, “New Contributions to the Optics of Intensely Light-Scattering Materials. Part I,” J. Opt. Soc. Am. 38, 448 (1948).
[CrossRef] [PubMed]

J. W. Ryde, “The Scattering of Light by Turbid Media. Part I,” Proc. R. Soc. A 131, 451 (1931).
[CrossRef]

J. W. Ryde, B. S. Cooper, “The Scattering of Light by Turbid Media. Part II,” Proc. R. Soc. A 131, 464 (1931).
[CrossRef]

1927

L. Silberstein, “The Transparency of Turbid Media,” Phil. Mag. 4, 1291 (1927).

1905

A. Schuster, “Radiation through a Foggy Atmosphere,” Astrophys. J. 21, 1 (1905).
[CrossRef]

Anderson, D. E.

Atkins, J. T.

J. K. Beasley, J. T. Atkins, F. W. Billmeyer, “Scattering and Absorption of Light in Turbid Media,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, (Gordon and Breach, New York, 1967).

Beasley, J. K.

J. K. Beasley, J. T. Atkins, F. W. Billmeyer, “Scattering and Absorption of Light in Turbid Media,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, (Gordon and Breach, New York, 1967).

Beckett, P.

P. Beckett, P. J. Foster, V. Hutson, R. L. Moss, “Radiative Transfer for a Cylindrical Beam Scattered Isotropically,” J. Quant. Spectrosc. Radiat. Transfer 14, 1115 (1974).
[CrossRef]

Billmeyer, F. W.

J. K. Beasley, J. T. Atkins, F. W. Billmeyer, “Scattering and Absorption of Light in Turbid Media,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, (Gordon and Breach, New York, 1967).

Brewster, M. Q.

M. Q. Brewster, C. L. Tien, “Examination of the Two-Flux Model for Radiative Transfer in Particular Systems,” Int. J. Heat Mass Transfer, 25, 1905 (1982).
[CrossRef]

Carter, L. L.

L. L. Carter, H. G. Horak, M. T. Sandford, “An Adjoint Monte Carlo Treatment of the Equations of Radiative Transfer for Polarized Light,” J. Comp. Phys. 26, 119 (1978).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, “On the Diffuse Reflection of a Pencil of Radiation by a Plane-Parallel Atmosphere,” Proc. Natl. Acad. Sci. U.S.A. 44, 933 (1958).
[CrossRef] [PubMed]

S. Chandrasekhar, Radiative Transfer (Oxford U.P., London, 1950).

Cooper, B. S.

J. W. Ryde, B. S. Cooper, “The Scattering of Light by Turbid Media. Part II,” Proc. R. Soc. A 131, 464 (1931).
[CrossRef]

Crosbie, A. L.

A. L. Crosbie, R. L. Dougherty, “Two-Dimensional Isotropic Scattering in a Finite Thick Cylindrical Medium Exposed to a Laser Beam,” J. Quant. Spectrosc. Radiat. Transfer, 27, 149 (1982).
[CrossRef]

D. C. Look, H. F. Nelson, A. L. Crosbie, J. Heat Transfer 103, 127 (1981).
[CrossRef]

Dougherty, R. L.

A. L. Crosbie, R. L. Dougherty, “Two-Dimensional Isotropic Scattering in a Finite Thick Cylindrical Medium Exposed to a Laser Beam,” J. Quant. Spectrosc. Radiat. Transfer, 27, 149 (1982).
[CrossRef]

Enamoto, H.

W. Ishihama, H. Enamoto, “New Experimental Method for Studies of Dust Explosions,” Combust. Flame 21, 177 (1973).
[CrossRef]

Foster, P. J.

P. Beckett, P. J. Foster, V. Hutson, R. L. Moss, “Radiative Transfer for a Cylindrical Beam Scattered Isotropically,” J. Quant. Spectrosc. Radiat. Transfer 14, 1115 (1974).
[CrossRef]

Gouesbet, G.

G. Gouesbet, G. Gréhan, B. Maheu, “Single Scattering Characteristics of Volume Elements in Coal Clouds,” Appl. Opt. 22, 2038 (1983).
[CrossRef] [PubMed]

G. Gréhan, G. Gouesbet, “Mie Theory Calculations: New Progress, with Emphasis on Particle Sizing,” Appl. Opt. 18, 3489 (1979).
[CrossRef] [PubMed]

G. Gouesbet, “Optical Sizing, with Emphasis on Simultaneous Measurement of Velocities and Sizes of Particles Embedded in Flows: a Plenary Lecture,” in Proceedings, Fifteenth International Symposium on Heat and Mass Transfer., 5–9 September 1983, Dubrovnik, Yugoslavia.

B. Maheu, G. Gouesbet, “Four-Flux Models to Solve the Scattering Transfer Equation in Terms of Lorenz-Mie Parameters: the Formalisms,” Internal Report MADO/MG/5/83/II.

B. Maheu, G. Gouesbet, “Four-flux Models to Solve the Scattering Transfer Equation in Terms of Lorenz-Mie Parameters,” Internal Report MADO/MG/1/84/I.

Gréhan, G.

Gruninger, J.

R. B. Lyons, J. Wormhoudt, J. Gruninger, “Scattering of Radiation by Particles in Low-Altitude Plumes,” in Proceedings AIAA Sixteenth Thermophysics Conference, 23–25 June, Palo Alto, Calif. (1981), paper AIAA-81-1053.

Horak, H. G.

L. L. Carter, H. G. Horak, M. T. Sandford, “An Adjoint Monte Carlo Treatment of the Equations of Radiative Transfer for Polarized Light,” J. Comp. Phys. 26, 119 (1978).
[CrossRef]

Hutson, V.

P. Beckett, P. J. Foster, V. Hutson, R. L. Moss, “Radiative Transfer for a Cylindrical Beam Scattered Isotropically,” J. Quant. Spectrosc. Radiat. Transfer 14, 1115 (1974).
[CrossRef]

Ishihama, W.

W. Ishihama, H. Enamoto, “New Experimental Method for Studies of Dust Explosions,” Combust. Flame 21, 177 (1973).
[CrossRef]

Kattawar, G. W.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

King, M. D.

M. D. King, “A Method for Determining the Single Scattering Albedo of Clouds through Observation of the Internal Scattered Radiation Field,” J. Atmos. Sci. 38, 2031 (1981).
[CrossRef]

Kortüm, G.

G. Kortüm, Reflectance Spectroscopy (Springer, New York, 1969).
[CrossRef]

Kubelka, P.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeits. Tech. Phys. 11a, 593 (1931), and P. Kubelka, “New Contributions to the Optics of Intensely Light-Scattering Materials. Part I,” J. Opt. Soc. Am. 38, 448 (1948).
[CrossRef] [PubMed]

Lee, J. S.

Look, D. C.

D. C. Look, H. F. Nelson, A. L. Crosbie, J. Heat Transfer 103, 127 (1981).
[CrossRef]

Lyons, R. B.

R. B. Lyons, J. Wormhoudt, J. Gruninger, “Scattering of Radiation by Particles in Low-Altitude Plumes,” in Proceedings AIAA Sixteenth Thermophysics Conference, 23–25 June, Palo Alto, Calif. (1981), paper AIAA-81-1053.

Maheu, B.

G. Gouesbet, G. Gréhan, B. Maheu, “Single Scattering Characteristics of Volume Elements in Coal Clouds,” Appl. Opt. 22, 2038 (1983).
[CrossRef] [PubMed]

B. Maheu, G. Gouesbet, “Four-flux Models to Solve the Scattering Transfer Equation in Terms of Lorenz-Mie Parameters,” Internal Report MADO/MG/1/84/I.

B. Maheu, G. Gouesbet, “Four-Flux Models to Solve the Scattering Transfer Equation in Terms of Lorenz-Mie Parameters: the Formalisms,” Internal Report MADO/MG/5/83/II.

Meier, R. R.

Moss, R. L.

P. Beckett, P. J. Foster, V. Hutson, R. L. Moss, “Radiative Transfer for a Cylindrical Beam Scattered Isotropically,” J. Quant. Spectrosc. Radiat. Transfer 14, 1115 (1974).
[CrossRef]

Mudgett, P. S.

Munk, F.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeits. Tech. Phys. 11a, 593 (1931), and P. Kubelka, “New Contributions to the Optics of Intensely Light-Scattering Materials. Part I,” J. Opt. Soc. Am. 38, 448 (1948).
[CrossRef] [PubMed]

Nelson, H. F.

D. C. Look, H. F. Nelson, A. L. Crosbie, J. Heat Transfer 103, 127 (1981).
[CrossRef]

Plass, G. N.

Pollack, J. B.

C. Sagan, J. B. Pollack, “Anisotropic Nonconservative Scattering and the Clouds of Venus,” J. Geophys. Res. 72, 469 (1967).
[CrossRef]

Richards, L. W.

Rybicki, G. B.

G. B. Rybicki, “The Searchlight Problem with Isotropic Scattering,” J. Quant. Spectrosc. Radiat. Transfer 11, 827 (1971).
[CrossRef]

Ryde, J. W.

J. W. Ryde, “The Scattering of Light by Turbid Media. Part I,” Proc. R. Soc. A 131, 451 (1931).
[CrossRef]

J. W. Ryde, B. S. Cooper, “The Scattering of Light by Turbid Media. Part II,” Proc. R. Soc. A 131, 464 (1931).
[CrossRef]

Sagan, C.

C. Sagan, J. B. Pollack, “Anisotropic Nonconservative Scattering and the Clouds of Venus,” J. Geophys. Res. 72, 469 (1967).
[CrossRef]

Sandford, M. T.

L. L. Carter, H. G. Horak, M. T. Sandford, “An Adjoint Monte Carlo Treatment of the Equations of Radiative Transfer for Polarized Light,” J. Comp. Phys. 26, 119 (1978).
[CrossRef]

Schuster, A.

A. Schuster, “Radiation through a Foggy Atmosphere,” Astrophys. J. 21, 1 (1905).
[CrossRef]

Silberstein, L.

L. Silberstein, “The Transparency of Turbid Media,” Phil. Mag. 4, 1291 (1927).

Tien, C. L.

M. Q. Brewster, C. L. Tien, “Examination of the Two-Flux Model for Radiative Transfer in Particular Systems,” Int. J. Heat Mass Transfer, 25, 1905 (1982).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, “The Spherical Albedo of a Planet Covered with a Homogeneous Cloud Layer,” Astron. Astrophys. 35, 209 (1974).

H. C. van de Hulst, Multiple Light Scattering, Tables, Formulas and Applications, Vols. 1 and 2 (Academic, New York, 1980).

Wormhoudt, J.

R. B. Lyons, J. Wormhoudt, J. Gruninger, “Scattering of Radiation by Particles in Low-Altitude Plumes,” in Proceedings AIAA Sixteenth Thermophysics Conference, 23–25 June, Palo Alto, Calif. (1981), paper AIAA-81-1053.

Appl. Opt.

Astron. Astrophys.

H. C. van de Hulst, “The Spherical Albedo of a Planet Covered with a Homogeneous Cloud Layer,” Astron. Astrophys. 35, 209 (1974).

Astrophys. J.

A. Schuster, “Radiation through a Foggy Atmosphere,” Astrophys. J. 21, 1 (1905).
[CrossRef]

Combust. Flame

W. Ishihama, H. Enamoto, “New Experimental Method for Studies of Dust Explosions,” Combust. Flame 21, 177 (1973).
[CrossRef]

Int. J. Heat Mass Transfer

M. Q. Brewster, C. L. Tien, “Examination of the Two-Flux Model for Radiative Transfer in Particular Systems,” Int. J. Heat Mass Transfer, 25, 1905 (1982).
[CrossRef]

J. Atmos. Sci.

M. D. King, “A Method for Determining the Single Scattering Albedo of Clouds through Observation of the Internal Scattered Radiation Field,” J. Atmos. Sci. 38, 2031 (1981).
[CrossRef]

J. Comp. Phys.

L. L. Carter, H. G. Horak, M. T. Sandford, “An Adjoint Monte Carlo Treatment of the Equations of Radiative Transfer for Polarized Light,” J. Comp. Phys. 26, 119 (1978).
[CrossRef]

J. Geophys. Res.

C. Sagan, J. B. Pollack, “Anisotropic Nonconservative Scattering and the Clouds of Venus,” J. Geophys. Res. 72, 469 (1967).
[CrossRef]

J. Heat Transfer

D. C. Look, H. F. Nelson, A. L. Crosbie, J. Heat Transfer 103, 127 (1981).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

G. B. Rybicki, “The Searchlight Problem with Isotropic Scattering,” J. Quant. Spectrosc. Radiat. Transfer 11, 827 (1971).
[CrossRef]

P. Beckett, P. J. Foster, V. Hutson, R. L. Moss, “Radiative Transfer for a Cylindrical Beam Scattered Isotropically,” J. Quant. Spectrosc. Radiat. Transfer 14, 1115 (1974).
[CrossRef]

A. L. Crosbie, R. L. Dougherty, “Two-Dimensional Isotropic Scattering in a Finite Thick Cylindrical Medium Exposed to a Laser Beam,” J. Quant. Spectrosc. Radiat. Transfer, 27, 149 (1982).
[CrossRef]

Phil. Mag.

L. Silberstein, “The Transparency of Turbid Media,” Phil. Mag. 4, 1291 (1927).

Proc. Natl. Acad. Sci. U.S.A.

S. Chandrasekhar, “On the Diffuse Reflection of a Pencil of Radiation by a Plane-Parallel Atmosphere,” Proc. Natl. Acad. Sci. U.S.A. 44, 933 (1958).
[CrossRef] [PubMed]

Proc. R. Soc. A

J. W. Ryde, “The Scattering of Light by Turbid Media. Part I,” Proc. R. Soc. A 131, 451 (1931).
[CrossRef]

J. W. Ryde, B. S. Cooper, “The Scattering of Light by Turbid Media. Part II,” Proc. R. Soc. A 131, 464 (1931).
[CrossRef]

Zeits. Tech. Phys.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeits. Tech. Phys. 11a, 593 (1931), and P. Kubelka, “New Contributions to the Optics of Intensely Light-Scattering Materials. Part I,” J. Opt. Soc. Am. 38, 448 (1948).
[CrossRef] [PubMed]

Other

B. Maheu, G. Gouesbet, “Four-Flux Models to Solve the Scattering Transfer Equation in Terms of Lorenz-Mie Parameters: the Formalisms,” Internal Report MADO/MG/5/83/II.

B. Maheu, G. Gouesbet, “Four-flux Models to Solve the Scattering Transfer Equation in Terms of Lorenz-Mie Parameters,” Internal Report MADO/MG/1/84/I.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

G. Kortüm, Reflectance Spectroscopy (Springer, New York, 1969).
[CrossRef]

G. Gouesbet, “Optical Sizing, with Emphasis on Simultaneous Measurement of Velocities and Sizes of Particles Embedded in Flows: a Plenary Lecture,” in Proceedings, Fifteenth International Symposium on Heat and Mass Transfer., 5–9 September 1983, Dubrovnik, Yugoslavia.

R. B. Lyons, J. Wormhoudt, J. Gruninger, “Scattering of Radiation by Particles in Low-Altitude Plumes,” in Proceedings AIAA Sixteenth Thermophysics Conference, 23–25 June, Palo Alto, Calif. (1981), paper AIAA-81-1053.

J. K. Beasley, J. T. Atkins, F. W. Billmeyer, “Scattering and Absorption of Light in Turbid Media,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, (Gordon and Breach, New York, 1967).

S. Chandrasekhar, Radiative Transfer (Oxford U.P., London, 1950).

H. C. van de Hulst, Multiple Light Scattering, Tables, Formulas and Applications, Vols. 1 and 2 (Academic, New York, 1980).

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Figures (9)

Fig. 1
Fig. 1

Geometry.

Fig. 2
Fig. 2

Sketch of the boundary conditions at Z for collimated radiation.

Fig. 3
Fig. 3

Sketch of the boundary conditions at Q for collimaited radiation.

Fig. 4
Fig. 4

Sketch of the conditions at (S) to compute Ec.

Fig. 5
Fig. 5

Sketch of the conditions at the slab (O · S) for the diffuse radiation.

Fig. 6
Fig. 6

Reflectances of an opaque cloud of coal particles versus diameter of the particle at λ = 337 μm (continuous line, diffuse illumination; dashed line, collimated illumination).

Fig. 7
Fig. 7

Extinction efficiency factor of a coal particle versus diameter of the particle at a visible wavelength (λ = 0.5145 μm) for extreme values of the complex refractive index m.

Fig. 8
Fig. 8

Extinction efficiency factor of a coal particle versus diameter of the particle at IR wavelength (λ = 337 μm) for extreme values of the complex refractive index m.

Fig. 9
Fig. 9

Reflectances of an opaque cloud of coal particles vs diameter d (continuous line, present work; dashed line, van de Hulst or King).

Tables (1)

Tables Icon

Table I Number Density of Homogeneous Coal Clouds Measured by Double-Turbidity Measurements

Equations (95)

Equations on this page are rendered with MathJax. Learn more.

k = N C abs = N ( C ext - C sca ) = N C ext ( 1 - a ) ,
s = N C sca = a N C ext ,
a = C sca C ext = Q sca Q ext = s k + s ,
I c ( Z ) = ( 1 - r c ) I c Z + r c J c ( Z ) ,
I d ( Z ) = ( 1 - r d e ) I d Z + r d i J d ( Z ) .
J c ( o ) = A 6 I c ( o ) ,
J d ( o ) = A 7 I d ( o ) ,
A 6 = r c ( 1 - 2 r c b ) + r c b 1 - r c r c b ,
A 7 = r d i + r d b ( 1 - r d e - r d i ) 1 - r d b r d e .
τ = τ c c + τ c d + τ d d ,
R = R c c + R c d + R d d .
d I c d z = ( k + s ) I c ,
d J c d z = - ( k + s ) J c ,
d I d d z = ɛ k I d + ɛ ( 1 - ζ ) s I d - ɛ ( 1 - ζ ) s J d - ζ s I c - ζ s I c ,
d J d d z = - ɛ k J d - ɛ ( 1 - ζ ) s J d + ɛ ( 1 - ζ ) s I d + ( 1 - ζ ) s I c + ζ s J c .
I c = C 1 exp [ ( k + s ) z ] ,
J c = C 2 exp [ - ( k + s ) z ] .
d ( I d + J d ) d z = [ ɛ k + 2 ɛ ( 1 - ζ ) s ] ( I d - J d ) + s ( 1 - 2 ζ ) ( I c - J c ) ,
d ( I d - J d ) d z = ɛ k ( I d + J d ) - s ( I c + J c ) .
d 2 ( I d + J d ) d z 2 = [ ɛ k + 2 ɛ ( 1 - ζ ) s ] [ ɛ k ( I d + J d ) - s ( I c + J c ) ] + s ( k + s ) ( 1 - 2 ζ ) ( I c + J c ) ,
d 2 ( I d - J d ) d z 2 = ɛ k [ ɛ k + 2 ɛ ( 1 - ζ ) s ] ( I d - J d ) + ɛ k s ( 1 - 2 ζ ) ( I c - J c ) - s ( k + s ) ( I c - J c ) .
- d 2 I d d z 2 + A 1 I d = A 2 I c + A 3 J c ,
- d 2 J d d z 2 + A 1 J d = A 3 I c + A 2 J c ,
A 1 = ɛ 2 k [ k + 2 ( 1 - ζ ) s ] ,
A 2 = s [ ɛ k ζ + ɛ s ( 1 - ζ ) + ζ ( k + s ) ] ,
A 3 = s ( 1 - ζ ) ( k + s ) ( ɛ - 1 ) .
- d 2 I d d z 2 + A 1 I d = A 2 C 1 exp [ ( k + s ) z ] + A 3 C 2 exp [ - ( k + s ) z ] .
I d = C 3 exp ( A 1 · z ) + C 4 exp ( - A 1 · z ) + C 5 exp [ ( k + s ) z ] + C 6 exp [ - ( k + s ) z ] ,
C 5 = A 2 C 1 A 1 - ( k + s ) 2 ,
C 6 = A 3 C 2 A 1 - ( k + s ) 2 .
J d = C 7 exp [ A 1 · z ] + C 8 exp [ - A 1 · z ] + C 9 exp [ ( k + s ) z ] + C 10 exp [ - ( k + s ) z ] ,
C 9 = A 3 C 1 A 1 - ( k + s ) 2 ,
C 10 = A 2 C 2 A 1 - ( k + s ) 2 .
{ C 3 A 1 - ɛ C 3 [ k + ( 1 - ζ ) s ] + ɛ ( 1 - ζ ) s C 7 } exp [ A 1 · z ] + { - C 4 A 1 - ɛ C 4 [ k + ( 1 - ζ ) s ] + ɛ ( 1 - ζ ) s C 8 } exp [ - A 1 · z ] = 0 ,
C 7 = A 4 - A 1 A 5 C 3 ,
C 8 = A 4 + A 1 A 5 C 4 ,
A 4 = ɛ [ k + ( 1 - ζ ) s ] ,
A 5 = ɛ ( 1 - ζ ) s .
C 1 = ( 1 - r c ) I c Z exp [ - ( k + s ) Z ] 1 - r c A 6 exp [ - 2 ( k + s ) Z ] ,
C 2 = ( 1 - r c ) A 6 I c Z exp [ - ( k + s ) Z ] 1 - r c A 6 exp [ - 2 ( k + s ) Z ] .
C 3 exp ( A 1 · Z ) + C 4 exp ( - A 1 · Z ) + C 5 exp [ ( k + s ) Z ) + C 6 exp [ - ( k + s ) Z ] = ( 1 - r d e ) I d Z + r d i { C 7 exp ( A 1 Z ) + C 8 exp ( - A 1 Z ) + C 9 exp [ ( k + s ) Z ] + C 10 exp [ - ( k + s ) Z ] } ,
C 7 + C 8 + C 9 + C 10 = A 7 ( C 3 + C 4 + C 5 + C 6 ) .
C 3 = A 10 A 13 - A 8 A 12 A 10 A 11 - A 9 A 12 ,
C 4 = A 8 A 11 - A 9 A 13 A 10 A 11 - A 9 A 12 ,
A 8 = C 9 + C 10 - A 7 ( C 5 + C 6 ) ,
A 9 = A 7 - A 4 - A 1 A 5 ,
A 10 = A 7 - A 4 + A 1 A 5 ,
A 11 = ( 1 - r d i A 4 - A 1 A 5 ) exp ( A 1 Z ) ,
A 12 = ( 1 - r d i A 4 + A 1 A 5 ) exp ( - A 1 Z ) ,
A 13 = ( 1 - r d e ) I d Z - ( C 5 - r d i C 9 ) exp [ ( k + s ) Z ] - ( C 6 - r d i C 10 ) × exp [ - ( k + s ) Z ] .
E c = ( 1 - r c ) ( 1 - r c b ) 1 - r c r c b I c ( o ) .
I c ( o ) = J c ( o ) + E c ,
τ c c = ( 1 - r c ) ( 1 - r c b ) 1 - r c r c b τ c I c ( o ) I c Z + I d Z ,
τ c c = ( 1 - r c b ) τ c 1 - r c r c b · τ c c 0
τ c c = τ c ( 1 - r c b ) ( 1 - r c ) 2 exp [ - ( k + s ) Z ] ( 1 - r c r c b ) - r c ( r c + r c b - 2 r c r c b ) exp [ - 2 ( k + s ) Z ] · I c Z I c Z + I d Z .
I c ( o ) = I c Z exp [ - ( k + s ) Z ] .
τ d t = τ c d + τ d d .
E d = ( 1 - r d b ) ( 1 - r d i ) I d ( o ) I - r d e r d b .
I d ( o ) = J d ( o ) + E d .
τ d t = ( 1 - r d b ) ( 1 - r d i ) τ d I d ( o ) ( 1 - r d e r d b ) ( I c Z + I d Z ) ,
τ d t = ( 1 - r d b ) τ d 1 - r d e r d b τ d t 0 ,
τ d t = ( 1 - r d b ) ( 1 - r d i ) τ d ( 1 - r d e r d b ) ( I c Z + I d Z ) · [ A 10 A 13 + A 8 A 11 - A 8 A 12 - A 9 A 13 A 10 A 11 - A 9 A 12 + A 2 C 1 + A 3 C 2 A 1 - ( k + s ) 2 ] .
τ c d = ( 1 - r d b ) ( 1 - r d i ) τ d ( 1 - r d e r d b ) ( I c Z + I d Z ) ( A 10 - A 9 A 10 A 11 - A 8 A 12 { ( r d i C 9 - C 5 ) × exp [ ( k + s ) Z ] + ( r d i C 10 - C 6 ) exp [ - ( k + s ) Z ] } + A 8 ( A 11 - A 12 ) A 10 A 11 - A 9 A 12 + A 2 C 1 + A 3 C 2 A 1 - ( k + s ) 2 ) ,
τ d d = ( 1 - r d b ) ( 1 - r d i ) ( 1 - r d e ) τ d ( 1 - r d e r d b ) · A 10 - A 9 A 10 A 11 - A 9 A 12 · I d Z I c Z + I d Z .
τ c d = ( 1 - r d b ) ( 1 - r d i ) ( 1 - r c ) τ d · exp [ - ( k + s ) Z ] ( 1 - r d e r d b ) [ A 1 - ( k + s ) 2 ] { 1 - r c A 6 exp [ - 2 ( k + s ) Z ] } · I c Z I c Z + I d Z · N D ,
N = A 1 [ r d i A 3 - A 2 + A 6 ( r d i A 2 - A 3 ) ] c h ( A 1 Z ) + [ ( A 5 - r d i A 4 ) ( A 3 + A 2 A 6 ) - ( A 4 - r d i A 5 ) ( A 2 + A 3 A 6 ) ] s h ( A 1 Z ) + A 1 { ( A 2 - r d i A 3 ) exp [ ( k + s ) Z ] + A 6 ( A 3 - r d i A 2 ) × exp [ - ( k + s ) ] }
D = A 1 ( r d i A 7 - 1 ) c h ( A 1 Z ) + [ A 7 ( A 5 - r d i A 4 ) + r d i A 5 - A 4 ] s h ( A 1 Z )
τ d d = ( 1 - r d b ) ( 1 - r d i ) ( 1 - r d e ) τ d 1 - r d e r d b · A 1 { A 1 ( 1 - r d i A 7 ) c h ( A 1 Z ) + [ A 4 ( 1 + r d i A 7 ) - A 5 ( A 7 + r d i ) ] s h ( A 1 Z ) } · I d Z I c Z + I d Z .
R c c = r c I c Z + ( 1 - r c ) J c ( Z ) I c Z + I d Z .
R c c = { r c + ( 1 - r c ) 2 ( r c + r c b - 2 r c r c b ) exp [ - 2 ( k + s ) Z ] ( 1 - r c r c b ) - r c ( r c + r c b - 2 r c r c b ) exp [ - 2 ( k + s ) Z ] } · I c Z I c Z + I d Z .
R d t = R c d + R d d ,
R d t = r d e I d Z + ( 1 - r d i ) J d ( Z ) I c Z + I d Z .
R d t = r d e I d Z I c Z + I d Z + 1 - r d i I c Z + I d Z · { C 7 exp ( A 1 Z ) + C 8 exp ( - A 1 Z ) + C 9 exp [ ( k + s ) Z ] + C 10 exp [ - ( k + s ) Z ] } .
R c d = ( 1 - r d i ) I c Z + I d Z { 1 A 5 ( A 10 A 11 - A 9 A 12 [ - ( A 4 - A 1 ) · ( A 10 { ( C 5 - r d i C 9 ) · exp [ ( k + s ) Z ] + ( C 6 - r d i C 10 ) exp [ - ( k + s ) Z ] } + A 8 A 12 ) exp ( A 1 Z ) + ( A 4 + A 1 ) ( A 8 A 11 + A 9 { ( C 5 - r d i C 9 ) exp [ ( k + s ) Z ] + ( C 6 - r d i C 10 ) exp [ - ( k + s ) Z ] } ) · exp ( - A 1 Z ) ] + C 9 exp [ ( k + s ) Z ] + C 10 exp [ - ( k + s ) Z ] } ,
R d d = { r d e + ( 1 - r d i ) ( 1 - r d e ) A 5 ( A 10 A 11 - A 9 A 12 ) [ ( A 4 - A 1 ) A 10 exp ( A 1 Z ) - ( A 4 + A 1 ) A 9 exp ( - A 1 Z ) ] } · I d Z I c Z + I d Z .
R c d = ( 1 - r d i ) ( 1 - r c ) exp [ - ( k + s ) Z ] [ A 1 - ( k + s ) 2 ] { 1 - r c A 6 exp [ - 2 ( k + s ) Z ] } · I c Z I c Z + I d Z · 1 A 1 ( r d i A 7 - 1 ) c h ( A 1 Z ) + [ A 5 ( A 7 + r d i ) - A 4 ( 1 + r d i A 7 ) ] s h ( A 1 Z ) · ( A 1 [ A 3 + A 2 A 6 - A 7 ( A 2 + A 3 A 6 ) ] + { A 1 ( A 2 A 7 - A 3 ) c h ( A 1 · Z ) + [ A 2 ( A 5 - A 4 A 7 ) + A 3 ( A 5 A 7 - A 4 ) ] s h ( A 1 Z ) } exp [ ( k + s ) Z ] + A 6 { A 1 ( A 3 A 7 - A 2 ) c h ( A 1 Z ) + [ A 3 ( A 5 - A 4 A 7 ) + A 2 ( A 5 A 7 - A 4 ) ] s h ( A 1 Z ) } exp [ - ( k + s ) Z ] )
R d d = { r d e + ( 1 - r d i ) ( 1 - r d e ) [ A 7 A 1 c h ( A 1 Z ) + ( A 5 - A 4 A 7 ) s h ( A 1 Z ) ] A 1 ( 1 - r d i A 7 ) c h ( A 1 Z ) + [ A 4 ( 1 + r d i A 7 ) - A 5 ( A 7 + r d i ) ] s h ( A 1 Z ) } I d Z I c Z + I d Z .
R = R d d = { r e d e + ( 1 - r d i ) ( 1 - r d e ) [ A 7 A 1 c h ( A 1 Z ) + ( A 5 - A 4 A 7 ) s h ( A 1 Z ) ] A 1 ( 1 - r d i A 7 ) c h ( A 1 Z ) + [ A 4 ( 1 + r d i A 7 ) - A 5 ( A 7 + r d i ) ] s h ( A 1 Z ) } .
R = r d b A 1 c h ( A 1 Z ) + ( A 5 - r d b A 4 ) s h ( A 1 Z ) A 1 c h ( A 1 Z ) + ( A 4 - r d b A 5 ) s h ( A 1 Z ) .
χ = 1 - a ζ a ( 1 - ζ ) ,
R = r d b χ 2 - 1 c h ( S χ 2 - 1 Z ) + ( 1 - r d b χ ) s h ( S χ 2 - 1 Z ) χ 2 - 1 c h ( S χ 2 - 1 Z ) + ( χ - r d b ) s h ( S χ 2 - 1 Z ) ,
S = A 5 = 2 ( 1 - ζ ) s .
R = R = r d b ( χ - χ 2 - 1 ) - 1 r d b - χ - χ 2 - 1 .
R = χ - χ 2 - 1 = 1 χ + χ 2 - 1 .
R = 1 - 1 - a 1 + 1 - a .
R = A 3 A 5 - A 2 ( A 4 - A 1 ) [ A 1 - ( k + s ) 2 ] · A 5 .
R = 2 a - 3 ( 1 - 1 - a 2 a - 3 ( 2 a - 1 ) .
τ c c = exp ( - N C ext Z ) ,
L n τ c c ( V ) = - N π d 2 2 Z ,
L n τ c c ( S ) = - N f ( d ) · π d 2 4 Z ,
L n τ c c ( V ) L n τ c c ( S ) = 2 f ( d ) ,
N = - 2 L n τ c c ( V ) π d 2 Z = - 4 L n τ c c ( S ) π d 2 f ( d ) Z .
R = ( 1 + a 1 s ) ( 1 - s ) ( 1 + a 2 s ) .
s = 1 - a .
R = ( 1 - s ) ( 1 + s ) .

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