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Applied Optics

Applied Optics


  • Vol. 37, Iss. 13 — May. 1, 1998
  • pp: 2615–2623

Generalized four-flux radiative transfer model

William E. Vargas  »View Author Affiliations

Applied Optics, Vol. 37, Issue 13, pp. 2615-2623 (1998)

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General solutions for a four-flux radiative transfer model, derived from the radiative transfer equation and based on Lorenz–Mie scattering and absorption parameters, have been obtained. Forward and backward average path-length parameters have been considered as well as forward-scattering ratios for diffuse anisotropic radiation going into the forward and the backward hemispheres. The reported solutions are generalizations of those obtained by Maheu et al. [Appl. Opt. 23, 3353–3362 (1984)]. Compared with the generalized solutions, numerical calculations indicate that the δ-Eddington approximation and the standard four-flux model of Maheu et al. overestimate the collimated–diffuse reflectance of particulate coatings, whereas these models give similar results in the case of collimated–diffuse transmittance.

© 1998 Optical Society of America

Original Manuscript: August 25, 1997
Revised Manuscript: December 17, 1997
Published: May 1, 1998

William E. Vargas, "Generalized four-flux radiative transfer model," Appl. Opt. 37, 2615-2623 (1998)

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  1. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).
  2. J. I. Frankel, “Computational attributes of the integral form of the equation of transfer,” J. Quant. Spectrosc. Radiat. Transfer 46, 329–342 (1991). [CrossRef]
  3. W. Hartel, “Zur Theorie der Lichtstreuung durch trübe Schichten besonders Trübgläser,” Licht 10, 141–143, 165, 190, 191, 214, 215, 232–234 (1940).
  4. A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905). [CrossRef]
  5. P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).
  6. J. Reichman, “Determination of absorption and scattering coefficients for nonhomogeneous media. 1: Theory,” Appl. Opt. 12, 1811–1815 (1973). [CrossRef] [PubMed]
  7. P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971). [CrossRef] [PubMed]
  8. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  9. C. M. Chu, S. W. Churchill, “Numerical solution of problems in multiple scattering of electromagnetic radiation,” Multiple Scatt. Electromag. Radiation 59, 855–863 (1955).
  10. B. Maheu, J. N. Lotoulouzan, G. Gouesbet, “Four-flux models to solve the scattering transfer equation in terms of Lorenz–Mie parameters,” Appl. Opt. 23, 3353–3362 (1984). [CrossRef] [PubMed]
  11. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  12. 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, eds. (Gordon & Breach, New York, 1967), pp. 765–785.
  13. W. E. Vargas, G. A. Niklasson, “Forward scattering ratios and average path-length parameters in radiative transfer models,” J. Phys. Condens. Matter 9, 9083–9096 (1997). [CrossRef]
  14. W. E. Vargas, G. A. Niklasson, “Average path-length parameter in radiative transfer models,” Appl. Opt. 36, 3735–3738 (1997). [CrossRef] [PubMed]
  15. W. E. Vargas, G. A. Niklasson, “Generalized method for evaluating scattering parameters used in radiative transfer models,” J. Opt. Soc. Am. A 14, 2243–2252 (1997). [CrossRef]
  16. W. E. Vargas, G. A. Niklasson, “Intensity of diffuse radiation in particulate media,” J. Opt. Soc. Am. A 14, 2253–2262 (1997). [CrossRef]
  17. K. Klier, “Absorption and scattering in plane parallel turbid media,” J. Opt. Soc. Am. 62, 882–885 (1972). [CrossRef]
  18. G. V. Efimov, W. von Waldenfels, R. Wehrse, “Analytical solution of the nondiscretized radiative transfer equation for a slab of finite optical depth,” J. Quant. Spectrosc. Radiat. Transfer 53, 59–74 (1995). [CrossRef]
  19. T. Kunitomo, Y. Tsuboi, S. Iwashita, H. M. Shafey, “Theoretical study on spectrally selective paint coatings,” in Proceedings of the Eighth Biennial Congress International Solar Energy Society, Vol. 3, S. V. Szokolay, ed. (Pergamon, Oxford, UK, 1983), pp. 1943–1947.
  20. S. Ito, “Optical wave propagation in discrete random media with large particles: a treatment of the phase function,” Appl. Opt. 32, 1652–1656 (1993). [CrossRef] [PubMed]
  21. F. B. Yurevich, L. A. Konyukh, “Radiation attenuation by disperse media,” Int. J. Heat Mass Transfer 18, 819–829 (1975). [CrossRef]
  22. E. P. Shettle, J. A. Weinman, “The transfer of solar irradiance through inhomogeneous turbid atmospheres evaluated by Eddington’s approximation,” J. Atmos. Sci. 27, 1048–1055 (1970). [CrossRef]
  23. J. H. Joseph, W. J. Wiscombe, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976). [CrossRef]
  24. F. Liu, J. Swithenbank, E. S. Garbett, “The boundary condition of the PN-approximation used to solve the radiative transfer equation,” Int. J. Heat Mass Transfer 35, 2043–2052 (1992). [CrossRef]
  25. S. Karanjai, M. Talukder, “Solution of the equation of transfer with general phase function by a modified spherical-harmonic method,” Astrophys. Space Sci. 197, 309–336 (1992). [CrossRef]
  26. W. E. Meador, W. R. Weaver, “Two-stream approximations to radiative transfer in planetary atmospheres: a unified description of existing methods and a new improvement,” J. Atmos. Sci. 37, 630–643 (1980). [CrossRef]
  27. Harshvardhan, M. D. King, “Comparative accuracy of diffuse radiative properties computed using selected multiple scattering approximations,” J. Atmos. Sci. 50, 247–259 (1993). [CrossRef]
  28. G. A. Niklasson, T. S. Eriksson, “Radiative cooling with pigmented polyethylene foils,” in Optical Materials Technology for Energy Efficiency and Solar Energy Conversion VII, C. Grandqvist, C. M. Lampert, eds., Proc. SPIE1016, 89–99 (1988). [CrossRef]
  29. W. E. Vargas, G. A. Niklasson, “Pigment mass density and refractive index determination from optical measurements,” J. Phys. Condens. Matter 9, 1661–1670 (1997). [CrossRef]
  30. L. Fukshansky, N. Fukshansky-Kazarinova, A. M. Remisowsky, “Estimation of optical parameters in a living tissue by solving the inverse problem of the multiflux radiative transfer,” Appl. Opt. 30, 3145–3153 (1991). [CrossRef] [PubMed]
  31. H. R. Wilson, W. Eck, “Transmission variation using scattering/transparent switching films,” Sol. Energy Mat. Sol. Cells 31, 197–214 (1993). [CrossRef]
  32. W. E. Vargas, G. A. Niklasson, “Applicability conditions of the Kubelka–Munk theory,” Appl. Opt. 36, 5580–5586 (1997). [CrossRef] [PubMed]
  33. B. Maheu, G. Gouesbet, “Four-flux models to solve the scattering transfer equation: special cases,” Appl. Opt. 25, 1122–1128 (1986). [CrossRef] [PubMed]

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