Abstract

A modified two-flux approximation is suggested for calculating the hemispherical transmittance and reflectance of a refracting, absorbing, and scattering medium in the case of collimated irradiation of the sample along the normal to the interface. The Fresnel reflection is taken into account in this approach. It is shown that the new approximation is rather accurate for the model transport scattering function. For an arbitrary scattering medium, the error of the modified two-flux approximation is estimated by comparison with the exact numerical calculations for the Henyey–Greenstein scattering function in a wide range of albedos and optical thicknesses. Possible applications of the derived analytical solution to identification problems are discussed.

© 2006 Optical Society of America

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  1. 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]
  2. J.-F. Sacadura and D. Baillis, "Experimental characterization of thermal radiation properties of disperse media," Int. J. Therm. Sci. 41, 699-707 (2002).
    [CrossRef]
  3. M. F. Modest, Radiative Heat Transfer, 2nd ed. (Academic, 2003).
  4. K. S. Adzerikho, E. F. Nogotov, and V. P. Trofimov, Radiative Heat Transfer in Two-Phase Media (CRC Press, 1992).
  5. L. A. Dombrovsky, Radiation Heat Transfer in Disperse Systems (Begell House, 1996).
  6. B. Davison, Neutron Transport Theory (Oxford U. Press, 1957).
  7. B. H. J. McKellar and M. A. Box, "The scaling group of the radiative transfer equation," J. Atmos. Sci. 38, 1063-1068 (1981).
    [CrossRef]
  8. H. Lee and R. O. Buckius, "Scaling anisotropic scattering in radiation heat transfer for a planar medium," ASME J. Heat Transfer 104, 68-75 (1982).
    [CrossRef]
  9. L. A. Dombrovsky, "Approximate methods for calculating radiation heat transfer in dispersed systems," Therm. Eng. 43, 235-243 (1996).
  10. H. T. K. Tagne and D. Baillis, "Isotropic scaling limits for one-dimensional radiative heat transfer with collimated incidence," J. Quant. Spectrosc. Radiat. Transf. 93, 103-113 (2005).
    [CrossRef]
  11. V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, 1975).
  12. M. Caldas and V. Semião, "A new approximate phase function for isolated particles and polydispersions," J. Quant. Spectrosc. Radiat. Transf. 68, 521-542 (2001).
    [CrossRef]
  13. P. J. Coelho, "Bounded skew high order resolution schemes for the discrete ordinates method," J. Comput. Phys. 175, 412-437 (2002).
    [CrossRef]
  14. B.-T. Liou and C.-Y. Wu, "Radiative transfer in a multi-layer medium with Fresnel interfaces," Heat Mass Transfer 32, 103-107 (1996).
    [CrossRef]
  15. C. Muresan, R. Vaillon, C. Menezo, and R. Morlot, "Discrete ordinates solution of coupled conductive radiative heat transfer in a two-layer slab with Fresnel interfaces subject to diffuse and obliquely collimated irradiation," J. Quant. Spectrosc. Radiat. Transf. 84, 551-562 (2004).
    [CrossRef]
  16. L. A. Dombrovsky, "Thermal radiation from nonisothermal spherical particle," Int. J. Heat Mass Transfer 43, 1661-1672 (2000).
    [CrossRef]
  17. L. A. Dombrovsky, "A modified differential approximation for thermal radiation of semitransparent nonisothermal particles: application to optical diagnostics of plasma spraying," J. Quant. Spectrosc. Radiat. Transf. 73, 433-441 (2002).
    [CrossRef]
  18. L. Pilon and R. Viskanta, "Radiation characteristics of glass containing bubbles," J. Am. Ceram. Soc. 86, 1313-1320 (2003).
    [CrossRef]
  19. 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]
  20. L. Dombrovsky, J. Randrianalisoa, D. Baillis, and L. Pilon, "Use of Mie theory to analyze experimental data to identify infrared properties of fused quartz containing bubbles," Appl. Opt. (to be published).
  21. A. J. Martin and J. Pidorenko, "Insulation microspheres and method of manufacture," U.S. patent 5,713,974 (February 3, 1998).
  22. M. S. Allen, R. G. Baumgartner, J. E. Fesnire, and S. D. Augustynowicz, "Advances in microsphere insulation systems," AIP Conf. Proc. 710, 619-626 (2004).
    [CrossRef]
  23. M. L. German and P. S. Grinchuk, "Mathematical model for calculating the heat-protection properties of the composite coating 'ceramic microspheres-binder'," J. Eng. Phys. Thermophys. 75, 1301-1313 (2002).
    [CrossRef]
  24. L. A. Dombrovsky, "Modeling of thermal radiation of a polymer coating containing hollow microspheres," High Temp. 43, 247-258 (2005).
    [CrossRef]

2005 (2)

H. T. K. Tagne and D. Baillis, "Isotropic scaling limits for one-dimensional radiative heat transfer with collimated incidence," J. Quant. Spectrosc. Radiat. Transf. 93, 103-113 (2005).
[CrossRef]

L. A. Dombrovsky, "Modeling of thermal radiation of a polymer coating containing hollow microspheres," High Temp. 43, 247-258 (2005).
[CrossRef]

2004 (3)

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]

M. S. Allen, R. G. Baumgartner, J. E. Fesnire, and S. D. Augustynowicz, "Advances in microsphere insulation systems," AIP Conf. Proc. 710, 619-626 (2004).
[CrossRef]

C. Muresan, R. Vaillon, C. Menezo, and R. Morlot, "Discrete ordinates solution of coupled conductive radiative heat transfer in a two-layer slab with Fresnel interfaces subject to diffuse and obliquely collimated irradiation," J. Quant. Spectrosc. Radiat. Transf. 84, 551-562 (2004).
[CrossRef]

2003 (1)

L. Pilon and R. Viskanta, "Radiation characteristics of glass containing bubbles," J. Am. Ceram. Soc. 86, 1313-1320 (2003).
[CrossRef]

2002 (4)

L. A. Dombrovsky, "A modified differential approximation for thermal radiation of semitransparent nonisothermal particles: application to optical diagnostics of plasma spraying," J. Quant. Spectrosc. Radiat. Transf. 73, 433-441 (2002).
[CrossRef]

M. L. German and P. S. Grinchuk, "Mathematical model for calculating the heat-protection properties of the composite coating 'ceramic microspheres-binder'," J. Eng. Phys. Thermophys. 75, 1301-1313 (2002).
[CrossRef]

P. J. Coelho, "Bounded skew high order resolution schemes for the discrete ordinates method," J. Comput. Phys. 175, 412-437 (2002).
[CrossRef]

J.-F. Sacadura and D. Baillis, "Experimental characterization of thermal radiation properties of disperse media," Int. J. Therm. Sci. 41, 699-707 (2002).
[CrossRef]

2001 (1)

M. Caldas and V. Semião, "A new approximate phase function for isolated particles and polydispersions," J. Quant. Spectrosc. Radiat. Transf. 68, 521-542 (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]

L. A. Dombrovsky, "Thermal radiation from nonisothermal spherical particle," Int. J. Heat Mass Transfer 43, 1661-1672 (2000).
[CrossRef]

1996 (1)

B.-T. Liou and C.-Y. Wu, "Radiative transfer in a multi-layer medium with Fresnel interfaces," Heat Mass Transfer 32, 103-107 (1996).
[CrossRef]

1982 (1)

H. Lee and R. O. Buckius, "Scaling anisotropic scattering in radiation heat transfer for a planar medium," ASME J. Heat Transfer 104, 68-75 (1982).
[CrossRef]

1981 (1)

B. H. J. McKellar and M. A. Box, "The scaling group of the radiative transfer equation," J. Atmos. Sci. 38, 1063-1068 (1981).
[CrossRef]

Adzerikho, K. S.

K. S. Adzerikho, E. F. Nogotov, and V. P. Trofimov, Radiative Heat Transfer in Two-Phase Media (CRC Press, 1992).

Allen, M. S.

M. S. Allen, R. G. Baumgartner, J. E. Fesnire, and S. D. Augustynowicz, "Advances in microsphere insulation systems," AIP Conf. Proc. 710, 619-626 (2004).
[CrossRef]

Augustynowicz, S. D.

M. S. Allen, R. G. Baumgartner, J. E. Fesnire, and S. D. Augustynowicz, "Advances in microsphere insulation systems," AIP Conf. Proc. 710, 619-626 (2004).
[CrossRef]

Baillis, D.

H. T. K. Tagne and D. Baillis, "Isotropic scaling limits for one-dimensional radiative heat transfer with collimated incidence," J. Quant. Spectrosc. Radiat. Transf. 93, 103-113 (2005).
[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]

J.-F. Sacadura and D. Baillis, "Experimental characterization of thermal radiation properties of disperse media," Int. J. Therm. Sci. 41, 699-707 (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]

L. Dombrovsky, J. Randrianalisoa, D. Baillis, and L. Pilon, "Use of Mie theory to analyze experimental data to identify infrared properties of fused quartz containing bubbles," Appl. Opt. (to be published).

Baumgartner, R. G.

M. S. Allen, R. G. Baumgartner, J. E. Fesnire, and S. D. Augustynowicz, "Advances in microsphere insulation systems," AIP Conf. Proc. 710, 619-626 (2004).
[CrossRef]

Box, M. A.

B. H. J. McKellar and M. A. Box, "The scaling group of the radiative transfer equation," J. Atmos. Sci. 38, 1063-1068 (1981).
[CrossRef]

Buckius, R. O.

H. Lee and R. O. Buckius, "Scaling anisotropic scattering in radiation heat transfer for a planar medium," ASME J. Heat Transfer 104, 68-75 (1982).
[CrossRef]

Caldas, M.

M. Caldas and V. Semião, "A new approximate phase function for isolated particles and polydispersions," J. Quant. Spectrosc. Radiat. Transf. 68, 521-542 (2001).
[CrossRef]

Coelho, P. J.

P. J. Coelho, "Bounded skew high order resolution schemes for the discrete ordinates method," J. Comput. Phys. 175, 412-437 (2002).
[CrossRef]

Davison, B.

B. Davison, Neutron Transport Theory (Oxford U. Press, 1957).

Dombrovsky, L.

L. Dombrovsky, J. Randrianalisoa, D. Baillis, and L. Pilon, "Use of Mie theory to analyze experimental data to identify infrared properties of fused quartz containing bubbles," Appl. Opt. (to be published).

Dombrovsky, L. A.

L. A. Dombrovsky, "Modeling of thermal radiation of a polymer coating containing hollow microspheres," High Temp. 43, 247-258 (2005).
[CrossRef]

L. A. Dombrovsky, "A modified differential approximation for thermal radiation of semitransparent nonisothermal particles: application to optical diagnostics of plasma spraying," J. Quant. Spectrosc. Radiat. Transf. 73, 433-441 (2002).
[CrossRef]

L. A. Dombrovsky, "Thermal radiation from nonisothermal spherical particle," Int. J. Heat Mass Transfer 43, 1661-1672 (2000).
[CrossRef]

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

L. A. Dombrovsky, "Approximate methods for calculating radiation heat transfer in dispersed systems," Therm. Eng. 43, 235-243 (1996).

Fesnire, J. E.

M. S. Allen, R. G. Baumgartner, J. E. Fesnire, and S. D. Augustynowicz, "Advances in microsphere insulation systems," AIP Conf. Proc. 710, 619-626 (2004).
[CrossRef]

German, M. L.

M. L. German and P. S. Grinchuk, "Mathematical model for calculating the heat-protection properties of the composite coating 'ceramic microspheres-binder'," J. Eng. Phys. Thermophys. 75, 1301-1313 (2002).
[CrossRef]

Gomez, R.

Grinchuk, P. S.

M. L. German and P. S. Grinchuk, "Mathematical model for calculating the heat-protection properties of the composite coating 'ceramic microspheres-binder'," J. Eng. Phys. Thermophys. 75, 1301-1313 (2002).
[CrossRef]

Lee, H.

H. Lee and R. O. Buckius, "Scaling anisotropic scattering in radiation heat transfer for a planar medium," ASME J. Heat Transfer 104, 68-75 (1982).
[CrossRef]

Liou, B.-T.

B.-T. Liou and C.-Y. Wu, "Radiative transfer in a multi-layer medium with Fresnel interfaces," Heat Mass Transfer 32, 103-107 (1996).
[CrossRef]

Martin, A. J.

A. J. Martin and J. Pidorenko, "Insulation microspheres and method of manufacture," U.S. patent 5,713,974 (February 3, 1998).

McKellar, B. H.

B. H. J. McKellar and M. A. Box, "The scaling group of the radiative transfer equation," J. Atmos. Sci. 38, 1063-1068 (1981).
[CrossRef]

Menezo, C.

C. Muresan, R. Vaillon, C. Menezo, and R. Morlot, "Discrete ordinates solution of coupled conductive radiative heat transfer in a two-layer slab with Fresnel interfaces subject to diffuse and obliquely collimated irradiation," J. Quant. Spectrosc. Radiat. Transf. 84, 551-562 (2004).
[CrossRef]

Modest, M. F.

M. F. Modest, Radiative Heat Transfer, 2nd ed. (Academic, 2003).

Morlot, R.

C. Muresan, R. Vaillon, C. Menezo, and R. Morlot, "Discrete ordinates solution of coupled conductive radiative heat transfer in a two-layer slab with Fresnel interfaces subject to diffuse and obliquely collimated irradiation," J. Quant. Spectrosc. Radiat. Transf. 84, 551-562 (2004).
[CrossRef]

Muresan, C.

C. Muresan, R. Vaillon, C. Menezo, and R. Morlot, "Discrete ordinates solution of coupled conductive radiative heat transfer in a two-layer slab with Fresnel interfaces subject to diffuse and obliquely collimated irradiation," J. Quant. Spectrosc. Radiat. Transf. 84, 551-562 (2004).
[CrossRef]

Nogotov, E. F.

K. S. Adzerikho, E. F. Nogotov, and V. P. Trofimov, Radiative Heat Transfer in Two-Phase Media (CRC Press, 1992).

Pidorenko, J.

A. J. Martin and J. Pidorenko, "Insulation microspheres and method of manufacture," U.S. patent 5,713,974 (February 3, 1998).

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]

L. Pilon and R. Viskanta, "Radiation characteristics of glass containing bubbles," J. Am. Ceram. Soc. 86, 1313-1320 (2003).
[CrossRef]

L. Dombrovsky, J. Randrianalisoa, D. Baillis, and L. Pilon, "Use of Mie theory to analyze experimental data to identify infrared properties of fused quartz containing bubbles," Appl. Opt. (to be published).

Randrianalisoa, H.

Randrianalisoa, J.

L. Dombrovsky, J. Randrianalisoa, D. Baillis, and L. Pilon, "Use of Mie theory to analyze experimental data to identify infrared properties of fused quartz containing bubbles," Appl. Opt. (to be published).

Sacadura, J.-F.

J.-F. Sacadura and D. Baillis, "Experimental characterization of thermal radiation properties of disperse media," Int. J. Therm. Sci. 41, 699-707 (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]

Semião, V.

M. Caldas and V. Semião, "A new approximate phase function for isolated particles and polydispersions," J. Quant. Spectrosc. Radiat. Transf. 68, 521-542 (2001).
[CrossRef]

Sobolev, V. V.

V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, 1975).

Tagne, H. T.

H. T. K. Tagne and D. Baillis, "Isotropic scaling limits for one-dimensional radiative heat transfer with collimated incidence," J. Quant. Spectrosc. Radiat. Transf. 93, 103-113 (2005).
[CrossRef]

Trofimov, V. P.

K. S. Adzerikho, E. F. Nogotov, and V. P. Trofimov, Radiative Heat Transfer in Two-Phase Media (CRC Press, 1992).

Vaillon, R.

C. Muresan, R. Vaillon, C. Menezo, and R. Morlot, "Discrete ordinates solution of coupled conductive radiative heat transfer in a two-layer slab with Fresnel interfaces subject to diffuse and obliquely collimated irradiation," J. Quant. Spectrosc. Radiat. Transf. 84, 551-562 (2004).
[CrossRef]

Viskanta, R.

Wu, C.-Y.

B.-T. Liou and C.-Y. Wu, "Radiative transfer in a multi-layer medium with Fresnel interfaces," Heat Mass Transfer 32, 103-107 (1996).
[CrossRef]

AIP Conf. Proc. (1)

M. S. Allen, R. G. Baumgartner, J. E. Fesnire, and S. D. Augustynowicz, "Advances in microsphere insulation systems," AIP Conf. Proc. 710, 619-626 (2004).
[CrossRef]

ASME J. Heat Transfer (1)

H. Lee and R. O. Buckius, "Scaling anisotropic scattering in radiation heat transfer for a planar medium," ASME J. Heat Transfer 104, 68-75 (1982).
[CrossRef]

Heat Mass Transfer (1)

B.-T. Liou and C.-Y. Wu, "Radiative transfer in a multi-layer medium with Fresnel interfaces," Heat Mass Transfer 32, 103-107 (1996).
[CrossRef]

High Temp. (1)

L. A. Dombrovsky, "Modeling of thermal radiation of a polymer coating containing hollow microspheres," High Temp. 43, 247-258 (2005).
[CrossRef]

Int. J. Heat Mass Transfer (1)

L. A. Dombrovsky, "Thermal radiation from nonisothermal spherical particle," Int. J. Heat Mass Transfer 43, 1661-1672 (2000).
[CrossRef]

Int. J. Therm. Sci. (1)

J.-F. Sacadura and D. Baillis, "Experimental characterization of thermal radiation properties of disperse media," Int. J. Therm. Sci. 41, 699-707 (2002).
[CrossRef]

J. Am. Ceram. Soc. (1)

L. Pilon and R. Viskanta, "Radiation characteristics of glass containing bubbles," J. Am. Ceram. Soc. 86, 1313-1320 (2003).
[CrossRef]

J. Atmos. Sci. (1)

B. H. J. McKellar and M. A. Box, "The scaling group of the radiative transfer equation," J. Atmos. Sci. 38, 1063-1068 (1981).
[CrossRef]

J. Comput. Phys. (1)

P. J. Coelho, "Bounded skew high order resolution schemes for the discrete ordinates method," J. Comput. Phys. 175, 412-437 (2002).
[CrossRef]

J. Eng. Phys. Thermophys. (1)

M. L. German and P. S. Grinchuk, "Mathematical model for calculating the heat-protection properties of the composite coating 'ceramic microspheres-binder'," J. Eng. Phys. Thermophys. 75, 1301-1313 (2002).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiat. Transf. (5)

M. Caldas and V. Semião, "A new approximate phase function for isolated particles and polydispersions," J. Quant. Spectrosc. Radiat. Transf. 68, 521-542 (2001).
[CrossRef]

H. T. K. Tagne and D. Baillis, "Isotropic scaling limits for one-dimensional radiative heat transfer with collimated incidence," J. Quant. Spectrosc. Radiat. Transf. 93, 103-113 (2005).
[CrossRef]

L. A. Dombrovsky, "A modified differential approximation for thermal radiation of semitransparent nonisothermal particles: application to optical diagnostics of plasma spraying," J. Quant. Spectrosc. Radiat. Transf. 73, 433-441 (2002).
[CrossRef]

C. Muresan, R. Vaillon, C. Menezo, and R. Morlot, "Discrete ordinates solution of coupled conductive radiative heat transfer in a two-layer slab with Fresnel interfaces subject to diffuse and obliquely collimated irradiation," J. Quant. Spectrosc. Radiat. Transf. 84, 551-562 (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]

Other (8)

L. A. Dombrovsky, "Approximate methods for calculating radiation heat transfer in dispersed systems," Therm. Eng. 43, 235-243 (1996).

M. F. Modest, Radiative Heat Transfer, 2nd ed. (Academic, 2003).

K. S. Adzerikho, E. F. Nogotov, and V. P. Trofimov, Radiative Heat Transfer in Two-Phase Media (CRC Press, 1992).

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

B. Davison, Neutron Transport Theory (Oxford U. Press, 1957).

V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, 1975).

L. Dombrovsky, J. Randrianalisoa, D. Baillis, and L. Pilon, "Use of Mie theory to analyze experimental data to identify infrared properties of fused quartz containing bubbles," Appl. Opt. (to be published).

A. J. Martin and J. Pidorenko, "Insulation microspheres and method of manufacture," U.S. patent 5,713,974 (February 3, 1998).

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Figures (9)

Fig. 1
Fig. 1

Effect of the scattering function on directional-hemispherical transmittance and reflectance. CDOM calculations for nonrefracting medium: a, transport approximation; b, Henyey–Greenstein approximation for μ ¯ = 0.8 , 1, τ t r 0 = 0.2 ; 2, 0.5; 3, 1.0; 4, 2.0; 5, 5.0.

Fig. 2
Fig. 2

Effect of the scattering function on directional-hemispherical transmittance and reflectance. CDOM calculations for n = 1.4 : a, transport approximation; b, Henyey–Greenstein approximation for μ ¯ = 0.8 , 1, τ t r 0 = 0.2 ; 2, 0.5; 3, 1.0; 4, 2.0; 5, 5.0.

Fig. 3
Fig. 3

Relative error of the transport approximation for the case of the Henyey–Greenstein scattering function ( μ ¯ = 0.8 ) : (a), (b) n = 1 ; (c), (d) n = 1.4 , 1, τ t r 0 = 0.2 ; 2, 0.5; 3, 1.0; 4, 2.0.

Fig. 4
Fig. 4

Directional-hemispherical transmittance and reflectance for the case of nonrefracting medium. Comparison of calculations by use of the modified two-flux approximation (dashed curves) with the exact numerical solution for the transport scattering function (solid curves). 1, τ t r 0 = 0.2 ; 2, 0.5; 3, 1.0; 4, 2.0; 5, 5.0.

Fig. 5
Fig. 5

Directional-hemispherical transmittance and reflectance for the case of refracting medium ( n = 1.4 ) . Comparison of calculations by use of the modified two-flux approximation (dashed curves) with the exact numerical solution for the transport scattering function (solid curves). 1, τ t r 0 = 0.2 ; 2, 0.5; 3, 1.0; 4, 2.0; 5, 5.0.

Fig. 6
Fig. 6

Dependence of the transport optical thickness on the difference of directional-hemispherical transmittance and reflectance calculated by use of the modified two-flux approximation: (a) n = 1 , (b) n = 1.4 , 1, ω t r = 0 ; 2, 0.25; 3, 0.5; 4, 0.75; 5, 1.

Fig. 7
Fig. 7

Dependence of the transport optical thickness on the difference of directional-hemispherical transmittance and reflectance. Comparison of the analytical solution for nonscattering medium (1) with the numerical data for the transport (2, 3) and Henyey–Greenstein (4, 5, μ ¯ = 0.8 ) scattering functions: (a) n = 1 , (b) n = 1.4 ; 2, 4, ω t r = 0.5 ; 3, 5, ω t r = 1 .

Fig. 8
Fig. 8

Dependence of the transport albedo on the sum of directional-hemispherical transmittance and reflectance. Comparison of the analytical solution in the modified two-flux approximation (I) with the numerical data for the transport (II) and Henyey–Greenstein (III, μ ¯ = 0.8 ) scattering functions: (a) n = 1 , (b) n = 1.4 , 1, τ t r 0 = 0.2 ; 2, 0.5; 3, 1.0.

Fig. 9
Fig. 9

Directional-hemispherical reflectance of optically thick refracting and scattering medium. Comparison of the analytical solution in the modified two-flux approximation (1) with the numerical data for the transport (2) and Henyey–Greenstein (3, μ ¯ = 0.8 ) scattering functions: a, n = 1.4 ; b, n = 1.6 .

Equations (36)

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

Φ ( μ 0 ) = 1 μ ¯ + 2 μ ¯ δ ( 1 μ 0 ) ,
μ I ¯ τ + I ¯ = ω 4 π 1 1 I ¯ ( τ , μ ) [ 0 2 π Φ ( μ 0 ) d ψ ] d μ ,
I ¯ ( 0 , μ ) = R I ¯ ( 0 , μ ) + ( 1 R ) δ ( 1 μ ) ,
I ¯ ( τ 0 , μ ) = R I ¯ ( τ 0 , μ ) , μ > 0 ,
μ I ¯ τ t r + I ¯ = ω t r 2 1 1 I ¯ d μ ,
I ¯ ( 0 , μ ) = R I ¯ ( 0 , μ ) + ( 1 R ) δ ( 1 μ ) ,
I ¯ ( τ t r 0 , μ ) = R I ¯ ( τ t r 0 , μ ) , μ > 0 .
I ¯ = J ¯ + 1 R 1 1 R 1 C t r { exp ( τ t r ) δ ( 1 μ ) + C t r exp ( τ t r ) δ ( 1 + μ ) } ,
μ J ¯ τ t r + J ¯ = ω t r 2 [ 1 1 J ¯ d μ + 1 R 1 1 R 1 C { exp ( τ t r ) + C t r exp ( τ t r ) } ] ,
J ¯ ( 0 , μ ) = R ( μ ) J ¯ ( 0 , μ ) , J ¯ ( τ t r 0 , μ ) = R ( μ ) J ¯ ( τ t r 0 , μ ) , μ > 0 .
R d h = R d h 0 + 0 1 [ 1 R ( μ ) ] J ¯ ( 0 , μ ) μ d μ ,
T d h = T d h 0 + 0 1 [ 1 R ( μ ) ] J ¯ ( τ t r 0 , μ ) μ d μ ,
R d h 0 = R 1 + ( 1 R 1 ) 2 C t r 1 R 1 C t r , T d h 0 = ( 1 R 1 ) 2 1 R 1 C t r exp ( τ t r 0 ) .
I ¯ = J ¯ + 1 R 1 1 R 1 C { exp ( τ ) δ ( 1 μ ) + C exp ( τ ) δ ( 1 + μ ) } ,
μ J ¯ τ + J ¯ = ω 2 [ 1 1 J ¯ ( μ ) f ( μ , μ ) d μ + 1 R 1 1 R 1 C { Φ ( μ ) exp ( τ ) + Φ ( μ ) C exp ( τ ) } ] ,
f ( μ , μ ) = 1 2 π 0 2 π Φ ( μ 0 ) d ψ , C = R 1 exp ( 2 τ 0 ) .
Φ ( μ 0 ) = ( 1 μ ¯ 2 ) ( 1 + μ ¯ 2 2 μ ¯ μ 0 ) 3 2 .
J ¯ ( τ t r , μ ) = { φ 0 ( τ t r ) , 1 μ < μ c ψ 0 ( τ t r ) , μ c < μ < μ c , μ c = ( 1 1 n 2 ) 1 2 φ 0 + ( τ t r ) , μ c < μ 1 } .
g 0 + κ 2 g 0 = κ 2 χ [ exp ( τ t r ) + C t r exp ( τ t r ) ] ,
( 1 + μ c ) g 0 ( 0 ) = 2 γ g 0 ( 0 ) , ( 1 + μ c ) g 0 ( τ t r 0 ) = 2 γ g 0 ( τ t r 0 ) ,
κ 2 = 4 ( 1 + μ c ) 2 1 ω t r 1 ω t r μ c , γ = 1 R 1 1 + R 1 , χ = ω t r 1 ω t r 1 R 1 1 R 1 C t r .
R d h = R d h 0 + γ ( 1 μ c 2 ) g 0 ( 0 ) 2 ,
T d h = T d h 0 + γ ( 1 μ c 2 ) g 0 ( τ t r 0 ) 2 .
g 0 * = τ t r 2 χ [ exp ( τ t r ) + C t r exp ( τ t r ) ] when κ = 1 ,
g 0 * = κ 2 κ 2 1 χ [ exp ( τ t r ) + C t r exp ( τ t r ) ] when κ 1 .
R d h = R d h 0 + D 1 B , T d h = T d h 0 + D 1 [ A + τ t r 0 ( 1 + R 1 ) E t r ] , D 1 = γ ( 1 μ c 2 ) χ 2 ,
A = k 1 ( φ s + c ) E t r + k 2 ( 1 + φ 2 ) s + 2 φ c , B = k 1 E t r + k 2 ( φ s + c ) ( 1 + φ 2 ) s + 2 φ c ,
k 1 = ( 1 + R 1 ) τ t r 0 ( 1 R 1 ) ( 1 + φ τ t r 0 ) , k 2 = 1 C t r .
R d h = R d h 0 + D ( 1 + B κ + C t r ) ,
T d h = T d h 0 + D [ A κ + ( 1 + R 1 ) E t r ] ,
D = D 1 κ 2 ( κ 2 1 ) ,
k 1 = ( 1 2 γ ¯ ) ( 1 + 2 γ ¯ ) R 1 , k 2 = ( 1 2 γ ¯ ) C t r ( 1 + 2 γ ¯ ) .
φ = 2 γ ¯ κ , γ ¯ = γ ( 1 + μ c ) ,
E t r = exp ( τ t r 0 ) , s = sinh ( κ τ t r 0 ) , c = cosh ( κ τ t r 0 ) .
τ t r 0 = ln ( 1 R 1 ) 2 R 1 + ( T d h R d h ) .
R d h = R 1 + ( 1 R 1 ) γ 2 n 2 ω t r 1 ω t r κ 2 ( 1 + κ ) ( 2 γ ¯ + κ ) .

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