Abstract

The scattering of radiation from collimated irradiation is accurately treated via normalization of phase function. This approach is applicable to any numerical method with directional discretization. In this study it is applied to the transient discrete-ordinates method for ultrafast collimated radiative transfer analysis in turbid media. A technique recently developed by the authors, which conserves a phase-function asymmetry factor as well as scattered energy for the Henyey–Greenstein phase function in steady-state diffuse radiative transfer analysis, is applied to the general Legendre scattering phase function in ultrafast collimated radiative transfer. Heat flux profiles in a model tissue cylinder are generated for various phase functions and compared to those generated when normalization of the collimated phase function is neglected. Energy deposition in the medium is also investigated. Lack of conservation of scattered energy and the asymmetry factor for the collimated scattering phase function causes overpredictions in both heat flux and energy deposition for highly anisotropic scattering media. In addition, a discussion is presented to clarify the time-dependent formulation of divergence of radiative heat flux.

© 2012 Optical Society of America

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References

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  1. S. L. Jacques, “Time resolved propagation of ultrashort laser pulses within turbid tissues,” Appl. Opt. 28, 2223–2229 (1989).
    [CrossRef]
  2. Y. Yamada, “Light-tissue interaction and optical imaging in biomedicine,” Annu. Rev. Heat Transfer 6, 1–59 (1995).
  3. K. Mitra and S. Kumar, “Development and comparison of models for light-pulse transport through scattering-absorbing media,” Appl. Opt. 38, 188–196 (1999).
    [CrossRef]
  4. K. H. Kim and Z. Guo, “Ultrafast radiation heat transfer in laser tissue welding and soldering,” Num. Heat Transfer A 46, 23–40 (2004).
    [CrossRef]
  5. H. Quan and Z. Guo, “Fast 3-D optical imaging with transient fluorescence signals,” Opt. Express 12, 449–457 (2004).
    [CrossRef]
  6. K. H. Kim and Z. Guo, “Multi-time-scale heat transfer modeling of turbid tissues exposed to short-pulsed irradiations,” Comput. Meth. Prog. Bio. 86, 112–123 (2007).
    [CrossRef]
  7. M. Jaunich, S. Raje, K. H. Kim, K. Mitra, and Z. Guo, “Bio-heat transfer analysis during short pulse laser irradiation of tissues,” Int. J. Heat Mass Transfer 51, 5511–5521 (2008).
    [CrossRef]
  8. J. C. Chai, H. S. Lee, and S. V. Patankar, “Finite volume method for radiation heat transfer,” J. Thermophys. Heat Transfer 8, 419–425 (1994).
    [CrossRef]
  9. Z. Guo and S. Kumar, “Radiation element method for transient hyperbolic radiative transfer in plane parallel inhomogeneous media,” Numer. Heat Transfer B 39, 371–387 (2001).
  10. P-F. Hsu, Z. Tan, and J. P. Howell, “Radiative transfer by the YIX method in non-homogeneous, scattering and non-gray medium,” J. Thermophys. Heat Transfer 7, 487–495 (1993).
    [CrossRef]
  11. B. G. Carlson and K. D. Lathrop, “Transport theory—the method of discrete-ordinates,” in Computing Methods in Reactor Physics, H. Greenspan, C. N. Kelber, and D. Okrent, eds. (Gorden and Breach, 1968).
  12. Z. Guo and S. Kumar, “Discrete-ordinates solution of short-pulsed laser transport in two-dimensional turbid media,” Appl. Opt. 40, 3156–3163 (2001).
    [CrossRef]
  13. Z. Guo and S. Kumar, “Three-dimensional discrete ordinates method in transient radiative transfer,” J. Thermophys. Heat Transfer 16, 289–296 (2002).
    [CrossRef]
  14. J. Jiao and Z. Guo, “Modeling of ultrashort pulsed laser ablation in water and biological tissues in cylindrical coordinates,” Appl. Phys. B 103, 195–205 (2011).
    [CrossRef]
  15. M. Akamatsu and Z. Guo, “Ultrafast radiative heat transfer in three-dimensional highly-scattering media subjected to pulse train irradiation,” Num. Heat Transfer A 59, 653–671 (2011).
    [CrossRef]
  16. T. K. Kim and H. Lee, “Effect of anisotropic scattering on radiative heat transfer in two-dimensional rectangular enclosures,” Int. J. Heat Mass Transfer 31, 1711–1721 (1988).
    [CrossRef]
  17. W. J. Wiscombe, “On initialization, error and flux conservation in the doubling method,” J. Quant. Spectrosc. Radiat. Transfer 16, 637–658 (1976).
    [CrossRef]
  18. P. Boulet, A. Collin, and J. L. Consalvi, “On the finite volume method and the discrete ordinates method regarding radiative heat transfer in acute forward anisotropic scattering media,” J. Quant. Spectrosc. Radiat. Transfer 104, 460–473 (2007).
    [CrossRef]
  19. B. Hunter and Z. Guo, “Conservation of asymmetry factor in phase function normalization for radiative transfer analysis in anisotropic scattering media,” Int. J. Heat Mass Transfer 55, 1544–1552 (2012).
    [CrossRef]
  20. B. Hunter and Z. Guo, “Reduction of angle splitting and computational time for the finite volume method in radiative transfer analysis via phase function normalization,” Int. J. Heat Mass Transfer 55, 2449–2460 (2012).
    [CrossRef]
  21. P. Rath and S. K. Mahapatra, School of Mechanical Sciences, Indian Institute of Technology, Bhubaneswar, Bhubaneswar, Odisha—751 013, India (personal communication, 2011).
  22. M. F. Modest, Radiative Heat Transfer2nd ed. (Academic, 2003).
  23. B. Hunter and Z. Guo, “Comparison of discrete-ordinates method and finite volume method for steady-state and ultrafast radiative transfer analysis in cylindrical coordinates,” Num. Heat Transfer B 59, 339–359 (2011).
    [CrossRef]
  24. W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron 26, 2166–2185 (1990).
    [CrossRef]
  25. T. K. Kim and H. Lee, “Scaled isotropic results for two-dimensional anisotropic scattering media,” J. Heat Transfer 112, 721–727 (1990).
    [CrossRef]
  26. H. Lee and R. O. Buckius, “Scaling anisotropic scattering in radiation heat transfer for a planar medium,” J. Heat Transfer 104, 68–75 (1982).
    [CrossRef]
  27. Z. Guo and S. Kumar, “Equivalent isotropic scattering formulation for transient short-pulse radiative transfer in anisotropic scattering planar media,” Appl. Opt. 39, 4411–4417 (2000).
    [CrossRef]
  28. S. Jendoubi, H. S. Lee, and T. K. Kim, “Discrete ordinates solutions for radiatively participating media in a cylindrical enclosure,” J. Thermophys. Heat Transfer 7, 213–219 (1993).
    [CrossRef]

2012 (2)

B. Hunter and Z. Guo, “Conservation of asymmetry factor in phase function normalization for radiative transfer analysis in anisotropic scattering media,” Int. J. Heat Mass Transfer 55, 1544–1552 (2012).
[CrossRef]

B. Hunter and Z. Guo, “Reduction of angle splitting and computational time for the finite volume method in radiative transfer analysis via phase function normalization,” Int. J. Heat Mass Transfer 55, 2449–2460 (2012).
[CrossRef]

2011 (3)

B. Hunter and Z. Guo, “Comparison of discrete-ordinates method and finite volume method for steady-state and ultrafast radiative transfer analysis in cylindrical coordinates,” Num. Heat Transfer B 59, 339–359 (2011).
[CrossRef]

J. Jiao and Z. Guo, “Modeling of ultrashort pulsed laser ablation in water and biological tissues in cylindrical coordinates,” Appl. Phys. B 103, 195–205 (2011).
[CrossRef]

M. Akamatsu and Z. Guo, “Ultrafast radiative heat transfer in three-dimensional highly-scattering media subjected to pulse train irradiation,” Num. Heat Transfer A 59, 653–671 (2011).
[CrossRef]

2008 (1)

M. Jaunich, S. Raje, K. H. Kim, K. Mitra, and Z. Guo, “Bio-heat transfer analysis during short pulse laser irradiation of tissues,” Int. J. Heat Mass Transfer 51, 5511–5521 (2008).
[CrossRef]

2007 (2)

K. H. Kim and Z. Guo, “Multi-time-scale heat transfer modeling of turbid tissues exposed to short-pulsed irradiations,” Comput. Meth. Prog. Bio. 86, 112–123 (2007).
[CrossRef]

P. Boulet, A. Collin, and J. L. Consalvi, “On the finite volume method and the discrete ordinates method regarding radiative heat transfer in acute forward anisotropic scattering media,” J. Quant. Spectrosc. Radiat. Transfer 104, 460–473 (2007).
[CrossRef]

2004 (2)

K. H. Kim and Z. Guo, “Ultrafast radiation heat transfer in laser tissue welding and soldering,” Num. Heat Transfer A 46, 23–40 (2004).
[CrossRef]

H. Quan and Z. Guo, “Fast 3-D optical imaging with transient fluorescence signals,” Opt. Express 12, 449–457 (2004).
[CrossRef]

2002 (1)

Z. Guo and S. Kumar, “Three-dimensional discrete ordinates method in transient radiative transfer,” J. Thermophys. Heat Transfer 16, 289–296 (2002).
[CrossRef]

2001 (2)

Z. Guo and S. Kumar, “Discrete-ordinates solution of short-pulsed laser transport in two-dimensional turbid media,” Appl. Opt. 40, 3156–3163 (2001).
[CrossRef]

Z. Guo and S. Kumar, “Radiation element method for transient hyperbolic radiative transfer in plane parallel inhomogeneous media,” Numer. Heat Transfer B 39, 371–387 (2001).

2000 (1)

1999 (1)

1995 (1)

Y. Yamada, “Light-tissue interaction and optical imaging in biomedicine,” Annu. Rev. Heat Transfer 6, 1–59 (1995).

1994 (1)

J. C. Chai, H. S. Lee, and S. V. Patankar, “Finite volume method for radiation heat transfer,” J. Thermophys. Heat Transfer 8, 419–425 (1994).
[CrossRef]

1993 (2)

P-F. Hsu, Z. Tan, and J. P. Howell, “Radiative transfer by the YIX method in non-homogeneous, scattering and non-gray medium,” J. Thermophys. Heat Transfer 7, 487–495 (1993).
[CrossRef]

S. Jendoubi, H. S. Lee, and T. K. Kim, “Discrete ordinates solutions for radiatively participating media in a cylindrical enclosure,” J. Thermophys. Heat Transfer 7, 213–219 (1993).
[CrossRef]

1990 (2)

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron 26, 2166–2185 (1990).
[CrossRef]

T. K. Kim and H. Lee, “Scaled isotropic results for two-dimensional anisotropic scattering media,” J. Heat Transfer 112, 721–727 (1990).
[CrossRef]

1989 (1)

1988 (1)

T. K. Kim and H. Lee, “Effect of anisotropic scattering on radiative heat transfer in two-dimensional rectangular enclosures,” Int. J. Heat Mass Transfer 31, 1711–1721 (1988).
[CrossRef]

1982 (1)

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

1976 (1)

W. J. Wiscombe, “On initialization, error and flux conservation in the doubling method,” J. Quant. Spectrosc. Radiat. Transfer 16, 637–658 (1976).
[CrossRef]

Akamatsu, M.

M. Akamatsu and Z. Guo, “Ultrafast radiative heat transfer in three-dimensional highly-scattering media subjected to pulse train irradiation,” Num. Heat Transfer A 59, 653–671 (2011).
[CrossRef]

Boulet, P.

P. Boulet, A. Collin, and J. L. Consalvi, “On the finite volume method and the discrete ordinates method regarding radiative heat transfer in acute forward anisotropic scattering media,” J. Quant. Spectrosc. Radiat. Transfer 104, 460–473 (2007).
[CrossRef]

Buckius, R. O.

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

Carlson, B. G.

B. G. Carlson and K. D. Lathrop, “Transport theory—the method of discrete-ordinates,” in Computing Methods in Reactor Physics, H. Greenspan, C. N. Kelber, and D. Okrent, eds. (Gorden and Breach, 1968).

Chai, J. C.

J. C. Chai, H. S. Lee, and S. V. Patankar, “Finite volume method for radiation heat transfer,” J. Thermophys. Heat Transfer 8, 419–425 (1994).
[CrossRef]

Cheong, W. F.

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron 26, 2166–2185 (1990).
[CrossRef]

Collin, A.

P. Boulet, A. Collin, and J. L. Consalvi, “On the finite volume method and the discrete ordinates method regarding radiative heat transfer in acute forward anisotropic scattering media,” J. Quant. Spectrosc. Radiat. Transfer 104, 460–473 (2007).
[CrossRef]

Consalvi, J. L.

P. Boulet, A. Collin, and J. L. Consalvi, “On the finite volume method and the discrete ordinates method regarding radiative heat transfer in acute forward anisotropic scattering media,” J. Quant. Spectrosc. Radiat. Transfer 104, 460–473 (2007).
[CrossRef]

Guo, Z.

B. Hunter and Z. Guo, “Reduction of angle splitting and computational time for the finite volume method in radiative transfer analysis via phase function normalization,” Int. J. Heat Mass Transfer 55, 2449–2460 (2012).
[CrossRef]

B. Hunter and Z. Guo, “Conservation of asymmetry factor in phase function normalization for radiative transfer analysis in anisotropic scattering media,” Int. J. Heat Mass Transfer 55, 1544–1552 (2012).
[CrossRef]

M. Akamatsu and Z. Guo, “Ultrafast radiative heat transfer in three-dimensional highly-scattering media subjected to pulse train irradiation,” Num. Heat Transfer A 59, 653–671 (2011).
[CrossRef]

J. Jiao and Z. Guo, “Modeling of ultrashort pulsed laser ablation in water and biological tissues in cylindrical coordinates,” Appl. Phys. B 103, 195–205 (2011).
[CrossRef]

B. Hunter and Z. Guo, “Comparison of discrete-ordinates method and finite volume method for steady-state and ultrafast radiative transfer analysis in cylindrical coordinates,” Num. Heat Transfer B 59, 339–359 (2011).
[CrossRef]

M. Jaunich, S. Raje, K. H. Kim, K. Mitra, and Z. Guo, “Bio-heat transfer analysis during short pulse laser irradiation of tissues,” Int. J. Heat Mass Transfer 51, 5511–5521 (2008).
[CrossRef]

K. H. Kim and Z. Guo, “Multi-time-scale heat transfer modeling of turbid tissues exposed to short-pulsed irradiations,” Comput. Meth. Prog. Bio. 86, 112–123 (2007).
[CrossRef]

K. H. Kim and Z. Guo, “Ultrafast radiation heat transfer in laser tissue welding and soldering,” Num. Heat Transfer A 46, 23–40 (2004).
[CrossRef]

H. Quan and Z. Guo, “Fast 3-D optical imaging with transient fluorescence signals,” Opt. Express 12, 449–457 (2004).
[CrossRef]

Z. Guo and S. Kumar, “Three-dimensional discrete ordinates method in transient radiative transfer,” J. Thermophys. Heat Transfer 16, 289–296 (2002).
[CrossRef]

Z. Guo and S. Kumar, “Discrete-ordinates solution of short-pulsed laser transport in two-dimensional turbid media,” Appl. Opt. 40, 3156–3163 (2001).
[CrossRef]

Z. Guo and S. Kumar, “Radiation element method for transient hyperbolic radiative transfer in plane parallel inhomogeneous media,” Numer. Heat Transfer B 39, 371–387 (2001).

Z. Guo and S. Kumar, “Equivalent isotropic scattering formulation for transient short-pulse radiative transfer in anisotropic scattering planar media,” Appl. Opt. 39, 4411–4417 (2000).
[CrossRef]

Howell, J. P.

P-F. Hsu, Z. Tan, and J. P. Howell, “Radiative transfer by the YIX method in non-homogeneous, scattering and non-gray medium,” J. Thermophys. Heat Transfer 7, 487–495 (1993).
[CrossRef]

Hsu, P-F.

P-F. Hsu, Z. Tan, and J. P. Howell, “Radiative transfer by the YIX method in non-homogeneous, scattering and non-gray medium,” J. Thermophys. Heat Transfer 7, 487–495 (1993).
[CrossRef]

Hunter, B.

B. Hunter and Z. Guo, “Conservation of asymmetry factor in phase function normalization for radiative transfer analysis in anisotropic scattering media,” Int. J. Heat Mass Transfer 55, 1544–1552 (2012).
[CrossRef]

B. Hunter and Z. Guo, “Reduction of angle splitting and computational time for the finite volume method in radiative transfer analysis via phase function normalization,” Int. J. Heat Mass Transfer 55, 2449–2460 (2012).
[CrossRef]

B. Hunter and Z. Guo, “Comparison of discrete-ordinates method and finite volume method for steady-state and ultrafast radiative transfer analysis in cylindrical coordinates,” Num. Heat Transfer B 59, 339–359 (2011).
[CrossRef]

Jacques, S. L.

Jaunich, M.

M. Jaunich, S. Raje, K. H. Kim, K. Mitra, and Z. Guo, “Bio-heat transfer analysis during short pulse laser irradiation of tissues,” Int. J. Heat Mass Transfer 51, 5511–5521 (2008).
[CrossRef]

Jendoubi, S.

S. Jendoubi, H. S. Lee, and T. K. Kim, “Discrete ordinates solutions for radiatively participating media in a cylindrical enclosure,” J. Thermophys. Heat Transfer 7, 213–219 (1993).
[CrossRef]

Jiao, J.

J. Jiao and Z. Guo, “Modeling of ultrashort pulsed laser ablation in water and biological tissues in cylindrical coordinates,” Appl. Phys. B 103, 195–205 (2011).
[CrossRef]

Kim, K. H.

M. Jaunich, S. Raje, K. H. Kim, K. Mitra, and Z. Guo, “Bio-heat transfer analysis during short pulse laser irradiation of tissues,” Int. J. Heat Mass Transfer 51, 5511–5521 (2008).
[CrossRef]

K. H. Kim and Z. Guo, “Multi-time-scale heat transfer modeling of turbid tissues exposed to short-pulsed irradiations,” Comput. Meth. Prog. Bio. 86, 112–123 (2007).
[CrossRef]

K. H. Kim and Z. Guo, “Ultrafast radiation heat transfer in laser tissue welding and soldering,” Num. Heat Transfer A 46, 23–40 (2004).
[CrossRef]

Kim, T. K.

S. Jendoubi, H. S. Lee, and T. K. Kim, “Discrete ordinates solutions for radiatively participating media in a cylindrical enclosure,” J. Thermophys. Heat Transfer 7, 213–219 (1993).
[CrossRef]

T. K. Kim and H. Lee, “Scaled isotropic results for two-dimensional anisotropic scattering media,” J. Heat Transfer 112, 721–727 (1990).
[CrossRef]

T. K. Kim and H. Lee, “Effect of anisotropic scattering on radiative heat transfer in two-dimensional rectangular enclosures,” Int. J. Heat Mass Transfer 31, 1711–1721 (1988).
[CrossRef]

Kumar, S.

Lathrop, K. D.

B. G. Carlson and K. D. Lathrop, “Transport theory—the method of discrete-ordinates,” in Computing Methods in Reactor Physics, H. Greenspan, C. N. Kelber, and D. Okrent, eds. (Gorden and Breach, 1968).

Lee, H.

T. K. Kim and H. Lee, “Scaled isotropic results for two-dimensional anisotropic scattering media,” J. Heat Transfer 112, 721–727 (1990).
[CrossRef]

T. K. Kim and H. Lee, “Effect of anisotropic scattering on radiative heat transfer in two-dimensional rectangular enclosures,” Int. J. Heat Mass Transfer 31, 1711–1721 (1988).
[CrossRef]

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

Lee, H. S.

J. C. Chai, H. S. Lee, and S. V. Patankar, “Finite volume method for radiation heat transfer,” J. Thermophys. Heat Transfer 8, 419–425 (1994).
[CrossRef]

S. Jendoubi, H. S. Lee, and T. K. Kim, “Discrete ordinates solutions for radiatively participating media in a cylindrical enclosure,” J. Thermophys. Heat Transfer 7, 213–219 (1993).
[CrossRef]

Mahapatra, S. K.

P. Rath and S. K. Mahapatra, School of Mechanical Sciences, Indian Institute of Technology, Bhubaneswar, Bhubaneswar, Odisha—751 013, India (personal communication, 2011).

Mitra, K.

M. Jaunich, S. Raje, K. H. Kim, K. Mitra, and Z. Guo, “Bio-heat transfer analysis during short pulse laser irradiation of tissues,” Int. J. Heat Mass Transfer 51, 5511–5521 (2008).
[CrossRef]

K. Mitra and S. Kumar, “Development and comparison of models for light-pulse transport through scattering-absorbing media,” Appl. Opt. 38, 188–196 (1999).
[CrossRef]

Modest, M. F.

M. F. Modest, Radiative Heat Transfer2nd ed. (Academic, 2003).

Patankar, S. V.

J. C. Chai, H. S. Lee, and S. V. Patankar, “Finite volume method for radiation heat transfer,” J. Thermophys. Heat Transfer 8, 419–425 (1994).
[CrossRef]

Prahl, S. A.

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron 26, 2166–2185 (1990).
[CrossRef]

Quan, H.

Raje, S.

M. Jaunich, S. Raje, K. H. Kim, K. Mitra, and Z. Guo, “Bio-heat transfer analysis during short pulse laser irradiation of tissues,” Int. J. Heat Mass Transfer 51, 5511–5521 (2008).
[CrossRef]

Rath, P.

P. Rath and S. K. Mahapatra, School of Mechanical Sciences, Indian Institute of Technology, Bhubaneswar, Bhubaneswar, Odisha—751 013, India (personal communication, 2011).

Tan, Z.

P-F. Hsu, Z. Tan, and J. P. Howell, “Radiative transfer by the YIX method in non-homogeneous, scattering and non-gray medium,” J. Thermophys. Heat Transfer 7, 487–495 (1993).
[CrossRef]

Welch, A. J.

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron 26, 2166–2185 (1990).
[CrossRef]

Wiscombe, W. J.

W. J. Wiscombe, “On initialization, error and flux conservation in the doubling method,” J. Quant. Spectrosc. Radiat. Transfer 16, 637–658 (1976).
[CrossRef]

Yamada, Y.

Y. Yamada, “Light-tissue interaction and optical imaging in biomedicine,” Annu. Rev. Heat Transfer 6, 1–59 (1995).

Annu. Rev. Heat Transfer (1)

Y. Yamada, “Light-tissue interaction and optical imaging in biomedicine,” Annu. Rev. Heat Transfer 6, 1–59 (1995).

Appl. Opt. (4)

Appl. Phys. B (1)

J. Jiao and Z. Guo, “Modeling of ultrashort pulsed laser ablation in water and biological tissues in cylindrical coordinates,” Appl. Phys. B 103, 195–205 (2011).
[CrossRef]

Comput. Meth. Prog. Bio. (1)

K. H. Kim and Z. Guo, “Multi-time-scale heat transfer modeling of turbid tissues exposed to short-pulsed irradiations,” Comput. Meth. Prog. Bio. 86, 112–123 (2007).
[CrossRef]

IEEE J. Quantum Electron (1)

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron 26, 2166–2185 (1990).
[CrossRef]

Int. J. Heat Mass Transfer (4)

M. Jaunich, S. Raje, K. H. Kim, K. Mitra, and Z. Guo, “Bio-heat transfer analysis during short pulse laser irradiation of tissues,” Int. J. Heat Mass Transfer 51, 5511–5521 (2008).
[CrossRef]

T. K. Kim and H. Lee, “Effect of anisotropic scattering on radiative heat transfer in two-dimensional rectangular enclosures,” Int. J. Heat Mass Transfer 31, 1711–1721 (1988).
[CrossRef]

B. Hunter and Z. Guo, “Conservation of asymmetry factor in phase function normalization for radiative transfer analysis in anisotropic scattering media,” Int. J. Heat Mass Transfer 55, 1544–1552 (2012).
[CrossRef]

B. Hunter and Z. Guo, “Reduction of angle splitting and computational time for the finite volume method in radiative transfer analysis via phase function normalization,” Int. J. Heat Mass Transfer 55, 2449–2460 (2012).
[CrossRef]

J. Heat Transfer (2)

T. K. Kim and H. Lee, “Scaled isotropic results for two-dimensional anisotropic scattering media,” J. Heat Transfer 112, 721–727 (1990).
[CrossRef]

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

J. Quant. Spectrosc. Radiat. Transfer (2)

W. J. Wiscombe, “On initialization, error and flux conservation in the doubling method,” J. Quant. Spectrosc. Radiat. Transfer 16, 637–658 (1976).
[CrossRef]

P. Boulet, A. Collin, and J. L. Consalvi, “On the finite volume method and the discrete ordinates method regarding radiative heat transfer in acute forward anisotropic scattering media,” J. Quant. Spectrosc. Radiat. Transfer 104, 460–473 (2007).
[CrossRef]

J. Thermophys. Heat Transfer (4)

J. C. Chai, H. S. Lee, and S. V. Patankar, “Finite volume method for radiation heat transfer,” J. Thermophys. Heat Transfer 8, 419–425 (1994).
[CrossRef]

P-F. Hsu, Z. Tan, and J. P. Howell, “Radiative transfer by the YIX method in non-homogeneous, scattering and non-gray medium,” J. Thermophys. Heat Transfer 7, 487–495 (1993).
[CrossRef]

S. Jendoubi, H. S. Lee, and T. K. Kim, “Discrete ordinates solutions for radiatively participating media in a cylindrical enclosure,” J. Thermophys. Heat Transfer 7, 213–219 (1993).
[CrossRef]

Z. Guo and S. Kumar, “Three-dimensional discrete ordinates method in transient radiative transfer,” J. Thermophys. Heat Transfer 16, 289–296 (2002).
[CrossRef]

Num. Heat Transfer A (2)

K. H. Kim and Z. Guo, “Ultrafast radiation heat transfer in laser tissue welding and soldering,” Num. Heat Transfer A 46, 23–40 (2004).
[CrossRef]

M. Akamatsu and Z. Guo, “Ultrafast radiative heat transfer in three-dimensional highly-scattering media subjected to pulse train irradiation,” Num. Heat Transfer A 59, 653–671 (2011).
[CrossRef]

Num. Heat Transfer B (1)

B. Hunter and Z. Guo, “Comparison of discrete-ordinates method and finite volume method for steady-state and ultrafast radiative transfer analysis in cylindrical coordinates,” Num. Heat Transfer B 59, 339–359 (2011).
[CrossRef]

Numer. Heat Transfer B (1)

Z. Guo and S. Kumar, “Radiation element method for transient hyperbolic radiative transfer in plane parallel inhomogeneous media,” Numer. Heat Transfer B 39, 371–387 (2001).

Opt. Express (1)

Other (3)

B. G. Carlson and K. D. Lathrop, “Transport theory—the method of discrete-ordinates,” in Computing Methods in Reactor Physics, H. Greenspan, C. N. Kelber, and D. Okrent, eds. (Gorden and Breach, 1968).

P. Rath and S. K. Mahapatra, School of Mechanical Sciences, Indian Institute of Technology, Bhubaneswar, Bhubaneswar, Odisha—751 013, India (personal communication, 2011).

M. F. Modest, Radiative Heat Transfer2nd ed. (Academic, 2003).

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Figures (9)

Fig. 1.
Fig. 1.

Conservation of scattered energy and asymmetry factor after collimated phase-function normalization.

Fig. 2.
Fig. 2.

Legendre polynomial phase-function distributions.

Fig. 3.
Fig. 3.

Comparison of radial wall heat flux versus axial location determined from numerical predictions and literature [28].

Fig. 4.
Fig. 4.

Steady-state radial wall heat flux versus axial location for various asymmetry factors and various normalization cases.

Fig. 5.
Fig. 5.

Transient radial wall heat flux versus axial location for g=0.927323 with and without collimated phase-function normalization.

Fig. 6.
Fig. 6.

Transient bottom wall heat flux versus radial location for g=0.927323 with and without collimated phase-function normalization.

Fig. 7.
Fig. 7.

Energy absorption rate versus radial location for various asymmetry factors.

Fig. 8.
Fig. 8.

Energy absorption rate at the radial centerline for various times and asymmetry factors with and without collimated phase-function normalization.

Fig. 9.
Fig. 9.

Effect of propagation term on radial centerline divergence of radiative heat flux.

Equations (22)

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

1cδI(r,s^,t)δt+s^·I(r,s^,t)=βI(r,s^,t)+σaIb(r,t)+σs4π4πI(r,s^,t)Φ(s^,s^)dΩ,
1cδIlδt+μlrδδr[rIl]1rδδϕ[ηlIl]+ξlδIlδz=βIl+βSl.
Sl=(1ω)Ib+ω4πl=1MwlΦllIl+ω4πICΦlClexp(τz),
Iwl=(1ρw,sl)Ibw+ρw,slIl,
ρw,sl={12[tan2(θilθrl)tan2(θil+θrl)+sin2(θilθrl)sin2(θil+θrl)],θil<θcr,1,θilθcr,
Iwl=(1ρw,d)Ibw+ρw,dπ[ICexp(τH)+l,ξl<0wlIl|ξl|],ξl>0,
G=l=1MwlIl+ICexp(τz).
Qr=l=1MμlwlIl,Qz=l=1MξlwlIl+ICexp(τz),
Φll=12[i=0NCiPi(cosΘ1ll)+i=0NCiPi(cosΘ2ll)],
{cos(Θ1ll)=μlμl+ηlηl+ξlξlcos(Θ2ll)=μlμlηlηl+ξlξl.
ΦlCl=i=0NCiPiξl.
14πl=1MΦllwl=1.
14πl=1MΦllcos(Θll)wl=g,
Φllcos(Θll)=12[i=0NCiPi(cosΘ1ll)cos(Θ1ll)+i=0NCiPi(cosΘ2ll)cos(Θ2ll)].
Φ¯ll=(1+All)Φll,
Φ¯lCl=(1+AClCl)ΦlCl.
Δt*min(Δr*,Δz*),
grad=β(1ω)(G4πIb),
grad*=β(1ω)G4πIC/H.
1cδδt4πI(r,s^,t)dΩ+·4πI(r,s^,t)s^dΩ=β4πI(r,s^,t)dΩ+σa4πIb(r,t)dΩ+σs4π4πI(r,s^,t)(4πΦ(s^,s^)dΩ)dΩ.
1cδδt4πI(r,s^,t)dΩ+·qrad=β4πI(r,s^,t)dΩ+4πβ(1ω)Ib(r,t)+βω4πI(r,s^,t)dΩ.
·qrad=β(1ω)(4πIbG)1cδGδt.

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