Advertisement

Radiative Transfer Equation and Solutions

  • Junming M. Zhao
  • Linhua H. Liu
Reference work entry

Abstract

Radiative transfer equation (RTE) is the governing equation of radiation propagation in participating media, which plays a central role in the analysis of radiative transfer in gases, semitransparent liquids and solids, porous materials, and particulate media, and is important in many scientific and engineering disciplines. There are different forms of RTEs that are suitable for different applications, including the RTE under different coordinate systems, the transformed RTE having good numerical properties, the RTE for refractive media, etc. This chapter gives a comprehensive overview and introduction of the different forms of RTEs. Furthermore, several fundamental numerical methods for solving RTEs are introduced with the focus on the deterministic methods, such as the spherical harmonics method, discrete-ordinate method, finite volume method, and finite element method. The understanding of the numerical errors for solving the RTEs, including their origin and effects on numerical results, and the related accuracy improvement strategies are reviewed and discussed.

Notes

Acknowledgements

The authors thank the supports by National Nature Science Foundation of China (Nos. 51336002, 51421063). The support by the Fundamental Research Funds for the Central Universities (Grant No. HIT. NSRIF. 2013094, HIT.BRETIII.201415) are also greatly acknowledged.

References

  1. Agrafiotis CC, Mavroidis I, Konstandopoulos AG, Hoffschmidt B, Stobbe P, Romero M, Fernandez-Quero V (2007) Evaluation of porous silicon carbide monolithic honeycombs as volumetric receivers/collectors of concentrated solar radiation. Sol Energy Mater Sol Cells 91(6):474–488CrossRefGoogle Scholar
  2. An W, Ruan LM, Qi H, Liu LH (2005) Finite element method for radiative heat transfer in absorbing and anisotropic scattering media. J Quant Spectrosc Radiat Transf 96(3–4):409–422CrossRefGoogle Scholar
  3. Asllanaj F, Fumeron S (2010) Modified finite volume method applied to radiative transfer in 2D complex geometries and graded index media. J Quant Spectrosc Radiat Transf 111(2):274–279CrossRefGoogle Scholar
  4. Ben Abdallah P, Le Dez V (2000a) Thermal emission of a semi-transparent slab with variable spatial refractive index. J Quant Spectrosc Radiat Transf 67(3):185–198CrossRefGoogle Scholar
  5. Ben Abdallah P, Le Dez V (2000b) Thermal field inside an absorbing-emitting semitransparent slab at radiative equilibrium with variable spatial refractive index. J Quant Spectrosc Radiat Transf 65(4):595–608CrossRefGoogle Scholar
  6. Ben Abdallah P, Charette A, Le Dez V (2001) Influence of a spatial variation of the thermo-optical constants on the radiative transfer inside an absorbing–emitting semi-transparent sphere. J Quant Spectrosc Radiat Transf 70(3):341–365CrossRefGoogle Scholar
  7. Benoit H, Spreafico L, Gauthier D, Flamant G (2016) Review of heat transfer fluids in tube-receivers used in concentrating solar thermal systems: properties and heat transfer coefficients. Renew Sust Energ Rev 55:298–315CrossRefGoogle Scholar
  8. Berberoglu H, Yin J, Pilon L (2007) Light transfer in bubble sparged photobioreactors for H2 production and CO2 mitigation. J Quant Spectrosc Radiat Transf 32(13):2273–2285Google Scholar
  9. Born M, Wolf E (1970) Principles of optics, 7th edn. Cambridge University Press, CambridgeGoogle Scholar
  10. Carlson BG, Lathrop KD (1965) Transport theory: the method of discrete ordinates. In: Greenspan H, Kelber CN, Okrent D (eds) Computational methods in reactor physics. Gordon and Breach, New York, pp 171–270Google Scholar
  11. Chai JC, Lee HS (1994) Finite-volume method for radiation heat transfer. J Thermophys Heat Transf 8(32):419–425CrossRefGoogle Scholar
  12. Chai JC, Lee HO, Patankar SV (1993) Ray effect and false scattering in the discrete ordinates method. Numer Heat Transf B 24(4):373–389CrossRefGoogle Scholar
  13. Chai JC, Lee HS, Patankar SV (2000a) Finite-volume method for radiation heat transfer. Adv Numer Heat Transf 2:109–141Google Scholar
  14. Chai JC, Lee HS, Patankar SV (2000b) Finite-volume method for radiation heat transfer. In: Minkowycz WJ, Sparrow EM (eds) Advances in numerical heat transfer, vol 2. Taylor & Francis, New York, pp 109–141Google Scholar
  15. Chai JL, Cheng Q, Song JL, Wang ZC, Zhou HC (2015) The DRESOR method for one-dimensional transient radiative transfer in graded index medium coupled with BRDF surface. Int J Therm Sci 91:96–104CrossRefGoogle Scholar
  16. Chandrasekhar S (1960) Radiative transfer. Dover Publications, New YorkzbMATHGoogle Scholar
  17. Cheong KB, Song TH (1997) An alternative discrete ordinates method with interpolation and source differencing for two-dimensional radiative transfer problems. Numer Heat Transf B 32(1):107–125CrossRefGoogle Scholar
  18. Chui EH, Raithby GD (1992) Implicit solution scheme to improve convergence rate in radiative transfer problems. Numer Heat Transf B 22(3):251–272CrossRefGoogle Scholar
  19. Coelho PJ (2002a) Bounded skew high-order resolution schemes for the discrete ordinates method. J Comput Phys 175(2):412–437zbMATHCrossRefGoogle Scholar
  20. Coelho PJ (2002b) The role of ray effects and false scattering on the accuracy of the standard and modified discrete ordinates methods. J Quant Spectrosc Radiat Transf 73:231–238CrossRefGoogle Scholar
  21. Coelho PJ (2004) A modified version of the discrete ordinates method for radiative heat transfer modelling. Comput Mech 33(5):375–388MathSciNetzbMATHCrossRefGoogle Scholar
  22. Coelho PJ (2005) A hybrid finite volume/finite element discretization method for the solution of the solution of the radiative heat transfer equation. J Quant Spectrosc Radiat Transf 93(1–3):89–101CrossRefGoogle Scholar
  23. Coelho PJ (2014) Advances in the discrete ordinates and finite volume methods for the solution of radiative heat transfer problems in participating media. J Quant Spectrosc Radiat Transf 145:121–146CrossRefGoogle Scholar
  24. Farmer JT, Howell JR (1994) Monte Carlo prediction of radiative heat transfer in inhomogeneous, anisotropic, nongray media. J Thermophys Heat Transf 8(1):133–139CrossRefGoogle Scholar
  25. Fiveland WA (1984) Discrete-ordinates solution of the radiative transport equation for rectangular enclosures. J Heat Transf 106(4):699–706CrossRefGoogle Scholar
  26. Fiveland WA (1988) Three-dimensional radiative heat-transfer solutions by the discrete-ordinates method. J Thermophys Heat Transf 2(4):309–316CrossRefGoogle Scholar
  27. Fiveland WA, Jessee JP (1994) Finite element formulation of the discrete-ordinates method for multidimensional geometries. J Thermophys Heat Transf 8(3):426–433CrossRefGoogle Scholar
  28. Fiveland WA, Jessee JP (1995) Comparison of discrete ordinates formulations for radiative heat transfer in multidimensional geometries. J Thermophys Heat Transf 9(1):47–54CrossRefGoogle Scholar
  29. Granate P, Coelho PJ, Roger M (2016) Radiative heat transfer in strongly forward scattering media using the discrete ordinates method. J Quant Spectrosc Radiat Transf 172:110–120CrossRefGoogle Scholar
  30. Hou M-F, Wu C-Y, Hong Y-B (2015) A closed-form solution of differential approximation for radiative transfer in a planar refractive medium. Int J Heat Mass Transf 83:229–234CrossRefGoogle Scholar
  31. Howell JR (1968) Application of Monte Carlo to heat transfer problems. Adv Heat Transf 5:1–54Google Scholar
  32. Howell JR, Siegel R, Menguc MP (2011) Thermal radiation heat transfer, 5th edn. CRC Press, New YorkGoogle Scholar
  33. Huang Y, Xia XL, Tan HP (2002a) Radiative intensity solution and thermal emission analysis of a semitransparent medium layer with a sinusoidal refractive index. J Quant Spectrosc Radiat Transf 74(2):217–233CrossRefGoogle Scholar
  34. Huang Y, Xia XL, Tan HP (2002b) Temperature field of radiative equilibrium in a semitransparent slab with a linear refractive index and gray walls. J Quant Spectrosc Radiat Transf 74(2):249–261CrossRefGoogle Scholar
  35. Huang ZF, Zhou HC, Hsu P (2011) Improved discrete ordinates method for ray effects mitigation. J Heat Transf 133(4):044502CrossRefGoogle Scholar
  36. Huang Y, Shi G-D, Zhu K-Y (2016) Runge–Kutta ray tracing technique for solving radiative heat transfer in a two-dimensional graded-index medium. J Quant Spectrosc Radiat Transf 176:24–33CrossRefGoogle Scholar
  37. Hunter B, Guo Z (2012a) Conservation of asymmetry factor in phase function discretization for radiative transfer analysis in anisotropic scattering media. Int J Heat Mass Transf 55(5):1544–1552zbMATHCrossRefGoogle Scholar
  38. Hunter B, Guo Z (2012b) Phase-function normalization in the 3-D discrete-ordinates solution of radiative transfer – part II: benchmark comparisons. Numer Heat Transf B 62(4):223–242CrossRefGoogle Scholar
  39. Hunter B, Guo Z (2014) A new and simple technique to normalize the HG phase function for conserving scattered energy and asymmetry factor. Numer Heat Transf B 65(3):195–217CrossRefGoogle Scholar
  40. Hunter B, Guo Z (2015) Numerical smearing, ray effect, and angular false scattering in radiation transfer computation. Int J Heat Mass Transf 81:63–74CrossRefGoogle Scholar
  41. Klose AD, Netz U, Beuthan J, Hielscher AH (2002) Optical tomography using the time-independent equation of radiative transfer – part 1: forward model. J Quant Spectrosc Radiat Transf 72(5):691–713CrossRefGoogle Scholar
  42. Koch R, Becker R (2004) Evaluation of quadrature schemes for the discrete ordinates method. J Quant Spectrosc Radiat Transf 84(4):423–435CrossRefGoogle Scholar
  43. Larsen EW, Thommes G, Klar A, Seaid M, Gotz T (2002) Simplified PN approximations to the equations of radiative heat transfer and applications. J Comput Phys 183(2):652–675MathSciNetzbMATHCrossRefGoogle Scholar
  44. Lathrop KD (1968) Ray effects in discrete ordinates equations. Nucl Sci Eng 32:357–369CrossRefGoogle Scholar
  45. Lemonnier D, Le Dez V (2002) Discrete ordinates solution of radiative transfer across a slab with variable refractive index. J Quant Spectrosc Radiat Transf 73(2–5):195–204CrossRefGoogle Scholar
  46. Li H-S, Flamant G, Lu J-D (2003) Mitigation of ray effects in the discrete ordinates method. Numer Heat Transf B 43(5):445–466CrossRefGoogle Scholar
  47. Li B-W, Sun Y-S, Yu Y (2008) Iterative and direct Chebyshev collocation spectral methods for one-dimensional radiative heat transfer. Int J Heat Mass Transf 51(25–26):5887–5894zbMATHCrossRefGoogle Scholar
  48. Liou KN (2002) An introduction to atmospheric radiation. Academic pressGoogle Scholar
  49. Liu LH (2004a) Discrete curved ray-tracing method for radiative transfer in an absorbing-emitting semitransparent slab with variable spatial refractive index. J Quant Spectrosc Radiat Transf 83(2):223–228MathSciNetCrossRefGoogle Scholar
  50. Liu LH (2004b) Finite element simulation of radiative heat transfer in absorbing and scattering media. J Thermophys Heat Transf 18(4):555–557MathSciNetCrossRefGoogle Scholar
  51. Liu LH (2006) Finite volume method for radiation heat transfer in graded index medium. J Thermophys Heat Transf 20(1):59–66MathSciNetCrossRefGoogle Scholar
  52. Liu LH, Tan HP (2006) Numerical simulation of radiative transfer in graded index media. Science Press, BeijingGoogle Scholar
  53. Liu LH, Tan JY (2007) Least-squares collocation meshless approach for radiative heat transfer in absorbing and scattering media. J Quant Spectrosc Radiat Transf 103(3):545–557CrossRefGoogle Scholar
  54. Liu LH, Ruan LM, Tan HP (2002) On the discrete ordinates method for radiative heat transfer in anisotropically scattering media. Int J Heat Mass Transf 45(15):3259–3262zbMATHCrossRefGoogle Scholar
  55. Liu LH, Zhang L, Tan HP (2006) Radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems. J Quant Spectrosc Radiat Transf 97(3):446–456CrossRefGoogle Scholar
  56. Liu LH, Zhao JM, Tan HP (2008) The finite element method and spectral element method for numerical simulation of radiative transfer equation. Science Press, BeijingGoogle Scholar
  57. Mahian O, Kianifar A, Kalogirou SA, Pop I, Wongwises S (2013) A review of the applications of nanofluids in solar energy. Int J Heat Mass Transf 57(2):582–594CrossRefGoogle Scholar
  58. Maruyama S (1993) Radiation heat transfer between arbitrary three-dimensional bodies with specular and diffuse surfaces. Numer Heat Transf A Appl 24(2):181–196CrossRefGoogle Scholar
  59. Mengüç MP, Iyer RK (1988) Modeling of radiative transfer using multiple spherical harmonics approximations. J Quant Spectrosc Radiat Transf 39(6):445–461CrossRefGoogle Scholar
  60. Mengüç MP, Viskanta R (1985) Radiative transfer in three-dimensional rectangular enclosures containing inhomogeneous, anisotropically scattering media. J Quant Spectrosc Radiat Transf 33(6):533–549CrossRefGoogle Scholar
  61. Modest MF (2013) Radiative heat transfer, 3rd edn. Academic Press, New YorkGoogle Scholar
  62. Modest MF, Haworth DC (2016) Radiative heat transfer in turbulent combustion systems: theory and applications. Springer International Publishing, ChamzbMATHCrossRefGoogle Scholar
  63. Murthy JY, Mathur SR (1998) Finite volume method for radiative heat transfer using unstructured meshes. J Thermophys Heat Transf 12(3):313–321CrossRefGoogle Scholar
  64. Pilon L, Berberoglu H, Kandilian R (2011) Radiation transfer in photobiological carbon dioxide fixation and fuel production by microalgae. J Quant Spectrosc Radiat Transf 112(17):2639–2660CrossRefGoogle Scholar
  65. Raithby GD, Chui EH (1990) A finite-volume method for predicting a radiant heat transfer in enclosures with participating media. J Heat Transf 112:415–423CrossRefGoogle Scholar
  66. Ramankutty MA, Crosbie AL (1997) Modified discrete ordinates solution of radiative transfer in two-dimensional rectangular enclosures. J Quant Spectrosc Radiat Transf 57(11):107–140CrossRefGoogle Scholar
  67. Sadat H (2006) On the use of a meshless method for solving radiative transfer with the discrete ordinates formulations. J Quant Spectrosc Radiat Transf 101(2):263–268CrossRefGoogle Scholar
  68. Siegel R, Howell JR (2002) Thermal radiation heat transfer, 4th edn. Taylor & Francis, New YorkGoogle Scholar
  69. Simmons FS (2000) Rocket exhaust plume phenomenology. Aerospace CorporationGoogle Scholar
  70. Song TH, Park CW 1992 Formulation and application of the second-order discrete ordinate method. In: Wang B-X (ed) Transport phenomena and science. Higher Education Press, Beijing, pp 833–841Google Scholar
  71. Sun Y-S, Li B-W (2009) Chebyshev collocation spectral method for one-dimensional radiative heat transfer in graded index media. Int J Therm Sci 48:691–698CrossRefGoogle Scholar
  72. Tagne Kamdem HT (2015) Ray effects elimination in discrete ordinates and finite volume methods. J Thermophys Heat Transf 29(2):306–318CrossRefGoogle Scholar
  73. Thurgood CP, Pollard A, Becker HA (1995) The TN quadrature set for the discrete ordinates method. J Heat Transf 117(4):1068–1070CrossRefGoogle Scholar
  74. Truelove JS (1988) Three-dimensional radiation in absorbing-emitting-scattering media using discrete-ordinate approximation. J Quant Spectrosc Radiat Transf 39(1):27–31CrossRefGoogle Scholar
  75. Viskanta R, Mengüç MP (1987) Radiation heat transfer in combustion systems. Prog Energy Combust Sci 13(2):97–160CrossRefGoogle Scholar
  76. Wang Z, Cheng Q, Wang G, Zhou H (2011) The DRESOR method for radiative heat transfer in a one-dimensional medium with variable refractive index. J Quant Spectrosc Radiat Transf 112(18):2835–2845CrossRefGoogle Scholar
  77. Wu C-Y, Hou M-F (2012) Solution of integral equations of intensity moments for radiative transfer in an anisotropically scattering medium with a linear refractive index. Int J Heat Mass Transf 55(7–8):1863–1872CrossRefGoogle Scholar
  78. Xia XL, Huang Y, Tan HP (2002) Thermal emission and volumetric absorption of a graded index semitransparent medium layer. J Quant Spectrosc Radiat Transf 74(2):235–248CrossRefGoogle Scholar
  79. Zhang L, Zhao JM, Liu LH (2010) Finite element approach for radiative transfer in multi-layer graded index cylindrical medium with Fresnel surfaces. J Quant Spectrosc Radiat Transf 111(3):420–432CrossRefGoogle Scholar
  80. Zhang L, Zhao JM, Liu LH, Wang SY (2012) Hybrid finite volume/finite element method for radiative heat transfer in graded index media. J Quant Spectrosc Radiat Transf 113(14):1826–1835CrossRefGoogle Scholar
  81. Zhang Y, Yi H-L, Tan H-P (2015) Analysis of transient radiative transfer in two-dimensional scattering graded index medium with diffuse energy pulse irradiation. Int J Therm Sci 87:187–198CrossRefGoogle Scholar
  82. Zhang L, Zhao JM, Liu LH (2016) A new stabilized finite element formulation for solving radiative transfer equation. J Heat Transf 138(6):064502–064502CrossRefGoogle Scholar
  83. Zhao JM, Liu LH (2006) Least-squares spectral element method for radiative heat transfer in semitransparent media. Numer Heat Transf B 50(5):473–489MathSciNetCrossRefGoogle Scholar
  84. Zhao JM, Liu LH (2007a) Second order radiative transfer equation and its properties of numerical solution using finite element method. Numer Heat Transf B 51:391–409CrossRefGoogle Scholar
  85. Zhao JM, Liu LH (2007b) Solution of radiative heat transfer in graded index media by Least Square spectral element method. Int J Heat Mass Transf 50:2634–2642zbMATHCrossRefGoogle Scholar
  86. Zhao JM, Tan JY, Liu LH (2012a) A deficiency problem of the least squares finite element method for solving radiative transfer in strongly inhomogeneous media. J Quant Spectrosc Radiat Transf 113(12):1488–1502CrossRefGoogle Scholar
  87. Zhao JM, Tan JY, Liu LH (2012b) On the derivation of vector radiative transfer equation for polarized radiative transport in graded index media. J Quant Spectrosc Radiat Transf 113(3):239–250CrossRefGoogle Scholar
  88. Zhao JM, Tan JY, Liu LH (2013) A second order radiative transfer equation and its solution by meshless method with application to strongly inhomogeneous media. J Comput Phys 232(1):431–455MathSciNetzbMATHCrossRefGoogle Scholar
  89. Zhou H-C, Cheng Q (2004) The DRESOR method for the solution of the radiative transfer equation in gray plane-parallel media. In: Mengüç MP, Selçuk N (eds) Proceedings of the fourth international symposium on radiative transfer, Istanbul, pp 181–190Google Scholar
  90. Zhou H-C, Lou C, Cheng Q, Jiang Z, He J, Huang B, Pei Z, Lu C (2005) Experimental investigations on visualization of three-dimensional temperature distributions in a large-scale pulverized-coal-fired boiler furnace. Proc Combust Inst 30(1):1699–1706CrossRefGoogle Scholar
  91. Zhu K-Y, Huang Y, Wang J (2011) Curved ray tracing method for one-dimensional radiative transfer in the linear-anisotropic scattering medium with graded index. J Quant Spectrosc Radiat Transf 112(3):377–383CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Energy Science and EngineeringHarbin Institute of TechnologyHarbinChina

Section editors and affiliations

  • Pinar Mengüç
    • 1
  1. 1.Çekmeköy CampusÖzyeğin UniversityÇekmeköy - IstanbulTurkey

Personalised recommendations