Radiative Heat Transfer and Applications for Glass Production Processes

  • Martin FrankEmail author
  • Axel Klar
Part of the Lecture Notes in Mathematics book series (LNM, volume 2010)


In glass manufacturing, a hot melt of glass is cooled down to room temperature. The annealing has to be monitored carefully in order to avoid excessive temperature differences which may affect the quality of the product or even lead to cracks in the material. In order to control this process it is, therefore, of interest to have a mathematical model that accurately predicts the temperature evolution. The model will involve the direction-dependent thermal radiation field because a significant part of the energy is transported by photons. Unfortunately, this fact makes the numerical solution of the radiative transfer equations much more complex, especially in higher dimensions, since, besides position and time variables, the directional variables also have to be accounted for. Therefore, approximations of the full model that are computationally less time consuming but yet sufficiently accurate have to be sought. It is our purpose to present several recent approaches to this problem that have been co-developed by the authors.


Radiative Transfer Half Space Transfer Equation Radiative Transfer Equation Discrete Ordinate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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We wish to thank all our collaborators and co-authors, in particular B. Dubroca, T. Götz, J. Lang, E.W. Larsen, M. Seaïd, G. Thömmes, R. Turpault and R. Pinnau. Parts of this work have been taken from the articles [18, 23, 24, 25, 27, 38, 47, 48, 76, 85]. This work was supported by German Research Foundation DFG under grants KL 1105/7 and 1105/14.


  1. 1.
    Adams, M.L., Larsen, E.W.: Fast iterative methods for deterministic particle transport computations. Prog. Nucl. Energy 40, 3–159 (2002)CrossRefGoogle Scholar
  2. 2.
    Adams, M.L.: Subcell balance formulations for radiative transfer on arbitrary grids. Transp. Theory and Stat. Phys. 26, 385–431 (1997)zbMATHCrossRefGoogle Scholar
  3. 3.
    Alcouffe, R., Brandt, A., Dendy, J., Painter, J.: The multigrid method for diffusion equations with strongly discontinuous coefficients. SIAM J. Sci. Stat. Comp. 2, 430–454 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Alcouffe, R.: Diffusion synthetic acceleration methods for the diamond-differenced discrete-ordinates equations. Nucl. Sci. Eng. 64, 344–355 (1977)Google Scholar
  5. 5.
    Agoshkov, V.: On the existence of traces of functions in spaces used in transport theory problems. Sov. Math. Dokl. 33, 628–632 (1986)zbMATHGoogle Scholar
  6. 6.
    Agoshkov, V.: Boundary value problems for transport equations. Birkhäuser, Boston (1998)zbMATHGoogle Scholar
  7. 7.
    Anile, A.M., Pennisi, S., Sammartino, M.: A thermodynamical approach to Eddington factors. J. Math. Phys. 32, 544–550 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Atkinson, K.E.: Iterative variants of the Nyström method for the numerical solution of integral equations. Numer. Math. 22, 17–31 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Brantley, P.S., Larsen, E.W.: The simplified P 3 approximation. Nucl. Sci. Eng. 134, 1–21 (2000)Google Scholar
  10. 10.
    Brown, P.N.: A linear algebraic development of diffusion synthetic acceleration for three-dimensional transport equations. SIAM. J. Numer. Anal. 32, 179–214 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Brunner, T.A., Holloway, J.P.: One-dimensional Riemann solvers and the maximum entropy closure. J. Quant. Spectrosc. Radiat. Transfer 69, 543–566 (2001)CrossRefGoogle Scholar
  12. 12.
    Cheng, P.: Dynamics of a radiating gas with applications to flow over a wavy wall. AIAA J. 4, 238–245 (1966)CrossRefGoogle Scholar
  13. 13.
    Clause, P.-J., Mareschal, M.: Heat transfer in a gas between parallel plates: Moment method and molecular dynamics. Phys. Rev. A 38, 4241–4252 (1988)CrossRefGoogle Scholar
  14. 14.
    Davison, B.: Neutron transport theory. Oxford University Press, Oxford (1958)Google Scholar
  15. 15.
    Dreyer, W.: Maximisation of the entropy in non-equilibrium. J. Phys. A 20, 6505–6517 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Dubroca, B., Feugeas, J.L.: Entropic moment closure hierarchy for the radiative transfer equation. C. R. Acad. Sci. Paris Ser. I 329, 915–920 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Dubroca, B.: Thèse d’Etat, Dept. of Mathematics, University of Bordeaux (2000)Google Scholar
  18. 18.
    Dubroca, B., Frank, M., Klar, A., Thömmes, G.: Half space moment approximation to the radiative heat transfer equations. Z. Angew. Math. Mech. 83, 853–858 (2003)zbMATHCrossRefGoogle Scholar
  19. 19.
    Dubroca, B., Klar, A.: Half moment closure for radiative transfer equations. J. Comput. Phys. 180, 584–596 (2002)zbMATHCrossRefGoogle Scholar
  20. 20.
    Eddington, A.: The Internal Constitution of the Stars. Dover, New York (1926)zbMATHGoogle Scholar
  21. 21.
    Fischer, A.E., Marsden, J.E.: The Einstein evolution equations as a first-order quasi-linear hyperbolic system I. Commun. Math. Phys. 26, 1–38 (1972)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Fiveland, W.A.: The selection of discrete ordinate quadrature sets for anisotropic scattering. ASME HTD. Fundam. Radiat. Heat Transf. 160, 89–96 (1991)Google Scholar
  23. 23.
    Frank, M.: Partial Moment Models for Radiative Transfer. PhD thesis, TU Kaiserslautern (2005)Google Scholar
  24. 24.
    Frank, M.: Approximate models for radiative transfer. Bull. Inst. Math. Acad. Sinica (New Series) 2, 409–432 (2007)Google Scholar
  25. 25.
    Frank, M., Dubroca, B., Klar, A.: Partial moment entropy approximation to radiative transfer. J. Comput. Phys. 218, 1–18 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Frank, M., Pinnau, R.: Analysis of a half moment model for radiative heat transfer equations. Appl. Math. Lett. 20, 189–193 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Frank, M., Seaïd, M., Janicka, J., Klar, A., Pinnau, R.: A comparison of approximate models for radiation in gas turbines. Prog. Comput. Fluid Dyn. 4, 191–197 (2004)CrossRefGoogle Scholar
  28. 28.
    Golse, F., Perthame, B.: Generalized solution of the radiative transfer equations in a singular case. Commun. Math. Phys. 106, 211–239 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Greenbaum, A.: Iterative Methods for Solving Linear Systems. SIAM, Philadelphia (1997)zbMATHCrossRefGoogle Scholar
  30. 30.
    Hackbusch, W.: Multi-Grid Methods and Applications. Springer Series in Computational Mathematics, vol. 4. Springer, New York (1985)Google Scholar
  31. 31.
    Howell, R., Siegel, J.R.: Thermal Radiation Heat Transfer, 3rd edn. Taylor & Francis, NewYork (1992)Google Scholar
  32. 32.
    Huang, K.: Introduction to Statistical Physics. Taylor and Francis, New York (2001)zbMATHGoogle Scholar
  33. 33.
    Jeans, J.H.: The equations of radiative transfer of energy. Mon. Not. R. Astron. Soc. 78, 28–36 (1917)Google Scholar
  34. 34.
    Junk, M.: Domain of definition of levermore’s five-moment system. J. Stat. Phys. 93, 1143–1167 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia (1995)zbMATHCrossRefGoogle Scholar
  36. 36.
    Kelley, C.T.: Multilevel Source Iteration Accelerators for the Linear Transport Equation in Slab Geometry. Transp. Theory Stat. Phys. 24, 679–707 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Kelley, C.T.: Existence and uniqueness of solutions of nonlinear systems of conductive radiative heat transfer equations. Transp. Theory Stat. Phys. 25, 249–260 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Klar, A., Lang, J., Seaid, M.: Adaptive solutions of SPN-Approximations to radiative heat transfer in glass. Int. J. Therm. Sci. 44, 1013–1023 (2005)CrossRefGoogle Scholar
  39. 39.
    Klar, A., Schmeiser, C.: Numerical passage from radiative heat transfer to nonlinear diffusion models. Math. Mod. Meth. Appl. Sci. 11, 749–767 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Klar, A., Siedow, N.: Boundary layers and domain decomposition for radiative heat transfer and diffusion equations: Applications to glass manufacturing processes. Eur. J. Appl. Math. 9–4, 351–372 (1998)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Korganoff, V.: Basic Methods in Transfer Problems. Dover, New York (1963)Google Scholar
  42. 42.
    Krook, M.: On the solution of equations of transfer. Astrophys. J. 122, 488 (1955)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Laitinen, M.T., Tiihonen, T.: Integro-differential equation modelling heat transfer in conducting, radiating and semitransparent materials. Math. Meth. Appl. Sci. 21, 375–392 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Laitinen, M.T., Tiihonen, T.: Conductive-radiative heat transfer in grey materials. Quart. Appl. Math. 59 737–768 (2001)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Larsen, E.W., Keller, J.B.: Asymptotic solution of neutron transport problems for small mean free path. J. Math. Phys. 15, 75 (1974)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Larsen, E.W., Pomraning, G., Badham, V.C.: Asymptotic analysis of radiative transfer problems. J. Quant. Spectr. Radiati. Transf. 29, 285–310 (1983)CrossRefGoogle Scholar
  47. 47.
    Larsen, E.W., Thömmes, G., Klar, A., Seaïd, M., Götz, T.: Simplified P N approximations to the equations of radiative heat transfer in glass. J. Comput. Phys. 183, 652–675 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Larsen, E.W., Thoemmes, G., Klar, A. : New frequency-averaged approximations to the equations of radiative heat transfer. SIAM Appl. Math. 64 565–582 (2003)zbMATHGoogle Scholar
  49. 49.
    Lentes, F.T., Siedow, N.: Three-dimensional radiative heat transfer in glass cooling processes. Glastech. Ber. Glass Sci. Technol. 72, 188–196 (1999)Google Scholar
  50. 50.
    Levermore, C.D.: Relating Eddington factors to flux limiters. J. Quant. Spectroscop. Radiat. Transf. 31, 149–160 (1984)CrossRefGoogle Scholar
  51. 51.
    Levermore, C.D.: Moment closure hierarchies for kinetic theories. J.Stat.Phys. 83 (1996)Google Scholar
  52. 52.
    Lewis, E.E., Miller, W.F. Jr., Computational Methods of Neutron Transport. Wiley, New York (1984)Google Scholar
  53. 53.
    Lopez-Pouso, O.: Trace theorem and existence in radiation. Adv. Math. Sci. Appl. 10, 757–773 (2000)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Mark, J.C.: The spherical harmonics method, part I. Tech. Report MT 92, National Research Council of Canada (1944)Google Scholar
  55. 55.
    Mark, J.C.: The spherical harmonics method, part II. Tech. Report MT 97, National Research Council of Canada (1945)Google Scholar
  56. 56.
    Marshak, R.E.: Note on the spherical harmonic method as applied to the milne problem for a sphere. Phys. Rev. 71, 443–446 (1947)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Mengüc, M.P., Iyer, R.K.: Modeling of radiative transfer using multiple spherical harmonics approximations. J. Quant. Spectrosc. Radiat. Transf. 39 (1988), 445–461.CrossRefGoogle Scholar
  58. 58.
    Mercier, B.: Application of accretive operators theory to the radiative transfer equations. SIAM J. Math. Anal. 18, 393–408 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Minerbo, G.N.: Maximum entropy Eddington factors. J. Quant. Spectrosc. Radiat. Transf. 20, 541–545 (1978)CrossRefGoogle Scholar
  60. 60.
    Modest, M.F.: Radiative Heat Transfer, 2nd edn. Academic, San Diego (1993)Google Scholar
  61. 61.
    Müller, I., Ruggeri, T.: Rational extended thermodynamics. Springer, New York (1998)zbMATHCrossRefGoogle Scholar
  62. 62.
    Murray, R.L.: Nuclear reactor physics. Prentice Hall, New Jersey (1957)Google Scholar
  63. 63.
    Ore, A.: Entropy of radiation. Phys. Rev. 98, 887 (1955)Google Scholar
  64. 64.
    Özisik, M.N., Menning, J., Hälg, W.: Half-range moment method for solution of the transport equation in a spherical symmetric geometry. J. Quant. Spectrosc. Radiat. Transf. 15, 1101–1106 (1975)CrossRefGoogle Scholar
  65. 65.
    Planck, M.: Distribution of energy in the spectrum. Ann. Phys. 4, 553–563 (1901)zbMATHCrossRefGoogle Scholar
  66. 66.
    Pomraning, G.C.: The equations of radiation hydrodynamics. Pergamon, New York (1973)Google Scholar
  67. 67.
    Pomraning, G.C.: Initial and boundary conditions for equilibrium diffusion theory. J. Quant. Spectrosc. Radiat. Transf. 36, 69 (1986)CrossRefGoogle Scholar
  68. 68.
    Pomraning, G.C.: Asymptotic and variational derivations of the simplified P N equations. Ann. Nucl. Energy 20, 623 (1993)CrossRefGoogle Scholar
  69. 69.
    Porzio, M.M., Lopez-Pouso, O.: Application of accretive operators theory to evolutive combined conduction, convection and radiation. Rev. Mat. Iberoam. 20, 257–275 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    Rosen, P.: Entropy of radiation. Phys. Rev. 96, 555 (1954)Google Scholar
  71. 71.
    Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM. J. Sci. Statist. Comput. 7, 856–869 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Schäfer, M., Frank, M., Pinnau, R.: A hierarchy of approximations to the radiative heat transfer equations: Modelling, analysis and simulation. Math. Meth. Mod. Appl. Sci. 15, 643–665 (2005)zbMATHCrossRefGoogle Scholar
  73. 73.
    Schuster, A.: Radiation through a foggy atmosphere. Astrophys. J. 21, 1–22 (1905)CrossRefGoogle Scholar
  74. 74.
    Schwarzschild, K.: Über das Gleichgewicht von Sonnenatmosphären, Akad. Wiss. Göttingen. Math. Phys. Kl. Nachr. 195, 41–53 (1906)Google Scholar
  75. 75.
    Seaïd, M.: Notes on Numerical Methods for Two-Dimensional Neutron Transport Equation, Technical Report Nr. 2232, TU Darmstadt (2002)Google Scholar
  76. 76.
    Seaid, M., Klar, A.: Efficient Preconditioning of Linear Systems Arising from the Discretization of Radiative Transfer Equation, Challenges in Scientific Computing. Springer, Berlin (2003)Google Scholar
  77. 77.
    Sherman, M.P.: Moment methods in radiative transfer problems. J. Quant. Spectrosc. Radiat. Transf. 7, 89–109 (1967)CrossRefGoogle Scholar
  78. 78.
    Struchtrup, H.: On the number of moments in radiative transfer problems. Ann. Phys. 266, 1–26 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    Sykes, J.B.: Approximate integration of the equation of transfer. Mon. Not. R. Astron. Soc. 111, 377 (1951)MathSciNetzbMATHGoogle Scholar
  80. 80.
    Tomašević, D.I., Larsen, E.W.: The Simplified P 2 Approximation. Nucl. Sci. Eng. 122, 309–325 (1996)Google Scholar
  81. 81.
    Turek, S.: An efficient solution technique for the radiative transfer equation. IMPACT, Comput. Sci. Eng. 5, 201–214 (1993)Google Scholar
  82. 82.
    Turek, S.: A generalized mean intensity approach for the numerical solution of the radiative transfer equation. Computing 54, 27–38 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    Turpault, R.: Construction d’une modèle M1-multigroupe pour les équations du transfert radiatif. C. R. Acad. Sci. Paris Ser. I 334, 1–6 (2002)MathSciNetCrossRefGoogle Scholar
  84. 84.
    Turpault, R.: A consistent multigroup model for radiative transfer and its underlying mean opacities. J. Quant. Spectrosc. Radiat. Transf. 94, 357–371 (2005)CrossRefGoogle Scholar
  85. 85.
    Turpault, R., Frank, M., Dubroca, B., Klar, A.: Multigroup half space moment approximations to the radiative heat transfer equations. J. Comput. Phys. 198, 363–371 (2004)zbMATHCrossRefGoogle Scholar
  86. 86.
    Van der Vorst, H.A.: BI-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems. SIAM. J. Sci. Statist. Comput. 13, 631–644 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  87. 87.
    Viskanta, R., Anderson, E.E.: Heat transfer in semitransparent solids. Adv. Heat Transf. 11, 318 (1975)Google Scholar
  88. 88.
    Viskanta, R., Mengüc, M.P.: Radiation heat transfer in combustion systems. Prog. Energy Combust. Sci. 13, 97–160 (1987)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.University of KaiserslauternKaiserslauternGermany
  2. 2.Fraunhofer ITWMKaiserslauternGermany

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