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Radiative Heat Transfer and Applications for Glass Production Processes

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Mathematical Models in the Manufacturing of Glass

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2010))

Abstract

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.

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References

  1. Adams, M.L., Larsen, E.W.: Fast iterative methods for deterministic particle transport computations. Prog. Nucl. Energy 40, 3–159 (2002)

    Article  Google Scholar 

  2. Adams, M.L.: Subcell balance formulations for radiative transfer on arbitrary grids. Transp. Theory and Stat. Phys. 26, 385–431 (1997)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alcouffe, R.: Diffusion synthetic acceleration methods for the diamond-differenced discrete-ordinates equations. Nucl. Sci. Eng. 64, 344–355 (1977)

    Google Scholar 

  5. Agoshkov, V.: On the existence of traces of functions in spaces used in transport theory problems. Sov. Math. Dokl. 33, 628–632 (1986)

    MATH  Google Scholar 

  6. Agoshkov, V.: Boundary value problems for transport equations. Birkhäuser, Boston (1998)

    MATH  Google Scholar 

  7. Anile, A.M., Pennisi, S., Sammartino, M.: A thermodynamical approach to Eddington factors. J. Math. Phys. 32, 544–550 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Atkinson, K.E.: Iterative variants of the Nyström method for the numerical solution of integral equations. Numer. Math. 22, 17–31 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brantley, P.S., Larsen, E.W.: The simplified P 3 approximation. Nucl. Sci. Eng. 134, 1–21 (2000)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  12. Cheng, P.: Dynamics of a radiating gas with applications to flow over a wavy wall. AIAA J. 4, 238–245 (1966)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  14. Davison, B.: Neutron transport theory. Oxford University Press, Oxford (1958)

    Google Scholar 

  15. Dreyer, W.: Maximisation of the entropy in non-equilibrium. J. Phys. A 20, 6505–6517 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dubroca, B.: Thèse d’Etat, Dept. of Mathematics, University of Bordeaux (2000)

    Google Scholar 

  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)

    Article  MATH  Google Scholar 

  19. Dubroca, B., Klar, A.: Half moment closure for radiative transfer equations. J. Comput. Phys. 180, 584–596 (2002)

    Article  MATH  Google Scholar 

  20. Eddington, A.: The Internal Constitution of the Stars. Dover, New York (1926)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. Frank, M.: Partial Moment Models for Radiative Transfer. PhD thesis, TU Kaiserslautern (2005)

    Google Scholar 

  24. Frank, M.: Approximate models for radiative transfer. Bull. Inst. Math. Acad. Sinica (New Series) 2, 409–432 (2007)

    Google Scholar 

  25. Frank, M., Dubroca, B., Klar, A.: Partial moment entropy approximation to radiative transfer. J. Comput. Phys. 218, 1–18 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Frank, M., Pinnau, R.: Analysis of a half moment model for radiative heat transfer equations. Appl. Math. Lett. 20, 189–193 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  28. Golse, F., Perthame, B.: Generalized solution of the radiative transfer equations in a singular case. Commun. Math. Phys. 106, 211–239 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  29. Greenbaum, A.: Iterative Methods for Solving Linear Systems. SIAM, Philadelphia (1997)

    Book  MATH  Google Scholar 

  30. Hackbusch, W.: Multi-Grid Methods and Applications. Springer Series in Computational Mathematics, vol. 4. Springer, New York (1985)

    Google Scholar 

  31. Howell, R., Siegel, J.R.: Thermal Radiation Heat Transfer, 3rd edn. Taylor & Francis, NewYork (1992)

    Google Scholar 

  32. Huang, K.: Introduction to Statistical Physics. Taylor and Francis, New York (2001)

    MATH  Google Scholar 

  33. Jeans, J.H.: The equations of radiative transfer of energy. Mon. Not. R. Astron. Soc. 78, 28–36 (1917)

    Google Scholar 

  34. Junk, M.: Domain of definition of levermore’s five-moment system. J. Stat. Phys. 93, 1143–1167 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia (1995)

    Book  MATH  Google Scholar 

  36. Kelley, C.T.: Multilevel Source Iteration Accelerators for the Linear Transport Equation in Slab Geometry. Transp. Theory Stat. Phys. 24, 679–707 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  39. Klar, A., Schmeiser, C.: Numerical passage from radiative heat transfer to nonlinear diffusion models. Math. Mod. Meth. Appl. Sci. 11, 749–767 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  41. Korganoff, V.: Basic Methods in Transfer Problems. Dover, New York (1963)

    Google Scholar 

  42. Krook, M.: On the solution of equations of transfer. Astrophys. J. 122, 488 (1955)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  44. Laitinen, M.T., Tiihonen, T.: Conductive-radiative heat transfer in grey materials. Quart. Appl. Math. 59 737–768 (2001)

    MathSciNet  MATH  Google Scholar 

  45. Larsen, E.W., Keller, J.B.: Asymptotic solution of neutron transport problems for small mean free path. J. Math. Phys. 15, 75 (1974)

    Article  MathSciNet  Google Scholar 

  46. Larsen, E.W., Pomraning, G., Badham, V.C.: Asymptotic analysis of radiative transfer problems. J. Quant. Spectr. Radiati. Transf. 29, 285–310 (1983)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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. Levermore, C.D.: Relating Eddington factors to flux limiters. J. Quant. Spectroscop. Radiat. Transf. 31, 149–160 (1984)

    Article  Google Scholar 

  51. Levermore, C.D.: Moment closure hierarchies for kinetic theories. J.Stat.Phys. 83 (1996)

    Google Scholar 

  52. Lewis, E.E., Miller, W.F. Jr., Computational Methods of Neutron Transport. Wiley, New York (1984)

    Google Scholar 

  53. Lopez-Pouso, O.: Trace theorem and existence in radiation. Adv. Math. Sci. Appl. 10, 757–773 (2000)

    MathSciNet  MATH  Google Scholar 

  54. Mark, J.C.: The spherical harmonics method, part I. Tech. Report MT 92, National Research Council of Canada (1944)

    Google Scholar 

  55. Mark, J.C.: The spherical harmonics method, part II. Tech. Report MT 97, National Research Council of Canada (1945)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  Google Scholar 

  58. Mercier, B.: Application of accretive operators theory to the radiative transfer equations. SIAM J. Math. Anal. 18, 393–408 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  59. Minerbo, G.N.: Maximum entropy Eddington factors. J. Quant. Spectrosc. Radiat. Transf. 20, 541–545 (1978)

    Article  Google Scholar 

  60. Modest, M.F.: Radiative Heat Transfer, 2nd edn. Academic, San Diego (1993)

    Google Scholar 

  61. Müller, I., Ruggeri, T.: Rational extended thermodynamics. Springer, New York (1998)

    Book  MATH  Google Scholar 

  62. Murray, R.L.: Nuclear reactor physics. Prentice Hall, New Jersey (1957)

    Google Scholar 

  63. Ore, A.: Entropy of radiation. Phys. Rev. 98, 887 (1955)

    Google Scholar 

  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)

    Article  Google Scholar 

  65. Planck, M.: Distribution of energy in the spectrum. Ann. Phys. 4, 553–563 (1901)

    Article  MATH  Google Scholar 

  66. Pomraning, G.C.: The equations of radiation hydrodynamics. Pergamon, New York (1973)

    Google Scholar 

  67. Pomraning, G.C.: Initial and boundary conditions for equilibrium diffusion theory. J. Quant. Spectrosc. Radiat. Transf. 36, 69 (1986)

    Article  Google Scholar 

  68. Pomraning, G.C.: Asymptotic and variational derivations of the simplified P N equations. Ann. Nucl. Energy 20, 623 (1993)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  70. Rosen, P.: Entropy of radiation. Phys. Rev. 96, 555 (1954)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  73. Schuster, A.: Radiation through a foggy atmosphere. Astrophys. J. 21, 1–22 (1905)

    Article  Google Scholar 

  74. Schwarzschild, K.: Über das Gleichgewicht von Sonnenatmosphären, Akad. Wiss. Göttingen. Math. Phys. Kl. Nachr. 195, 41–53 (1906)

    Google Scholar 

  75. Seaïd, M.: Notes on Numerical Methods for Two-Dimensional Neutron Transport Equation, Technical Report Nr. 2232, TU Darmstadt (2002)

    Google Scholar 

  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. Sherman, M.P.: Moment methods in radiative transfer problems. J. Quant. Spectrosc. Radiat. Transf. 7, 89–109 (1967)

    Article  Google Scholar 

  78. Struchtrup, H.: On the number of moments in radiative transfer problems. Ann. Phys. 266, 1–26 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  79. Sykes, J.B.: Approximate integration of the equation of transfer. Mon. Not. R. Astron. Soc. 111, 377 (1951)

    MathSciNet  MATH  Google Scholar 

  80. Tomašević, D.I., Larsen, E.W.: The Simplified P 2 Approximation. Nucl. Sci. Eng. 122, 309–325 (1996)

    Google Scholar 

  81. Turek, S.: An efficient solution technique for the radiative transfer equation. IMPACT, Comput. Sci. Eng. 5, 201–214 (1993)

    Google Scholar 

  82. Turek, S.: A generalized mean intensity approach for the numerical solution of the radiative transfer equation. Computing 54, 27–38 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  84. Turpault, R.: A consistent multigroup model for radiative transfer and its underlying mean opacities. J. Quant. Spectrosc. Radiat. Transf. 94, 357–371 (2005)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  87. Viskanta, R., Anderson, E.E.: Heat transfer in semitransparent solids. Adv. Heat Transf. 11, 318 (1975)

    Google Scholar 

  88. Viskanta, R., Mengüc, M.P.: Radiation heat transfer in combustion systems. Prog. Energy Combust. Sci. 13, 97–160 (1987)

    Article  Google Scholar 

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Acknowledgements

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.

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  1. University of Kaiserslautern, Erwin-Schrödinger-Strasse, 67663, Kaiserslautern, Germany

    Martin Frank & Axel Klar

  2. Fraunhofer ITWM, Fraunhofer Platz 1, 67663, Kaiserslautern, Germany

    Axel Klar

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  1. Martin Frank
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  1. , Dipto. di Matematica "Ulisse Dini", Università degli Studi di Firenze, Viale Morgagni 67/A, Firenze, 50134, Italy

    Antonio Fasano

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Frank, M., Klar, A. (2011). Radiative Heat Transfer and Applications for Glass Production Processes. In: Fasano, A. (eds) Mathematical Models in the Manufacturing of Glass. Lecture Notes in Mathematics(), vol 2010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15967-1_2

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