Radiative Heating of a Glass Plate

Luc Paquet[1]; Raouf El Cheikh[2]; Dominique Lochegnies[2]; Norbert Siedow[3]

  • [1] Univ. Lille Nord de France UVHC-ISTV, LAMAV-EDP FR no 2956, 59313 Valenciennes, France (Author to whom all correspondence should be addressed)
  • [2] Univ. Lille Nord de France UVHC-ISTV, TEMPO, 59313 Valenciennes, France
  • [3] Fraunhofer Institute for Industrial Mathematics, ITWM, 67663 Kaiserlautern, Germany

MathematicS In Action (2012)

  • Volume: 5, Issue: 1, page 1-30
  • ISSN: 2102-5754

Abstract

top
This paper aims to prove existence and uniqueness of a solution to the coupling of a nonlinear heat equation with nonlinear boundary conditions with the exact radiative transfer equation, assuming the absorption coefficient κ ( λ ) to be piecewise constant and null for small values of the wavelength λ as in the paper of N. Siedow, T. Grosan, D. Lochegnies, E. Romero, “Application of a New Method for Radiative Heat Tranfer to Flat Glass Tempering”, J. Am. Ceram. Soc., 88(8):2181-2187 (2005). An important observation is that for a fixed value of the wavelength λ , Planck function is a Lipschitz function with respect to the temperature. Using this fact, we deduce that the solution is at most unique. To prove existence of a solution, we define a fixed point problem related to our initial boundary value problem to which we apply Schauder theorem in a closed convex subset of the Banach separable space L 2 ( 0 , t f ; C ( [ 0 , l ] ) ) . We use also Stampacchia truncation method to derive lower and upper bounds on the solution.

How to cite

top

Paquet, Luc, et al. "Radiative Heating of a Glass Plate." MathematicS In Action 5.1 (2012): 1-30. <http://eudml.org/doc/251048>.

@article{Paquet2012,
abstract = {This paper aims to prove existence and uniqueness of a solution to the coupling of a nonlinear heat equation with nonlinear boundary conditions with the exact radiative transfer equation, assuming the absorption coefficient $\kappa (\lambda )$ to be piecewise constant and null for small values of the wavelength $\lambda $ as in the paper of N. Siedow, T. Grosan, D. Lochegnies, E. Romero, “Application of a New Method for Radiative Heat Tranfer to Flat Glass Tempering”, J. Am. Ceram. Soc., 88(8):2181-2187 (2005). An important observation is that for a fixed value of the wavelength $\lambda $, Planck function is a Lipschitz function with respect to the temperature. Using this fact, we deduce that the solution is at most unique. To prove existence of a solution, we define a fixed point problem related to our initial boundary value problem to which we apply Schauder theorem in a closed convex subset of the Banach separable space $L^\{2\}(0,t_\{f\};C([0,l]))$. We use also Stampacchia truncation method to derive lower and upper bounds on the solution.},
affiliation = {Univ. Lille Nord de France UVHC-ISTV, LAMAV-EDP FR no 2956, 59313 Valenciennes, France (Author to whom all correspondence should be addressed); Univ. Lille Nord de France UVHC-ISTV, TEMPO, 59313 Valenciennes, France; Univ. Lille Nord de France UVHC-ISTV, TEMPO, 59313 Valenciennes, France; Fraunhofer Institute for Industrial Mathematics, ITWM, 67663 Kaiserlautern, Germany},
author = {Paquet, Luc, El Cheikh, Raouf, Lochegnies, Dominique, Siedow, Norbert},
journal = {MathematicS In Action},
keywords = {elementary pencil of rays; Planck function; radiative transfer equation; glass plate; nonlinear heat-conduction equation; Stampacchia truncation method; Schauder theorem; Vitali theorem},
language = {eng},
number = {1},
pages = {1-30},
publisher = {Société de Mathématiques Appliquées et Industrielles},
title = {Radiative Heating of a Glass Plate},
url = {http://eudml.org/doc/251048},
volume = {5},
year = {2012},
}

TY - JOUR
AU - Paquet, Luc
AU - El Cheikh, Raouf
AU - Lochegnies, Dominique
AU - Siedow, Norbert
TI - Radiative Heating of a Glass Plate
JO - MathematicS In Action
PY - 2012
PB - Société de Mathématiques Appliquées et Industrielles
VL - 5
IS - 1
SP - 1
EP - 30
AB - This paper aims to prove existence and uniqueness of a solution to the coupling of a nonlinear heat equation with nonlinear boundary conditions with the exact radiative transfer equation, assuming the absorption coefficient $\kappa (\lambda )$ to be piecewise constant and null for small values of the wavelength $\lambda $ as in the paper of N. Siedow, T. Grosan, D. Lochegnies, E. Romero, “Application of a New Method for Radiative Heat Tranfer to Flat Glass Tempering”, J. Am. Ceram. Soc., 88(8):2181-2187 (2005). An important observation is that for a fixed value of the wavelength $\lambda $, Planck function is a Lipschitz function with respect to the temperature. Using this fact, we deduce that the solution is at most unique. To prove existence of a solution, we define a fixed point problem related to our initial boundary value problem to which we apply Schauder theorem in a closed convex subset of the Banach separable space $L^{2}(0,t_{f};C([0,l]))$. We use also Stampacchia truncation method to derive lower and upper bounds on the solution.
LA - eng
KW - elementary pencil of rays; Planck function; radiative transfer equation; glass plate; nonlinear heat-conduction equation; Stampacchia truncation method; Schauder theorem; Vitali theorem
UR - http://eudml.org/doc/251048
ER -

References

top
  1. A. Ancona, “Continuité des contractions dans les espaces de Dirichlet”, 563 (1976), Springer-Verlag Zbl0341.31006MR588389
  2. H. Brézis, “Analyse fonctionnelle Théorie et applications”, (1993), Masson, Paris Zbl0511.46001MR697382
  3. D. Clever, J. Lang, Optimal Control of Radiative Heat Transfer in Glass Cooling with Restrictions on the Temperature Gradient, Optim. Control Appl. Meth. (2011) Zbl1258.49066MR2910090
  4. L. Dautray, J.-L. Lions, “Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques Volume 8 Evolution: semi-groupe, variationnel”, (1988), Masson, Paris Zbl0749.35004
  5. F. Desvignes, “Rayonnements optiques Radiométrie-Photométrie”, (1991), Masson, Paris 
  6. P.-E. Druet, Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and arbitrary p-summable right-hand side, Applications of Mathematics 55 (2010), 111-149 Zbl1224.35057MR2600939
  7. L.C. Evans, “Partial Differential Equations”, GSM 19 (1999), AMS Providence, Rhode Island Zbl0902.35002MR1625845
  8. A. Friedman, “Partial Differential Equations”, (1976), Robert E. Krieger Publishing Company MR454266
  9. M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich, “Optimization with PDE constraints”, (2009), Springer Zbl1167.49001MR2516528
  10. H.C. Hottel, A.F. Sarofim, “Radiative Transfer”, (1967), McGraw-Hill Book Company 
  11. D. Kinderlehrer, G. Stampacchia, “An Introduction to Variational Inequalities and their Applications”, (1980), Academic Press Zbl0457.35001MR567696
  12. O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Uralceva, “Linear and Quasilinear Equations of Parabolic Type”, 23 (1968), Translations of Mathematical Monographs, AMS Providence MR241822
  13. L.T. Laitinen, “Mathematical modelling of conductive-radiative heat transfer”, (2000), Finland 
  14. M.T. Laitinen, T. Tiihonen, Integro-differential equation modelling heat transfer in conducting, radiating and semitransparent materials, Math. Methods Appl. Sci. 21 (1998), 375-392 Zbl0958.80003MR1608072
  15. J. Lang, Adaptive computation for boundary control of radiative heat transfer in glass, Journal of Computational and Applied Mathematics 183 (2005), 312-326 Zbl1100.65073MR2158803
  16. J.-L. Lions, “Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires”, (1967), Dunod, Paris Zbl0189.40603
  17. A. Maréchal, Optique Géométrique Générale, in Encyclopedia of Physics, volume XXIV: Fundamentals of Optics with 761 figures, edited by S. Flügge, 24 (1956), 44-170 
  18. C.-M. Marle, “Mesures et probabilités”, (1974), Hermann Zbl0306.28001MR486378
  19. C. Meyer, P. Philip, F. Tröltzsch, Optimal control of a semilinear PDE with nonlocal radiation interface conditions, SIAM J. Control Optim. 45 (2006), 699-721 Zbl1109.49026MR2246096
  20. M.F. Modest, “Radiative Heat Transfer”, (2003), Academic Press Zbl1042.76592
  21. D. Pascali, S. Sburlan, “Non linear mappings of monotone type”, (1978), Editura Academiei, Bucuresti, Romania, Sijthoff & Noordhoff International Publishers Zbl0423.47021
  22. R. Pinnau, Analysis of optimal boundary control for radiative heat ransfer modeled by the SP 1 -system, Communications in Mathematical Sciences 5,4 (2007), 951-969 Zbl1145.49015MR2375055
  23. M. Planck, “The theory of Heat Radiation”, (1991), Dover Publications Inc. Zbl0127.21701MR111466
  24. M. Reed, B. Simon, “Methods of Modern Mathematical Physics I: Functional analysis”, (1972), Academic Press Zbl0459.46001MR493419
  25. N. Siedow, T. Grosan, D. Lochegnies, E. Romero, Application of a New Method for Radiative Heat Tranfer to Flat Glass Tempering, J. Am. Ceram. Soc. 88 (2005), 2181-2187 
  26. L. Soudre, “Etude numérique et expérimentale du thermoformage d’une plaque de verre”, (2009), France 
  27. J. Taine, E. Iacona, J.-P. Petit, “Transferts Thermiques Introduction aux transferts d’énergie”, (2008), Dunod 
  28. J. Taine, J.-P. Petit, “Heat Transfer”, (1993), Prentice Hall Zbl0818.73001
  29. K. Yosida, “Functional Analysis”, (1971), Springer-Verlag Zbl0217.16001

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.