Radiative Heating of a Glass Plate

Paquet, Luc; El Cheikh, Raouf; Lochegnies, Dominique; Siedow, Norbert

MathematicS In Action, Tome 5 (2012) no. 1, pp. 1-30.

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 $\kappa \left(\lambda \right)$ 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 $L^2(0,t_f;C\left([0,l]\right))$. We use also Stampacchia truncation method to derive lower and upper bounds on the solution.

Classification : 35K20, 35K55, 35K58, 35K90, 35Q20, 35Q60, 35Q80
Mots-clés : elementary pencil of rays, Planck function, radiative transfer equation, glass plate, nonlinear heat-conduction equation, Stampacchia truncation method, Schauder theorem, Vitali theorem.

[1] Ancona, A. “Continuité des contractions dans les espaces de Dirichlet”, 563, Springer-Verlag, 1976 | MR | Zbl

[2] Brézis, H. “Analyse fonctionnelle Théorie et applications”, Masson, Paris, 1993

[3] Clever, D.; Lang, J. Optimal Control of Radiative Heat Transfer in Glass Cooling with Restrictions on the Temperature Gradient, Optim. Control Appl. Meth. (2011) | Zbl

[4] Dautray, L.; Lions, J.-L. “Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques Volume 8 Evolution: semi-groupe, variationnel”, Masson, Paris, 1988 | Zbl

[5] Desvignes, F. “Rayonnements optiques Radiométrie-Photométrie”, Masson, Paris, 1991

[6] Druet, P.-E. Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and arbitrary p-summable right-hand side, Applications of Mathematics, Volume 55 (2010) no. 2, pp. 111-149 | DOI | MR | Zbl

[7] Evans, L.C. “Partial Differential Equations”, GSM 19, AMS Providence, Rhode Island, 1999

[8] Friedman, A. “Partial Differential Equations”, Robert E. Krieger Publishing Company, 1976

[9] Hinze, M.; Pinnau, R.; Ulbrich, M.; Ulbrich, S. “Optimization with PDE constraints”, Springer, 2009 | Zbl

[10] Hottel, H.C.; Sarofim, A.F. “Radiative Transfer”, McGraw-Hill Book Company, 1967

[11] Kinderlehrer, D.; Stampacchia, G. “An Introduction to Variational Inequalities and their Applications”, Academic Press, 1980 | Zbl

[12] Ladyzenskaja, O.A.; Solonnikov, V.A.; Uralceva, N.N. “Linear and Quasilinear Equations of Parabolic Type”, 23, Translations of Mathematical Monographs, AMS Providence, 1968 | MR

[13] Laitinen, L.T. “Mathematical modelling of conductive-radiative heat transfer”, University of Jyväskylä, Finland (2000) (Ph. D. Thesis)

[14] Laitinen, M.T.; Tiihonen, T. Integro-differential equation modelling heat transfer in conducting, radiating and semitransparent materials, Math. Methods Appl. Sci., Volume 21 (1998) no. 5, pp. 375-392 | DOI | MR | Zbl

[15] Lang, J. Adaptive computation for boundary control of radiative heat transfer in glass, Journal of Computational and Applied Mathematics, Volume 183 (2005), pp. 312-326 | DOI | MR | Zbl

[16] Lions, J.-L. “Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires”, Dunod, Paris, 1967 | Zbl

[17] Maréchal, A. Optique Géométrique Générale, in Encyclopedia of Physics, volume XXIV: Fundamentals of Optics with 761 figures, edited by S. Flügge, Volume 24 (1956), pp. 44-170

[18] Marle, C.-M. “Mesures et probabilités”, Hermann, 1974 | Zbl

[19] Meyer, C.; Philip, P.; Tröltzsch, F. Optimal control of a semilinear PDE with nonlocal radiation interface conditions, SIAM J. Control Optim., Volume 45 (2006), pp. 699-721 | DOI | MR | Zbl

[20] Modest, M.F. “Radiative Heat Transfer”, Academic Press, 2003

[21] Pascali, D.; Sburlan, S. “Non linear mappings of monotone type”, Editura Academiei, Bucuresti, Romania, Sijthoff & Noordhoff International Publishers, 1978 | Zbl

[22] Pinnau, R. Analysis of optimal boundary control for radiative heat ransfer modeled by the SP${}_1$-system, Communications in Mathematical Sciences, Volume 5,4 (2007), pp. 951-969 | DOI

[23] Planck, M. “The theory of Heat Radiation”, Dover Publications Inc., 1991 | Zbl

[24] Reed, M.; Simon, B. “Methods of Modern Mathematical Physics I: Functional analysis”, Academic Press, 1972

[25] Siedow, N.; Grosan, T.; Lochegnies, D.; Romero, E. Application of a New Method for Radiative Heat Tranfer to Flat Glass Tempering, J. Am. Ceram. Soc., Volume 88 (2005) no. 8, pp. 2181-2187 | DOI

[26] Soudre, L. “Etude numérique et expérimentale du thermoformage d’une plaque de verre”, Nancy-Université Henri Poincaré, laboratoire LEMTA,, France (2009) (Ph. D. Thesis)

[27] Taine, J.; Iacona, E.; Petit, J.-P. “Transferts Thermiques Introduction aux transferts d’énergie”, Dunod, 2008

[28] Taine, J.; Petit, J.-P. “Heat Transfer”, Prentice Hall, 1993 | Zbl

[29] Yosida, K. “Functional Analysis”, Springer-Verlag, 1971 | DOI | Zbl