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Radiative heating of a glass plate

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

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MathematicS in Action 5 (2012), Nr.1, S.1-30
ISSN: 2102-5754
Englisch
Zeitschriftenaufsatz, Elektronische Publikation
Fraunhofer ITWM ()

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 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 L2 (0, t f ; C ( [ 0 , l ] ) ) . We use also Stampacchia truncation method to derive lower and upper bounds on the solution.

: http://publica.fraunhofer.de/dokumente/N-356193.html