Radiative Heating of a Glass Plate

Luc Paquet

Description

Niveau: Supérieur, Licence, Bac+2
Radiative Heating of a Glass Plate Luc Paquet?, 59313 Valenciennes France, Raouf El Cheikh, 59313 Valenciennes France, Dominique Lochegnies, 59313 Valenciennes France, Norbert Siedow, 67663 Kaiserslautern Deutchland. Abstract This paper aims to prove existence and uniqueness of a solution to the coupling of a nonlinear heat equation with nonlinear boundary condi- tions with the exact radiative transfer equation, assuming the absorption coefficient ?(?) to be piecewise constant and null for small values of the wavelength ? 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 ?, Planck's 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's theorem in a closed convex subset of the Banach separable space L2(0, tf ;C([0, l])). We use also Stampacchia's truncation method to derive lower and upper bounds on the solution. Keywords: elementary pencil of rays, Planck's function, radiative transfer equation, glass plate, nonlinear heat-conduction equation, Stampacchia's trun- cation method, Schauder's theorem, Vitali's theorem.

• face xg


• hemispher- ical emittance


• radiative intensity


• glass plate


• horizontal area


• rays after


• angle d?


• flat glass


• equation

Sujets

• Paquet
• Photographic plate
• Flat glass
• Équation

Informations

• Publié par: profil-vieg-2012
• Nombre de lectures: 20
• Langue: English

Extrait

Radiative

Heating

of a Glass Plate

Luc Paquet ∗ , 59313 Valenciennes France, Raouf El Cheikh, 59313 Valenciennes France, Dominique Lochegnies, 59313 Valenciennes France, Norbert Siedow, 67663 Kaiserslautern Deutchland.

Abstract

This paper aims to prove existence and uniqueness of a solution to the coupling of a nonlinear heat equation with nonlinear boundary condi- tions with the exact radiative transfer equation, assuming the absorption coefficientκ(λ) to be piecewise constant and null for small values of the wavelengthλas 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”, Ceram. Soc.,J. Am.88[8] 2181-2187 (2005). An important observation is that for a fixed value of the wavelengthλ, Planck’s 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’s theorem in a closed convex subset of the Banach separable spaceL 2 (0, t f ;C([0, l])). We use also Stampacchia’s truncation method to derive lower and upper bounds on the solution.

Keywords: elementary pencil of rays, Planck’s function, radiative transfer equation, glass plate, nonlinear heat-conduction equation, Stampacchia’s trun- cation method, Schauder’s theorem, Vitali’s theorem.

Mathematics Subject Classification (2010): 35K20,35K55,35K58,35K90, 35Q20, 35Q60, 35Q80.

1 Introduction and statement of the problem

We consider an infinite plane horizontal glass plate of widthl, laid down on its lower facex g = 0, on a black sheet-metal maintained at absolute ambient tem- peratureT=T a . Thexis directed upward orthogonally to the glass plate-axis ∗ Author  luc.paquet@univ- e-mail:to whom all correspondence should be addressed. valenciennes.fr, [e-mail] (complete addresses of the authors are written at the end of the manuscript).

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so that the upper (resp. lower) face of the glass-plate hasx g =l(resp.x g = 0) for equation. An infinite plane black sheet metalS, at absolute temperature T S (t) at timet, placed above the glass plate emits radiation in every direction, whose radiative intensity at wavelengthλin the (dry) air gap between is given by Planck’s function:

2C 1 B(T  λ) = −) w 1 hitT=T S (t).(1) λ 5 (e C 2 λT C 1 =hc 2 0 = 0.595531 10 −16 W.m 2 /sr andC 2 = ch k B 0 = 1.438786 10 −2 m. ◦ K ([5], p.98). For a thermal ray, we denote byθits polar angle with thex−axis and byµ:= cos(θ). After refraction at the interfacex g =lbetween air and glass, some part of the radiative energy emitted by the sourceS, will be absorbed i.e. converted into heat in the glass producing in such a way an increase of the tem- peratureT(x t assume independency with respect to the) in the glass plate. We coordinatesyandz radiative heating of the Thisof all the dependent variables. glass plate is modelized by the following equations and boundary conditions in the glass plate in the unknownsI k (x t µ) (k= 1 . . .  M) and the temperature T(x t) (0≤x≤l0≤t≤t f ),t f denoting the final time of the heating process. Let us explain before what are the unknownsI k (x t µ) (k= 1 . . .  M). Firstly, given an electromagnetic wave, we denote byλthe wavelength of the wave in vacuum or (dry) air, the corresponding wavelength in glass being nλ g wheren g means the refractory index of the glass (n g ≈1.46) (this wavelength is denoted λ 0  Likein [19] p.8).in [24], [12], [3], we assume that the absorption coefficient  κ(λ) is piecewise constant: κ(λ) =κ k ∈R ∗ + forλ∈[λ k  λ k+1 [ k= 1 . . .  M(2) where theMintervals [λ k  λ k+1 [,k= 1 . . .  Mform a partition of the glass semi-tranparent region in the electromagnetic wave spectrum. ByI(x t µ λ), we denote the spectral radiative intensity (to be correct, we should say “specific intensity”) at any pointPof elevationx, timet, wavelengthλin a direction at pointPmaking a polar angleθ=Arccos(µ)∈[0 π] with the parallel to thex-axis at pointPazimuthal symmetry of the spectral radiative assume . We intensity with respect to the direction. The spectral radiative intensity is defined in photometry in the following way: considering a small areadAwith normal ~nat the pointP∈dA, the radiative energydEwith wavelength in the interval [λ λ+dλ] which flows throughdA, during the time interval [t t+dt], in directions confined to a narrow cone of solid angledΩ whose mean axiss~makes an angle θwith~n, is given by the formula ([10], p. 7) 13): ([22], p.

dE=I(λPt~s) cos(θ)dA dt dΩdλ

(3)

I(P~stλcoefficient of proportionality called the spectral radiative) being a intensity at pointP, timet, in directions~belonging to the unit sphere with center at pointP, and at wavelengthλ to our azimuthal symmetry hy-. Due pothesis of the spectral radiative intensity with respect to the direction~s,dE

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in the left-hand side of formula (3), does not depend on the azimuthal angle of the mean axis of the narrow cone of directions of solid angledΩ. ByI k (x t µ) (k= 1 . . .  M), we mean: λ k+1 I k (x t µ) := Z I(x t µ λ)dλ.

λ k Similarly for a given absolute temperatureT, we define: λ k+1 B k (T) := Z B(T  λ)dλ k= 1 . . .  M

λ k whereB(T  λ us notePlanck function defined by formula (1). Let) denotes the thatB(T  λdoes not depend on the direction nor directly on) x; this radia- tive intensity of pencil of rays emitted in vacuum, or in dry air, by a black- body at absolute temperatureTdepends only on the absolute temperatureT and on the wavelengthλ also denote by. WeB g (T  λ) :=n g 2 B(T  λ) and by B gk (T) :=n g 2 B k (T) (k= 1 . . .  M both sides of the radiative). Integrating transfer equation in the glass plate, assuming no scattering in the glass plate ([10], p.343, p.9):

µ dI(x,t,µ,λ dx ) +κ(λ)I(x t µ λ) =κ(λ)B g (T(x t) λ) (0≤x≤l0≤t≤t f −1< µ <1 > λ0) (4) and using (2), we obtain the following system ofMequations in the glass plate only coupled by the temperatureT(x t): µ dI k ( dx xt,µ,) +κ k I k (x t µ ≤ ) t = f κ  k − B 1 gk < (T µ ( < tx 1) )) k  = 1 . . .  M (5) (0≤x≤l0≤t  one equation for each wavelength band [λ k  λ k+1 [,k= 1 . . .  M. Now let us derive the boundary conditions for the equations (5). Considering an elementary pencil of rays ([22], p.18, second paragraph) emitted by the black sourceSof radiative intensityB(T S (t) λin (dry) air, a balance of energy shows that after) refraction in the “direction”µ(−1< µ <0) at the interfacex g =lbetween air and glass, that its radiative intensity will become (1−ρ g (µ))n 2 g B(T S (t) λ) = (1−ρ g (µ))B g (T S (t) λ)(6) whereρ g (µ) denotes the reflectivity coefficient given by Fresnel’s relation ([19], formula (2.96) p. 47). The factorn 2 g appearing in the left-hand side of equation (6) is due to the conservation of the optical outspread of an elementary pencil of rays after refraction (see [10] p.8 or [5] formula (29) p.35) (a particular case ofthesocalledClausius’relationbyA.Mare´chal:[16],pp.56-59),thatisof the product of the square of the refractive index, times the cross-sectional area

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normal to the elementary pencil of rays, times its solid angle of directionsdΩ. Making once again a balance of energy, in an elementary pencil of rays diverging from a small horizontal areadA([22], p.18, second paragraph) contained in the interfacex g =lin a narrow cone with solid angle, in directions lying dΩ, its mean axis making a polar angleθ=Arccos(µ) (−1< µ <0) with thex-axis, we obtain the following boundary condition on the surfacex g =lof the glass plate (somewhat similar to [22], p.33-34):

I(l t µ λ) =ρ g (µ)I(l t−µ λ) + (1−ρ g (µ))B g (T S (t) λ) for−1< µ <00≤t≤t f  λ >0. After integration of both sides of this equation with respect to wavelengthλ fromλ k toλ k+1 , we obtain the following boundary condition on the surface x g =lof the glass plate for equation (5): I k (l t µ) =ρ g (µ)I k (l t−µ) + (1−ρ g (µ))B gk (T S (t))−1< µ <00≤t≤t f  (7) fork= 1 . . .  M the lower face. Onx g = 0 of the glass plate in contact with the black-sheet metal maintained at ambient absolute temperatureT=T a , we have the simple boundary condition for the “integrated” radiative intensity I k (x t µ): I k (0 t µ) =B gk (T a )0< µ <10≤t≤t f .(8) The heat source in the heat-conduction equation is given by minus times the divergence of the radiative flux ([27], p.221-222), ([26], p.354-355):

+∞+1 q(x t) = 2π Z Z µ I(x t µ λ)dµ dλ

0−1

Using the radiative transfer equation (4) and our hypothesis (2) on the absorp- tion coefficientκ(λ), we have:

+∞+1 ∂ ∂ q x (x t) =−2π Z Z µ ∂ I∂ x (x t µ λ)dµ dλ − 0−1 +∞+∞+1 =−4π Z κ(λ)B g (T(x t) λ)dλ+ 2π Z Z κ(λ)I(x t µ λ)dµ dλ (9) 0 0−1 k=M k=M +1 =− X 4πκ k B gk (T(x t)) + X 2πκ k Z I k (x t µ)dµ. k=1k 1 1= −

+1 Thus the quantities of interest are Z I k (x t µ)dµ(k= 1 . . .  M) for which −1 we shall give an explicit formula by solving explicitely for eachk∈ {1 . . .  M}

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