Conductive-radiative heat transfer in grey materials

Laitinen, M.; Tiihonen, Timo

doi:10.1090/qam/1866555

Cited by 43 publications

(61 citation statements)

References 15 publications

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“…Analogous to (2.1) problems arise in conductive-radiative heat transfer, see, for instance, M. Laitinen and T. Tiihonen [9], K. Birgelis [3], C. Mayer and others [10] and references therein.…”

Section: Consider the Problem (With Respect Tomentioning

confidence: 99%

“…In [3,9] it was shown that if functions f 0 , g, u * belong to a given set M 0 , elements of which are of practical interest, then the solutions u 0 of (2.1) are bounded and there exists a constant u max such that…”

Section: Modified Problemmentioning

confidence: 99%

“…We mention only few of them: glass fabric production [3,4]; glass melting and fining [8]; crystal growth in the production of semiconductors [10]. The corresponding mathematical model for conductive-radiative steady-state heat transfer was suggested and investigated by M. Laitinen and T. Tiihonen in [9], where the solvability of the state equation and boundedness of solutions for practically interesting data was given (see also [3,10]). …”

Section: Introductionmentioning

confidence: 99%

“…1) according to [9] is given by the variational equality Here (x) ∈ (0, 1] is emissivity distribution on Γ ∪ Γ h , I is the identity operator and K ∈ L(L 5/4 (Γ ∪ Γ h ) → L 5/4 (Γ ∪ Γ h )) is an integral operator defined as…”

Section: Introductionmentioning

confidence: 99%

“…The key assumptions in our approach are that the operator H, caused by the radiative process, is a contraction in L 1 (Γ ) due to "openness" of the surface Γ (see, for instance, [9]) and that the "good geometry" of the system ensures necessary solvability of the Neumann problem for the Poisson equation in Ω with right hand sides in spaces L p (Ω; R 3 ) with p ∈ [4/3, 4].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Strictly Convergent Algorithm for an Elliptic Equation With Nonlocal and Nonlinear Boundary Conditions

Birgelis

Raitums

2012

Mathematical Modelling and Analysis

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Abstract. The paper describes a formally strictly convergent algorithm for solving a class of elliptic problems with nonlinear and nonlocal boundary conditions, which arise in modeling of the steady-state conductive-radiative heat transfer processes. The proposed algorithm has two levels of iterations, where inner iterations by means of the damped Newton method solve an appropriate elliptic problem with nonlinear, but local boundary conditions, and outer iterations deal with nonlocal terms in boundary conditions.

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Section: Consider the Problem (With Respect Tomentioning

confidence: 99%

Section: Modified Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Strictly Convergent Algorithm for an Elliptic Equation With Nonlocal and Nonlinear Boundary Conditions

Birgelis

Raitums

2012

Mathematical Modelling and Analysis

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Weak solutions to a stationary heat equation with nonlocal radiation boundary condition and right‐hand side in L^p (p⩾1)

Druet

2008

Math Methods in App Sciences

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SUMMARYAccurate modelling of heat transfer in high-temperature situations requires accounting for the effect of heat radiation. In complex industrial applications involving dissipative heating, we hardly can expect from the mathematical theory that the heat sources will be in a better space than L 1 . In this paper, we focus on a stationary heat equation with nonlocal boundary conditions and L p right-hand side, with p 1 being arbitrary. Thanks to new coercivity results, we are able to produce energy estimates that involve only the L p norm of the heat sources and to prove the existence of weak solutions. Copyright q 2008 John Wiley & Sons, Ltd.KEY WORDS: existence of generalized solutions; nonlinear PDE of elliptic type; radiative heat transfer; nonlocal boundary condition; right-hand side L 1 INTRODUCTIONAccurate modelling of heat transfer in high-temperature situations requires accounting for the effect of heat radiation. In the field of industrial applications, crystal growth, for example, has motivated a lot of mathematical work on this topic [1][2][3][4][5][6]. For this type of applications, situations are relevant in which a transparent medium is enclosed by one or several opaque or diffuse * Correspondence to: Pierre-Étienne Druet, Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany. † E-mail: druet@wias-berlin.de The heat radiation incoming in a part of the boundary of the transparent cavity 0 depends on the radiation outgoing at the other parts of the surface that it can see. This effect has to be modelled by means of nonlocal radiation boundary conditions for the conductive heat flux (see, for example, [2,6]).From the point of view of mathematical analysis, an important result was attained in the paper [7], in which existence and uniqueness of generalized solutions were proved for the heat equation with radiation boundary conditions and heat sources in the class [W 1,2 ] * .In the present paper, we want to extend these results to the case where the heat source density might be less regular. In many applications, the heat sources have to be computed from Maxwell's equations (resistive/inductive heating) or from the Navier-Stokes equations (heat-conducting fluids). From the viewpoint of the presently available regularity theory, this leads in complex situations (temperature-dependent coefficients, nonsmooth surfaces) to heat source densities that belong only to L 1 , or at most to L 1+ . The latter observations have motivated research on elliptic problems with L 1 right-hand sides (see, for example, [8,9]). An L 1 -theory is also particularly attractive for the heat equation in that it leads to natural energy estimates, that is, estimates in terms of the total heating power, the quantity which is actually controlled in applications. The mathematical problemWe assume that 1 , . . . , m are disjoint bounded domains in R 3 , separated from each other by a transparent medium 0 . They represent opaque bodies with different material properties. The bounded domain...

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Unique solvability of a stationary radiative‐conductive heat transfer problem in a system consisting of an absolutely black body and several semitransparent bodies

Амосов

2021

Math Methods in App Sciences

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We consider a stationary boundary value problem describing a radiative‐conductive heat transfer in a system consisting of one absolutely black body and several semitransparent bodies. To describe the radiative transfer, the integro‐differential radiative transfer equation is used. We do not take into account the dependence of the radiation intensity and the properties of semitransparent materials on the radiation frequency. We proved at the first time the unique solvability of this problem. Besides, we proved the comparison theorems and established the results on improving the properties of solutions with increasing exponents of data summability.

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Conductive-radiative heat transfer in grey materials

Cited by 43 publications

References 15 publications

Strictly Convergent Algorithm for an Elliptic Equation With Nonlocal and Nonlinear Boundary Conditions

Strictly Convergent Algorithm for an Elliptic Equation With Nonlocal and Nonlinear Boundary Conditions

Weak solutions to a stationary heat equation with nonlocal radiation boundary condition and right‐hand side in L^p (p⩾1)

Unique solvability of a stationary radiative‐conductive heat transfer problem in a system consisting of an absolutely black body and several semitransparent bodies

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Conductive-radiative heat transfer in grey materials

Cited by 43 publications

References 15 publications

Strictly Convergent Algorithm for an Elliptic Equation With Nonlocal and Nonlinear Boundary Conditions

Strictly Convergent Algorithm for an Elliptic Equation With Nonlocal and Nonlinear Boundary Conditions

Weak solutions to a stationary heat equation with nonlocal radiation boundary condition and right‐hand side in Lp (p⩾1)

Unique solvability of a stationary radiative‐conductive heat transfer problem in a system consisting of an absolutely black body and several semitransparent bodies

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Weak solutions to a stationary heat equation with nonlocal radiation boundary condition and right‐hand side in L^p (p⩾1)