Transient Conductive–radiative  Heat Transfer: Discrete Existence and Uniqueness for a Finite Volume Scheme

Klein, Olaf; Philip, Peter

doi:10.1142/s0218202505000340

Cited by 13 publications

(24 citation statements)

References 18 publications

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“…Since in view of (38), we have T 2 − T 1 = 0 almost everywhere on 3 , we conclude that T 2 − T 1 ∈ V 2,∞ ( ). Therefore, T 2 − T 1 can be directly inserted in (36).…”

Section: Lemma 36mentioning

confidence: 71%

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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“…Since in view of (38), we have T 2 − T 1 = 0 almost everywhere on 3 , we conclude that T 2 − T 1 ∈ V 2,∞ ( ). Therefore, T 2 − T 1 can be directly inserted in (36).…”

Section: Lemma 36mentioning

confidence: 71%

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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“…The used scheme, including the discretization of nonlocal terms stemming from the modeling of diffuse-gray radiation, was previously described in Refs. [15,18]. The convergence of the scheme has been verified numerically for stationary cases in Refs.…”

Section: Numerical Methods and Implementationmentioning

confidence: 93%

Transient numerical study of temperature gradients during sublimation growth of SiC: Dependence on apparatus design

Geiser

Klein

Philip

2006

Journal of Crystal Growth

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Using transient and stationary mathematical heat transfer models including heat conduction, radiation, and radio frequency (RF) induction heating, we numerically investigate the time evolution of temperature gradients in axisymmetric growth apparatus during the sublimation growth of silicon carbide (SiC) bulk single crystals by physical vapor transport (PVT) (modified Lely method). Temperature gradients in the bulk and on the surface of the growing crystal can cause defects. Here, the evolution of these gradients is studied numerically during the heating, growth, and cooling stages, varying the apparatus design, namely the amount of the source powder charge as well as the size of the upper blind hole used for cooling of the seed. Our results show that a smaller upper blind hole can reduce the temperature gradients both in the bulk and on the surface of the crystal without reducing the surface temperature itself. r

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“…The employed scheme, including the discretization of nonlocal terms stemming from the modeling of diffuse-gray radiation, was previously described in Ref. [13]; modifications to allow for the anisotropic thermal conductivity are treated in Ref. [14].…”

Section: Modeling and Numerical Methodsmentioning

confidence: 99%

Numerical simulation of temperature fields during the sublimation growth of SiC single crystals, using WIAS-HiTNIHS

Geiser

Klein

Philip

2007

Journal of Crystal Growth

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We present numerical computations of the temperature fields in axisymmetric growth apparatus for sublimation growth of silicon carbide (SiC) bulk single crystals by physical vapor transport (PVT) (modified Lely method). The results are computed using our software WIAS-HiTNIHS, the WIAS High Temperature Numerical Induction Heating Simulator; pronunciation: $hit-nice, by solving the energy balance in the entire growth apparatus, taking into account the heat conduction in the solid parts as well as in gas cavities, and also accounting for the radiative heat transfer between the surfaces of the gas cavities. The insulation in a PVT growth apparatus usually consists of graphite felt, where the fibers are aligned in one particular direction, resulting in an anisotropic thermal conductivity. We show that neglecting this anisotropy can overestimate the SiC crystal's temperature by 70 K or underestimate the required heating power by 800 W. r

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Transient Conductive–radiative Heat Transfer: Discrete Existence and Uniqueness for a Finite Volume Scheme

Cited by 13 publications

References 18 publications

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

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

Transient numerical study of temperature gradients during sublimation growth of SiC: Dependence on apparatus design

Numerical simulation of temperature fields during the sublimation growth of SiC single crystals, using WIAS-HiTNIHS

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Transient Conductive–radiative Heat Transfer: Discrete Existence and Uniqueness for a Finite Volume Scheme

Cited by 13 publications

References 18 publications

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

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

Transient numerical study of temperature gradients during sublimation growth of SiC: Dependence on apparatus design

Numerical simulation of temperature fields during the sublimation growth of SiC single crystals, using WIAS-HiTNIHS

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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)

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