Error Estimate for Discrete Approximation of the Radiative-Conductive Heat Transfer Problem in a System of Absolutely Bl
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Journal of Mathematical Sciences, Vol. 251, No. 6, December, 2020
ERROR ESTIMATE FOR DISCRETE APPROXIMATION OF THE RADIATIVE-CONDUCTIVE HEAT TRANSFER PROBLEM IN A SYSTEM OF ABSOLUTELY BLACK RODS A. A. Amosov ∗ National Research University “Moscow Power Engineering Institute” 14, Krasnokazarmennaya St., Moscow 111250, Russia [email protected]
N. E. Krymov National Research University “Moscow Power Engineering Institute” 14, Krasnokazarmennaya St., Moscow 111250, Russia [email protected]
UDC 517.95
√ We prove an error estimate of order O( ε/λ) for a discrete approximation of the radiative-conductive heat transfer problem in a system of absolutely black rods bundled in a square box; here, ε is the rod diameter and λ is the heat transfer coefficient. Bibliography: 7 titles. Illustrations: 1 figure.
1
Introduction. Statement of the Problem
The paper continues the series of works [1]–[5] devoted to constructing and justifying special discrete, semidiscrete, and asymptotic approximations of the radiative-conductive heat transfer problem in a periodic system of heat-conductive elements separated by the vacuum. We use the discrete approximation proposed in [5] and designated for the stationary radiative-conductive heat transfer problem in a system consisting of n2 absolutely black rods with circular sections of diameter ε = 1/n. It is assumed that the system of rods is regularly bundled in a square box with boundary Γ (cf. Figure below). With each rod we associate an open disc Gi,j with diameter ε and center xi,j = ((i − 1/2)ε, (j − 1/2)ε), 1 i n, 1 j n. We set G=
n n
Gi,j .
i=1j=1
The stationary heat transfer in the system G without inner sources or sinks of heat is governed by the boundary value problem − div (λ∇u) = 0, ∗
x ∈ G,
(1.1)
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 107, 2020, pp. 3-14. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2516-0773
773
λ
∂u + h(u) = ∂n
h(u(ξ))ϕ(ξ, x) dσ(ξ) +
h(uΓ (ξ))ϕ(ξ, x) dσ(ξ),
x ∈ ∂G,
(1.2)
Γ
∂G
where u(x) = u(x1 , x2 ) is the unknown absolute temperature and λ = const > 0 is the heatconductivity coefficient. The boundary condition (1.2) describes the heat exchange by radiation on the boundary. The function h(u) = σ0 |u|3 u with u > 0 is the density of the heat radiation flow, σ0 > 0 is the Stefan–Boltzmann constant, uΓ is the temperature given on Γ, dσ(x) is the natural measure on ∂G ∪ Γ, and ⎧ ⎨ (n(ξ), x − ξ)(n(x), ξ − x) , [x, ξ] ∩ G = ∅, 2|x − ξ|3 ϕ(ξ, x) = ⎩ 0, [x, ξ] ∩ G = ∅; here, n(x) is the outward normal to ∂G if x ∈ ∂G or the inward normal to Γ if x ∈ Γ.
Γ
Figure. The system of rods.
2
Properties of Solutions to Problem (1.1), (1.2)
We note that the set G is not connected. Respectively, by W 1,2 (G) we understand the space = {u ∈ L2 (G) | u ∈ W 1,2 (Gi,j ), 1 i n, 1 j n}, where W 1,2 (Gi,j ) are the classical Sobolev spaces. Throughout the paper, we assume that uΓ ∈ L∞ (Γ) and
W 1,2 (G)
umin uΓ (x) umax ,
x ∈ Γ,
(2.1)
where umin and umax are pos
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