Résolutiondestransfertsconducto1 radiatifspar,la,méthodede...
Transcript of Résolutiondestransfertsconducto1 radiatifspar,la,méthodede...
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Résolution des transferts conducto-‐radiatifs par la méthode de Monte Carlo en
milieux poreux Cyril CaliotResearcher CNRS (HDR)[email protected]
Toulouse UniversityS. BLANCOR. FOURNIER
B. PIAUD, C. COUSTETV. EYMET, V. FOREST
M. El HAFI
JERT 2017CEMHTI Orléans
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Outline• Introduction
• Stochastic model
• Simulation configuration
• Results and discussion
• Conclusion and future work
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Solving combined conduction-radiation problems
• Deterministic methods => Need a mesh– Finite differences/volumes/elements– Lattice boltzmannCompute the temperature field
• Stochastic methods => Meshless– Monte CarloCompute the local temperature
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Résolution de la conduction par Monte Carlo
Figs. fromTalebi et al. Prog. Nuc. Energy 96 (2017)
𝝏𝑻𝝏𝒕 + ∆𝑻 = 𝒇 et CI + CL• Fixed random walk
(Curtiss IBM Corp. 1949 ; Emery and Carson ASME JHT 1968)
• Semi floating random walk(Talebi et al. Prog. Nuc. E. 2017)
• Floating random walk (Walk-‐On-‐Sphere)(Haji-‐Sheikh and Sparrow ASME JHT 1966, Grigoriu ASME JHT 2000)λ hétérogène (Burmeister ASME JHT 2002 ; Bahadori et al. IJHMT 2017)
• Brownian motion (Itô processes)(Grigoriu ASCE JEM 1997, ASME JHT 2000)
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Monte Carlo pour les transferts couplés Cond-Ray
• Calcul de la conductivité effective (cond-‐ray) dans un milieu poreux solide-‐gaz au stationnaire– Conduction (Itô-‐Taylor) et MC pour ETR
• Solide opaque : Vignoles IJHMT 2016CL avec linéarisation du transfert radiatif• Solide semi-‐transparent : Dauvois Thèse 2016Couplage non-‐linéaire de la conduction et du rayonnement : itérations
• Calcul de T locale dans un solide :– Transitoire, cond-‐conv-‐ray : Fournier et al.
Eurotherm 2015– Stationnaire, cond-‐ray (Caliot et al. SFT 2017)
𝝏𝑻𝝏𝒕 + ∆𝑻 = 𝒇 + ETR dans des milieux semi-‐transparents et CI + CL
From Vignoles IJHMT 2016
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Stochastic method
𝑇 𝒙𝒃 = 𝜆/𝛿/𝜆𝛿/+ ℎ1
𝑇 𝒙𝒃 − 𝛿/𝒏 + ℎ1
𝜆𝛿/+ ℎ1
𝑇145 𝒙𝒃
ℎ1 = 4𝜖8𝜎𝑇1:;<
𝑝5>;; =𝜆/𝛿/𝜆𝛿/+ ℎ1
𝛿5>;; =𝛿/2
02
2
2
2
2
2
=¶¶
+¶¶
+¶¶
zT
yT
xTSolid: Opaque, diffuse,
homogeneous, complex geometry in vacuum
Conduction-‐radiation flux balance at the boundary: 𝛿/ and ℎ1
Stationnary Conduction-‐Radiation problem
𝑇(𝑥, 𝑦, 𝑧) = F 𝑝G 𝑇 𝑟 𝑑𝐴G
(r) 𝑝G =LG
𝑇(𝑥, 𝑦, 𝑧) ≅ 1𝑁P𝑇/,>
Q
>RL
Dirichlet BC
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Linearization of the radiative exchange
𝑞LT = 𝜎 𝑇LU − 𝑇TU
𝑞LTV = 4𝜎𝑇1:;< 𝑇L − 𝑇T 𝑅𝑒𝑙. 𝐸𝑟𝑟 = 𝑞LT − 𝑞LTV𝑞LT
𝑻𝒓𝒆𝒇 = 𝑻𝟏 + 𝑻𝟐
𝟐
𝑇L [K]
𝑇T [K]
𝑅𝑒𝑙. 𝐸𝑟𝑟
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Simulation configuration
Case 𝝀, W.m-‐1.K-‐1 𝒑𝒅𝒊𝒇𝒇 Tmin -‐Tmax, K
1a 40 ~1 300-‐310
1b 10-‐3 ~0.65 300-‐310
2a 4.2 10-‐3 0.1 1000-‐1500
2b 3.765 10-‐2 0.9 1000-‐1500
Objective: compute the average temperature along the plane
𝑇 = F 𝑝G 𝑇 𝒙 𝑑𝒙G
Plate2 mm
Plate2 mm 3 Kelvin’s cells
3*4 mm
𝛿/ = 0.1 𝑚𝑚 ; 𝛿5>;; = 𝛿/2
Strutdiameter: 0.6 mm
Strutemissivity:0.85
Stochastic method:C++ code Startherm (GPL)
Deterministic method:ANSYS Fluent -‐ Energy balance eq.
2nd order upwind-‐ Radiative transfer eq.
Discrete ordinates-‐ 1st order upwind,
6*6 disc. Octant, pixelation 6*6
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Results: Low temperature difference 10 K
𝜆 = 40 W.m-‐1.K-‐1 𝑝5>;;~ 1 𝜆 = 10-‐3 W.m-‐1.K-‐1 𝑝5>;;~ 0.65
High conductivity: no influence of radiation Low conductivity: Temperature homogeneization by radiation
Plate2 mm
Plate2 mm
The stochastic method isvalidated with low temperaturedifferences
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Results: High temperature difference 500 K
𝜆 = 3.765 10-‐2 W.m-‐1.K-‐1 𝑝5>;;~ 0.9 𝜆 = 4.2 10-‐3 W.m-‐1.K-‐1 𝑝5>;;~ 0.1
High influence of radiation
The stochastic method isvalidated with high temperaturedifferences
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Conclusion• A Monte-‐Carlo algorithm was established to solve the
combined conduction and radiation heat transfers in complex geometries and at the stationary regime.
• A comparison was conducted with the results obtained with the finite volume method (ANSYS Fluent).
• When conduction or radiation dominates the heat transfers, the stochastic method reproduces well the results of the finite volume method and it is considered numerically validated.
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Future work• Accelerate the diffusion (conduction) random path:
using the Walk-‐On-‐Rectangle technique.• Extend the algorithm to a non-‐stationary regime• Introduce the convection heat transfer mode
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Thanks for your attention