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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
Outline• Introduction
• Stochastic model
• Simulation configuration
• Results and discussion
• Conclusion and future work
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
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)
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
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
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
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
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
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
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.
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
13
Thanks for your attention
