HOMOGENIZATION OF A CONVECTIVE, CONDUCTIVE AND RADIATIVE HEAT - - PowerPoint PPT Presentation

Homogenization of a heat transfer problem 1

• G. Allaire

HOMOGENIZATION OF A CONVECTIVE, CONDUCTIVE AND RADIATIVE HEAT TRANSFER PROBLEM

Work partially supported by CEA

Gr´ egoire ALLAIRE, Ecole Polytechnique Zakaria HABIBI, CEA Saclay.

• 1. Introduction and model
• 2. Homogenization
• 3. Numerical results

Multiscale Simulation & Analysis in Energy and the Environment, December 12-16, 2011, Linz

Homogenization of a heat transfer problem 2

• G. Allaire
• I- INTRODUCTION

✃ Motivation: gas cooled nuclear reactor core. ✃ Heat transfer by convection, conduction and radiation. ✃ Very heterogenous periodic porous medium Ω: fluid part ΩF

ǫ , solid part ΩS ǫ .

✃ Small parameter ǫ = ratio between period and macroscopic size. ✃ Interface Γǫ between solid and fluid where the radiative operator applies. Goals: define a macroscopic or effective model (not obvious), propose a multiscale numerical algorithm.

Homogenization of a heat transfer problem 3

• G. Allaire

Homogenization of a heat transfer problem 4

• G. Allaire

✄ ✂

• ✁

Model of radiative transfer ✃ Radiative transfer takes place only in the gas (assumed to be transparent). ✃ Model = non-linear and non-local boundary condition on the interface Γ. ✃ For simplicity we assume black walls (emissivity e = 1). ✃ Single radiation frequency. On Γ, continuity of the temperature and of the total heat flux T S = T F and − KS∇T S · n = −KF ∇T F · n + σG

• (T F )4
• n Γ

with σ the Stefan-Boltzmann constant and F(s, x) the view factor G(T 4) = (Id − ζ)(T 4) and ζ(T 4)(s) =

• Γ

T 4(x)F(s, x)dx

Homogenization of a heat transfer problem 5

• G. Allaire

✄ ✂

• ✁

Formula for the view factor F 3D(s, x) = nx · (s − x)ns · (x − s) π|x − s|4 , F 2D(s′, x′) = n′

x · (s′ − x′)n′ s · (x′ − s′)

2|x′ − s′|3

Homogenization of a heat transfer problem 6

• G. Allaire

✞ ✝ ☎ ✆ Properties of the radiative operator ✄ The view factor F(s, x) satisfies (for a closed surface Γ) F(s, x) ≥ 0, F(s, x) = F(x, s),

• Γ

F(s, x)ds = 1 ✄ The kernel of G = (Id − ζ) is made of all constant functions ker(Id − ζ) = R ✄ As an operator from L2 into itself, ζ ≤ 1 ✄ The radiative operator G is self-adjoint on L2(Γ) and non-negative in the sense that

• Γ

G(f) f ds ≥ 0 ∀ f ∈ L2(Γ)

Homogenization of a heat transfer problem 7

• G. Allaire

✄ ✂

• ✁

Scaled model                        − div(KS

ǫ ∇T S ǫ )

= f in ΩS

ǫ

− div(ǫKF

ǫ ∇T F ǫ ) + Vǫ · ∇T F ǫ

= 0 in ΩF

ǫ

−KS

ǫ ∇T S ǫ · n

= −ǫKF

ǫ ∇T F ǫ · n + σ

ǫ Gǫ(T F

ǫ )4

• n Γǫ

T S

ǫ

= T F

ǫ

• n Γǫ

Tǫ = 0

• n ∂Ω.

f is the source term (due to nuclear fission, only in the solid part). Vǫ is the (given) incompressible fluid velocity. KS

ǫ , ǫKF ǫ are the thermal conductivities.

Homogenization of a heat transfer problem 8

• G. Allaire

✞ ✝ ☎ ✆ Modelling issues ✍ The solid part ΩS

ǫ is a connected domain.

✍ The fluid part ΩF

ǫ is the union of parallel cylinders.

✍ The cylinders boudaries Γǫ,i are disjoint and are not closed surfaces Gǫ(Tǫ)(s) = Tǫ(s) −

• Γǫ,i

Tǫ(x)F(s, x)dx = (Id − ζǫ)Tǫ(s) ∀ s ∈ Γǫ,i and

• Γǫ,i

F(s, x)dx < 1 Some radiations are escaping at the top and bottom of the cylinders. ✍ The fluid thermal conductivity is very small so it is scaled like ǫ (this is not crucial). ✍ The radiative operator is scaled like 1/ǫ to ensure a perfect balance between conduction and radiation at the microscopic scale y.

Homogenization of a heat transfer problem 9

• G. Allaire

✞ ✝ ☎ ✆ Geometry of Ω Vertical fluid cylinders. x = (x′, x3) with x′ ∈ R2.

Homogenization of a heat transfer problem 10

• G. Allaire

✞ ✝ ☎ ✆ Geometry of the unit cell 2-D unit cell ! Microscopic variable y′ ∈ Λ = ΛS ∪ ΛF .

Homogenization of a heat transfer problem 11

• G. Allaire

✞ ✝ ☎ ✆ Assumptions on the coefficients Given fluid velocity Vǫ(x) = V (x, x′ ǫ ) in ΩF

ǫ ,

with a smooth vector field V (x, y′), defined in Ω × ΛF , periodic with respect to y′ and satisfying the two incompressibility constraints divxV = 0 and divy′V = 0 in ΛF , and V · n = 0 on γ. A typical example is V = (0, 0, V3). Conductivities KS

ǫ (x) = KS(x, x′

ǫ ) in ΩS

ǫ ,

ǫKF

ǫ (x) = ǫKF (x, x′

ǫ ) in ΩF

ǫ ,

where KS(x, y′), KF (x, y′) are periodic symmetric positive definite tensors defined in Ω × Λ.

Homogenization of a heat transfer problem 12

• G. Allaire
• II- HOMOGENIZATION RESULT

By the method of formal two-scale asymptotic expansions Tǫ = T0(x) + ǫ T1(x, x′ ǫ ) + ǫ2 T2(x, x′ ǫ ) + O(ǫ3) we can obtain the homogenized and cell problems (in the non-linear case). A rigorous justification by the method of two-scale convergence has been

• btained in the linear case (upon linearization of the radiative operator).

Homogenization of a heat transfer problem 13

• G. Allaire
• Theorem. T0 is the solution of a non-linear homogenized problem

   − div(K∗(x, T 3

0 )∇T0(x)) + V ∗(x) · ∇T0(x)

= θ f(x) in Ω T0(x) = 0

• n ∂Ω

with the porosity factor θ = |ΛS| / |Λ| and the homogenized velocity V ∗ = 1 |Λ|

• ΛF V (x, y′) dy′.

The corrector term T1 is given by T1(x, y′) =

3

• j=1

ωj(x, T 3

0 (x), y′)∂T0

∂xj (x) where

• ωj(x, T 3

0 (x), y′)

• 1≤j≤3 are the solutions of the cell problems.

Homogenization of a heat transfer problem 14

• G. Allaire

✞ ✝ ☎ ✆ Cell problems

• ωj(x, T 3

0 (x), y′)

• 1≤j≤3 are the solutions of the 2-D cell problems

                       − divy′ KS(x, y′)(ej + ∇yωS

j (y′))

• = 0

in ΛS −KS(y′, x3)(ej + ∇yωS

j (y′)) · n = 4σT 3 0 (x)G(ωS j (y′) + yj)

• n γ

− divy′ KF (x, y′)(ej + ∇yωF

j (y′))

• + V (x, y′) · (ej + ∇yωF

j (y′)) = 0

in ΛF ωF

j (y′) = ωS j (y′)

• n γ

y′ → ωj(y′) is Λ-periodic, First we solve for ωS

j in the solid part with a linearized radiative boundary

condition. Second we solve for ωF

j in the fluid part with a Dirichlet boundary condition.

Homogenization of a heat transfer problem 15

• G. Allaire

✞ ✝ ☎ ✆ Homogenized conductivity coefficients The homogenized conductivity is given by its entries, for j, k = 1, 2, 3, K∗

j,k(x, T 3 0 ) = 1

|Λ|

• ΛS KS(x, y′)(ej + ∇yωj(y′)) · (ek + ∇yωk(y′))dy′

+ 4σT 3

0 (x)

• γ

G(ωk(y′) + yk)(ωj(y′) + yj) + 2σT 3

0 (x)

• γ
• γ

F 2D(s′, y′)|s′ − y′|2dy′ds′ δj3δk3

• The above last term is due to radiation losses at both end of the cylinders.

Note that the cell solutions ωj and the effective conductivity depend on T 3

0 .

Homogenization of a heat transfer problem 16

• G. Allaire

✄ ✂

• ✁

Remarks ✗ Radiative transfer appears only in the cell problems. ✗ Space dimension reduction (3-D to 2-D): the cell problems are 2-D. ✗ Additional vertical diffusivity due to radiation losses. ✗ The 2-D case was simpler (A. and El Ganaoui, SIAM MMS 2008). ✗ Even the formal method of two-scale ansatz is not obvious because the radiative operator has a singular ǫ-scaling. ✗ A naive method of volume averaging does not work. ✗ Numerical multiscale approximation Tǫ ≈ T0(x) + ǫ

3

• j=1

ωj(x, T 3

0 (x), x′

ǫ )∂T0 ∂xj (x) Big CPU gain because of the 3-D to 2-D reduction of the integral operator.

Homogenization of a heat transfer problem 17

• G. Allaire

✞ ✝ ☎ ✆ Key ideas of the proof

• 1. Do not plug the ansatz in the strong form of the equations !
• 2. Rather use the variational formulation (following an idea of J.-L. Lions).
• 3. Periodic oscillations occur only in the horizontal variables x′/ǫ.
• 4. Perform a 3-D to 2-D limit in the radiative operator.
• 5. Transform a Riemann sum over the periodic surfaces Γǫ,i into a volume

integral over Ω.

Homogenization of a heat transfer problem 18

• G. Allaire

✄ ✂

• ✁

Variational two-scale ansatz

• ΩS

ǫ

KS

ǫ (x)∇Tǫ(x) · ∇φǫ(x)dx + ǫ

• ΩF

ǫ

KF

ǫ (x)∇Tǫ(x) · ∇φǫ(x)dx

+

• ΩF

ǫ

Vǫ(x) · ∇Tǫ(x)φǫ(x)dx + σ ǫ

• Γǫ

Gǫ(Tǫ)(x)φǫ(x)ds =

• ΩS

ǫ

f(x)φǫ(x)dx ∀ φǫ ∈ H1

0(Ω)

Take φǫ(x) = φ0(x) + ǫ φ1(x, x′ ǫ ) + ǫ2 φ2(x, x′ ǫ ) and assume Tǫ = T0(x) + ǫ T1(x, x′ ǫ ) + ǫ2 T2(x, x′ ǫ ) + O(ǫ3)

Homogenization of a heat transfer problem 19

• G. Allaire

✞ ✝ ☎ ✆ Singular radiative term The radiative term seems to blow up σ ǫ

• Γǫ

Gǫ(Tǫ)(x)φǫ(x)ds because convergence takes place for lim

ǫ→0 ǫ

• Γǫ

ψ(x, x′ ǫ )ds = 1 |Λ|

• Ω
• γ

ψ(x, y′) dx dsy′ However, using the fact that ker Gǫ = R and performing a Taylor expansion of the test function around the center of each cylinder Γǫ,i, one can gain a ǫ2 factor.

Homogenization of a heat transfer problem 20

• G. Allaire

✞ ✝ ☎ ✆ 3-D to 2-D asymptotic of the view factor

• Lemma. For any given s3 ∈ (0, L),

L F 3D(s, x)dx3 = F 2D(s′, x′) + O( ǫ2 L3 ) For any function g ∈ C3(0, L) with compact support in (0, L), L g(x3)F 3D(s, x)dx3 = F 2D(s′, x′)

• g(s3) + |x′ − s′|2

2 g′′(s3) + O(ǫ3| log ǫ|)

• ,

where g′′ denotes the second derivative of g. ✗ Note that |x′ − s′|2 = O(ǫ2). ✗ The corrector term, proportional to g′′(s3), is the cause of the additional vertical diffusion.

Homogenization of a heat transfer problem 21

• G. Allaire

✄ ✂

• ✁

Corrector To obtain the corrector term in the fluid part, we perform an ansatz of the variational formulation aǫ(Tǫ, φǫ) = Lǫ(φǫ) ∀ φǫ ∈ H1

0(Ωǫ)

as a0(T0, T1, φ0, φ1) + ǫa1(T0, T1, T2, φ0, φ1, φ2) = L0(φ0, φ1) + ǫL1(φ0, φ1, φ2) + O(ǫ2) The zero-order equality a0(T0, T1, φ0, φ1) = L0(φ0, φ1) gives the homogenized equation and the cell problem in the solid part. The first-order equality a1(T0, T1, T2, φ0, φ1, φ2) = L1(φ0, φ1, φ2) yields the cell problem in the fluid part.

Homogenization of a heat transfer problem 22

• G. Allaire
• III- NUMERICAL RESULTS

Geometry of a typical fuel assembly for a gas-cooled nuclear reactor Ω = 3

j=1(0, Lj) with L3 = 0.025m and, for j = 1, 2, Lj = Njℓjǫ where N1 = 3,

N2 = 4 and ℓ1 = 0.04m, ℓ2 = 0.07m.

Homogenization of a heat transfer problem 23

• G. Allaire

✞ ✝ ☎ ✆ Numerical parameters ✍ Reference computation for ǫ = ǫ0 = 1

4.

✍ Each (2-D) periodicity cell contains 2 circular holes with radius 0.0035m. ✍ No source term, f = 0. ✍ Periodic boundary conditions in the x1 direction and non-homogeneous Dirichlet boundary conditions in the other directions Tǫ = 3200x1 + 400x2 + 800 ✍ Conductivities KS = 30Wm−1K−1 and ǫ0KF = 0.3Wm−1K−1. ✍ Constant vertical velocity V = 80 e3 ms−1.

Homogenization of a heat transfer problem 24

• G. Allaire

✞ ✝ ☎ ✆ Standard homogenization in a fixed domain Ω

Homogenization of a heat transfer problem 25

• G. Allaire

✞ ✝ ☎ ✆ Rescaled process of homogenization with a constant periodicity cell The domain is increasing as ǫ−1Ω.

Homogenization of a heat transfer problem 26

• G. Allaire

✞ ✝ ☎ ✆ Solutions of the cell problems for T0 = 800K

Homogenization of a heat transfer problem 27

• G. Allaire

Homogenized conductivities K∗

• T0=50K

=     25.90 0. 0. 0. 25.91 0. 0. 0. 30.05     , K∗

• T0=20000K

=     49.80 0. 0. 0. 49.71 0. 0. 0. 3680.     . Homogenized velocity V ∗ =     15.13     ms−1.

Homogenization of a heat transfer problem 28

• G. Allaire

✞ ✝ ☎ ✆ Homogenized conductivities as a function of T0

Homogenization of a heat transfer problem 29

• G. Allaire

Homogenization of a heat transfer problem 30

• G. Allaire

Homogenization of a heat transfer problem 31

• G. Allaire

✞ ✝ ☎ ✆ To avoid boundary layers: smaller domain

Homogenization of a heat transfer problem 32

• G. Allaire

✞ ✝ ☎ ✆ To avoid boundary layers: smaller domain

Homogenization of a heat transfer problem 33

• G. Allaire

✄ ✂

• ✁

Relative error as a function of ǫ

Homogenization of a heat transfer problem 34

• G. Allaire

✄ ✂

• ✁

Relative error as a function of ǫ

Homogenization of a heat transfer problem 35

• G. Allaire

The same approach works for unsteady problems too.