Homogenization of a heat transfer problem 1 G. Allaire

HOMOGENIZATION OF A CONVECTIVE,

CONDUCTIVE AND RADIATIVE

HEAT TRANSFER PROBLEM

Work partially supported by CEA

Grégoire 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 ΩFǫ , solid part ΩSǫ .

✃ 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

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�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

TS = TF and −KS∇TS · n = −KF∇TF · n+ σG((TF )4

)on 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

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�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

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�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

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�Scaled model

− div(KSǫ ∇TSǫ ) = f in Ω

Sǫ

− div(ǫKFǫ ∇TFǫ ) + Vǫ · ∇T

Fǫ = 0 in Ω

Fǫ

−KSǫ ∇TSǫ · n = −ǫK

Fǫ ∇T

Fǫ · n+

σ

ǫGǫ(T

Fǫ )

4 on Γǫ

TSǫ = TFǫ on Γǫ

Tǫ = 0 on ∂Ω.

f is the source term (due to nuclear fission, only in the solid part).

Vǫ is the (given) incompressible fluid velocity.

KSǫ , ǫKFǫ are the thermal conductivities.

Homogenization of a heat transfer problem 8 G. Allaire

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�Modelling issues

✍ The solid part ΩSǫ is a connected domain.

✍ The fluid part ΩFǫ 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

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�Geometry of Ω

Vertical fluid cylinders.

x = (x′, x3) with x′ ∈ R2.

Homogenization of a heat transfer problem 10 G. Allaire

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�Geometry of the unit cell

2-D unit cell !

Microscopic variable y′ ∈ Λ = ΛS ∪ ΛF .

Homogenization of a heat transfer problem 11 G. Allaire

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�Assumptions on the coefficients

Given fluid velocity

Vǫ(x) = V (x,x′

ǫ) in ΩFǫ ,

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

KSǫ (x) = KS(x,

x′

ǫ) in ΩSǫ , ǫK

Fǫ (x) = ǫK

F (x,x′

ǫ) in ΩFǫ ,

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

obtained 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 30 )∇T0(x)) + V∗(x) · ∇T0(x) = θ f(x) in Ω

T0(x) = 0 on ∂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, T3

0 (x), y′)∂T0∂xj

(x)

where(ωj(x, T

30 (x), y

′))

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

Homogenization of a heat transfer problem 14 G. Allaire

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�Cell problems

(ωj(x, T

30 (x), y

′))

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

− divy′(KS(x, y′)(ej +∇yω

Sj (y

′)))= 0 in ΛS

−KS(y′, x3)(ej +∇yωSj (y

′)) · n = 4σT 30 (x)G(ωSj (y

′) + yj) on γ

− divy′(KF (x, y′)(ej +∇yω

Fj (y

′)))+ V (x, y′) · (ej +∇yω

Fj (y

′)) = 0 in ΛF

ωFj (y′) = ωSj (y

′) on γ

y′ 7→ ωj(y′) is Λ-periodic,

First we solve for ωSj in the solid part with a linearized radiative boundary

condition.

Second we solve for ωFj in the fluid part with a Dirichlet boundary condition.

Homogenization of a heat transfer problem 15 G. Allaire

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�Homogenized conductivity coefficients

The homogenized conductivity is given by its entries, for j, k = 1, 2, 3,

K∗j,k(x, T3

0 ) =1

|Λ|

[∫

ΛS

KS(x, y′)(ej +∇yωj(y′)) · (ek +∇yωk(y

′))dy′

+ 4σT 30 (x)

∫

γ

G(ωk(y′) + yk)(ωj(y

′) + yj)

+ 2σT 30 (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 T30 .

Homogenization of a heat transfer problem 16 G. Allaire

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�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) + ǫ3∑

j=1

ωj(x, T3

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

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�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

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�Variational two-scale ansatz

∫

ΩSǫ

KSǫ (x)∇Tǫ(x) · ∇φǫ(x)dx+ ǫ

∫

ΩFǫ

KFǫ (x)∇Tǫ(x) · ∇φǫ(x)dx

+

∫

ΩFǫ

Vǫ(x) · ∇Tǫ(x)φǫ(x)dx+σ

ǫ

∫

Γǫ

Gǫ(Tǫ)(x)φǫ(x)ds

=

∫

ΩSǫ

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

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�Singular radiative term

The radiative term seems to blow up

σ

ǫ

∫

Γǫ

Gǫ(Tǫ)(x)φǫ(x)ds

because convergence takes place for

limǫ→0

ǫ

∫

Γǫ

ψ(x,x′

ǫ)ds =

1

|Λ|

∫

Ω

∫

γ

ψ(x, y′) dx dsy′

However, using the fact that kerGǫ = 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

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�3-D to 2-D asymptotic of the view factor

Lemma. For any given s3 ∈ (0, L),

∫ L

0

F 3D(s, x)dx3 = F2D(s′, x′) +O(

ǫ2

L3)

For any function g ∈ C3(0, L) with compact support in (0, L),

∫ L

0

g(x3)F3D(s, x)dx3 = F

2D(s′, x′)(

g(s3) +|x′ − s′|2

2g′′(s3) +O(ǫ

3| 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

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�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) = L

0(φ0, φ1)+ ǫL1(φ0, φ1, φ2)+O(ǫ

2)

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

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�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 e3ms−1.

Homogenization of a heat transfer problem 24 G. Allaire

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�Standard homogenization in a fixed domain Ω

Homogenization of a heat transfer problem 25 G. Allaire

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�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

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�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 ∗ =

0

0

15.13

ms−1.

Homogenization of a heat transfer problem 28 G. Allaire

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�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

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�To avoid boundary layers: smaller domain

Homogenization of a heat transfer problem 32 G. Allaire

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�To avoid boundary layers: smaller domain

Homogenization of a heat transfer problem 33 G. Allaire

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�Relative error as a function of ǫ

Homogenization of a heat transfer problem 34 G. Allaire

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�Relative error as a function of ǫ

Homogenization of a heat transfer problem 35 G. Allaire

The same approach works for unsteady problems too.