HOMOGENIZATION OF A CONVECTIVE, CONDUCTIVE AND RADIATIVE HEAT TRANSFER PROBLEM

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1 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

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

3 Homogenization of a heat transfer problem 3 G. Allaire

4 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 K S T S n = K F T F n+σg ( (T F ) 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) = T 4 (x)f(s,x)dx Γ

5 Homogenization of a heat transfer problem 5 G. Allaire Formula for the view factor F 3D (s,x) = n x (s x)n s (x s), F 2D (s,x ) = n x (s x )n s (x s ) π x s 4 2 x s 3

6 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), 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 L 2 into itself, ζ 1 The radiative operator G is self-adjoint on L 2 (Γ) and non-negative in the sense that G(f)f ds 0 f L 2 (Γ) Γ Γ

7 Homogenization of a heat transfer problem 7 G. Allaire Scaled model div(k S ǫ T S ǫ ) = f in Ω S ǫ div(ǫkǫ F Tǫ F )+V ǫ Tǫ F = 0 in Ω F ǫ Kǫ S Tǫ S n = ǫkǫ F Tǫ F n+ σ ǫ G ǫ(tǫ F ) 4 on Γ ǫ Tǫ S = Tǫ F on Γ ǫ T ǫ = 0 on Ω. f is the source term (due to nuclear fission, only in the solid part). V ǫ is the (given) incompressible fluid velocity. K S ǫ,ǫk F ǫ are the thermal conductivities.

8 Homogenization of a heat transfer problem 8 G. Allaire Modelling issues The solid part Ω S ǫ is a connected domain. The fluid part Ω F ǫ is the union of parallel cylinders. The cylinders boudaries Γ ǫ,i are disjoint and are not closed surfaces G ǫ (T ǫ )(s) = T ǫ (s) T ǫ (x)f(s,x)dx = (Id ζ ǫ )T ǫ (s) s Γ ǫ,i Γ ǫ,i and F(s,x)dx < 1 Γ ǫ,i 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.

9 Homogenization of a heat transfer problem 9 G. Allaire Geometry of Ω Vertical fluid cylinders. x = (x,x 3 ) with x R 2.

10 Homogenization of a heat transfer problem 10 G. Allaire Geometry of the unit cell 2-D unit cell! Microscopic variable y Λ = Λ S Λ F.

11 Homogenization of a heat transfer problem 11 G. Allaire Assumptions on the coefficients Given fluid velocity V ǫ (x) = V(x, x ǫ ) in ΩF ǫ, with a smooth vector field V(x,y ), defined in Ω Λ F, periodic with respect to y and satisfying the two incompressibility constraints div x V = 0 and div y V = 0 in Λ F, and V n = 0 on γ. A typical example is V = (0,0,V 3 ). Conductivities K S ǫ (x) = K S (x, x ǫ ) in ΩS ǫ, ǫk F ǫ (x) = ǫk F (x, x ǫ ) in ΩF ǫ, where K S (x,y ),K F (x,y ) are periodic symmetric positive definite tensors defined in Ω Λ.

12 Homogenization of a heat transfer problem 12 G. Allaire -II- HOMOGENIZATION RESULT By the method of formal two-scale asymptotic expansions T ǫ = T 0 (x)+ǫ T 1 (x, x ǫ )+ǫ2 T 2 (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).

13 Homogenization of a heat transfer problem 13 G. Allaire Theorem. T 0 is the solution of a non-linear homogenized problem div(k (x,t0) T 3 0 (x))+v (x) T 0 (x) = θf(x) in Ω T 0 (x) = 0 with the porosity factor θ = Λ S / Λ and the homogenized velocity V = 1 V(x,y )dy. Λ Λ F The corrector term T 1 is given by on Ω T 1 (x,y ) = 3 j=1 ω j (x,t 3 0(x),y ) T 0 x j (x) where ( ω j (x,t 3 0(x),y ) ) 1 j 3 are the solutions of the cell problems.

14 Homogenization of a heat transfer problem 14 G. Allaire Cell problems ( ωj (x,t0(x),y 3 ) ) are the solutions of the 2-D cell problems 1 j 3 ( div y K S (x,y )(e j + y ωj S(y )) ) = 0 in Λ S K S (y,x 3 )(e j + y ωj S(y )) n = 4σT0(x)G(ω 3 j S(y )+y j ) on γ ( div y K F (x,y )(e j + y ωj F(y )) ) +V(x,y ) (e j + y ωj F(y )) = 0 in Λ F ω F j (y ) = ω S j (y ) y ω j (y ) is Λ-periodic, on γ First we solve for ωj S condition. in the solid part with a linearized radiative boundary Second we solve for ω F j in the fluid part with a Dirichlet boundary condition.

15 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, Kj,k(x,T 0) 3 = 1 [ K S (x,y )(e j + y ω j (y )) (e k + y ω k (y ))dy Λ Λ S +4σT0(x) 3 G(ω k (y )+y k )(ω j (y )+y j ) + 2σT0(x) 3 γ γ γ F 2D (s,y ) s y 2 dy ds δ j3 δ k3 ] The above last term is due to radiation losses at both end of the cylinders. Note that the cell solutions ω j and the effective conductivity depend on T 3 0.

16 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 T ǫ T 0 (x)+ǫ 3 j=1 ω j (x,t 3 0(x), x ǫ ) T 0 x j (x) Big CPU gain because of the 3-D to 2-D reduction of the integral operator.

17 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 Γ ǫ,i into a volume integral over Ω.

18 Homogenization of a heat transfer problem 18 G. Allaire Variational two-scale ansatz Ω S ǫ Take and assume Kǫ S (x) T ǫ (x) φ ǫ (x)dx+ǫ + Ω F ǫ Ω F ǫ V ǫ (x) T ǫ (x)φ ǫ (x)dx+ σ ǫ = Ω S ǫ f(x)φ ǫ (x)dx φ ǫ H 1 0(Ω) K F ǫ (x) T ǫ (x) φ ǫ (x)dx Γ ǫ G ǫ (T ǫ )(x)φ ǫ (x)ds φ ǫ (x) = φ 0 (x)+ǫ φ 1 (x, x ǫ )+ǫ2 φ 2 (x, x ǫ ) T ǫ = T 0 (x)+ǫ T 1 (x, x ǫ )+ǫ2 T 2 (x, x ǫ )+O(ǫ3 )

19 Homogenization of a heat transfer problem 19 G. Allaire Singular radiative term The radiative term seems to blow up σ G ǫ (T ǫ )(x)φ ǫ (x)ds ǫ Γ ǫ because convergence takes place for lim ǫ ψ(x, x ǫ 0 Γ ǫ ǫ )ds = 1 Λ Ω γ ψ(x,y )dxds y However, using the fact that kerg ǫ = R and performing a Taylor expansion of the test function around the center of each cylinder Γ ǫ,i, one can gain a ǫ 2 factor.

20 Homogenization of a heat transfer problem 20 G. Allaire 3-D to 2-D asymptotic of the view factor Lemma. For any given s 3 (0,L), L 0 F 3D (s,x)dx 3 = F 2D (s,x )+O( ǫ2 L 3) For any function g C 3 (0,L) with compact support in (0,L), L 0 g(x 3 )F 3D (s,x)dx 3 = F 2D (s,x ) (g(s 3 )+ x s 2 ) g (s 3 )+O(ǫ 3 logǫ ), 2 where g denotes the second derivative of g. Note that x s 2 = O(ǫ 2 ). The corrector term, proportional to g (s 3 ), is the cause of the additional vertical diffusion.

21 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 as a ǫ (T ǫ,φ ǫ ) = L ǫ (φ ǫ ) φ ǫ H 1 0(Ω ǫ ) a 0 (T 0,T 1,φ 0,φ 1 )+ǫa 1 (T 0,T 1,T 2,φ 0,φ 1,φ 2 ) = L 0 (φ 0,φ 1 )+ǫl 1 (φ 0,φ 1,φ 2 )+O(ǫ 2 ) The zero-order equality a 0 (T 0,T 1,φ 0,φ 1 ) = L 0 (φ 0,φ 1 ) gives the homogenized equation and the cell problem in the solid part. The first-order equality a 1 (T 0,T 1,T 2,φ 0,φ 1,φ 2 ) = L 1 (φ 0,φ 1,φ 2 ) yields the cell problem in the fluid part.

22 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,L j) with L 3 = 0.025m and, for j = 1,2, L j = N j l j ǫ where N 1 = 3, N 2 = 4 and l 1 = 0.04m, l 2 = 0.07m.

23 Homogenization of a heat transfer problem 23 G. Allaire Numerical parameters Reference computation for ǫ = ǫ 0 = 1 4. Each (2-D) periodicity cell contains 2 circular holes with radius m. No source term, f = 0. Periodic boundary conditions in the x 1 direction and non-homogeneous Dirichlet boundary conditions in the other directions T ǫ = 3200x x Conductivities K S = 30Wm 1 K 1 and ǫ 0 K F = 0.3Wm 1 K 1. Constant vertical velocity V = 80e 3 ms 1.

24 Homogenization of a heat transfer problem 24 G. Allaire Standard homogenization in a fixed domain Ω

25 Homogenization of a heat transfer problem 25 G. Allaire Rescaled process of homogenization with a constant periodicity cell The domain is increasing as ǫ 1 Ω.

26 Homogenization of a heat transfer problem 26 G. Allaire Solutions of the cell problems for T 0 = 800K

27 Homogenization of a heat transfer problem 27 G. Allaire Homogenized conductivities }{{} K = T 0 =50K , K }{{} T 0 =20000K = Homogenized velocity V = ms 1.

28 Homogenization of a heat transfer problem 28 G. Allaire Homogenized conductivities as a function of T 0

29 Homogenization of a heat transfer problem 29 G. Allaire

30 Homogenization of a heat transfer problem 30 G. Allaire

31 Homogenization of a heat transfer problem 31 G. Allaire To avoid boundary layers: smaller domain

32 Homogenization of a heat transfer problem 32 G. Allaire To avoid boundary layers: smaller domain

33 Homogenization of a heat transfer problem 33 G. Allaire Relative error as a function of ǫ

34 Homogenization of a heat transfer problem 34 G. Allaire Relative error as a function of ǫ

35 Homogenization of a heat transfer problem 35 G. Allaire The same approach works for unsteady problems too.

Homogenization of a Conductive, Convective and Radiative Heat Transfer Problem in a Heterogeneous Domain

Homogenization of a Conductive, Convective and Radiative Heat Transfer Problem in a Heterogeneous Domain Grégoire Allaire, Zakaria Habibi To cite this version: Grégoire Allaire, Zakaria Habibi. Homogenization