G R Égoire Allaire

Marc Briane

Robert Brizzi

Yves Capdeboscq

TWO ASYMPTOTIC MODELS FOR ARRAYS OF UNDERGROUND WASTE CONTAINERS

Keywords: Homogenization, diffusion, porous media. 2000 Mathematics Subject Classification: 35B27, 74Q10

We study the homogenization of two models of an underground nuclear waste repository. The nuclear waste cells are periodically stored in the middle of a geological layer and are the only source terms in a parabolic evolution problem. The diffusion constants have a very large contrast between the fuel repository and the soil. It is thus a combined problem of homogenization and singular perturbation. For two different asymptotic contrasts we give the homogenized limit problem which is rigorously justified by using two-scale convergence. Eventually we perform 2-d numerical computations to show the effectiveness of using the limit model instead of the original one.

Introduction

In this paper, we consider a simplified model used to describe the diffusion process of radioactive elements from a repository into surrounding geological layers. It is a parabolic evolution problem with very contrasted coefficients along a thin strip. Namely, we study the following model ( 1)

         ρ ε (x) ∂u ε ∂t -div (A ε (x)∇u ε ) + ω ε (x) u ε = f ε (t, x)
in Ω for a.e. t > 0, u ε = 0 on ∂Ω for a.e. t > 0,

u ε (t = 0) = 0 in Ω,
where Ω is a smooth open set of R d , with d = 2 or 3, which intersects the horizontal hyperplane passing through the origin, i.e., Ω ∪ {x d = 0} = ∅ (for any x ∈ R d we write x = (x ′ , x d ) with x ′ ∈ R d-1 ). The unknown is u ε (t, x) ∈ L 2 (R + ; H 1 0 (Ω)). The repository, where the inhomogeneities and the source term f ε are located, is an horizontal strip of thickness ε and midplane {x d = 0}. Furthermore, the small parameter ε is the periodicity of the repository heterogeneities too. Outside the repository, the diffusion tensor A ε and the scalar porosities ω ε and ρ ε are functions of the macroscopic or "slow" variable x only. More precisely, we assume that there exist functions µ 0 , ω 0 and ρ 0 in L ∞ (Ω) such that [START_REF] Allaire | Homogenization and two-scale convergence[END_REF] for any x = (x ′ , x d ) ∈ Ω with |x d | > ε/2, f ε (t, x) = 0, ∀t ≥ 0 and A ε (x) (resp. ω ε (x), ρ ε (x)) = µ 0 (x) (resp. ω 0 (x), ρ 0 (x)).

Inside the repository, we assume that all coefficients, as well as the source term, are periodic in the first (d -1) variables. We furthermore consider a highly contrasted diffusion tensor compared to µ 0 . Namely, we assume [START_REF] Allaire | Boundary layers in the homogenization of a spectral problem in fluid-solid structures[END_REF] for any x = (x ′ , x d ) ∈ Ω with

|x d | < ε/2, f ε (t, x) = ε -1 f 1 t, x ε , ∀t ≥ 0 and A ε (x) = ε a µ 1 x ε , ω ε (x) = ω 1 x ε and ρ ε (x) = ρ 1 x ε , with periodic functions µ 1 ∈ L ∞ #,d (Y ) d×d , ω 1 , ρ 1 ∈ L ∞ #,d (Y ) and f 1 ∈ L ∞ (R + ; L ∞ #,d (Y )
). We assume that ω 0 , ρ 0 ≥ c > 0 (resp. ω 1 , ρ 1 ≥ c > 0) where the constant c is independent of ε and that both diffusion tensors µ 0 and µ 1 are uniformly elliptic and bounded, that is, there exists two positive constants β ≥ α > 0 such that, for all ξ ∈ R d , (4)

α |ξ| 2 ≤ µ i ξ • ξ ≤ β |ξ| 2 i = 0, 1.
The exponent a in (3) is either -1 or 2. The periodicity cell is the unit cube Y = (-1/2, +1/2) d , and by L ∞ #,d (Y ) we denote the following space

L ∞ #,d (Y ) = f ∈ L ∞ (R d ), s.t. ∀y = (y ′ , y d ) ∈ R d and ∀p ∈ Z d-1 , f (y ′ + p, y d ) = f (y ′ , y d ) .
Note that the scaling of the source term is such that its integral over the repository is of order one. We denote by χ ε the characteristic function of the repository, that is,

χ ε (x) = χ x d ε
where χ is the indicator function of the interval (-1/2, +1/2). We denote by T > 0 the finite time-scale for which the model under consideration is valid. We also use the notations Ω ′ = Ω ∩ {x d = 0}, Ω + = Ω ∩ {x d > 0}, Ω -= Ω ∩ {x d < 0}, and Y ′ = (-1/2, +1/2) d-1 .

The motivation for this work is the study on the long time behavior of an underground waste repository in the presence of leakage. A sketch of the medium under consideration is presented in Figure 1. The spent fuel cells are stored in the center of the middle geological layer. Inside each module, 10 to 15 spent fuel cells are stored horizontally (normally to the section represented). A description of the geometry is presented in Section 6.1. This model problem, introduced as a benchmark case for the COUPLEX exercise [START_REF] Bourgeat | Talandier The COUPLEX Test Cases: Nuclear Waste Disposal Simulation[END_REF], has been previously studied by Bourgeat et al. in [START_REF] Bourgeat | Mathematical modeling of an array of underground waste containers[END_REF], [START_REF] Pino | Asymptotic analysis and layer decomposition for the couplex exercise[END_REF], [START_REF] Bourgeat | Modelling of an underground waste disposal site by upscaling[END_REF], [START_REF] Bourgeat | A homogenized model of an underground waste repository including a disturbed zone[END_REF], [START_REF] Bourgeat | Modelling of an underground waste disposal site by upscaling, simulation with domain decomposition method[END_REF]. The novelty here is that we allow a strong contrast between the fuel repository and the soil. The first case is that of a very small permeability of the engineered structure, a = 2, where the repository is designed as a sealed containment area. The second case, is that of a large permeability, a = -1, where the repository structure is the weak point of the apparatus, the surrounding soil being a better natural barrier. This last case is what happens with a concrete structure is a clay layer. The source term represents a leakage (which would happen in these worst case scenario in every module).

The first part of the paper, sections 3 to 5, is devoted to the asymptotic study of two models (a = -1 and a = 2) in the limit as ε tends to zero. After establishing a priori estimates and recalling the notion of two scale convergence for boundary layers [START_REF] Allaire | Boundary layers in the homogenization of a spectral problem in fluid-solid structures[END_REF] in Section 3, we derive the homogenized limits for the case a = 2 in Section 4 and for the case a = -1 in Section 5. A related result appeared in [START_REF] Neuss-Radu | Effective transmission conditions for reaction diffusion processes in domains separated by an interface[END_REF] with a different application to transmission through thin membrane.

The second part of the paper is dedicated to the application of this study in the nuclear repository context. The model used by the so-called MoMaS group 1 , in charge of the mathematical aspects of the long term underground waste repository behavior predictions, is slightly more complex than the one studied in the first part, and is exposed first. Because the coefficients are measured physical quantities, the relevant small parameter ε has to be identified, and evaluated. With that in mind, we conduct a dimensional analysis of the available data, and express the MoMaS group model in this setting. The two cases we have considered correspond to limit situations, with very strong or very weak diffusion. The corresponding numerical results are presented in Section 6.

Main Results

The first result addresses the case when a = 2, i.e. of very weak diffusion in the repository zone. In such a case, we need an additional technical assumption on the smoothness of the coefficients in the vicinity of the repository boundaries, namely

(5) ∃ x d 0 ∈ (0, 1/2) and r > 0 such that µ 1 (y) ∈ C 0,r # Y ∩ {|x d | > x d 0 } d×d .
Our result describes the limit behavior of u ε both outside the fuel repository and inside. Theorem 1. Under assumptions [START_REF] Allaire | Homogenization and two-scale convergence[END_REF][START_REF] Allaire | Boundary layers in the homogenization of a spectral problem in fluid-solid structures[END_REF][START_REF] Bourgeat | Mathematical modeling of an array of underground waste containers[END_REF][START_REF] Bourgeat | Modelling of an underground waste disposal site by upscaling[END_REF], when a = 2, the solution u ε of ( 1) is such that (1χ ε ) u ε converges strongly in L 2 (0, T ); L 2 (Ω) to u, given by u := u + in Ω + and u := u -in Ω -, and

(1 -χ ε ) ∇u ε converges weakly in L 2 (0, T ); L 2 (Ω) d to ξ, with ξ = ∇u ± in Ω ±
, where u ± is the unique solution of (6)

               ρ 0 (x) ∂u ± ∂t -div (µ 0 (x)∇u ± ) + ω 0 (x) u ± = 0 in Ω ± , u ± = 0 on ∂Ω ∩ ∂Ω ± , ∓µ 0 (x) ∇u ± • e d = s ± on Ω ′ , u ± (t = 0) = 0 in Ω.
The effective source terms s + and s -are given by

(7) s ± (t) = ± Y ′ µ 1 (y ′ , ±1/2) ∇u 2 (t, y ′ , ±1/2) • e d dy ′
, where u 2 is the limit of εu ε in the sense of two-scale boundary layers convergence (see Proposition 5). The limit u 2 (t, y) is the unique solution in L 2 (0,

T ); H 1 (Y ) of the following problem (8)            ρ 1 (y) ∂u 2 ∂t -div y (µ 1 (y)∇ y u 2 ) + ω 1 (y)u 2 = f 1 (t, y) in (0, T ) × Y, u 2 (t, y) = 0 on (0, T ) × {|y d | = 1/2}, (y 1 , . . . , y d-1 ) → u 2 (t, y) Y ′ -periodic, u 2 (t = 0) = 0.
Asymptotically, the effect of the strip is therefore to transform the source term f 1 into two effective time-dependent boundary conditions s + or s -for the normal derivatives given by [START_REF] Bourgeat | Modelling of an underground waste disposal site by upscaling, simulation with domain decomposition method[END_REF]. Note that system (8) is independent of system (6). Such a limit problem is the inspiration for an approximate numerical method presented in Section 6. Theorem 1 is related to a recent result in [START_REF] Neuss-Radu | Effective transmission conditions for reaction diffusion processes in domains separated by an interface[END_REF], for nonlinear reactiondiffusion systems. However the scaling, as well as the analysis, are quite different.

We now address the second case when a = -1.

Theorem 2. Under assumptions [START_REF] Allaire | Homogenization and two-scale convergence[END_REF][START_REF] Allaire | Boundary layers in the homogenization of a spectral problem in fluid-solid structures[END_REF][START_REF] Bourgeat | Mathematical modeling of an array of underground waste containers[END_REF], when a = -1, the solution u ε of ( 1) is such that it converges strongly in L 2 (0, T ); L 2 (Ω) to u and ∇u ε converges weakly in L 2 (0, T ); L 2 (Ω) d to ∇u, where u ∈ H 1 (Ω) is the unique solution of the coupled system (9)

           ρ 0 ∂u ∂t -div (µ 0 ∇u) + ω 0 u = 0 in Ω + ∪ Ω - u(t = 0) = 0 in Ω + ∪ Ω -, u = 0 on ∂Ω, u = u -1 on Ω ′ = Ω ∩ {x d = 0}, and
u -1 is the solution in H 1 (Ω ′ ) of (10) -div x ′ (µ * ∇ x ′ u -1 (t, x)) + [µ 0 ∇u • n] = f ,
where [j] stands for the jump of a quantity j between Ω + and Ω -. The effective coefficients µ * and f are given by

(11) f (t) = Y f 1 (t, y) dy, and (12)
µ * = Y µ 1 (y) (I d + P (y)) dy.

For each i = 1, . . . , d, P (•)e i is given by P (•)e i = ∇ϕ i where

ϕ i ∈ H 1 #,d (Y ) is Y ′ -periodic in its (d -1
) first variables and is the unique solution, up to an additive constant, of

(13)    div (µ 1 (∇y i + ∇ y ϕ i )) = 0 in Y, µ 1 (∇y i + ∇ y ϕ i ) • e d = 0 on Y ′ × {+1/2}, µ 1 (∇y i + ∇ y ϕ i ) • e d = 0 on Y ′ × {-1/2}.
Of course, the two partial differential equations ( 9) and ( 10) are strongly coupled. Asymptotically, the effect of the strip is therefore to transform the source term f 1 into an effective time-dependent tangential diffusion along the interface given by (10). The homogenized problem ( 9)-( 10) is the inspiration for an approximate numerical method in Section 6. For a homogeneous thin strip, Theorem 2 was already obtained in [START_REF] Huy | Phénomènes de transmission à travers des couches minces de conductivité élevée[END_REF].

A priori estimates and two-scale boundary layer convergence

A prerequisite for the proofs of Theorems 1 and 2 is to check that the solution u ε of (1) is uniformly bounded in appropriate norms. The necessary a priori estimates are established in the following Proposition.

Proposition 3. Assume the coefficients A ε , ω ε , ρ ε satisfy assumptions [START_REF] Allaire | Homogenization and two-scale convergence[END_REF][START_REF] Allaire | Boundary layers in the homogenization of a spectral problem in fluid-solid structures[END_REF][START_REF] Bourgeat | Mathematical modeling of an array of underground waste containers[END_REF].

(i) If, instead of assumption (3), we suppose that

f ε (t, x) = ε b χ ε x d ε f 1 t, x ε with f 1 ∈ L ∞ (R + ; L ∞ #,d (Y ′ × (-1/2, 1/2))), and b ∈ R, we have u ε L ∞ ((0,T );L 2 (Ω)) + (1 -χ ε )∇u ε L 2 ((0,T );L 2 (Ω) d ) +ε a 2 χ ε ∇u ε L 2 ((0,T );L 2 (Ω) d ) ≤ Cε 1 2 +b min 1, ε -a 2 , ( 14
) (ii) Alternatively, instead of assumption (3) assume that f ε (t, x) = s ε (t, x ′ ) δ {|x d |=ε/2} x d ε where s ε ∈ C R + × R d-1 is bounded above and below independently of ε. Then we have (15) u ε L ∞ (0,T,L 2 (Ω)) + (1 -χ ε )∇u ε L 2 ((0,T );L 2 (Ω) d ) +ε a 2 χ ε ∇u ε L 2 ((0,T );L 2 (Ω) d ) ≤ C.
Corollary 4. Assume the coefficients A ε , ω ε , ρ ε satisfy assumptions (2, 3 and 4). If a = -1, then we have

(16) u ε L ∞ ((0,T );L 2 (Ω)) + (1 -χ ε )∇u ε L ∞ ((0,T );L 2 (Ω) d ) ≤ C, and
(17) χ ε u ε L 2 ((0,T );L 2 (Ω)) + χ ε ∇u ε L 2 ((0,T );L 2 (Ω) d ) ≤ C √ ε. If a = 2, then w ε = εu ε satisfies (18) w ε L ∞ ((0,T );L 2 (Ω)) + (1 -χ ε )∇w ε L 2 ((0,T );L 2 (Ω) d ) +ε χ ε ∇w ε L 2 ((0,T );L 2 (Ω) d ) ≤ C √ ε
which, in particular, implies that

(19) χ ε w ε L ∞ ((0,T );L 2 (Ω)) + ε χ ε ∇w ε L 2 ((0,T );L 2 (Ω) d ) ≤ C √ ε.
Proof of Corollary 4. Estimates ( 16), ( 18) and ( 19) are obvious consequences of Proposition 3. Inequality ( 17) is easily deduced from ( 14) and ( 22).

Proof of Proposition 3. Testing (1) against u ε we obtain, after integration by parts with respect to the space variable x,

1 2 d dt Ω ρ ε u 2 ε dx + Ω (1 -χ ε )µ 0 (x) ∇u ε • ∇u ε dx + Ω ω x ε u 2 ε dx +ε a Ω χ ε µ 1 x ε ∇u ε • ∇u ε dx = Ω χ ε f ε u ε dx. (20) Let us first address the case (i): there is a constant M such that f ε L ∞ (R + ×Ω) ε -b < M < ∞ for any ε.
Then (20) yields, because of the positivity of ρ ε , ω ε and the coercivity of µ 0 and µ 1 , that

d dt ρ 1 2 ε u ε 2 L 2 (Ω) ≤ C χ ε f ε L 2 (Ω) χ ε ρ 1 2 ε u ε L 2 (Ω) ≤ Cε 1 2 +b χ ε ρ 1 2 ε u ε L 2 (Ω)
, which in turns shows that u ε (t, •) L 2 (Ω) ≤ Ctε 1 2 +b , the constant C, depending a priori on min Ω |ρ ε |, and M being chosen independently of ε. Integrating (20) with respect to time, and using Cauchy-Schwarz inequality again we obtain,

(21) (1 -χ ε )∇u ε 2 L 2 ((0,T );L 2 (Ω) d ) + ε a χ ε ∇u ε 2 L 2 ((0,T );L 2 (Ω) d ) ≤ Cε 1+2b .
This implies [START_REF] Neuss-Radu | Effective transmission conditions for reaction diffusion processes in domains separated by an interface[END_REF]. On the other hand, thanks to the Dirichlet boundary condition satisfied by u ε in (1), we have, for all t > 0, the following Poincaré inequality

(22) χ ε u ε 2 L 2 (Ω) ≤ d Ω χ ε ∇u ε 2 L 2 (Ω) d
, where d Ω depends on Ω only. This can be proved, for example, by integrating u ε along a vertical or horizontal path connecting x and the boundary of the domain within the repository. Consequently, we have, for all t > 0, and all λ > 0, 23) in (20) we obtain that for all t > 0, 1 2 14) is proved, and the proof of the first part of Proposition 3 is finished.

Ω χ ε f ε u ε dx ≤ ε -a d Ω 4λα χ ε f ε 2 L 2 (Ω) + ε a αλ d Ω χ ε u ε 2 L 2 (Ω) , ≤ ε -a d Ω 4αλ χ ε f ε 2 L 2 (Ω) + ε a λ Ω χ ε µ 1 ( x ε )∇u ε • ∇u ε dx. (23) For λ = 1, this last inequality combined with (20) yields 1 2 d dt Ω ρ ε u 2 ε dx ≤ d Ω 4α ε -a χ ε f ε 2 L 2 (Ω) ≤ Cε -a+1+2b , (24) which shows that u ε (t, •) L 2 (Ω) ≤ Cε 1 2 +b-1 2 a . For λ = 1/2, using (
d dt Ω ρ ε u 2 ε dx + Ω (1 -χ ε )µ 0 (x) ∇u ε • ∇u ε dx + 1 2 ε a Ω χ ε µ 1 x ε ∇u ε • ∇u ε dx ≤ Cε -a+1+2b , which in turn implies that (1 -χ ε )∇u ε 2 L 2 ((0,T );L 2 (Ω) d ) + ε a χ ε ∇u ε 2 L 2 ((0,T );L 2 (Ω) d ) ≤ Cε -a+1+2b . Estimate (
Let us now turn to case (ii), and suppose that

f ε = s ε (t, x ′ ) δ {|x d |=ε/2} with s ε ∈ C(R + × R d-1 ), and s ε L ∞ < M where the constant M is independent of ε. Then Ω f ε (t, x)u ε dx ≤ M {x d =±ε/2} |u ε | dx ′ ≤ M 2 τ + Cτ {x d =±ε/2} |u ε | 2 dx ′ (25) ≤ M 2 τ + Ω (1 -χ ε )µ 0 (x)∇u ε • ∇u ε dx (26)
for a sufficiently small constant τ > 0. Therefore we obtain 1 2

d dt Ω ρ ε (x)u 2 ε dx ≤ M 2 τ ,
and arguing as above, we obtain [START_REF]Test cases definition: Homogenization of the source terms[END_REF].

Following the strategy introduced in [3], we shall derive the asymptotic limit of problem (1) using two-scale convergence for boundary layers. This notion of convergence generalizes the usual two-scale convergence [START_REF] Allaire | Homogenization and two-scale convergence[END_REF], [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF] to the case when the test functions periodically oscillate in the first (d -1) space directions and simply decay at infinity in the last direction. The following proposition summarizes the definition and properties of two-scale convergence in the sense of boundary layers. We denote by G the band Y ′ × R. A generic point y ∈ G is denoted by y = (y ′ , y d ) with y ′ ∈ Y ′ and y d ∈ R. We introduce the space L 2 # (G) of square integrable functions on G which are periodic in the first d -1 variables, i.e.

L 2 # (G) = ϕ ∈ L 2 (G) s.t. y ′ → ϕ(y ′ , y d ) is Y ′ -periodic . Similarly, we define H 1 # (G) the Sobolev space of functions in H 1 (G) which are Y ′ - periodic, and C ∞ #,c (G) and C ∞ # (G)
the space of infinitely differentiable functions -with compact support in the first case-on G. We denote by C(Ω ′ ) the space of continuous functions on the closure of Ω ′ , a compact set in R d-1 .

Proposition 5. (1) Let v ε be a sequence in L 2 (Ω) such that v ε L 2 (Ω) ≤ C √ ε.
There exists a subsequence, still denoted by ε, and a limit v 0 (x ′ , y) ∈ L 2 Ω ′ ; L 2 # (G) such that v ε two-scale converges weakly in the sense of boundary layers to v 0 i.e.

lim ε→0 1 ε Ω v ε (x)ϕ x ′ , x ε dx = 1 |Y ′ | Ω ′ G v 0 (x ′ , y) ϕ (x ′ , y) dx ′ dy for all test functions ϕ (x ′ , y) ∈ C Ω ′ ; L 2 # (G) .
(2) Let v ε be a sequence which two-scale converges weakly in the sense of boundary layers to v 0 , and furthermore satisfies

lim ε→0 1 √ ε v ε L 2 (Ω) = 1 |Y ′ | 1 2 v 0 L 2 (Ω ′ ×G) .
Then, v ε is said to two-scale converge strongly in the sense of boundary layers to v 0 , which means that, for any sequence w ε in L 2 (Ω) which two-scale converges weakly in the sense of boundary layers to

w 0 (x ′ , y) ∈ L 2 (Ω ′ × G) one has lim ε→0 1 ε Ω v ε (x)w ε (x)ϕ x ′ , x ε dx = 1 |Y ′ | 1 2 Ω ′ G v 0 (x ′ , y)w 0 (x ′ , y)ϕ (x ′ , y) dx ′ dy, for all smooth functions ϕ (x ′ , y) ∈ C Ω ′ ; C #,c (G) .
Note that two-scale boundary layer convergence (called 2SBL in the sequel) can be also used to characterize the precise behavior of a non-vanishing function, restricted to a small domain, as it is shown by the following simple lemma.

Lemma 6. Let v ε be a bounded sequence in H 1 0 (Ω) converging weakly to a limit v. Assume that w ε = χ ε v ε admits a limit w in the sense of boundary layers convergence. Then w = χw and for almost every

x ′ ∈ Ω ′ , v(x ′ , 0) = Y w(x ′ , y) dy.
Proof. Recall that χ ε = χ( x ε ) is the characteristic function of the repository zone. The fact that w = χw is obvious. Note that the assumption on w ε having a twoscale limit in the sense of boundary layers is plausible since

, if v ε is a constant sequence v ∈ L ∞ (Ω), then w ε satisfies the a priori estimate w ε L 2 (Ω) ≤ C √ ε
which is required for the two-scale convergence in the sense of boundary layers. We have

1 ε Ω χ ε v ε (x ′ , 0)ϕ(x)dx - Ω ′ v ε (x ′ , 0)ϕ(x ′ , 0)dx ′ = 1 ε Ω χ ε v ε (x ′ , 0)ϕ(x ′ , 0)dx - Ω ′ v ε (x ′ , 0)ϕ(x ′ , 0)dx ′ + Ω v ε (x ′ , 0) 1 ε χ ε (ϕ(x ′ , 0) -ϕ(x)) dx.
The first term is exactly zero. For the second one, note that

1 ε χ ε (ϕ(x ′ , 0) -ϕ(x)) ≤ 1 ε ε/2 -ε/2 ∂ ∂x d ϕ(x ′ , t) dt ≤ ∇ϕ L ∞ (Ω) , therefore, Ω v ε (x ′ , 0) 1 ε χ ε (ϕ(x ′ , 0) -ϕ(x)) dx ≤ ∇ϕ L ∞ (Ω) ε Ω ′ v ε (x ′ , 0)dx ′ ≤ Cε ∇ϕ L ∞ (Ω) ∇v ε L 2 (Ω) d ,
where this last term is of order O(ε) since the sequence v ε is bounded in H 1 0 (Ω). Similarly,

Ω 1 ε χ ε (v ε (x ′ , 0) -v ε (x)) dx ≤ Ω χ ε ε ε/2 -ε/2 ∂ ∂x d v ε (x ′ , s d ) ds d dx ≤ C √ ε ∇v ε L 2 (Ω) d . As a consequence, Ω 1 ε χ ε v ε(x) ϕ(x)dx - Ω ′ v(x ′ , 0)ϕ(x ′ , 0)dx ′ ≤ Ω 1 ε χ ε v ε (x ′ , 0)ϕ(x)dx - Ω ′ v(x ′ , 0)ϕ(x ′ , 0)dx ′ + O( √ ε) ≤ Ω ′ |v ε (x ′ , 0) -v(x ′ , 0)| |ϕ(x ′ , 0)| dx ′ + O( √ ε) ≤ C(ϕ) v ε (•, 0) -u(•, 0) L 2 (Ω ′ ) + O( √ ε).
The trace of v ε on Ω ′ converge strongly in L 2 (Ω ′ ) to the trace of v, thus we have proved that

Ω ′ G w(x ′ , y)ϕ(x ′ , 0) dx ′ dy = lim ε→0 Ω 1 ε χ ε (x)v ε (x)ϕ(x ′ , 0) dx = lim ε→0 Ω 1 ε χ ε (x)v ε (x)ϕ(x) dx = Ω ′ v(x ′ , 0)ϕ(x ′ , 0) dx ′ .
which implies the desired result, namely that u(x ′ , 0) is the average on Y of w(x ′ , y).

The two scale convergence for boundary layers was introduced for time independent problems. It naturally extends to time dependent problems.

Proposition 7. (i) Suppose that a = -1. There exists a function u -1 (t, x ′ ) ∈ L 2 (0, T ); H 1 (Ω ′ ) and a function w(t, x ′ , y) ∈ L 2 (0, T ) × Ω ′ ; H 1 loc,# (G) such that, up to a subsequence,

χ ε u ε 2SBL ---⇀ χ(y d )u -1 (t, x ′ ) (27) χ ε ∇u ε 2SBL ---⇀ χ(y d ) (∇ x ′ u -1 (t, x ′ ) + ∇ y w(t, x ′ , y)) . ( 28
)
(ii) Suppose that a = 2. There exists a function u 2 (t, x, y) ∈ L 2 (0, T ) × Ω ′ ; H 1 loc,# (G) such that, up to a subsequence,

εu ε 2SBL ---⇀ u 2 (t, x ′ , y) (29) ε 2 ∇u ε 2SBL ---⇀ χ(y d )∇ y u 2 (t, x ′ , y) (30) Furthermore, (31)
(1χ(y d ))u 2 (t, x ′ , y) = 0 for a.e. y ∈ G.

Note. We use the notation

∇ x ′ = ∂ ∂x 1 , . . . , ∂ ∂x d-1 , 0 .
Similarly, for a d dimensional vector Ψ we shall write

Ψ ′ = (Ψ 1 , . . . , Ψ d-1 , 0).
Proof. Suppose a = 2. The bound (18) shows that w ε = εu ε satisfies

w ε L 2 ((0,T );L 2 (Ω)) + ε ∇w ε L 2 ((0,T );L 2 (Ω) d ) ≤ C √ ε.
Consequently, w ε (resp. ε∇w ε ) admits a limit u 2 (resp. ζ 2 ) in the sense of boundary layer two scale limits. It is a classical result [START_REF] Allaire | Homogenization and two-scale convergence[END_REF], [START_REF] Allaire | Boundary layers in the homogenization of a spectral problem in fluid-solid structures[END_REF] that ε∇w ε two scale converges in the sense of boundary layers to ζ 2 = ∇ y u 2 . We shall now prove (31). The bounds (18) show that the sequence w ε satisfies

(1 -χ ε )w ε L 2 (0,T ;L 2 (Ω)) + (1 -χ ε )∇w ε L 2 (0,T ;L 2 (Ω) d ) ≤ C √ ε.
Therefore, up to a subsequence, (1χ ε )∇w ε converges in the sense of boundary layers to ξ 2 (t, x ′ , y). Since χ ε converges strongly in the sense of two-scale boundary layers convergence to χ, we have

ξ 2 (t, x ′ , y) = (1 -χ(y d ))ξ 2 (t, x ′ , y). Let Ψ(x ′ , y) ∈ C ∞ c Ω ′ ; C ∞ #,c (G) d be a smooth test function such that Ψ(x ′ , y)(1 - χ(y d )) = Ψ(x ′ , y), and ϕ(t) ∈ C[0, T ]
. By an integration by parts we see that

lim ε→0 1 ε T 0 ϕ(t) Ω (1 -χ ε )∇w ε • Ψ x ′ , x ε dt dx = -lim ε→0 1 ε 2 T 0 ϕ(t) Ω w ε div y Ψ x ′ , x ε dt dx ′ dy -lim ε→0 1 ε T 0 ϕ(t) Ω w ε div x ′ Ψ x ′ , x ε dt dx ′ dy.
This implies that lim ε→0

1 ε T 0 ϕ(t) Ω w ε div y Ψ x ′ , x ε dt dx = 0, that is, (1 -χ)u 2 (t, x ′ , y) does not depend on y and is therefore 0, since y → u 2 (•, •, y) ∈ L 2 # (G).
Let us now turn to the case a = -1. The bound (17) shows that χ ε u ε and χ ε ∇u ε admit two scale limits in the sense of boundary layers convergence. Denoting by w -1 and ζ -1 these limits, we have for all ψ(x ′ , y)

∈ C ∞ c Ω ′ ; C ∞ c# (G) , Ψ(x ′ , y) ∈ C ∞ c Ω ′ ; C ∞ c# (G) d and ϕ(t) ∈ C[0, T ], lim ε→0 1 ε T 0 ϕ(t)dt Ω χ ε u ε ψ x ′ , x ε dx = T 0 ϕ(t)dt Ω ′ ×G w -1 (t, x ′ , y)ψ (x ′ , y) dx ′ dy, lim ε→0 1 ε T 0 ϕ(t)dt Ω χ ε ∇u ε • Ψ x ′ , x ε dx = T 0 ϕ(t)dt Ω ′ ×G ζ -1 (t, x ′ , y) • Ψ (x ′ , y) dx ′ dy.
We select a test function Ψ = (Ψ 1 , . . . , Ψ d ) such that Ψ(x ′ , y)χ(y d ) = Ψ(x ′ , y), integrating by parts and passing to the limit we obtain a condition on w -1 ,

T 0 ϕ(t)dt Ω ′ ×G
w -1 (t, x ′ , y)div y Ψ (x ′ , y) dx ′ dy = 0, which implies that the two-scale limit w -1 of χ ε u ε is independent of y within the support of χ:

w -1 (t, x ′ , y) = χ(y d )u -1 (t, x ′ ). Selecting now Ψ ′ = (Ψ 1 , . . . , Ψ d-1 , 0) = Ψ ′ (x ′ , y)χ(y d ) satisfying additionally that div y Ψ ′ = 0, we have T 0 ϕ(t)dt Ω ′ ×G ξ -1 (t, x ′ , y) • Ψ ′ (x ′ , y) dx ′ dy = - T 0 ϕ(t)dt Ω ′ ×G χ(y d )u -1 (t, x ′ )div x ′ Ψ ′ (x ′ , y) dx ′ dy = T 0 ϕ(t)dt Ω ′ ×G χ(y d )∇ x ′ u -1 (t, x ′ ) • Ψ ′ (x ′ , y) dx ′ dy.
Since the orthogonal set in L 2 (Ω ′ × Y ) of all Y ′ -periodic functions Ψ ′ satisfying the constraint div y Ψ ′ = 0 is precisely the set of all gradients ∇ y w(x ′ , y) with w ∈ L 2 (Ω ′ ; H 1 # (Y )) (recall that these last functions are not periodic in the last y d variable), we have proved that

ζ -1 (t, x ′ , y) = χ(y d )∇ x ′ u -1 (t, x ′ ) + χ(y d )∇ y w(t, x ′ , y).

Proof of Theorem 1

For any test function ψ(x ′ , x/ε) such that ψ(x ′ , y)χ(y d ) = ψ(x ′ , y), and any smooth function ϕ(t) such that ϕ(0) = ϕ(T ) = 0, we obtain from Proposition 7 that

ε T 0 ϕ(t)dt Ω ρ ε (x) ∂u ε ∂t ψ(x ′ , x/ε)dx = - T 0 dt Ω ′ G ψ (x ′ , y) ρ 1 (y)u 2 (t, x ′ , y) ∂ϕ ∂t dx ′ dy + o(1), T 0 ϕ(t)dt Ω ε 2 A ε ∇u ε • ∇ψ(x ′ , x/ε)dx = T 0 ϕdt Ω ′ G µ 1 (y) ∇ y u 2 (t, x ′ , y) • ∇ y ψ (x ′ , y) dx ′ dy + o(1), ε T 0 ϕ(t)dt Ω ω ε u ε ψ(x ′ , x/ε)dx = T 0 ϕ(t)dt Ω ′ G ω(y)u 2 (t, x ′ , y) ψ (x ′ , y) dx ′ dy + o(1), T 0 ϕ(t)dt Ω f ε (x)ψ(x ′ , x/ε)dx = T 0 ϕ(t)dt Ω ′ G f 1 (t, y)ψ(x ′ , y)dx ′ dy + o(1).
Here, as in Proposition 7, u 2 is the limit of εu ε . Combining these terms as in [START_REF]Dossier 2005 Argile -Évolution phénoménologique du stockage géologique[END_REF] we deduce that u 2 satisfies the limit system

ρ 1 (y) ∂u 2 ∂t -div y (µ 1 (y)∇ y u 2 ) + ω 1 (y)u 2 = f 1 (t, y) in Ω ′ × Y, for a.e. t u 2 (t, x ′ , y) = 0 on Ω ′ × Y ′ × {|y d | = 1/2}, for a.e. t (32) u 2 (t = 0) = 0.
Note that u 2 is independent of x ′ : thus, system (32) is in fact [START_REF] Bourgeat | Talandier The COUPLEX Test Cases: Nuclear Waste Disposal Simulation[END_REF], and our notations are consistent. In the sequel we shall write u 2 (t, y), the unique solution in C([0, T ]; H 1 # (Y )) of ( 8). This first result, the characterization of the limit behavior of εu ε , does not provide a detailed description of the asymptotic behavior of the solution of (1), since the support of u 2 (•, • ε ) shrinks to zero with ε. Consequently, we turn to the second order term, and define

(33) v ε (t, x) = u ε (t, x) - 1 ε u 2 t, x ε for a.e. (t, x) ∈ [0, T ] × Ω.
The following gives the equation satisfied by v ε .

Proposition 8. The solution u ε of problem (1) can be written

u ε (t, x) = 1 ε u 2 (t, x/ε) + v ε (t, x), for a.e. (t, x) ∈ [0, T ] × Ω,
where u 2 is the unique solution of system (32). The remainder term v ε satisfies (34)

       ρ ε (x) ∂v ε ∂t -div (A ε (x)∇v ε ) +ω ε (x) v ε = s t, x ε δ {x d =ε/2} -δ {x d =-ε/2}
in Ω,

u ε = 0 on ∂Ω, u ε (t = 0) = 0 in Ω, where (35) s(t
, y) = µ 1 (y) ∂u 2 ∂y d (t, y) .
Furthermore, v ε satisfies the following a priori estimate

(36) v ε L ∞ ((0,T );L 2 (Ω)) + (1 -χ ε )∇v ε L 2 ((0,T );L 2 (Ω) d ) + ε χ ε ∇v ε L 2 ((0,T );L 2 (Ω) d ) ≤ C,
where C is a constant independent of ε.

Proof. System (34) is obtained without difficulty from (1) and (32). From assumption (5) we deduce that there exists η > 0 such that

u 2 ∈ L ∞ (0, T ); C 1,η (Y ′ × [-1/2, -1/2 + η] ∪ [1/2 -η, 1/2])
(see e.g. [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]Thm 8.34]). As a consequence, s (t, x/ε) is continuous, and bounded around x d = ±ε/2. Since in the right hand side of (34) s is multiplied by Dirac functions located at x d = ±ε/2, we can assume without loss of generality that s(t, x/ε) ∈ C([0, T ] × Ω). Then, estimate (36) is a consequence of [START_REF]Test cases definition: Homogenization of the source terms[END_REF].

The conclusion of the proof of Theorem 1 is given by the following proposition.

Proposition 9. The solution v ε of (34) is such that (1 -χ ε )v ε converges strongly in L 2 ((0, T ) × Ω ± ) to u ± and (1 -χ ε ) ∇v ε converges weakly in L 2 ((0, T ) × Ω ± ) d to ∇u ±
, where u ± is the solution of the homogenized problem [START_REF] Bourgeat | A homogenized model of an underground waste repository including a disturbed zone[END_REF].

Proof. We focus on the convergence to u + , the convergence to u -can be proved by similar arguments. The bound (36) on v ε L ∞ (0,T,L 2 (Ω)) shows that, up to the possible extraction of a subsequence, there exists v 0 (t, x) ∈ L 2 (0, T ); L 2 (Ω + ) such that for v ε converges weakly in L 2 (0, T ); L 2 (Ω) to v 0 . Similarly, the bound ( 36)

on (1 -χ ε )∇v ε L 2 ((0,T );,L 2 (Ω) d )
shows that, up to the possible extraction of a subsequence, there exists ξ 0 (t, x) ∈ L 2 (0, T ); L 2 (Ω)

d such that for (1 -χ ε )∇v ε con- verges weakly in L 2 (0, T ); L 2 (Ω) d to ξ 0 . For any Φ ∈ C 1 c ((0, T ) × (Ω + ∪ Ω -)) d ,
one easily obtains that

T 0 Ω + ∪Ω - ξ 0 • Φ dtdx = - T 0 Ω + ∪Ω - v 0 div Φ dtdx,
or in other words that ξ 0 = ∇v 0 on Ω + ∪ Ω -. Let θ ∈ C 2 (R) be a non-negative one dimensional cut-off function, such that θ(s) = 1 for s ≥ 1/2 and θ(s) = 0 for s ≤ -1/2, and define

θ ε (x) = θ x d ε x ∈ R d .
Given a test function ϕ ∈ C 1 (0, T ); H 1 0 (Ω) , such that ϕ(T, •) = 0, we test system (34) against ϕθ ε to obtain

T 0 dt Ω + dx (1 -χ ε ) θ ε (x)v ε (x) -ρ 0 (x) ∂ϕ ∂t + ω 0 (x) ϕ + T 0 dt Ω + dx (1 -χ ε ) µ 0 (x)∇v ε (x) • ∇ϕ + T 0 dt Ω ′ dx ′ s t, x ′ ε , 1/2 ϕ (t, x ′ , ε) = T 0 dt Ω dx χ ε θ ε v ε ρ ε (x) ∂ϕ ∂t -ω ε (x) ϕ - T 0 dt Ω dx ε 2 χ ε µ 1 x ε ∇v ε • ∇ (θ ε ϕ) .
Thanks to (36), the first right-hand side term is bounded by

√ ε ρ 1 ∂ϕ ∂t -ω 1 ϕ L ∞ ((0,T );L 2 (Ω)) v ε L ∞ ((0,T );L 2 (Ω)) ≤ C √ ε,
whereas the second right-hand side term is bounded by

ε χ ε ∇v ε L 2 ((0,T );L 2 (Ω) d ) µ 1 L ∞ (R d ) ε ∇ϕ L ∞ ((0,T );L 2 (Ω) d ) + √ ε ϕ L ∞ ((0,T );L 2 (Ω)) χ∇θ L 2 (R d ) ≤ C √ ε.
Passing to the limit in the left-hand side, we obtain

T 0 Ω + v 0 -ρ 0 (x) ∂ϕ ∂t + ω 0 (x) ϕ + T 0 Ω + µ 0 (x)∇v 0 •∇ϕ- T 0 Ω ′ s + (t)ϕ(t, x ′ , 0)dx ′ = 0.
which is the system (6) satisfied by u + . The limit system for u -is obtained similarly.

Proof of Theorem 2

Thanks to the a priori estimate (16) we know that u ε is a bounded sequence in L ∞ ((0, T ); H 1 0 (Ω)). It therefore admits a weak limit u, up to the extraction of a subsequence, in L 2 ((0, T ); H 1 0 (Ω)). We can now pass to the limit in (1) in any subdomain ω ⊂ Ω not intersecting the plane x d = 0. Furthermore, thanks to Lemma 6, we know that for a.e. x ′ ∈ Ω ′ and every t > 0,

u(t, x ′ , 0) = G χ(y d )u -1 (t, x ′ )dy = u -1 (t, x ′ ).
We therefore have proved that, up to a subsequence, u satisfies

               ρ 0 (x) ∂u ∂t -div (µ 0 (x)∇u) + ω 0 (x) u = 0 in Ω, u(t, x ′ , 0) = u -1 (t, x ′ ) on Ω ′ , u(t = 0) = 0 in Ω, u = 0 on ∂Ω.
Let us now turn to the derivation of equation (10) to conclude the proof.

The following proposition determines w(t, x, y), defined in Proposition 7, in terms of u -1 (t, x ′ ).

Proposition 10. The two-scale limit of χ ε ∇u ε is precisely χ(y d ) (

I d + P (y)) ∇ x ′ u -1 (t, x ′ )
where I d is the identity matrix of R d , and P (y) is the matrix-valued function defined by P (•)e i = ∇ϕ i , for i = 1, . . . , d, where

ϕ i ∈ H 1 #,d (Y )
is the solution of (13).

Proof. For any

ψ(x ′ , y) ∈ C(Ω ′ ; L 2 # (G)) and ϕ(t) ∈ C 1 c (0, T ), testing (1) against εϕ(t)ψ(x ′ , x/ε) yields T 0 ϕ(t)dt Ω χ ε (x)µ 1 x ε ∇u ε • ∇ x ′ ψ + 1 ε ∇ y ψ = o(1).
Since µ 1 (y) is periodic, it converges strongly with respect to two-scale convergence and we have

1 ε T 0 ϕ(t)dt Ω µ 1 x ε χ ε (x)∇u ε • ∇ y ψ dx = T 0 ϕ(t)dt Ω ′ ×G µ 1 (y)χ(y d ) (∇ x ′ u -1 (t, x ′ ) + ∇ y w(t, x ′ , y)) • ∇ y ψ dx ′ dy + o(1).
The above limit is just the variational formulation, for a.e. x ∈ Ω ′ and t > 0, of

div y (µ 1 (y) (∇ x ′ u -1 (t, x ′ ) + ∇ y w(t, x ′ , y))) = 0 in y ∈ Y,
with Neumann boundary conditions on the lower and upper faces as in [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF].

Let us now test (1) against ϕ ∈ C 1 c ((0, T ) × Ω) with ϕ(T, •) = 0. We obtain In order to pass to the limit as ε tends to zero, remark that χ ε ϕ or χ ε ∇ϕ are admissible test functions for two-scale convergence in the sense of boundary layers.

T 0 dt Ω dx(1 -χ ε ) u ε -ρ 0 (x) ∂ϕ ∂t + ω 0 (x) ϕ + µ 0 (x)∇u ε • ∇ϕ + T 0 dt Ω dxχ ε u ε -ρ 1 x ε ∂ϕ ∂t + ω 1 x ε ϕ + 1 ε µ 1 x ε ∇u ε • ∇ϕ - T 0 dt Ω χ ε 1 ε ϕf 1 dx = 0.
As a consequence, we obtain

T 0 dt Ω dx u -ρ 0 (x) ∂ϕ ∂t (t, x ′ , x d ) + ω 0 (x) ϕ (t, x ′ , x d ) + µ 0 (x)∇u • ∇ϕ (t, x ′ , x d ) + T 0 dt Ω ′ G χ(y d ) (µ 1 (y) (I d + P (y))) ∇ x ′ u -1 (t, x ′ ) • ∇ x ′ ϕ (t, x ′ , x d ) dydx ′ - T 0 dt Ω ′ dx ′ G χ(y d )f 1 (t, y)ϕ (t, x ′ , x d ) dy = 0.
This last system is the weak formulation of ( 9)-( 10),

T 0 Ω F (u, ϕ)dtdx+ T 0 Ω ′ F ′ (u(t, x ′ , 0), ϕ(t, x ′ , 0))dtdx ′ = T 0 Ω ′ f (t)ϕ(t, x ′ , 0))dtdx ′ ,
where f (t) is given by ( 11), F and F ′ are the bilinear forms

F (u, v) = u -ρ 0 dv dt + ω 0 v + µ 0 ∇ x u • ∇ x v, F ′ (u, v) = µ * ∇ x ′ u • ∇ x ′ v
, with µ * being the (d -1)-dimensional periodic homogenized diffusion coefficient given by (12).

Numerical simulation of a nuclear waste storage

In this section, we revisit the scaling of our model [START_REF]Dossier 2005 Argile -Évolution phénoménologique du stockage géologique[END_REF]. Introducing appropriate scales, we perform an adimensionalization and introduce the small parameter ε. Then, using some experimental values of the diffusion coefficients, we show that the dimensionless derived model is of the same type as [START_REF]Dossier 2005 Argile -Évolution phénoménologique du stockage géologique[END_REF]. Relying on the previously obtained homogenized problems we perform numerical simulations of the long time behavior of the nuclear waste storage. These computations are done with the FreeFem++ software which may be downloaded from [10].

6.1. Nuclear waste repository. The storage of nuclear waste is achieved within glass containers in groups of 10, inside alveolus situated on both sides of 3 galleries. This set is called module, following the ANDRA (French National Radioactive Waste Management Agency) terminology.

A large number of disposal modules makes up a storage zone. The whole repository system lies at a depth of about four to five hundred meters in the 130 meters thick geological argillite layer called Callovo-Oxfordian formation. Above this layer, the Oxfordian formation (400 meters height) is considered as the first geological limestone underground layer. Bellow, the Dogger formation (150 meters height) is a limestone layer. The computational domain is made of these three geological layers. This zone is on the whole L z = 684 meters deep (see Figure 3). Modeling all the details of a repository is difficult and it is a common practice to use a simplified model based on the homogenization of the storage zone [START_REF]Dossier 2005 Argile -Évolution phénoménologique du stockage géologique[END_REF]. In our approach, the cylindrical unit of storage (1.5 meters height and 1 meter diameter) containing the radioactive waste is surrounded by material such as concrete, clay, argillite or bentonite. This is what the simplified elementary cell is composed of. In our 2-d setting the cell has height H = 3 meters and length L = 3 meters. A large number of these elementary cells, 3 meters distant from each other, lie horizontally, overlapping the reference plane (z = 0) located in the middle of the Callovo-Oxfordian.

Scale analysis of the diffusion problem.

In first approximation the radionucleide transport is governed by a pure diffusion process [START_REF]Dossier 2005 Argile -Évolution phénoménologique du stockage géologique[END_REF]. Thus, the radionucleide concentration C satisfies the diffusion equation

R ω( ∂C ∂t + µ C) -div{D ∇ C} = F in O
where R is the latency retardation factor, ω the effective porosity and µ is related to the half life time τ of the element by the relation µ = log2/τ . We denote by D the effective diffusion tensor. Here, O is the 2-d (Oxz) plane section of the repository surrounded with its geological layers and we only consider the section given by L x = γ L z with γ ≥ 1. All the containers are supposed to be cut by this plane (see Figure 3). The source term F is related to experimental data f (t)(moles/year) by the relation F (t) = f (t)/S where S is the surface occupied by the nuclear waste and is zero outside the containers.

In the following, we use the subscript ref for characteristic or reference values of the parameters and variables involved. Setting

x a = x γL ref , z a = z L ref , D a = D D ref , R a = R R ref , ω a = ω ω ref , t a = t T ref
and omitting the subscript "a" for the dimensionless parameters and variables, the equation for the transport of the radioactive elements in each medium takes the general form where

(37) R ref ω ref T ref R ω( ∂C ∂t + λ C)-div{ D ref L 2 ref D γ ∇ C} = F (T ref t).
D γ = D   γ -2 0 0 1   .
Here λ stands for µ T ref .

The concentration C takes its values in the rescaled domain Ω = ]0, 1[×]0, 1[ (see Figure 3 ) composed by the subdomains denoted Ω sup (limestone layer), Ω COX (argillite layer) and Ω inf (limestone layer).

After dividing by

R ref ω ref /T ref , equation (37) reads R ω( ∂C ∂t + λ C)-div{ D ref T ref L 2 ref R ref ω ref D γ ∇ C} = F T ref R ref ω ref in Ω.
The adimensionalization relation

T ref = L 2 ref ρ ref D ref
defines the reference time. The surface S occupied by the L x /L containers represents a percentage δ of the thin strip in which they are stored

S = δ γ L ref H.
Introducing the concentration reference value

C ref = max( f ) 1 δγD ref ,
the dimensionless transport equation for the rescaled concentration

U = C/C ref is R ω( ∂U ∂t + λ U )-div{D γ ∇ U } = L ref H f in Ω,
where f = f /max( f ) inside the containers and zero outside.

6.3. Small parametric dependences. We choose the "observation distance" L ref = L z as the reference distance value and we introduce the small parameter ε as the ratio of the repository height over the reference length: ε = H L ref (≃ 4.3810 -3 ). The L x /L containers are distributed periodically over Ω ′ with period ε. They are surrounded by a coating (or buffer) material.

Typical values for diffusion coefficients in the case of 129 I (Iodine 129) are 1.8910 -2 m 2 /year for concrete [START_REF]Test cases definition: Homogenization of the source terms[END_REF] and 9.48 10 -7 [START_REF] Bourgeat | Talandier The COUPLEX Test Cases: Nuclear Waste Disposal Simulation[END_REF] for the argillite. The choice of concrete diffusion coefficient, as reference value, gives a dimensionless coefficient for surrounding material equal to 5.01 10 -5 (which is of the same order of magnitude as ε 2 ≃ 1.910 -5 ) and corresponds to the first case (a = 2).

Another typical value for diffusion coefficient of the argillite (always in the case of 129 I and with the same value for concrete) is 1.5810 -4 m 2 /year [START_REF]Test cases definition: Homogenization of the source terms[END_REF]. With this value as reference value the dimensionless coefficient for concrete equal to 120 (which is of the same order of magnitude as ε -1 ≃ 228) and corresponds to the second case (a = -1).

This simple scaling computation is thus a justification of our choice of ε-scaling in [START_REF]Dossier 2005 Argile -Évolution phénoménologique du stockage géologique[END_REF]. In other words, the diffusion tensor A ε scales like ε -1 or ε 2 , i.e., takes the form

(38) A ε (x, z) = ε a A(x/ε, z/ε)   γ -2 0 0 1   with a = -1 or 2,
in the thin strip |x| < ε/2 where (39)

A(x/ε, z/ε) = α 1 χ ε W ((x, z) + α 2 (1 -χ ε W ((x, z)).
In (39), α 1 denotes the diffusion constant for the vitrified nuclear waste, α 2 the one of the buffer material and χ ε W (x, z) = χ W (( x ε , z ε ) the characteristic function of the waste into the computational domain. For |x| > ε/2, the physical values of the diffusion constants of the others geological layers define the diffusion tensor A ε of order ε 0 .

In the same way, we denote by ρ ε (resp. ω ε ) the bounded function corresponding to the products R ω (resp. ρ ε λ) in the different subdomains of Ω. Introducing a right hand side or source term, which is of average of order ε 0 ,

f ε = ε -1 f χ ε W ((x, z),
the adimensionalized diffusion equation reads

ρ ε ∂u ε ∂t -div{A ε ∇ u ε } + λ ε u ε = f ε in Ω
where the concentration in the different subdomains is denoted u ε . Of course, we assume perfect transmission conditions at the interfaces between geological layers, namely continuity of the concentration and of the normal flux. As a final remark, let us mention that we have ignored convection phenomena which can take place, most notably in the upper and lower geological layers. This is just for simplicity and there is no conceptual difficulty to include them in our model as well.

6.4. Numerical simulation of a nuclear waste storage. Our asymptotic analysis in the previous sections leads to two different homogenized problems according to the value of the exponent a = -1, 2. The diffusion coefficients for the storage zone are constant, equal to ε a with a = -1, 2. All other physical constants are taken to be equal to one. We make a comparison between direct simulations of the original model ( 1) and reconstructed solutions of the homogenized models. The simulations concern the direct problems and its asymptotic ones. The source term is a hat function which takes its maximum value (f = 1) in the middle of the time interval (0, 1):

f (t) =    2t for0 ≤ t ≤ 1/2 2 -2t for 1/2 ≤ t ≤ 1 0 for t ≥ 1
All computations are perform on a unit square domain Ω (γ = 1). The numerical approximation of such parabolic boundary value problems is classical: we use a P 1 finite element method to discretize the variational formulation in space and an Euler implicit scheme for the time integration in the FreeFem++ software [10]. Time integration is perform until t = 1 in 120 time-steps. For the two direct problems (a = -1, 2), the same triangular mesh is used (≃ 34000 degrees of freedom). The corresponding Callovo-Oxfordian zone, surrounding the waste, is meshed more finely than the rest of the computational domain.

6.4.1. The case of small diffusion coefficient: a = 2. In this case, the asymptotic model ( 6) has a time-dependent Neumann type boundary condition on the asymptotic limit surface occupied by the storage zone. This emitting surface is the support of effective boundary conditions which supply a source term for the partial differential equation describing the diffusion of the concentration of radionucleides into geological medium surrounding the repository. The numerical simulation of the direct problem (1) shows a decoupling (see Figure 4(a)) of the solution at the emitting surface between the upper and lower parts of the computational domain Ω.

The solution of the direct problem (1) remains highly concentrated in the storage zone (see Figure 4(b)) which weakly emits towards the rest of the domain.

We compare this solution with that of the asymptotic problem [START_REF] Bourgeat | A homogenized model of an underground waste repository including a disturbed zone[END_REF] where the source terms s + and s -, defined by [START_REF] Bourgeat | Modelling of an underground waste disposal site by upscaling, simulation with domain decomposition method[END_REF], are computed via the numerical integration of the parabolic problem (8) associated to u 2 . In the parabolic boundary value problem [START_REF] Bourgeat | Talandier The COUPLEX Test Cases: Nuclear Waste Disposal Simulation[END_REF], all coefficients are constant, equal to one. Eventually, u 2 is computed on a very fine mesh (see Figure 5) of the elementary cell (a unit square containing the nuclear waste) with a classical P 2 finite element method and s + (resp. s -) by a standard quadrature formula.

Then, using two different triangular meshes for, respectively, the upper and lower parts of the computational domain (with respectively ≃ 7600 and ≃ 3400 degrees of freedom), the computation of the solution of the asymptotic problem (see Figure 6) is achieved.

The decoupling phenomenon, appearing in the asymptotic problem formulation, can be illustrated by Figure 7 which allows to compare the truncated solution of the direct problem in the storage zone and the one of the asymptotic problem. Outside the repository, the asymptotic solution overestimates the concentration found in the numerical computation of the direct problem. The numerical simulation of the direct problem (1) shows isovalues (Figure 8(a)) in the shape of a "smiling configuration". The same shape (Figure 8(b)) is also found in the numerical simulation of the homogenized problem [START_REF] Pino | Asymptotic analysis and layer decomposition for the couplex exercise[END_REF]. This numerical approximation (≃ 11000 degrees of freedom) requires the knowledge of the effective coefficient µ * defined by [START_REF] Huy | Phénomènes de transmission à travers des couches minces de conductivité élevée[END_REF]. Since we choose constant coefficients in the unit cell, the cell solutions ϕ i , defined by [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF], are explicit. More precisely, the horizontal solution is ϕ 1 ≡ 0 and the vertical solution is ϕ 2 = -y 2 . This implies that µ * is just equal to the constant value of µ 1 . The cross sections of the two solutions are displayed on Figure 9 which shows that both solutions behave similarly.

Remark 11. It is worth mentioning that the retardation factors as well as the effective porosity inside the thin strip containing the repository do not play any part in the homogenized problem (10).

Figure 1 .

 1 Figure 1. Sketch of the medium.

Figure 2 .

 2 Figure 2. Nuclear waste repository and surrounding geological layers.

Figure 3 .

 3 Figure 3. The computational domain.

Figure 4 .

 4 Figure 4. Small diffusion coefficient (a = 2). In Figure (a), isovalues of the solution of the direct problem (t = 1/2). In Figure (b), Close-up of the solution of the direct problem (t = 1/2).

Figure 5 .

 5 Figure 5. Mesh (≃ 21000 degrees of freedom) of the unit cell containing the waste.

Figure 6 . 6 . 4 . 2 .

 6642 Figure 6. Small diffusion coefficient (a = 2): Isovalues of the solution of the asymptotic problem (6) in the two parts of the computational domain (t = 1/2).

Figure 7 .Figure 9 .

 79 Figure 7. Small diffusion coefficient (a = 2): Cross sections of the truncated (dashed) direct and asymptotic solutions at x = 0.5 (t = 1/2).

GR ÉGOIRE ALLAIRE, MARC BRIANE, ROBERT BRIZZI, YVES CAPDEBOSCQ

Acknowledgments. This work has been supported by the Groupement de Recherches MoMaS sponsored by ANDRA, BRGM, CEA, CNRS, EDF, and IRSN. G. Allaire is a member of the DEFI project at INRIA Saclay Ile de France.


Seth R Hogg

Satish Muthu

Michael O'callaghan

Jean-Francois Lahitte

Merlin L Bruening

published or not. The documents may come

Wet air oxidation of formic acid using nanoparticle-modified polysulfone hollow fibers as gas-liquid contactors

Open Archive TOULOUSE Archive Ouverte (OATAO)

OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 9989

To link to this article : DOI:10.1021/am201693e URL : http://dx.doi.org/10.1021/am201693e

To cite this version : Hogg, Seth R. and Muthu, Satish and O'Callaghan, Michael and Lahitte, Jean-Francois and Bruening, Merlin L. Wet air oxidation of formic acid using nanoparticlemodified polysulfone hollow fibers as gas-liquid contactors. (2012) ACS Applied Materials & Interfaces, vol. 4 (n° 3). pp. 1440-1448. ISSN 1944-8244 Any correspondance concerning this service should be sent to the repository administrator: staff-oatao@listes-diff.inp-toulouse.fr


Cristina Flox

Enric Brillas

André Savall

Karine Groenen-Serrano

Kinetic Study of the Electrochemical Mineralization of m-Cresol on a Boron-Doped Diamond Anode

Keywords: Electrooxidation, Boron-doped diamond, Cresol, Wastewater treatment

The kinetics of the electrooxidation of m-cresol in aqueous solution was investigated in a one-compartment flow electrochemical cell with a boron-doped diamond electrode (BDD). Cyclic voltammograms recorded on BDD revealed that cresol oxidation takes place at a potential very close to the discharge of water. Under potentiostatic conditions, at a working potential lower than water discharge, a passive layer was rapidly formed on the electrode surface due to cresol polymerization. The anode fouling was not observed during electrolysis performed with the flow electrochemical cell operating under galvanostatic conditions. In this case, the decay of mcresol concentration followed a pseudo-first-order kinetics. The abatement of chemical oxygen demand (COD) showed that the kinetics of m-cresol oxidation was limited by mass transfer and that a full mineralization was achieved. A good agreement between predicted and experimental COD and instantaneous current efficiency values was obtained, although some deviations were observed at high current since the experimental data decreased faster than those predicted ones. These deviations can be explained by the occurrence of oxygen evolution which increases the mass transfer coefficient.

INTRODUCTION

Aromatic compounds are generally toxic and the products having a quinonic structure are characterized by very strong toxicities. Aromatic structures are usually very stable and as a consequence, aqueous effluents polluted with such compounds should be decontaminated by effective physico-chemical techniques of oxidation. Cresols are classified by the US-EPA as persistent, priority and toxic chemicals, showing chronic effects at 12 mg L-1 of the quantitative structure-activity relationship [START_REF] Kavitha | Destruction of cresols by Fenton oxidation process[END_REF]. Unfortunately, the high contamination level that these compounds reach in industrial wastewaters, their seasonal production and the presence of other organic pollutants such as lipids usually render these effluents inappropriate for direct biological treatment. Alternative powerful oxidation technologies are then needed to be developed for achieving total destruction of cresols from wastewaters.

Several papers reported the removal of cresols by advanced oxidation processes (AOPs) [START_REF] Kavitha | Destruction of cresols by Fenton oxidation process[END_REF][START_REF] Hatipo Lu | An experimental and theoretical investigation of the photocatalytic degradation of meta-cresol in TiO2 suspensions: A model for the product distribution[END_REF][START_REF] Wang | The heterogeneous photocatalytic degradation, intermediates and mineralization for the aqueous solution of cresols and nitrophenols[END_REF][START_REF] Sivalingam | Kinetics of the photodegradation of substituted phenols by solution combustion synthesized TiO2[END_REF]. The main features of the electrochemical oxidation of cresols on BDD and PbO 2 were presented in a previous paper [START_REF] Flox | Electrochemical incineration of cresols: A comparative study between PbO2 and boron-doped diamond anodes[END_REF]. The use of Fenton's reagent for cresol oxidation is effective during the first steps of the process, but it suffers the formation of by-products like oxalic or acetic acids that limits the mineralization because of the formation of stable complexes with iron ions [START_REF] Brillas | Electro-Fenton process and related electrochemical technologies based on Fenton's reaction chemistry[END_REF].

Electrochemistry is a convenient technique for the treatment of industrial effluents containing organics. Mineralization of organic compounds can be fully achieved under conditions in which the transfer of oxygen atoms is effective. It is generally assumed that hydroxyl radical (•OH) resulting from water discharge by reaction [START_REF] Kavitha | Destruction of cresols by Fenton oxidation process[END_REF] is particularly active to carry out oxygen atoms transfer [START_REF] Wabner | Reactive intermediates during oxidation of water on lead dioxide and platinum electrodes[END_REF][START_REF] Fleszar | An attempt to define benzene and phenol electrochemical oxidation mechanism[END_REF][START_REF] Trasatti | Electrocatalysis by oxides -Attempt at a unifying approach[END_REF][START_REF] Vitt | The importance of anodic discharge of H2O in anodic oxygen-transfer reactions[END_REF]:

H 2 O • OH + H + + e (1)
The strong activity of this radical in aqueous solution results from the high value of its standard potential E 0 = 2.74 V/SHE [START_REF] Kläning | Standard Gibbs free energy of formation of the hydroxyl radical in aqueous solution. Rate constants for the reaction ClO2 -+ O3 O3 -+ ClO2[END_REF]. Organic molecules are degraded into smaller molecules by a succession of oxygen atom transfers until complete destruction of the carbon skeleton [START_REF] Vitt | The importance of anodic discharge of H2O in anodic oxygen-transfer reactions[END_REF]. In these last 20 years, the use of boron-doped diamond (BDD) has known an increasing interest in the electrochemical community because of its unique properties. BDD produces hydroxyl radicals that are free on its surface and can react massively close to the electrode allowing the complete mineralization of organic compounds.

Several papers reported the rapid removal of o-, m-and p-cresol from waters by electro-oxidation using a BDD anode [START_REF] Flox | Electrochemical incineration of cresols: A comparative study between PbO2 and boron-doped diamond anodes[END_REF][START_REF] Nava | Electrochemical incineration of p-cresol and o-cresol in the filter-press-type FM01-LC electrochemical cell using BDD electrodes in sulfate media at pH 0[END_REF][START_REF] Mascia | Electrochemical treatment of phenolic waters in presence of chloride with boron-doped diamond anodes: Experimental study and mathematical model[END_REF]. The behaviour of BDD was numerically predicted by mathematical model developed by Mascia et al. [START_REF] Mascia | Electrochemical treatment of phenolic waters in presence of chloride with boron-doped diamond anodes: Experimental study and mathematical model[END_REF] and the effect of the presence of chloride ions on the COD removal was studied.

A previous study [START_REF] Flox | Electrochemical incineration of cresols: A comparative study between PbO2 and boron-doped diamond anodes[END_REF] showed that the electrochemical incineration of m-cresol on a BDD anode occurs by giving by-products in very weak concentration. Furthermore, it was found that the PbO 2 anode is comparatively much less efficient due to the lower oxidation ability of hydroxyl radicals that it is able to produce under the same operating conditions.

The aim of this work is to gain a better insight into the kinetics of the electrochemical mineralization of m-cresol in a onecompartment flow electrochemical reactor with a BDD anode. The influence of operating parameters such as applied current, initial mcresol concentration and electrolyte flow rate was examined. The results obtained are presented and discussed in terms of m-cresol decay, current efficiency and chemical oxygen demand (COD) removal. Cyclic voltammetry was also used to ascertain the nature of the oxidation process as a function of the applied potential.

EXPERIMENTAL

Chemicals

m-Cresol (99% purity) was purchased from Sigma and used as received. Solutions were prepared with high-purity water obtained from a Millipore Milli-Q system (resistivity > 18 M cm at 25 ºC). Analytical grade sulfuric acid from Prolabo was used to adjust the initial pH to 4.0. Anhydrous sodium sulfate employed as supporting electrolyte was analytical grade supplied by Prolabo. Other chemicals and organic solvents were either HPLC or analytical grade from Sigma-Aldrich, Merck and Prolabo.

Cyclic Voltammetric Study

Cyclic voltammograms were carried out in a conventional three-electrode cell of 100 mL capacity using a computer controlled Autolab potentiostat Model 30. A BDD thin film of 0.196 cm 2 from Adamant was used as working electrode, a Pt wire as counter electrode and a Hg/Hg 2 Cl 2 ,Cl -(sat) electrode (SME) as the reference electrode. Before each experiment, the electrode was pre-treated using anodic polarisation at 2.84 V/SHE during 40 s into a 1 M H 2 SO 4 solution.

Electrochemical Reactor

Electrolyses were conducted in a one-compartment flow filterpress reactor under galvanostatic conditions (Fig. 1) The electrolyte was stored in a 1 liter thermoregulated glass reservoir (1) and circulated through the electrolytic cell using a centrifugal pump (2). Electrodes were two discs of 63 cm of active surface separated 10 mm. The BDD anode from Adamant was elaborated by chemical vapour deposition on a conductive substrate of silicium. The cathode was a 1 mm thick disc of zirconium. The current was supplied by an ELC AL 924 power supply.

Analysis Procedures

Samples withdrawn from treated solutions were filtered with Whatman 0.45 m PTFE filters before analysis. The mineralization process was monitored from COD decay. COD was determined by photometry by means of test tubes ready for use and of a pocket photometer (Dr Changes Lasa 50 system). m-Cresol concentration was followed by reversed-phase HPLC chromatography using a Hewlett-Packard 1100 Series liquid chromatograph, fitted with a Spherisorb ODS2 5 m, 150 mm x 4.6 mm (i.d.), column at room temperature, equipped with an UV/Vis detector selected at = 276 nm and controlled with an Agilent ChemStation software. These measurements were made by injecting 20 L aliquots and circulating a 50:45:5 (v/v/v) methanol/phosphate buffer (pH = 2.5)/pentanol mixture at 1.0 mL min -1 as mobile phase.

Kinetic Regimes

The kinetics of the electrode processes can be controlled either by mass transfer or by charge transfer. For a constant current density higher than the limiting current density, the process rate is limited by mass transport to the electrode. In that case, oxygen evolution is the main secondary reaction occurring on the anode. In contrast, for a constant current density lower than the limiting value, the global reaction is limited by charge transfer. The initial limiting current density is calculated as follows:

i lim 0 = z F k d C 0 ( 2
)
where z is the number of exchanged electrons for the electrode process, F is the Faraday constant (96,487 C mol -1 ), k d is the mass transfer coefficient (m s -1 ) and C 0 is the initial concentration of the substrate (mol m -3 ).

While the z number is not exactly known for an organic compound oxidized on a PbO 2 anode characterized by a high-oxygen overvoltage [START_REF] Weiss | A kinetic study of the electrochemical oxidation of maleic acid on boron doped diamond[END_REF][START_REF] Belhadj Tahar | Mechanistic aspects of phenol electrochemical degradation by oxidation on a Ta/PbO2 anode[END_REF], in the case of the BDD anode, the z number is known and the concentration C 0 in Eq. 2 can be replaced by the initial COD expressed in mol of O 2 m -3 [START_REF] Panizza | Anodic oxidation of 2-naphthol at boron-doped diamond electrodes[END_REF] as giving in Eq. 3:

i lim 0 = 4 F k d COD° (3)
Under galvanostatic conditions, two different operating regimes are defined:

o When the applied current (I) is lower than the limiting current (I lim ), the electrolysis is controlled by the charge transfer, the current efficiency is 100% and the rate of COD removal is constant and can be written as:

COD t ( ) = COD 0 1 t (4)
In Eq. 4, is defined as the ratio I/I lim , t is the electrolysis time and is a time characteristic of the device defined as = V/(k d A), where V the volume of the solution to be treated (m 3 ) and A is the electrode active area (m 2 ).

o If the applied current exceeds the limiting current ( > 1), the process is mass transfer controlled. In this regime, COD decreases exponentially with time according to Eq. 5.

COD t

( ) = COD 0 exp t (5)
Secondary reactions such as oxygen evolution take place and the instantaneous current efficiency (ICE) is lower than 100%. This parameter is calculated from COD values from Eq. [START_REF] Brillas | Electro-Fenton process and related electrochemical technologies based on Fenton's reaction chemistry[END_REF].

ICE = COD exp COD th = 4 F V COD t ( ) COD(t + t) I t M O 2 (6)
+ BDD Zr (1) (2) (3)
where COD(t) and COD(t + t) are the concentrations of COD expressed in g L -1 at times t and t + t, respectively, M O 2 is the dioxygen molar weight (32 g mol -1 ) and V is the volume of solution (L).

RESULTS AND DISCUSSION

Cyclic Voltammetric Study

The electrochemical behavior of m-cresol in sodium sulfate solution was studied by cyclic voltammetry on the BDD electrode. Fig. [START_REF] Hatipo Lu | An experimental and theoretical investigation of the photocatalytic degradation of meta-cresol in TiO2 suspensions: A model for the product distribution[END_REF] shows two successive scans recorded between -0.5 and 1.5 V at 25 °C. During the first scan towards positive potentials, a well defined peak which corresponds to m-cresol oxidation appears before water oxidation (see dashed line). Fig. (2) also shows that electrode passivation occurred after the first cycle since no oxidation peak was observed in the second cycle. This behavior is due to the formation of a passive layer of polymers at the surface of the working electrode.

Various cyclic voltammograms were also carried out at different concentrations of m-cresol. In each case, only the first scan gave an oxidation peak at similar potentials to those shown in Fig. [START_REF] Hatipo Lu | An experimental and theoretical investigation of the photocatalytic degradation of meta-cresol in TiO2 suspensions: A model for the product distribution[END_REF]. After each scan, the electrode activity was restored using anodic polarisation at 2.84 V during 40 s. The variation of the anodic peak current as function of m-cresol concentration is shown in Fig. (3). 3) depicts that at concentrations of m-cresol lower than 5 mM, the peak current varies linearly with the concentration in the solution. In contrast, at concentrations higher than 5 mM, the peak current is stabilized due to the formation of a polymer film spread on the electrode surface. This passive film blocks the electrode activity. Indeed, a visual examination of the electrode surface revealed the presence of an adherent film of polymer (see micrograph in Fig. 4).

I /mA [m-cresol] /mM

Similar observations have been noted using BDD anodes in the case of phenol [START_REF] Iniesta | Electrochemical oxidation of phenol at boron-doped diamond electrode[END_REF], 2-naphtol [START_REF] Panizza | Influence of anode material on the electrochemical oxidation of 2-naphthol: Part 1. Cyclic voltammetry and potential step experiments[END_REF] and 4-chlorophenol [START_REF] Gherardini | Electrochemical oxidation of 4-chlorophenol for wastewater treatment: Definition of normalized current efficiency[END_REF]. In these papers, it is assumed that the electron transfer to the phenolic ring leads to phenoxy radicals that react to form polymeric structures by coupling reactions. This behavior could also be considered for m-cresol oxidation at potentials lower than those of water discharge.

Electrolysis in the Flow Cell Under Galvanostatic Conditions

Fig. [START_REF] Flox | Electrochemical incineration of cresols: A comparative study between PbO2 and boron-doped diamond anodes[END_REF] shows that the kinetics of the oxidation of 534 ppm of m-cresol in 0.05 M Na 2 SO 4 using the flow electrochemical cell with a BDD anode is enhanced when the temperature rises. So, the compound was completely removed in 420 min at 25 ºC, but only in about 300 min at 60 ºC operating at 2.5 A and liquid flow rate of 126 L h -1 . Under these conditions, the BDD reached potentials in the water discharge region and no polymers were deposited on it.

According to our previous study [START_REF] Flox | Electrochemical incineration of cresols: A comparative study between PbO2 and boron-doped diamond anodes[END_REF], the above concentration decays were well-fitted to a pseudo-first-order kinetic equation. A quite similar kinetic behavior has also been described by Polcaro et al. [START_REF] Polcaro | Electrochemical oxidation of phenolic and other organic compounds at boron doped diamond electrodes for wastewater treatment: Effect of mass transfer[END_REF] for the removal of phenol solutions with BDD in galvanostatic electrolysis. The inset panel of Fig. [START_REF] Flox | Electrochemical incineration of cresols: A comparative study between PbO2 and boron-doped diamond anodes[END_REF] confirms the firstorder kinetics for the anodic oxidation of m-cresol, allowing deter-mining the apparent kinetic constant (k) for the above temperatures. The k-value measured for the same ratio electrode surface/volume treated was 1.69 10 -4 s -1 for 25 ºC and 3.19 10 -4 s -1 for 60 ºC. These data allow estimating an activation energy of 15 kJ mol -1 , which is in agreement with a kinetic process limited by mass transfer [START_REF] Benson | The Foundations of Chemical kinetics[END_REF][START_REF] Su | Kinetics of trichloroethene reduction by zerovalent iron and tin: Pretreatment effect, apparent activation energy and intermediate products[END_REF].

Fig. [START_REF] Brillas | Electro-Fenton process and related electrochemical technologies based on Fenton's reaction chemistry[END_REF] highlights the influence of the flow rate on COD decay at the same operating parameters. As can be seen, the COD removal becomes more effective as the flow rate of the electrolyte is higher, indicating that the process is mass transport limited. For the electrochemical reactor, the following relation between the flow rate of the solution (u, in L h -1 ) and the mass transfer coefficient (k d , in m s -1 ) was determined by means of the ferri/ferro redox system under the same hydrodynamic conditions [START_REF] Panizza | Anodic oxidation of 2-naphthol at boron-doped diamond electrodes[END_REF]: k d ==2.78 10 -6 u 0.38 [START_REF] Wabner | Reactive intermediates during oxidation of water on lead dioxide and platinum electrodes[END_REF] Note that at 25°C, for an electrolysis carried out at I = 2.5 A using a flow rate of 126 L h -1 , m-cresol decays in 87% at t = 200 min (Fig. 5) while the related COD removal is smaller, close to 75% (Fig. 6). Comparison of these m-cresol and COD removals confirms the almost direct conversion of the initial pollutant into carbon dioxide and water using a BDD anode, in agreement with the very few amounts of by-products reported by us previously [START_REF] Flox | Electrochemical incineration of cresols: A comparative study between PbO2 and boron-doped diamond anodes[END_REF]. Indeed,

42 L h -1 126 L h -1 252 L h -1
this study highlighted that the temporal evolution of the experimental dissolved organic carbon (DOC) obtained during galvanostatic electrolysis of a m-cresol solution using BDD was similar to that calculated from the m-cresol concentration. Consequently, it is convenient to use the global parameter of COD and ICE to monitor the detoxification of the solution during the electrolysis (Eq 4-6).

Fig. [START_REF] Wabner | Reactive intermediates during oxidation of water on lead dioxide and platinum electrodes[END_REF] shows the decrease of the normalized chemical oxygen demand of m-cresol solutions at different initial concentrations during electrolysis carried out at 2.5 A, 25 ºC and electrolyte flow rate of 126 L h -1 . The m-cresol concentrations and the constant current density were chosen in such a way that the process is limited by the mass transfer to the electrode from the beginning of the electrolysis (except for m-cresol = 1253 ppm, I = I lim at 168 min). Fig. [START_REF] Wabner | Reactive intermediates during oxidation of water on lead dioxide and platinum electrodes[END_REF] evidences the same variation of the normalized COD with time whatever the initial concentration of m-cresol in the solution, reaching always overall decontamination in about 500 min. Moreover, Fig. [START_REF] Wabner | Reactive intermediates during oxidation of water on lead dioxide and platinum electrodes[END_REF] shows a very good agreement between the experimental and theoretical (given by Eq. 4-5) COD values. Thus, this model allows predicting the change of COD for any m-cresol solution. The COD variation then follows a first-order kinetics whatever the initial concentration because a m-cresol molecule reaching the BDD anode is almost totally mineralized, without significant formation and diffusion of by-products towards the bulk. Fig. [START_REF] Fleszar | An attempt to define benzene and phenol electrochemical oxidation mechanism[END_REF] shows the variation of the m-cresol concentration as a function of time for the same experiments of Fig. [START_REF] Wabner | Reactive intermediates during oxidation of water on lead dioxide and platinum electrodes[END_REF]. The inset panel in Fig. [START_REF] Fleszar | An attempt to define benzene and phenol electrochemical oxidation mechanism[END_REF] depicts that the concentration decay always verifies a pseudo-first-order kinetics and that the rate of disappearance of m-cresol becomes faster for smaller initial concentration. The kvalues thus obtained were 2.16 10 -4 , 1.69 10 -4 , 1.50 10 -4 and 1.33 10 -4 s -1 for 246, 534, 790 and 1253 ppm, respectively. Nevertheless, at a given time, increasing concentrations yield the removal of more amount of m-cresol.

These results can be explained as follows. The decrease in mcresol concentration leads to an increasing part of the current used for water discharge by reaction (1). Thus, a higher production of oxygen bubbles occurs close to the electrode by the following sequence:

• OH + • OH k 1 H 2 O 2 (8)
H 2 O 2 + • OH k 2 H 2 O+ O 2 •- + H + (9) O 2 •
- + • OH k 3 O 2 + OH - (10)
The absolute constant rates for these reactions are k 1 = 5.2 10 9 M -1 s -1 [START_REF] Pagsberg | Far-ultraviolet spectra of hydrogen and hydroxyl radicals from pulse radiolysis of aqueous solutions. Direct measurement of the rate of H+H[END_REF][START_REF] Thomas | Rates of reaction of the hydroxyl radical[END_REF], k 2 = 2.7 10 7 M -1 s -1 [START_REF] Buxton | Critical review of rate constants for reactions of hydrated electrons, hydrogen atoms and hydroxyl radicals, (OH, O2 -) in aqueous solution[END_REF][START_REF] Christensen | Reactions of hydroxyl radicals with hydrogen peroxide at ambient and elevated temperatures[END_REF] and k 3 = 10 10 M -1 s -1 [START_REF] Elliot | Temperature dependence of the reactions OH + O2and OH + HO2 in water up to 200[END_REF][START_REF] Christensen | Temperature dependence of the reactions OH + O2and OH + HO2 in water up to 200 °C[END_REF]. Consequently, the mass transfer coefficient increases locally. Accordingly, the kinetic constant k = k d A/V increases with decreasing m-cresol concentration. It is important to compare the experimental variation of COD and the theoretical values predicted by theory (Eq. 4-6) for both operating regimes:

• I < I lim ° (1.76 A). For this case, Fig. 9 highlights the existence of two consecutive different kinetic zones. At the beginning of the electrolysis (specific charge (Q) < 0.78 Ah L -1 , t < 31 min) and according to electron transfer control, COD varies linearly according to Eq. 4. However, at times longer than t c = 31 min, COD decreases according to an exponential decay, characteristic of a process limited by mass transfer (Eq. 5). Moreover, theoretical and experimental values for both COD and ICE (inset panel of Fig. 9) are in quite good agreement.

• I > I lim °. Fig. 10 shows the variation of COD during galvanostatic electrolysis at a current widely higher than the initial limiting current. As can be seen, the experimental values are lower than those calculated from the model (Eq. 5, dashed line in Fig. 10). A similar behavior is observed for the instantaneous current efficiency (inset panel of Fig. 10). To explain these deviations, it can be taken into account that during the electrolysis, the depletion of organic matter leads to an increase of oxygen evolution at the anode, as pointed out above. Consequently, the rate of gas evolution close to the electrode has a great effect on mass transfer since a release of bubbles increases the mass transfer coefficient [START_REF] Vogt | Mass transfer at gas involving electrodes with superposition of hydrodynamic flow[END_REF]. Beck [START_REF] Beck | A contribution to the theory of electrolytic chlorate formation[END_REF] showed that the mass transfer coefficient k d is the sum of two contributions, the flow rate k d h and the convection due to bubbles k d b , according to Eq. 11:

k d = k d h + k d b (11)
Nevertheless, Vogt [START_REF] Vogt | Mass transfer at gas involving electrodes with superposition of hydrodynamic flow[END_REF] tested and proposed a more precise correlation given by Eq. 12, which takes into account the influence of the flow rate on the bubble radius before dropping out: ture. The average value for dioxygen in acid media is estimated to m=0.5 [START_REF] Beck | A contribution to the theory of electrolytic chlorate formation[END_REF][START_REF] Janssen | The effect of electrolytically evolved gas bubbles on the thickness of the diffusion layer[END_REF].

The gas flow production,

V G • S
, is a function of the part of the applied current which produces dioxygen, i O 2 , and the molar volume of dioxygen, V mol :

V G • A = i O 2 n F V mol ( 14
)
The constant of eq. 13 was estimated to 5.4 10 -3 with experimental data resulted from an electrolysis controlled by mass transfer à t=3h.

For Fig. [START_REF] Vitt | The importance of anodic discharge of H2O in anodic oxygen-transfer reactions[END_REF], a k d value of 3.5 10 -5 m s -1 was estimated from Eq. 12 to 14, instead of k d h = 1.75 10 -5 m s -1 . Taking this corrected k d value, the theoretical COD and ICE decays show an excellent fitting with the experimental points, as can be seen in Fig. [START_REF] Vitt | The importance of anodic discharge of H2O in anodic oxygen-transfer reactions[END_REF].

CONCLUSIONS

The anodic oxidation of m-cresol on BDD at a potential lower than the water discharge promotes the formation of polymeric films quickly fouling the electrode. The electron transfer leads to phenoxy radicals that react to yield polymeric structures by coupling reactions. The formation of this non conductive polymer is fast and makes that the current of its anodic peak reaches a limiting value for a concentration of 5 mm. The variation of m-cresol concentration during electrolysis performed in galvanostatic mode using a flow electrochemical cell with a BDD anode corresponds to a firstorder kinetics, in accordance with a process limited by mass transfer. Also, the effect of the electrolyte flow rate on the mass transfer coefficient shows that m-cresol oxidation is limited by diffusion. Experiments carried out at different temperatures allowed estimating an activation energy of 15 kj mol -1 , a rather low value which confirms a diffusion-controlled process. The COD variation evidences the excellent performance of BDD anode towards the mineralization of m-cresol. Practically all the flow of organic matter arriving at the electrode is instantaneously mineralized. The mass transfer is widely increased by oxygen evolution when the electrolysis is carried out at current higher than its limiting value.

Fig. ( 1 ).

 1 Fig. (1). Discontinuous process with a single compartment electrochemical reactor, (1): 1 L tank, (2): centrifugal pump and (3): electrochemical cell.

Fig. (

  Fig. (3) depicts that at concentrations of m-cresol lower than 5 mM, the peak current varies linearly with the concentration in the solution. In contrast, at concentrations higher than 5 mM, the peak current is stabilized due to the formation of a polymer film spread on the electrode surface. This passive film blocks the electrode activity. Indeed, a visual examination of the electrode surface revealed the presence of an adherent film of polymer (see micrograph in Fig.4).

Fig. ( 4

 4 Fig. (4). Optical micrograph of the BDD surface after the cyclic voltammograms of Fig. (2). The black zone corresponds to BDD deposit and the white spots to deposited polymers.

Fig. ( 2 ).

 2 Fig. (2). Consecutive cyclic voltammograms (scans 1 and 2) recorded for 10 mM m-cresol in 0.5 M Na2SO4. The dashed curve corresponds to the electrolyte solution. Working electrode: BDD (0.196 cm ), counter electrode: Pt and reference electrode: MSE. Scan rate: 50 mV s -1 .

Fig. ( 5 ).

 5 Fig. (5).Variation of m-cresol concentration during the electrolysis of 1 L of a 534 ppm m-cresol solution in 0.05 M Na2SO4 using a flow electrochemical cell with a BDD anode at 2.5 A, temperatures of 25 and 60 ºC and liquid flow rate of 126 L h -1 . The inset panel shows the kinetic analysis for the related experiments assuming a pseudo-first-order reaction for each temperature.

Fig. ( 6 )

 6 Fig.[START_REF] Brillas | Electro-Fenton process and related electrochemical technologies based on Fenton's reaction chemistry[END_REF]. COD abatement during the electrolysis of 1 L of a 534 ppm m-cresol solution in 0.05 M Na2SO4 using a flow electrochemical cell with a BDD anode at 2.5 A and 25 °C for electrolyte flow rates of 42, 126 and 252 L h -1 . The inset panel shows the kinetic analysis for the related experiments assuming a pseudo-first-order reaction for each flow rate.

Fig. ( 7 ). 7 .

 77 Fig. (7). Variation of normalized COD calculated (lines) and experimental (symbols) for the electrochemical oxidation of solutions containing m-cresol concentrations of 246, 534, 790 and 1253 ppm at 2.5 A, 25 º C and electrolyte flow rate of 126 L h -1 .

  In this equation, D is the gas diffusion coefficient in liquid phase, V G • is the volumetric rate of gas evolution and S area of the flow cross section. The constant value m is function of the gas na-

Fig. ( 9 ).

 9 Fig. (9). Variation of COD and ICE (inset panel) with specific current during the electrochemical oxidation of 534 ppm m-cresol in 0.05 M Na2SO4 on a BDD anode. Operating conditions: I = 1.5 A, Ilim°= 1.76 A, tc = 31 min, T = 25 °C and electrolyte flow rate = 126 L h -1 . Experimental points (symbols) and theoretical variation (solid line).

Fig. ( 10 ).

 10 Fig.[START_REF] Vitt | The importance of anodic discharge of H2O in anodic oxygen-transfer reactions[END_REF]. Change of COD and ICE (inset panel) with specific current during the electrochemical oxidation of 534 ppm m-cresol in 0.05 M Na2SO4 Na2SO4 on a BDD anode. Operating conditions: I = 5 A, T = 25 °C and electrolyte flow rate = 126 L h -1 . Experimental points (symbols) and theoretical variation (dashed line: kd = k h d = 1.75 10 5 m s -1 , solid line: kd corrected = 3.5 10 -5 m s -1 (from Eq. 12)).

ACKNOWLEDGEMENT

Declared none.

CONFLICT OF INTEREST

The author(s) confirm that this article content has no conflicts of interest.


domain: sde spi

text:
Loubna Qabbal

Zohir Younsi

Hassane Naji

Naji Hassane

Indoor air quality (IAQ) measurements in a tertiary building via a smart sensor connected to a Raspberry Pi card: application to a demonstrator building

Keywords: Comfort parameters, Indoor air quality (IAQ), Smart ventilation

The aim of this study is to improve indoor air quality (IAQ) using an intelligent ventilation for optimal air distribution and energy consumption. For this purpose, we developed a smart multi-sensor. This device measures air pollutants (CO2, CO,VOCs, formaldehyde, PM2.5, and benzene) and comfort parameters (temperature, humidity, noise level and lighting) in a demonstrating building. It should be noted that measurements are achieved instantly and temporally, and that data processing is done via an algorithm. This will effectively ventilate the building as needed. To go further, we conducted three sets of measurements on IAQ and hygrothermal comfort in winter, mid-season and summer in a demonstrator building room. The obtained results showed that the level of air quality is acceptable in terms of VOCs, PM2.5, formaldehyde and benzene. In addition, the CO2 rate turned out to be high during occupancy periods. In terms of hygrothermal comfort, the air was dry and very hot, especially during the winter. This is due to the current ventilation system which does not consider the variable "humidity" to predict the comfort. Over the three measurements, we considered the occupants' perception of IAQ and hygrothermal comfort using a survey that has been proposed to occupants (students).

Introduction

In France, indoor air quality (IAQ) in buildings has a particular attention over the past several years [START_REF] Annesi-Maesano | Indoor air quality and sources in schools and related health effects[END_REF]. Since we spend 90% of our time in closed spaces (housing, office, school, transportation, etc.), the air breathed every day must be of good quality. However, many studies show that indoor air is 8 times more polluted than outdoor air.

In recent years, the use of synthetic materials in building and furnishing, the adoption of new lifestyles and the intensive use of products for environmental cleaning and personal hygiene have contributed to the deterioration of quality of indoor air. As a result, they have introduced new sources of risk to humans [START_REF] Cao | Associated relationship between ventilation rates and indoor air quality[END_REF]. It becomes clear that indoor air pollution has important consequences for the health of individuals. Poor indoor air quality (IAQ) can cause health problems such as respiratory system conditions (rhinitis or bronchitis), headaches, fatigue, eye irritation, nausea, etc. In France, asthma affects 3.5 million people and severe respiratory failure affects 50,000 people. However, a good IAQ has a positive effect on the reduction of absenteeism, the well-being of occupants and their learning.

The ventilation is a well-recognized solution for reducing the variety of contaminants that can be found inside residential buildings. However, just as clean air is an essential factor for a healthy living and a healthy building, low energy consumption is significant for a healthy planet. Therefore, a good IAQ in schools is important to provide a safe, healthy, productive, and comfortable environment for occupants [START_REF] Fisk | The ventilation problem in schools; literature review[END_REF].

In France, the desire to deepen knowledge in the air quality field inside buildings has been specified in the first National Health and Environment Plan (PNSE1, 2004-2008).

In addition, the issue of IAQ was the subject of the third National Health Environment Plan (PNSE 3, 2013) to study concrete actions to ensure a healthy environment.

Health and economic issues related to IAQ are important. Thereby, it is important to find a solution that improves IAQ and saves energy at the same time. The ventilation has become an important approach in the handling of the building energy consumption [START_REF] Wargocki | The effects of moderately raised classroom temperatures and classroom ventilation rate on the performance of schoolwork by children (RP-1257)[END_REF][START_REF] Daisey | Indoor air quality, ventilation and health symptoms in schools: an analysis of existing information[END_REF]. It has been stated [START_REF] Myhrvold | Indoor environment in schools-pupils health and performance in regard to CO2 concentrations[END_REF] that when the ventilation flow is high in classrooms, students' performance increases by 15%. Indeed, high concentrations of CO2 indicate that ventilation is insufficient, which can lead to health symptoms [START_REF] Hänninen | Analysis of CO2 monitoring data demonstrates poor ventilation rates in Albanian schools during the cold season[END_REF]. It is worth noting that a European study of 800 children in schools revealed that student scores on concentration tests decreased when CO2 levels increased [START_REF] Cartieaux | Indoor air quality in schools[END_REF].

It appears that poor ventilation in schools is associated with the accumulation of indoorgenerated pollutants, which is associated with "stuffy" air. Thereby, there is a high risk of infectious diseases and learning outcomes may be impaired [START_REF] Au Yeung | Sick building syndrome-A case study[END_REF]. This shows that there is a correlation between pollutants concentrations and the appearance of learners' health disorders [START_REF] Ramalho | Air stuffiness and air exchange rate in French schools and day-care centres[END_REF]. In school settings, IAQ has been shown to influence academic performance. Thereby, finding a compromise between the energy and health aspects is very sensitive. For this purpose, we propose a solution based on an intelligent ventilation study, which helps to preserve the health of the occupants in terms of IAQ while optimizing the rate of the air renewal. This study is based on the automation of the existing ventilation system in the demonstrator building using a smart sensor "multisensor sensor" that has been developed in-situ. This sensor measures several air pollutants, namely CO2, VOCs, formaldehyde, benzene, CO, PM2.5, as well as comfort parameters (temperature, humidity, noise and brightness). The sensor has been developed via a Raspberry Pi 3-card that allows the connectivity of the case. The measurements are made instantly while allowing the building manager to know the comfort situation (thermal and sanitary) in each space of the building and act quickly in case of malfunction of the ventilation system.

Experimental approach description

To determine the atmospheric pollutants existing in the demonstrator building located in the Lille city, we conducted three campaigns to measure IAQ and comfort parameters. We measured the CO2, VOCs, formaldehyde, CO, benzene, ozone and PM2.5. The comfort parameters were evaluated in terms of temperature and humidity. Previously, one of the building's rooms (T201) was instrumented. It should be noted that this room can accommodate up to 56 students (Fig. 1 and Fig. 2).

Indoor air quality (IAQ) measurements in a tertiary building via a smart sensor connected to a Raspberry Pi card: application to a demonstrator building Loubna Qabbal, Zohir Younsi, Naji Hassane

Geometry of the case study

The sensor, which measures IAQ and comfort parameters, was placed in the middle of the wall behind occupants and at a height of 1.50 m above the floor (occupants' sitting height). Usually, the sensor should be placed near the pollution source. The instrumentation room (T201) is located on the 2nd floor of the demonstrator building whose dimensions are shown in Fig. 3.

Results and discussion

IAQ assessment in the T201-class

In this section, we analyze the results of the three IAQ measurement campaigns and comfort parameters in the T201-class.

• CO2-concentrations

During the three measurement campaigns, CO2 levels exceed 10 3 ppm (parts per million, which is the maximum value recommended by ASHRAE 62 (Fig. 4, Fig. 5 and Fig. 6 ). Recall that CO2 is produced by the human body during breathing. It is closely related to human occupation and to air renewal rates. Indeed, when the piece T201 is occupied, the CO2 content becomes important. During holidays, its concentration does T201-class Area = 69.412 m² Sensor position

T201-class

Indoor air quality (IAQ) measurements in a tertiary building via a smart sensor connected to a Raspberry Pi card: application to a demonstrator building Loubna Qabbal, Zohir Younsi, Naji Hassane not exceed 4.10 2 ppm. Moreover, after class periods, its concentration drops to a minimum value. It should be noted that during the first IAQ measurement period, CO2 concentrations were high, particularly during the week of 20-24 March 2017. This is due to the large number of occupants and the insufficient ventilation rate. Note that when the CO2 concentration is high, the air confinement becomes so high. Thereby, taking occupants into account becomes paramount as they influence IAQ through CO2 production.

• Relationship between CO2-concentrations and occupation

It is well known when breathing, we eat up oxygen and we emit carbon dioxide. A normal person emits 1kg of CO2/day (about 20 l of CO2 per hour). By its behavior, its uses, its living conditions and so on, the occupant plays a role as important as the technical performance of the building and systems. The CO2 contaminant is often considered as a tracer gas allowing to know the IAQ and the confinement level. Its concentration varies according to the number of occupants, the occupation time, the ventilation rate and the space volume. It is clear that poor ventilation leads to high levels of CO2 that will lead to a decrease in school skills. Indeed, the CO2-concentration is proportional to the occupation (Fig. 4, Fig. 5 and Fig. 6). In addition, high CO2concentration indicates that ventilation is insufficient, which can lead to health symptoms. Indoor air quality (IAQ) measurements in a tertiary building via a smart sensor connected to a Raspberry Pi card: application to a demonstrator building Loubna Qabbal, Zohir Younsi, Naji Hassane Fig. 6. CO2-concentrations and occupancy rate during the third IAQ measurement period.

CO2 concentrations (ppm) Occupants

• Concentrations of indoor air pollutants during measurement campaigns

Mean concentrations of volatile organic compounds (VOCs) in indoor air of T201-class and formaldehyde were found to be higher during the second period (May) (Fig. 7). This explains that VOCs are influenced by warm weather, especially when the sun is present. Fig. 9 shows that, during the month of May, indoor temperatures are higher than the first and third measurements range. This indicates that formaldehyde and VOCs levels were higher in the warm period (May). It turns out that average particulate matter with 2.5 of diameter concentrations (PM2.5) is higher during the third period (October-November) (Fig. 7). In winter, PM2.5 concentrations in the outdoor air increase, resulting in increased values in indoor air. This may explain the high concentrations of PM2.5 during the cold period. PM2.5 from outside air seeps into homes through leaky parts (of buildings) or through windows during aeration. However, the PM2.5 concentrations inside remains very low compared to outside. Indeed, coarse particles having a diameter of 2.5 to 10 micrometers, are deposited relatively quickly on surfaces and on the ground. As a result, indoor air usually contains fewer particles than outdoor air (Fig. 8).

Assessment of comfort parameters in the T201-class

•Hygrothermal comfort assessment: Humidex coefficient and bioclimatic digraph

To assess hygrothermal comfort, we involved two methods, viz. the Humidex coefficient and the bioclimatic digraph. Subsequently we conducted a survey to estimate the occupants' perception on the level of comfort felt in the T201-class. The Humidex is an index used to describe the occupants' sensation of hot or humid air. It combines temperature and humidity to reflect the perceived temperature (temperature felt). It takes both factors into account as they are decisive for summer comfort. That's why the Humidex is a better indicator of the overwhelming heat sensation than the temperature or humidity taken individually. The Humidex index was developed in Canada in 1979 by J. M. Masterton and F. A. Richardson (Eq. ( 1)). Fig. 10 shows that the actual temperature and the temperature felt (Humidex) follow the same profile, while exhibiting a slight shift. Note that the indoor temperature is always higher than the temperature set at 21 °C. This means that the air in the class is very hot and not humid. Subsequently, we assessed the comfort zone using the Humidex T e

                   (1)
C T C T RH RH T C C C T C T                                         (2)
with C2 = 17.081 and C3 = 234.175

Tair: air temperature (°C) and Tdew: dewpoint temperature (°C). Table 1. Humidex's Range and comfort scale [START_REF] Masterton | Humidex: a method of quantifying human discomfort due to excessive heat and humidity[END_REF].

Humidex index Comfort level 20 to 29

Little to no discomfort 30 to 39 Some discomfort 40 to 45 Great discomfort; avoid exertion Above 45 Dangerous; heat stroke quite possible Fig. 11 plots the comfort parameters (T °C and RH %) using the bioclimatic diagram and (Eqs. ( 3) -( 4)). The ambient humidity varies between 21 and 51 % (air being dry). This explains the results obtained when calculating the Humidex coefficient to evaluate the comfort temperature. This is because the regulation of the ventilation system is based only on the temperature measurement and not the humidity temperature. In addition, it is found that the temperature is too high (24 °C) in the room during the entire measurement period (offseason). Knowing that the comfort temperature for a seated person is 21 °C, comfort is not set up in this case and hygrothermal comfort conditions are not met.

   

0.622 101325

sat sat p RH SH p RH        (3)
p                       (4)
where SH: specific humidity (Kgwater/Kghumid-air), θ: temperature (°C), RH: relative humidity (%) and sat p : vapor pressure (Pa). A review of Fig. 11 shows that more than 60% of the points are located in zone 1 (drought zone) and less than 40% are located in comfort zone 4 (hygrothermal comfort polygon). The ambient humidity varies between 21 and 51% showing that the air is dry. It should be noted that the regulation of the current ventilation system is based only on the measurement of the temperature and not the temperature and the RH %, which explains why the air was dry during the month of March. In addition, our findings seem consistent with the March 2017 survey (the hygrothermal comfort zone being between 1 and 4). In fact, the analysis of the comfort questionnaire shows that 80% of occupants were warm and the air was dry (Fig. 12). From these results, we can state that working conditions are not comfortable. This leads to absenteeism and a drop in the productivity.

The smart sensor development

The results of the three sets of IAQ measures and comfort parameters show that ventilation is not effective. In other word, the high CO2-concentrations indicate that the ventilated airflow in the demonstrator building is not sufficient. Hence, to improve IAQ and comfort without increasing energy consumption, we proposed a solution that consists in automating the ventilation system to regulate the flow of blown air and extract it if needed via a smart sensor. The developed sensor will equip the demonstrator located in the Lille city (France) to monitor the variation of IAQ and comfort. The "smart sensor" is connected to a Raspberry Pi model 3B card whose the main purpose is to connect the smart sensor (IAQ and comfort parameters) with a ventilation system using the IoT technology (Internet of Things). Measurements are stored in a database via Wi-Fi. Fig. 13 shows the development approach of the sensor connected to the Raspberry Pi model 3B card as well as the air pollutant sensors that have been integrated. The smart sensor is in the design and improvement phase.

Conclusion

This study deals with measurements of indoor air quality (IAQ) and comfort parameters in a demonstrator building allowing to assess IAQ and hygrothermal comfort levels. The high CO2-concentrations that have been measured in a class demonstrates the effectiveness of the ventilation system. To achieve the aim seeked, we proposed a solution based on a development of a smart sensor that measures indoor air pollutants and comfort parameters (CO2, VOCs, CO, formaldehyde, benzene, PM2.5, humidity temperature, noise and brightness). The sensor measures atmospheric pollutants in real time and send information to the building managers in the event of malfunction of ventilation, air conditioning and heating systems. The ultimate goal is to install the smart sensor in the demonstrator building to map the quality of indoor environments (IAQ and comfort parameters) to identify the most polluted parts. Subsequently, the ventilation system will have to regulate the flow of air blown or extracted to improve IAQ. As a result, energy consumption will be better controlled by adapting the building to the occupants' needs while improving comfort

Indoor air quality (IAQ) measurements in a tertiary building via a smart sensor connected to a Raspberry Pi card: application to a demonstrator building Loubna Qabbal, Zohir Younsi, Naji Hassane conditions to improve occupant productivity. The final goal is to improve IAQ and ensure comfort while saving energy.

Fig. 1 .

 1 Fig. 1. The T201-class location within the demonstrator building.

Fig. 2 .

 2 Fig. 2. T201-class overview.

Fig. 3 .

 3 Fig. 3. T201-class dimensions and smart sensor position.

Fig. 4 .

 4 Fig. 4. CO2-concentrations and occupancy rate during the first period of IAQ measurement.

Fig. 5 .

 5 Fig. 5. CO2-concentrations and occupancy rate during the second period of IAQ measurement.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Average concentrations of indoor air pollutants during IAQ measurement campaigns in T201-class.

Fig. 9 .

 9 Fig. 9. Indoor temperatures during measurement campaigns.

Fig. 10 .

 10 Fig. 10. Temporal evolution of Humidex index and real temperature.11 5417.7530 273.16 273.15 0.5555 6.11 10 dew T air

Fig. 11 .

 11 Fig. 11. Comfort zone location in the T201-class (March 2017).

Fig. 12 .

 12 Fig. 12. Indoor temperature perception (March 2017).

Fig. 13 .

 13 Fig. 13. Development of the smart IAQ sensor and comfort parameters (CO2, VOCs, CO, Formaldehyde, Benzene, PM2.5, humidity, temperature, noise and brightness).

Indoor air quality (IAQ) measurements in a tertiary building via a smart sensor connected to a Raspberry Pi card: application to a demonstrator building Loubna Qabbal, Zohir Younsi, Naji Hassane bioclimatic

				17.61
	11.56	12.59
	0.14		0.15	0.25
	First_campaign	Second_campaign	Third_campaign
	PM2.5-Outside (μg/m3)	PM2.5-Intside (μg/m3)

diagram to locate the comfort zone of the T201-room under these conditions.

Indoor air quality (IAQ) measurements in a tertiary building via a smart sensor connected to a Raspberry Pi card: application to a demonstrator building Loubna Qabbal, Zohir Younsi, Naji Hassane

	sat	 	exp 23.3265	3 802.7 273.18	472.68 273.18	2

air quality (IAQ) measurements in a tertiary building via a smart sensor connected to a Raspberry Pi card: application to a demonstrator building Loubna Qabbal, Zohir Younsi, Naji Hassane

  ).

	Indoor
							Humidity
				Hygrometric degree		content (gr/Kg)
							Comfort
							zone
		Enthalpy
		h (KJ/Kg)
				Temperature (°C)
			44%
				28%
					15%
	3%	6%				3%	0%	0%
	Extremely	Very hot	Hot	Slightly	Slightly	Cold Very cold Extremely
	hot			warm	cold		cold