OPTIMAL RELATIONS BETWEEN THE PARAMETERS OF A P.E.M. FUEL CELL

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THE PARAMETERS OF A P.E.M. FUEL CELL

ANCA C ˘AP ˘AT¸ˆIN ˘A, HORIA ENE, GELU PAS¸A, DAN POLISEVSCHI and RUXANDRA STAVRE

We use a mathematical model of the combustion gas behavior during the passage through the anode of a PEM fuel cell. We study here the influence of the electro chemic parameters and of the fluid dynamics upon the qualitative and quantitative properties of the fields that characterize the problem. First, we briefly describe the physical problem and establish the mathematical model. We present the relations which insure the existence, uniqueness and regularity of the weak solution of our problem. Finally, we consider a specific optimal control problem; the main result consists of the necessary conditions of optimality that are obtained here. These are the relations that the free parameters (the difference of the concentration and the pressure between the entry and exit borders) have to satisfy in order to obtain a certain thermodynamic configuration.

AMS 2000 Subject Classification: 76S05, 80A20, 76R50.

Key words: PEM fuel cell, variational problem, existence of a weak solution, control problem.

1. INTRODUCTION

The anode of the PEM fuel cell is represented by a three-dimensional porous medium consisting of a solid parallelepiped crossed by thin channels.

The process is modeled by using the Boussinesq approximation of the Darcy law (for details see [2], [4], [6], [7]) and the boundary conditions which describe the absorption and the emission of hydrogen on the different parts of the anode surface (see also [3]).

It is well known that the efficaciousness of the PEM fuel cells depend strongly on the temperature distribution in the domain. The variation of the temperature in the anode of the PEM fuel cell is influenced by many parameters. Among them, the main part is played by the concentration and the pressure imposed on the absorption and the emission of hydrogen boundaries.

The domain in which we study the phenomenon is the anode of the PEM fuel cell, represented by the parallelepiped Ω = (0, l)×(0, a)×(0, b). The absorption of the hydrogen is made through the part of the lateral surface of

MATH. REPORTS10(60),4 (2008), 299–308

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the anode Σ1 defined by the equationx1= 0. During the oxidation process, in the presence of an electrolyte, the hydrogen is dissociated in ions and electrons.

Then, the emission is made through the part of the lateral surface Σ2 defined by the equation x1=l.

The unknown functions of our problem are: the velocity,uthe pressurep and the temperatureT of the hydrogen-electrolyte mixture in the anode while C is the concentration of the hydrogen in the mixture. After homogenizing the boundary conditions and denoting Σ = Σ1∪Σ2 si Γ =∂Ω−Σ, we obtain a system of equations and boundary conditions (one can find details of these procedures in [1]), namely,

(1)











µm

k u =−∇p+ [a_h−αC−βT]f in Ω,

div u= 0 in Ω,

div (Tu−D₁∇T) +u∇T_h=Q_h in Ω, div (Cu−D2∇C) +C2−C1

l u1= 0 in Ω,

C = 0 on Σ,

∂C

∂n = 0 on Γ,

T = 0 on∂Ω,

p=πi on Σi, i= 1,2,

u·n= 0 on Γ.

where the functions ah(x) = 1−α^C²^−C_l ¹x1−βTh and Qh are bounded with respect to hwhile T_h∈H²(Ω) has (see [5]) the property

kθ∇T_hk_(L2(Ω))³ ≤hk∇θk_(L2(Ω))³, ∀θ∈H₀¹(Ω).

Denoting δ = ^π²^−π_l ¹, d = ^C²^−C_l ¹ and takingp = (1− ^x_l¹)π₁+ ^x_l¹π₂+q withq = 0 on Σ, after dividing equation (1)1 µm

k and by keeping the notation for pand f, we obtain for (u, T, C, p) the system

(2)











u =−∇p+ [ah−αC−βT]f−δ(e)1 in Ω,

div u= 0 in Ω,

div (Tu−D₁∇T) +u∇T_h =Q_h in Ω, div (Cu−D2∇C) +d(u)1 = 0 in Ω,

C= 0 on Σ,

∂C

∂n = 0 on Γ,

T = 0 on∂Ω,

p= 0 on Σ,

u·n= 0 on Γ.

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2. EXISTENCE AND UNIQUENESS OF THE SOLUTION

In this section we give a variational formulation of problem (2). This formulation allows us to choose the parameters of the problem in order to get the existence and uniqueness of the solution. Denote

X₁ ={v∈[L²(Ω)]³; divv= 0 in Ω,v·n= 0 on Γ}, X2=H₀¹(Ω), X3 ={ϕ∈H¹(Ω); ϕ= 0 on Σ},

Y₁ =X₁, Y₂=W₀^1,4(Ω), Y₃ ={ϕ∈W^1,4(Ω);ϕ= 0 on Σ}, X=X₁×X₂×X₃, Y =Y₁×Y₂×Y₃.

We then obtain the following variational formulation of problem (2):

(VP)_d,δ

( Find (u, T, C)∈X such that

hG_d,δ(u, T, C),(v, θ, ϕ)i_Y⁰_,Y = 0, ∀(v, θ, ϕ)∈Y,

where for real parametersη andλthe operator G_d,δ :X→Y⁰ is defined by

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









hG_d,δ(u, T, C),(v, θ, ϕ)i_Y⁰_,Y

= Z

Ω

u·vdx+δ Z

Ω

(v)₁dx− Z

Ω

(a_h−αC−βT)f·vdx +η²D₁

Z

Ω

∇T · ∇θdx+η² Z

Ω

u· ∇T θdx+η² Z

Ω

u· ∇T_hθdx

−η² Z

Ω

Q_hθdx+λ²D₂ Z

Ω

∇C∇ϕdx+λ² Z

Ω

u· ∇Cϕdx +λ²d

Z

Ω

(u)₁ϕdx, ∀((u, T, C),(v, θ, ϕ))∈Y×Y.

We begin our study with the a priori estimation given below.

Proposition 1. If (u, T, C) is a solution of problem (VP)_d,δ, then

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









maxn

kuk²_(L2(Ω))³, η²D₁k∇Tk²_(L2(Ω))³, λ²D₂kCk²_(L2(Ω))³

o

≤4

max

x∈Ω¯

|a_h(x)| kfk_(L^∞_(Ω))3 +C(Ω)δ 2

+4η² D1

C²(Ω)kQ_hk²_L2(Ω), where C(Ω)is a positive constant only depending on Ω.

Proof. By using the same technique as in [5], we can prove that the solution of problem (2) has the regularity p ∈ H²(Ω),u ∈ (H¹(Ω))³. This allow us to set (v, θ, ϕ) = (u, T, C) in (VP)_d,δ, where (u, T, C) is a solution of

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problem (VP)_d,δ. So, we obtain

kuk²_(L2(Ω))³+η²D₁k∇Tk²_(L2(Ω))³ +λ²D₂kCk²_(L2(Ω))³ =

−α Z

Ω

Cf·udx−β Z

Ω

Tf·udx−η² Z

Ω

Tu· ∇T_hdx

−λ²d Z

Ω

C(u)₁dx+ Z

Ω

a_hf·udx−δ Z

Ω

(u)₁dx+η² Z

Ω

Q_hTdx

≤C(Ω)kfk_(L∞(Ω))³(αk∇Ck_(L2(Ω))³ +βk∇Tk_(L2(Ω))³)kuk_(L2(Ω))³

+η²hk∇Tk_(L2(Ω))³kuk_(L2(Ω))³ +λ²dC(Ω)k∇Ck_(L2(Ω))³kuk_(L2(Ω))³

+ max

x∈Ω¯

|a_h(x)| kfk_(L∞(Ω))³kuk_(L2(Ω))³ +C(Ω)δkuk_(L2(Ω))³

+η²C(Ω)kQ_hk_L2(Ω)k∇Tk_(L2(Ω))³.

It follows that there existd₀ >0 andh₀ >0 such that ford≤d₀,h≤h₀ and λand µproperly chosen we have

kuk²_(L2(Ω))³ +η²D₁k∇Tk²_(L2(Ω))³+λ²D₂kCk²_(L2(Ω))³

≤ 1

2kuk²_(L2(Ω))³+ 2

max

x∈Ω¯

|a_h(x)| kfk_(L∞(Ω))³ +C(Ω)δ 2

+η²D₁

2 k∇Tk²_(L2(Ω))³ +2η² D1

C²(Ω)kQ_hk²_L2(Ω). Hence

maxn1

2kuk²_(L2(Ω))³, η²D₁

2 k∇Tk²_(L2(Ω))³, λ²D₂

2 kCk²_(L2(Ω))³

o

≤2 max

x∈Ω¯

|a_h(x)| kfk_(L∞(Ω))³ +C(Ω)δ2

+2η²

D₁C²(Ω)kQ_hk²_L2(Ω). that is, (4).

We are now in a position to prove the following result of existence and uniquenes.

Theorem 1. (1) There existh0 >0 and d0 >0 such that for all h >0 andd∈R withh < h₀ and|d|< d₀one can chooseλ=λ(h, d)andη=η(h, d) such that problem (VP)_d,δ has at least one solution for all δ >0.

(2) If the solution of problem (VP)_d,δ is regular then, for d, h, λ and η chosen as above, there exists δ0 such that the problem (VP)_d,δ has a unique solution for all δ≤δ0.

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Proof.(1) In order to obtain this result we use a generalization of Gossez’

theorem as in [1] (see also [5]). We have to look for λ, η andr >0 such that (5) hG_d,δ(v, θ, ϕ),(v, θ, ϕ)i_Y⁰_,Y ≥0, ∀(v, θ, ϕ)∈Y,k(v, θ, ϕ)k_X =r.

Taking into account that the terms of order 3 vanish (see [1]), we have

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









hG_d,δ(v, θ, ϕ),(v, θ, ϕ)i_Y⁰_,Y =kvk²_(L2(Ω))³+η²D₁k∇θk²_(L2(Ω))³

λ²D₂kϕk²_(L2(Ω))³ +α Z

Ω

ϕf·vdx+β Z

Ω

θf·vdx +η²

Z

Ω

θv· ∇T_hdx+λ²d Z

Ω

ϕ(v)1dx− Z

Ω

a_hf·vdx +δ

Z

Ω

(v)1dx−η² Z

Ω

Q_hθdx.

Using the same technique as in [1] we obtain (5) by taking h₀ = D1

βkfk_(L∞(Ω))³

, d₀ = D2

αkfk_(L∞(Ω))³

.

(2) To prove the uniqueness of the solution, we consider two solutions (u_i, T_i, C_i),i= 1,2, of problem (VP)_d,δ. Let (u, T, C) = (u₁−u₂, T₁−T₂, C₁− C₂). By subtracting the equations satisfied by the two solutions and by setting (v, θ, ϕ) = (u, T, C) we obtain

Z

Ω

u²dx+η²D1

Z

Ω

∇T²dx+λ²D2

Z

Ω

∇C²dx=− Z

Ω

(αC+βT)f·udx +η²

Z

Ω

∇T₁u· ∇Tdx−η² Z

Ω

∇Tu· ∇T_hdx+λ² Z

Ω

C1u· ∇Cdx

−λ²d Z

Ω

C(u)1· ∇Cdx

≤C(Ω)kfk_(L^∞_(Ω))3(αk∇Ck_(L2(Ω))³ +βk∇Tk_(L2(Ω))³)kuk_(L2(Ω))³

+η²hk∇Tk_(L2(Ω))³kuk_(L2(Ω))³ +λ²dC(Ω)k∇Ck_(L2(Ω))³kuk_(L2(Ω))³

+η² Z

Ω

∇T₁u· ∇Tdx+λ² Z

Ω

C1u· ∇Cdx .

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For d0, h0 chosen as in Proposition 1, for d ≤ d0, h ≤ h0 and for λ, µ properly chosen, we have

kuk²_(L2(Ω))³ +η²D1k∇Tk²_(L2(Ω))³+λ²D2kCk²_(L2(Ω))³

≤2η² Z

Ω

T1u· ∇Tdx+ 2λ² Z

Ω

C1u· ∇Cdx .

One can prove (by using the same technique as in [5]) thatT1 ∈H²(Ω), C1 ∈H²(Ω), henceT, C ∈L^∞(Ω). From the above relation we get

kuk²_(L2(Ω))³ +η²D1k∇Tk²_(L2(Ω))³+λ²D2kCk²_(L2(Ω))³

≤(kT₁k_L^∞_(Ω)k∇Tk_(L2(Ω))³ +kC₁k_L^∞_(Ω)k∇Ck_(L2(Ω))³)kuk²_(L2(Ω))³. By using the estimates from Proposition 1 we obtain that (u, T, C) = (0,0,0) and the proof of the theorem is complete.

3. THE CONTROL PROBLEM

In this section we will consider a problem which allow us to determine the parameters dand δ in order to obtain an optimal temperature at the porous anode of the PEM fuel cell. Denote

(7) D= [d1, d2]×[δ1, δ2],

where di ∈(0, d0], δi ∈(0, δ0], i= 1,2 with d0 and δ0 given by Proposition 1.

In the sequel we suppose that the data and parameters of the problems are such that for all (d, δ)∈Dproblem (VP)_d,δhas a unique solution denoted by (ud,δ, Td,δ, Cd,δ) (i.e. the assumptions of Theorem 2.1 are satisfied).

Define the functionalJ :D→ Rby

(8) J(d, δ) = 1

2 Z

Ω

(T_d,δ−T0)²dx,

where T₀ ∈X₂ is the desired configuration for the temperature. Consider the optimal control problem

(CP) Find (d^∗, δ^∗)∈Dsuch thatJ(d^∗, δ^∗)≤J(d, δ), ∀(d, δ)∈D.

Proposition 2. The control problem (CP)has at least one solution.

Proof. We shall prove that the functional J is continuous. Indeed, let {(d_n, δn)}_n⊂Dand (d^∗, δ^∗)∈Dbe such that lim

n→∞dn=d^∗ and lim

n→∞δn=δ^∗. Let (u_n, T_n, C_n) be the unique solution corresponding to problem (VP)_(d_n_,δ_n₎. It follows from Proposition 1 that the sequence {(u_n, Tn, Cn)}_n is bounded

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in X, hence there exist a subsequence {(u_n_p, Tnp, Cnp)}_n_p and (u, T, C) ∈X such that

unp*u in (L²(Ω))³, Tnp* T inH₀¹(Ω), Cnp * C inH¹(Ω).

By passing to the limit in (VP)_d_np_,δ_np, we obtain that (u, T, C) is the unique solution of problem (VP)_d∗,δ^∗, which imply that T_n * T = T_d^∗_,δ^∗ in H¹(Ω) so T_n→T inL²(Ω). Therefore,

n→∞lim J(dn, δn) = 1 2

Z

Ω

(T−T0)²dx=J(d^∗, δ^∗).

The proof is now complete by taking into account thatDis compact.

In order to get the necessary conditions of optimality, we shall first prove the next result.

Lemma 1. Let (d^∗, δ^∗) be an optimal control for problem (CP). Then (9)

Z

Ω

(T^∗−T₀)T_d,δ^∗ dx≥0, ∀(d, δ)∈D,

where T^∗ is such that(u^∗, T^∗, C^∗)is the unique solution of problem(VP)_(d∗,δ^∗)

and, for all d ∈ [d₁, d₂] and δ ∈ [δ₁, δ₂], (u^∗_d,δ, T_d,δ^∗ , C_d,δ^∗ ) ∈ X is the unique solution of the problem

(Q^∗_d,δ)









 Z

Ω

u^∗_d,δ·vdx+ (δ−δ^∗) Z

Ω

(v)1dx+α Z

Ω

C_d,δ^∗ f·vdx

+β Z

Ω

T_d,δ^∗ f·vdx+η²D₁ Z

Ω

∇T_d,δ^∗ · ∇θdx+η² Z

Ω

u^∗_d,δ· ∇T^∗θdx

+η² Z

Ω

u^∗· ∇T_d,δ^∗ θdx+η² Z

Ω

u^∗_d,δ· ∇T_hθdx+λ²D₂ Z

Ω

∇C_d,δ^∗ · ∇ϕdx

+λ² Z

Ω

u^∗_d,δ· ∇C^∗ϕdx+λ² Z

Ω

u^∗· ∇C_d,δ^∗ ϕdx+λ²d^∗ Z

Ω

(u^∗_d,δ)1ϕdx

+λ²(d−d^∗) Z

Ω

(u)^∗₁ϕdx= 0 ∀(v, θ, ϕ)∈Y.

Proof. Lett∈[0,1], d∈[d₁, d₂] and δ∈[δ₁, δ₂]. Putd_t=d^∗+t(d−d^∗), δt=δ^∗+t(δ−δ^∗) and let (udt,δt, Tdt,δt, Cdt,δt) be the unique solution of problem (VP)_d

t,δt. An easy computation yields u_d_t_,δ_t = u^∗ +tu^∗_d

t,δt, T_d_t_,δ_t = T^∗+ tT_d^∗

t,δt, C_d_t_,δ_t =C^∗+C_d^∗

t,δt, where (u^∗_d

t,δt, T_d^∗

t,δt, C_d^∗

t,δt) is the unique solution

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(uniqueness is obtained in the same way as in the proof of Theorem 1 of the problem

(Q^∗_d

t,δt)









 Z

Ω

u^∗_d_t_,δ_t ·vdx+ (δ−δ^∗) Z

Ω

(v)₁dx+α Z

Ω

C_d^∗_t_,δ_tf·vdx +β

Z

Ω

T_d^∗_t_,δ_tf·vdx+η²D1

Z

Ω

∇T_d^∗

t,δt· ∇θdx +η²t

Z

Ω

u^∗_d_t_,δ_t· ∇T_d^∗

t,δtθdx+η² Z

Ω

u^∗_d_t_,δ_t· ∇T^∗θdx +η²

Z

Ω

u^∗· ∇T_d^∗_t_,δ_tθdx+η² Z

Ω

u^∗_d_t_,δ_t · ∇T_hθdx +λ²D₂

Z

Ω

∇C_d^∗_t_,δ_t · ∇ϕdx+λ²t Z

Ω

u^∗_d_t_,δ_t· ∇C_d^∗_t_,δ_tϕdx +λ²

Z

Ω

u^∗_d_t_,δ_t· ∇C^∗ϕdx+λ² Z

Ω

u^∗· ∇C_d^∗

t,δtϕdx +λ²t(d−d^∗)

Z

Ω

(u^∗_d_t_,δ_t)₁ϕdx+λ²d^∗ Z

Ω

(u^∗_d_t_,δ_t)₁ϕdx +λ²(d−d^∗)

Z

Ω

(u)^∗₁ϕdx= 0 ∀(v, θ, ϕ)∈Y.

By using the method in the proof of Proposition 1 we deduce the bounded- ness of the solution of this problem with respect tot. Hence, lettingt→0, we deduce that the limit (u^∗_d,δ, T_d,δ^∗ , C_d,δ^∗ ) is the unique solution of problem (Q^∗_d,δ).

Using these facts we get

(10)











J⁰(d^∗, δ^∗)·(d−d^∗, δ−δ^∗) = lim

t→0

J(d_t, δ_t)−J((d^∗, δ^∗)) t

= 1 2lim

t→0

Z

Ω

[(Tdt,δt−T0)²−(T^∗−T0)²] dx t

= 1 2lim

t→0

2

Z

Ω

T_d^∗_t_,δ_t(T^∗−T₀) dx+t Z

Ω

(T_d^∗_t_,δ_t)²dx

= Z

Ω

T_d,δ^∗ (T^∗−T₀) dx.

Taking into account that (d^∗, δ^∗) is a minimum point for J on D, relation (9) follows from (10).

The above result allows us to obtain the necessary conditions of optimality for problem (CP).

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Theorem 2. Let (d^∗, δ^∗) be an optimal control for problem(CP). Then there exist unique elements (u^∗, T^∗, C^∗) ∈ X and (ν^∗, τ^∗, γ^∗) such that the system below is verified:

(11) hG_d^∗_,δ^∗(u^∗, T^∗, C^∗),(v, θ, ϕ)i_Y⁰_,Y = 0, ∀(v, θ, ϕ)∈Y,

(12)









 Z

Ω

ν^∗·vdx+α Z

Ω

ν^∗·fϕdx+β Z

Ω

ν^∗·fθdx +η²D₁

Z

Ω

∇τ^∗· ∇θdx+η² Z

Ω

τ^∗∇T^∗·vdx+η² Z

Ω

τ^∗u^∗· ∇θdx +η²

Z

Ω

τ^∗∇T_h·vdx+λ²D₂ Z

Ω

∇γ^∗· ∇ϕdx+λ² Z

Ω

γ^∗∇C^∗·vdx +λ²

Z

Ω

γ^∗u^∗· ∇ϕdx+λ²d^∗ Z

Ω

γ^∗(v)₁dx− Z

Ω

(T^∗−T₀)θdx= 0,

∀(v, θ, ϕ)∈Y,

(13) λ²(d−d^∗) Z

Ω

γ^∗(u^∗)1dx+ (δ−δ^∗) Z

Ω

(ν^∗)1dx≤0, ∀(d, δ)∈D.

Proof. The existence and uniqueness of the element (u^∗, T^∗, C^∗) was proved in Theorem 1. By computations similar to those in the proof of Theo- rem 1, one can prove that problem (12) has a unique solution.

In order to obtain (13), we take (v, θ, ϕ) = (u^∗_d,δ, T_d,δ^∗ , C_d,δ^∗ ) (the unique solution of problem Q^∗_d,δ) in problem (12) and get

(14)









 Z

Ω

ν^∗·u^∗_d,δdx+α Z

Ω

ν^∗·fC_d,δ^∗ dx+β Z

Ω

ν^∗·fT_d,δ^∗ dx +η²D₁

Z

Ω

∇τ^∗· ∇T_d,δ^∗ dx+η² Z

Ω

τ^∗∇T^∗·u^∗_d,δdx+η² Z

Ω

τ^∗u^∗· ∇T_d,δ^∗ dx +η²

Z

Ω

τ^∗∇T_h·u^∗_d,δdx+λ²D₂ Z

Ω

∇γ^∗· ∇C_d,δ^∗ dx+λ² Z

Ω

γ^∗∇C^∗·u^∗_d,δdx +λ²

Z

Ω

γ^∗u^∗· ∇C_d,δ^∗ dx+λ²d^∗ Z

Ω

γ^∗(u^∗_d,δ)₁dx− Z

Ω

(T^∗−T₀)T_d,δ^∗ dx= 0.

Next, by taking in problem (Q^∗_d,δ) as test function the unique solution (ν^∗, τ^∗, γ^∗) of problem (12), we obtain

(10)

(15)









 Z

Ω

u^∗_d,δ·ν^∗dx+ (δ−δ^∗) Z

Ω

(v)1dx+α Z

Ω

C_d,δ^∗ f·ν^∗dx +β

Z

Ω

T_d,δ^∗ f ·ν^∗dx+η²D1

Z

Ω

∇T_d,δ^∗ · ∇τ^∗dx+η² Z

Ω

u^∗_d,δ· ∇T^∗τ^∗dx +η²

Z

Ω

u^∗· ∇T_d,δ^∗ τ^∗dx+η² Z

Ω

u^∗_d,δ· ∇T_hτ^∗dx+λ²D2

Z

Ω

∇C_d,δ^∗ · ∇γ^∗dx +λ²

Z

Ω

u^∗_d,δ· ∇C^∗γ^∗dx+λ² Z

Ω

u^∗· ∇C_d,δ^∗ γ^∗dx+λ²d^∗ Z

Ω

(u^∗_d,δ)₁γ^∗dx +λ²(d−d^∗)

Z

Ω

(u)^∗₁γ^∗dx= 0.

By substrating (15) from (14) we conclude that Z

Ω

(T^∗−T0)T_d,δ^∗ dx=−λ²(d−d^∗) Z

Ω

γ^∗(u^∗)1dx−(δ−δ^∗) Z

Ω

(ν^∗)1dx . The proof is now complete on account of the last equation and Lem-

ma 1.

Acknowledgements. This work was done with the support of Contract CEx 189/2006.

REFERENCES

[1] A. C˘ap˘at¸ˆın˘a, H. Ene, G. Pa¸sa, D. Polisevschi and R. Stavre,Mathematical model for the PEMfuel cells using sulphuretted hydrogen. To appear.

[2] G. Chavent and J. Jaffr´e,Mathematical Models and Finite Elements for Reservoirs Sim- ulation. North Holland, Amsterdam, 1986.

[3] A. Ciocanea,Modelarea matematica a pilelor de combustie. In:Proc. Conf. CIEM, Vol. 2, pp. 6–65. “Politehnica” Univ. Bucure¸sti, 2003.

[4] C.F. Curtiss and R.B. Bird,Multicomponent diffusion. Ind. Engrg. Chem. Res.38(1999), 322–332.

[5] H.I. Ene and D. Poliˇsevski,Thermal Flows in Porous Media. Reidel, Dordrecht, 1987.

[6] C.M. Marle,Multiphase Flow in Porous Media. Techniq, Paris, 1981.

[7] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis. North-Holland, Amsterdam, 1977.

Received 28 January 2008 Romanian Academy

“Simion Stoilow” Institute of Mathematics Calea Grivitei 21

014700 Bucharest, Romania Anca.Capatina@imar.ro