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SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Modelisation and simulation

• f sulphur dioxide aggression

to calcium carbonate stones

• D. Aregba–Driollet IMB, Bordeaux 1 university

Joint work with

• R. Natalini and F. Diele IAC-CNR

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Aggression of buildings by pollutants: industry, transportation, heating. Here we study a model of deterioration of calcium carbonate stones CaCO3 by sulphur dioxide SO2. Formation of gypsum CaSO4 · 2H2O. Simplified one-step reaction: CaCO3 + SO2 + 1 202 →H2O CaSO4 · 2H2O + CO2

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

The model

∂t(ϕ(c)s) = −

• A

mc

• ϕ(c)sc + d∇ · (ϕ(c)∇s)

∂tc = −

• A

ms

• ϕ(c)sc

c: density of CaCO3. s: porose concentration of SO2. ϕ: porosity. ϕ = intermediate limit of volume of void total volume ϕ(c) = αc + β, α, β > 0.

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

A, d: positive constants. mc, ms, mg: molecular mass of calcite, SO2 and gypsum. The amount of gypsum g is then given by c + mc mg g = c0 + mc mg g0

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Analytical results

• F. Guarguaglini and R. Natalini
• 1. Local existence is classical
• 2. Global existence is more difficult to degenerate parabolic

problem:

• no a priori H¨
• lder estimates
• no a priori Hs estimates
• Nonlinear term in the GRADIENT ∇s · ∇c: not only a L∞ estimate
• no coupling conditions (Kawashima-Shizuta)

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

             ∂t(ϕ(c)s) = div(ϕ(c)∇s) − ϕ(c)cs, ∂tc = −ϕ(c)cs, (1) for (x, t) ∈ [Ω × [0, T] (T > 0, Ω ⊂ RN). ϕ(c) = αc + β > 0 in [0, c0∞] min{β, αc0∞} + β} ≥ ϕm > 0 Initial conditions s(x, 0) = s0(x), c(x, 0) = c0(x) bdy conditions for s s(x, t) = ψ(x, t) for (x, t) ∈ ∂Ω × (0, T]

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Let P = N if N > 2, P > 2 if N = 2, P = 2 if N = 1. The data c0, s0, ψ are nonnegative functions such that s0 ∈ W2, P

2 (Ω) ∩ L∞(Ω) , c0 ∈ W1,P(Ω) ∩ L∞(Ω) ;

(2) ψ ∈ C([0, T]; W2, P

2 (Ω))∩C1([0, T]; L P 2 (Ω))∩W2, P 2 (QT) for all T > 0 ; (3)

the trace of the function ψ verifies ψ ∈ L∞(∂Ω × (0, T)) for all T > 0 . (4)

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Theorem Let s0, c0, ψ ≥ 0, s0, c0 ∈ W2,q(Ω), ψ ∈ C([0, T]; W2,q(Ω)) ∩ C1([0, T]; Lq(Ω))) ∩ W2, q

2 (QT) for all T > 0 and q > P.

Then there ex- ists a nonnegative bounded global weak solution to problem (1), with (s, c) ∈ (C([0, T]; W2,q(Ω)) ∩ C1([0, T]; Lq(Ω)))2 .

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Numerical study of the 1D scaled model ∂t(ϕ(c)s) − ∂x ϕ(c)∂xs = −ϕ(c)sc ∂tc = −ϕ(c)sc x ∈ Ω =]0, 1[ or ]0, +∞[, t > 0. Initial conditions: s(x, 0) = 0, c(x, 0) = c0 > 0. Boundary conditions: s(0, t) = s0(t), eventual Neumann condition for x = 1.

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Two approaches: finite differences and finite elements.

• 1. Finite differences:
• Simple, no linear system to solve, easy to modify the time

integration scheme.

• Problems:

– Meshing difficulties for 2D or 3D extensions. – Difficult to increase the accuracy.

• 2. Finite elements:
• Meshing, accuracy: great flexibility.
• There is a linear system to solve.

In both cases: semi-implicit treatment of the nonlinear source term No nonlinear algebraic system to solve.

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Finite differences Main unknowns: ρs = ϕ(c)s concentration of SO2, c . Approximation of of ∂x (a(x)∂xr): ∆m(a, r) := (am + am+1)(rm+1 − rm) − (am−1 + am)(rm − rm−1) 2∆x2 . Notation: S(ρs, c) = −ρsc

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

The scheme (θ ∈ [0, 1]):            cn+1

m

= cn

me−∆tρn

s,m

ρn+1

s,m − ρn s,m

∆t − ∆m

• ϕn, ρn

s

ϕn

• = S((1 − θ)ρn

s,m + θρn+1 s,m , cn+1 m ).

This is a semi-implicit first order scheme. As S is linear with respect to ρs it can be written explicitly. The differential term is not implicited: this would lead to a nonsymmetric linear system.

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Denoting u =        ρs c       : lim

∆t→0

un+1 − un ∆t = F(un, ∆x). Higher order in time by discretization of u′(t) = F(un, ∆x).

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Proposition For all n ≥ 0, all m = 1, . . . , N: ρn

s,m ≥ 0,

cn

m ∈ [0, c0]

under the time step restriction: ∆t ≤ β∆x2 ϕ0 + β(1 + ∆x2c0(1 − θ)) where ϕ0 = ϕ(c0).

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Finite elements P1 approximation for s P0 approximation for c. Denote σ(x, t) = s(x, t) − s(0, t): ∂t(ϕ(c)σ) − ∂x(ϕ(c)∂xσ) = F(σ, c, t) Functionnal space: H = {u ∈ H1(]0, 1[), u(0) = 0}

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Variational formulation Find (σ, c) ∈ C1([0, +∞[, H × L2(]0, 1[)) such that for all (p, q) ∈ H × L2(]0, 1[):                    ∂t

• ]0,1[

ϕσpdx +

• ]0,1[

ϕ∂xσ∂xpdx =

• ]0,1[

pFdx, ∂t

• ]0,1[

cqdx = −

• ]0,1[

ϕ(σ + s(0, .))cqdx.

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Discrete problem Regular mesh: [0, 1] = ∪1≤i≤N[xi, xi+1], xi = (i − 1)∆x, ∆x = 1/N. pi, i = 1, . . . , N + 1 : classical P1 basis functions. qi, i = 1, . . . , N:characteristic function of [xi, xi+1[. The solution (s, c) is approximated by sh(x, t) =

N+1

• i=1

ξi(t)pi(x), ch(x, t) =

N

• k=1

ηk(t)qk(x).

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

The discrete variational problem (without boundary condition):                ∂t(M(η)ξ) + K(η)ξ = 0, ∂tηk = −(ξk + ξk+1) 2 ηk(αηk + β) = −γkηk(αηk + β), k = 1, . . . , N. M(η) and K(η) are tridiagonal symmetric matrices. Mij(η) =

N

• k=1

xk+1

xk

ϕ(ηk)pipjdx, Kij(η) =

N

• k=1

xk+1

xk

ϕ(ηk)

• p′

ip′ j + ηkpipj

• dx

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Resolution

• 1. Fix ξn and solve exactly on [tn, tn+1] the equation for ηk,

k = 1, . . . , N: one obtains ηn+1.

• 2. Fix η = ηn+1 and solve the system for ξ by the θ method: one
• btains ξn+1.

Order two in time: by Heun scheme. One has to solve a linear system.

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Proposition Suppose that θ ∈]1/3, 1] and ∆x2 < 3(3θ − 1) c0 . If the time step satisfies the condition ∆x2 θ(6 − c0∆x2) < ∆t ≤ ∆x2 (1 − θ)(3 + c0∆x2) then for all x ∈ [0, 1], ch(x, .) is a non increasing function of t and for all t ≥ 0: ρs,h(x, t) ≥ 0, ch(x, t) ∈]0, c0]. Moreover the condition is not empty. No upper bound on ∆t for θ = 1. Uniform bound also for ρs with additional conditions.

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Numerical results

• 1. Finite difference and finite element methods give comparable

results

• 2. For the finite element method with θ = 1: good results with

∆t = ∆x

• 3. For θ = 1/2: the numerical order of accuracy is γ = 2. (Heun

scheme for FE)

• 4. Order 3 in time does not improve the numerical order of

accuracy for FD

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Finite Elements Finite Differences 9.985 9.99 9.995 10 10.005 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Finite Elements Finite Differences

SO2 concentration Calcite density Comparison among FEM and FDM, with △x = 0.003125 and θ = 1. ∆t = ∆x for FE, ∆t = C.∆x2 for FD.

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Finite Elements Finite Differences 9.985 9.99 9.995 10 10.005 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Finite Elements Finite Differences

SO2 concentration Calcite density Comparison among FEM and FDM, with △x = 0.0015625 and θ = 1. ∆t = ∆x for FE, ∆t = C.∆x2 for FD.

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

h γs ||ρs(h) − ρs(h/2)||1 γc ||c(h) − c(h/2)||1 0.2 3.50066927 0.0545620474 6.44459105 0.001924897 0.1 1.99293298 0.00482041247 1.99127884 2.21E-05 0.05 2.00834819 0.00121102077 2.01460837 5.5585E-06 0.025 2.02053637 0.000301008351 2.03328576 1.375625E-06 0.0125 2.03989415 7.41884825E-05 2.07065766 3.360625E-07 0.00625 2.0685759 1.80412722E-05 2.14077583 8.E-08 0.003125 —— 4.30094375E-06 —— 1.8140625E-08 Numerical order of accuracy and L1 errors for finite element, θ = 1/2.

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Solution from t = 0 to t = 0.5 with A = 1.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t=0 t=0.1 t=0.2 t=0.3 t=0.4 t=0.5 9.92 9.94 9.96 9.98 10 10.02 10.04 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t=0 t=0.1 t=0.2 t=0.3 t=0.4 t=0.5

SO2 concentration Calcite density.

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Solution from t = 0 to t = 0.5 with A = 10000.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t=0 t=0.1 t=0.2 t=0.3 t=0.4 t=0.5 2 4 6 8 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t=0 t=0.1 t=0.2 t=0.3 t=0.4 t=0.5

SO2 concentration Calcite density.

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

MultiD simulations

• F. Freddi, R. Natalini, C. Nitsch

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Left: initial mesh, right: mesh and external calcite density at time t = 0.015.

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Exterior calcite density; left: time t = 0.06, right: Time t = 0.12

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Calcite density on a section in the middle of the marble specimen; left: time t = 0.06, right: Time t = 0.12

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Left: exterior SO2 concentration right: SO2 concentration density on a section in the middle of the marble specimen both for time t = 0.06

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Qualitative behavior of solutions

             ∂t(ϕ(c)s) − ∂x(ϕ(c)∂xs) = −ϕ(c)sc, ∂tc = −ϕ(c)sc, Scaling: sk(x, t) = s(kx, k2t), ck(x, t) = c(kx, k2t)

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Qualitative behavior of solutions

             ∂t(ϕ(ck)sk) − ∂x(ϕ(ck)∂xsk) = −k2ϕ(ck)skck, ∂tck = −k2ϕ(ck)skck, Scaling: sk(x, t) = s(kx, k2t), ck(x, t) = c(kx, k2t)

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

The fast reaction limit (F. Guarguaglini-R. Natalini)

Asymptotic behavior There exist two functions Σ, Γ ∈ L1

loc(R+ × R+) such that

limk→+∞(sk, ck) = (Σ, Γ) in L1

loc(R+ × R+)

Σ(x, t) =        S(x, t) for x < ζ(t) for x > ζ(t) Γ(x, t) =        for x < ζ(t) c0 for x > ζ(t) S(x, t) and the curve x = ζ(t) solve the Stefan problem                          St = Sxx , S > 0 for x < ζ(t) S(0, t) = ψ(t), S(ζ(t), t) = 0 ζ′(t) = − Sx(ζ(t),t)

co

, ζ(0) = 0

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

ψ ≡ ψ0 = const. (Σ(x, t), Γ(x, t)) =                (ψ0 − ξ0c0

2

• x

√ t

e

1 4 (ξ2 0−η2)dη, 0) , x < ξ0

√ t), (0, c0) , x > ξ0 √ t, where ξ0 = G( 2ˆ

s c0 ) is the unique

solution of ξ0 ξ0 e

1 4 (ξ2 0−η2)dη =

2ψ0 c0

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Comparison between numerical results and the asymptotic profile

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 t=20 t=40 t=30 t=80 t=100 t=infinity 2 4 6 8 10 0.2 0.4 0.6 0.8 1 t=20 t=40 t=30 t=80 t=100 t=infinity

SO2 Calcium Carbonate

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Numerical estimate of the time evolution of the maximum error between solutions and asymptotic profiles cutting a region of size d around the front

0.2 0.4 0.6 0.8 1 1.2 20 40 60 80 100 120 140 160 180 200 time d=0 d=0.01 d=0.05 d=0.1 d=0.2 1 2 3 4 5 6 7 8 20 40 60 80 100 120 140 160 180 200 time d=0 d=0.01 d=0.05 d=0.1 d=0.2

SO2 Calcium Carbonate

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Numerical estimate of the time evolution of the L1-error between solutions and asymptotic profiles cutting a region of size d around the front

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 20 40 60 80 100 120 140 160 180 200 time d=0 d=0.01 d=0.05 d=0.1 d=0.2 1 2 3 4 5 20 40 60 80 100 120 140 160 180 200 time d=0 d=0.01 d=0.05 d=0.1 d=0.2

SO2 Calcium Carbonate

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Remarks

• connection with long time behavior for k = 1

Theorem Let (s, c) be the solution of the IBV and (Σ, Γ) the limit profiles. Then sup

t∈(0,∞)

|s(x, t) − Σ(x, t)| →t→∞ 0 , sup

t∈[a √ t,b √ t]

|c(x, t) − Γ(x, t)| →t→∞ 0 , for all a, b are such that the interval [a √ t, b √ t] does not contain any point of the free boundary x = ζ(t).

• The numerical results are not satisfactory since there is no evidence of

the convergence of ck (even for the model with constant porosity).

IMB, Bordeaux 1 university

• D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08

Conclusion

• The bases of the numerical approximation of sulphatation

phenomena are established.

• First 3D simulations have been performed.
• Qualitative behaviour of solutions: front propagation.
• Quantitative aspects are investigated.

IMB, Bordeaux 1 university

• D. Aregba-Driollet