Mathematical issues concerning evolving sharp-reaction interfaces in non-saturated reactive porous materials

Transcription

1 Mathematical issues concerning evolving sharp-reaction interfaces in non-saturated reactive porous materials Centre for Analysis, Scientific computing and Applications Supported by DFG-SPP1122 Prediction of the Course of Physicochemical Damage Processes Involving Mineral Materials Eindhoven, May 2007

2 Modeling corrosion of concrete via moving reaction interfaces

3 Outline of the Talk Modeling Carbonation via Moving Interfaces Introduction. Relevance. Motivation. Assumptions Basic Geometry Primary and Secondary Carbonation Effects Transport and Production Terms Moving-Interface Carbonation Models Analysis (Special Cases) Fixed-Domain Formulation Main Results: Local and Global Well-Posedness; Regularity; Useful Bounds Numerical Simulation (Special Cases) Scaling of the Moving-Boundary Models Simulation Results. Validation of the Models Summary. Further Research. Open Problems

4 The moving-boundary problem Industrial need: Prediction of the lifetime of concrete-based materials! What means carbonation of concrete? Why and how is a moving-boundaries (interfaces) approach applicable? Which mechanism drives the front position? Aims: better understanding of the process quantitative prediction

5 Carbonation of concrete Reaction of CO 2 (g) from the environment with Ca(OH) 2, KOH, NaOH, Mg(OH) 2, C-S-H and C-A-H phases, C 2S, C 3S, etc. The concrete sample is sufficiently wet CO 2 (g) penetrates the concrete via the air phase and disolves in the pore water (CO 2 (aq)) Reduced scenario: Here (!) CO 2 (aq) reacts in the aqueous phase with Ca(OH) 2 (aq) available by dissolution from the pore matrix CO 2 (g aq) + Ca(OH) 2 (s aq) H 2 O + CaCO 3 (aq s)

6 Specific goals: Prediction of the carbonation penetration depth Estimation of the final time of the process The tools are our (newly developed) carbonation models: moving sharp-interface model moving reaction-layer model moving two-reaction zones model isolines model [with reduced and extended chemistry] (carbonation of CH, CSH, C 2S, C 3S + concurrent hydration and pozzolanic reactions) The isolines model is used for validation/comparison.

7 Basic geometry Cross-section of the sample we deal with. Typical corner of a concrete structure. Our 1D geometry:

8 What is the carbonation front? I. Geometrical alternatives? (a) The front is thin and can be idealized by a surface Γ(t). (b) Γ(t) is the centerline of a reaction layer Ω ɛ(t). (c) Isolines models/the concept of carbonation degree II. Alternatives employing the chemistry of the sample? (e.g. using the drop in ph or a reactant profile) Secondary reaction effects! Each kinetics is proportional to [CO 2 ] p [Ca(OH) 2 ] q, p 1, q 1 (generalized mass-action law). u = (u 1, u 2,... ) t is the concentration vector and Λ M Λ is a set of parameters. Here η = η(u, Λ) is positive, locally Lipschitz and bounded in u 3.

9 Transport and productions by dissolution/precipitation/henry-like exchange Special cases: u 1 concentration of CO 2 (aq) u 2 concentration of CO 2 (g) u 3 concentration of Ca(OH) 2 (aq),... j i := D i u i Fickian flux of ith species Exchange at air/liquid interfaces (via Henry s law) f i,henry = ±P i (Q i u 2 u 1 ) Productions by dissolution/precipitation: f i,diss/prec = ±S i,diss/prec (u i,eq u i )

10 The reaction-diffusion system in the sharp-interface case Mass balance of CO 2(aq) in ]0, s(t)[: u 1 t Mass balance of CO 2(g) in ]0, s(t)[: u 2 t D 1 u 1 = P 1 (u 1 Q 1 u 2 ) D 2 u 2 = P 2 (u 1 Q 1 u 2 ) Mass balance of Ca(OH) 2(aq) in ]s(t), L[: u 3 t D 3 u 3 = S 3,diss (u 3,eq u 3 ) + PDEs in two phases for moisture + PDE for the precipitating CaCO 3 + initial and boundary conditions

11 Conditions across the moving interface Rankine-Hugoniot conditions at x = s(t) for each active concentration: D 1 u 1 = s (t)u 1 (s(t), t) + η(u(s(t), Λ)) D 2 u 2 = s (t)u 2 (s(t), t) D 3 u 3 = s (t)u 3 (s(t), t) η(u(s(t), Λ)) Evolution law for the velocity of the reaction interface: s (t) = Ψ Γ(t) (η(u(s(t), t), Λ), u 3 (s(t), t)), with s(0) = s 0.

12 Well-posedness: local vs. global results The working program is in some sense classical: (A) Local well-posedness of the model: Existence of locally in time positive weak solutions Uniqueness of weak solutions Continuous dependence of the weak solution w.r.t. initial data and model parameters Regularity (B) Further (practical) estimates: Strict lower bounds on all involved concentrations Lower and upper bounds on the speed of the reaction front Upper bound on the final time of the carbonation process Lower bound on the time needed to initiate corrosion (C) Now, (A) + (B) global well-posedness of the mbp models

13 Fixed-domain formulation. Landau s trick Freeze the moving boundaries by mapping Ω 2 (t) [0, T ] (x, t) (y, t) [1, 2] [0, T ] via y = 1 + x s(t) ɛ/2 L s(t) ɛ/2, e.g. Advantage vs. drawbacks: v t + Dv xx = f (v) in ]s(t) + ɛ/2, L[:= Ω 2 (t) becomes D s u t + u (L s(t) ɛ/2) 2 yy + yuy = f (u) in L s(t) ɛ/2 ]1, 2[. Geometrical restrictions: 0 < s 0 s(t) L < L for all t S T 0 < ɛ < L 2s 0 a priori given. The weak formulation will refer to the fixed domains [0, 1], [1, 2] (say [a, b]).

14 Theorem: Under some restrictions on the model parameters, structure of the production terms and geometry, there exists a positive weak solution on a sufficiently small time interval, i.e. δ > 0 such that w.r.t. S δ =]0, δ[ we have s W 1,4 (S δ ) with u W 1 2 (S δ, V, H) [S δ L (a, b)] nspec u(0) = u 0, s(0) = s 0, s (t) = Ψ Γ(t) (η(u(s(t), t), Λ), u 3 (s(t), t)), t S δ.

15 Estimates (e.g.): Positivity/Maximum Estimates: 0 u i (y, t) + λ i (t) k i for all y [a, b], i = 1,..., nspec a.e. t S δ (nonstandard) Energy Estimates (standard L 2 -setting) Corollary: Monotonicity of the reaction interface location The position s W 1, (S δ ) of the reaction interface Γ(t)is monotonically increasing for a.e. t S δ.

16 Construction of the fixed-point operator Let S δ =]0, δ[, δ > 0, σ > 0. Select (M(S δ ), ρ), where M(S δ ) := {r W 1,4 (S δ ) : r(t 0 ) = s 0, r(t) s 0 [0, δσ], and r (t) 0 a.e.t S δ, r L 4 (S δ ) σ} = ρ : M(S δ ) M(S δ ) R + : ρ(r 1, r 2 ) = r 2 r 1 L 4 (S δ ) r 1, r 2 M(S δ ). Solve j s = F s(s, u) (A) B(s)u + C(s, s, u, u y) = F u(s, u) (B) Set T : s i M(S δ ) solution u s,i of (B) r i W 1,4 (S δ ) cf. (A) (i = 1, 2) Prove: (i) T : M(S δ ) M(S δ ) is well-defined δ > 0 : S δ S T. (ii) χ > 0 : ρ(t s 2, T s 1 ) δχρ(s 2, s 1 ) with δχ < 1, s 1, s 2 M(S δ ). Repeat the procedure on S 2δ =]δ, 2δ[,S 3δ =]2δ, 3δ[,... When should we stop?

17 Stability estimates The application H W 1,2 (S δ ) nspec M Υ M Λ W 1 2 (S δ, V, H) W 1,4 (S δ ) which maps (u 0, λ, u eq, k,... ) (u, s) is Lipschitz continuous, i.e. c > 0 such that u 2 u 1 2 W 1 2 (S δ,v,h) L (S δ,h) + s 2 s 1 2 W 1,4 (S δ ) L (S δ ) c u 0,2 u 0,1 2 H L (a,b) nspec + λ 2 λ 1 2 W 1,2 (S δ ) nspec L (S δ ) nspec + «+ u eq,2 u eq,1 2 M Υ L (S δ ) + max k 2 k , M Λ where the choice of c depends on the maximum estimates and model parameters.

18 Global solvability of the MBP Working Program: local solutions on S δ to global solutions on S Tfin via additional estimates (similar arguments as for ode s) Example of estimates: (i) strict lower bounds on active concentrations: u u LOW > 0 (ii) on the interface position and its speed: η min < s (t) < η max (t S δ ) (iii) on the final time of the process: L 0 s 0 η max < T fin < L 0 s 0 η min Here η min and η max denote uniform lower and upper bounds of Ψ Γ(t) ( ). By (iii), we uniquelly extend the solution up to S Tfin. Input from analysis into modeling: revisit (ii) and (iii)!!!

19 Nondimensionalization of the models support for numerics distinguish between large and small contributions supplementary bonus In our case, bonus means: (1) s Φ 2, (2) ɛ 1 Φ 2, where Φ 2 := τ diff τ reac = characteristic diffusion time characteristic reaction time is Thiele modulus (or 2nd Damköhler number). Here Φ 2 1 fast reaction vs. slow diffusion scenario... Special case (for further work): Study ɛ 0, i.e. shrink the layer Ω ɛ(t) to Γ(t) (disappearence of the mushy region)

20 Simulation of carbonation penetration in concrete: case studies We focus on two different situations: accelerated carbonation tests (short term tests, most of the data is fairly well controlled ) natural carbonation tests (long term tests) Main output: Penetration depth vs. time curve!

21 Typical concentration profiles. Penetration depth vs. time CO 2(aq), Ca(OH) 2(aq), CaCO 3(aq) profiles vs. position in the sample. Each curve refers to time t = i years, i = 1,..., 18. Interface position vs. the experimental points after T fin = 18 years of exposure.

22 Internal layers Internal layer formation in the carbonation problem (isolines approach). Both Ca(OH) 2 and CSH can react with CO 2 to produce carbonates. See S.Meier, M.Peter, A.Muntean, M.Böhm: Modeling and simulation of concrete carbonation with internal layers. (Chem. Engng. Sci. 62(2007), )

23 Water barrier effect Numerically, we note that the production of water may locally fill air parts of the pores hindering in this way the transport of CO 2. This effect is visible especially in the simulation of accelerated tests. Prediction of moisture profiles under accelerated and natural conditions. Left: acc. test, CO 2 = 50%, RH=65%. Right: nat. test, CO 2 = 0.03%, RH=78%. See A. Muntean, S. Meier, M. Peter, M. Böhm, J. Kropp: A note on limitations of the use of accelerated concrete-carbonation tests for service-life predictions (2005).

24 Self-adaptivity around corners We are interested in capturing effects of the sample geometry on the speed of penetration. In the enclosed figure, we see that around corners the carbonation front moves faster. Tool: Self-Adaptive Finite Element Method Carbonation attack around a corner: self-adaptive approach See A. Schmidt, A. Muntean, M. Böhm: Numerical experiments with self-adaptive finite element simulations in 2D for the carbonation of concrete (2005) (Simulation with ALBERTA, Transcon/RILEM proceedings (2007)).

25 Summary. Further research/open problems Summary (1) New modeling methodology (2) Scaling, practical bounds, well-posedness of the models (3) The models provide results in good agreement with data from accelerated/natural tests. Further research (Modeling) improved modeling of CaOH) 2 dissolution What can be drawn from the short-time experiments w.r.t. long time ones? qualitative information (ok) / quantitative information (?) Suggest a carbonation scenario, where a t α -law is applicable and a correct extrapolation between accelerated and natural information is guaranteed?

26 Further research (Analysis) ɛ 0, Φ 2, slow diffusion limit(s) (D 2 0, D 3 0) Long-time behavior of concentrations and reaction location Stefan-Signorini conditions driving moving interfaces Dynamics of reaction interfaces near corners (?) Further research (Simulation) Error estimates in 1D for non-equilibrium models in 2 phases Quantitative a posteriori error estimates for 2D semi-linear RD systems (+ their use in ALBERTA) numerical framework for ɛ 0 and Φ 2 Stefan-Signorini conditions driving moving interfaces (1D)