A free-boundary problem for concrete carbonation: Front nucleation and rigorous justification of the $\sqrt{{t}}$-law of propagation

Abstract

We study a one-dimensional free-boundary problem describing the penetration of carbonation fronts (free reaction-triggered interfaces) in concrete. A couple of decades ago, it was observed experimentally that the penetration depth versus time curve (say s(t) vs. t) behaves like s(t) = C √ t for sufficiently large times t > 0 (with C a positive constant). Consequently, many fitting arguments solely based on this experimental law were used to predict the large-time behavior of carbonation fronts in real structures, a theoretical justification of the √ t-law being lacking until now.The aim of this paper is to fill this gap by justifying rigorously the experimentally guessed asymptotic behavior. We have previously proven the upper bound s(t) ≤ C √ t for some constant C ; now we show the optimality of the rate by proving the right nontrivial lower estimate, i.e. there exists C > 0 such that s(t) ≥ C √ t. Additionally, we obtain weak solutions to the free-boundary problem for the case when the measure of the initial domain vanishes. In this way, our mathematical model is now allowing for the appearance of a moving carbonation front -a scenario that until was hard to handle from the analysis point of view.

Design and Analysis of a Finite Volume Scheme for a Concrete Carbonation Model

“…Fig 2 shows the behavior of s for T = 100 in linear scale and for T = 1000 in logarithmic scale. These numerical experiments support the √ T -law of propagation given in [3,10]. The exact solutions u and v of (2) are not explicitly known.…”

“…Fig 1 shows the different profiles of v and u as a function of x ∈ [0, s(t)] for t ∈ {20, 40, 60, 80, 100}. We note that the profiles are similar to those given in [3,10]. Fig 2 shows the behavior of s for T = 100 in linear scale and for T = 1000 in logarithmic scale.…”

“…For this system Aiki and Muntean have shown in [1] the existence and the uniqueness of a global solution. They also have shown that the penetration depth follows a √ T -law of propagation in [2,3]. Our aim is to define a finite volume scheme for the concrete carbonation model and to show the convergence of the numerical scheme.…”

Large-time asymptotics of moving-reaction interfaces involving nonlinear Henry’s law and time-dependent Dirichlet data

“…To get a bit the flavor of mathematical investigations of the effects by Henry's law for this or closely related reactiondiffusion systems, we refer the reader to [3] (linear Henry's law) and [4,5] (micro-and micro-macro Henry-like laws). Essentially, we are able to present refined estimates that extend the proof of the rigorous large-time asymptotics beyond the settings that we have elucidated in [6,7]. In practical terms, we show that there exist two positive constants c * and C * , depending on all material parameters and initial and boundary data, such that c * √ t ≤ s(t) ≤ C * √ t + 1 for t ≥ 0.…”

Solving Moving Boundary Concrete Carbonation Models

“…Such conclusions were based on the Neumann solution of the two-phase Stefan problem, see Section 13.2.2 of [6]. In recent papers [1,2], the authors studied a one-dimensional free boundary problem modeling the carbonation process. The unknown CO 2 mass concentrations in air and water phases of pores are denoted by U (t, x) and V (t, x) respectively, depending on variables time t and space x.…”