A mathematical model of carbon dioxide transport in concrete carbonation process

Concrete carbonation is known as one of a phenomenon which is a big serious damage to concrete buildings. From the civil engineering point of view, its analysis of the dynamics is very important problem. In the paper, we report some mathematical results for a model of carbon dioxide transport in this phenomenon without the precise proof. First, we explain the concrete carbonation phenomenon. Concrete is harding of sand, gravel with cement, and the main ingredient of cement is calcium hydroxide Ca$(OH)_{2}$ which shows alkalinity. The exposure concrete always touchs air, and carbon dioxide $CO_{2}$ enters the porous structure of concrete-based materials, and $CO_{2}$ dissolves in the pore water by Henry’s law. Since Ca$(OH)_{2}$ also dissolves in this pore water, $CO_{2}$ reacts with available alkaline species Ca$(OH)_{2}$ , i.e.


Introduction
Concrete carbonation is known as one of a phenomenon which is a big serious damage to concrete buildings. From the civil engineering point of view, its analysis of the dynamics is very important problem. In the paper, we report some mathematical results for a model of carbon dioxide transport in this phenomenon without the precise proof.
In this reaction, by $OH^{-}$ is comsumed, alkalinity changes into acidly. In the case of the reinforced concrete, by this phenomenon, there is rust forming on the surface of the iron bar in the concrete, and by the volume expanding of the iron bar, the crack is formed. Therefore, the concrete carbonation gives a big effect to the durability and the persistence of the concrete buildings.
On a mathematical result of concrete carbonation, Muntean-B\"ohm [14] considered a mathematical model of concrete carbonation process as a free boundary problem of the carbonated front in one dimensional case, and proved the existence and uniqueness of a solution of this problem. Also, Aiki-Muntean [5,6] considered a reduction model of Muntean-B\"ohm, and proved the large time behavior of the free boundary. Our aim of this study is to construct a mathematical model of this process in three dimensional case and to analyze the dynamics of concrete carbonation process. As the first step of this study, Aiki-Kumazaki [1,3] proposed a mathematical model of moisture transport which contains the hysteresis operator due to Maekawa-Ishida-Kishi [13] and Maekawa-Chaube-Kishi [12], and proved the existence and uniqueness of a solution of this model. For the uniqueness of a solution of this model, Aiki-Krej\v{c}i-Kumazaki [4] proved in three dimensional case by using various techniques of [11].
In the paper, we focus on the carbon dioxide in this process. In [2,8], we proposed the following balanced law of carbon dioxide transport due to Maekawa-Chaube-Kishi [12] and Maekawa-Ishida-Kishi [13]: From the physical point of view, $\Omega$ is a domain occupied by concrete, and the unknown functions $v=v(t, x)$ and $u=u(t, x)$ represent the concentration of carbon dioxide in air and the concentration of carbon dioxide in water at a time $t$ and a position $x\in\Omega,$ respectively. In the equilibrium state, it is known that the relation $v=\rho_{0}u$ holds for a positive constant $\rho_{0}$ by Henry's law. Also, $\phi=\phi(z)$ represents the porosity, which is the ratio of the volume of the total pore spaces inside of concrete to the volume of the whole concrete, and $z$ is the ratio of the volume of consumed calcium hydroxide to the volume of the total calcium hydroxide. Moreover, $\mathcal{S}$ represents the degree of saturation corresponding to the relative humidity, and the relationship between the relative humidity and the degree of saturation is given as a hysteresis operator in [12,13]. In the flux term, $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ are positive constants. In the forcing term, $\kappa$ is a reaction rate and $w$ represents the concentration of calcium ion and this forcing term represents the consumed carbon dioxide in the concrete carbonation process, and is given by the reaction rate theory.
Here, we show that $w$ and $z$ has the form of (1.2) briefly. From the reaction rate theory we have  2 Mathematical results

Large time behavior of a solution
Large time behavior of a solution, in particular the convergences of a solution of (P) to a solution of the steady state problem of (P) is important to see that how much the concrete is carbonated finally. In fact, by $0\leq u$ a.e. on $Q(T)$ in Theorem 1 $v(t)=e^{-\int_{0}^{t}u(\tau)d\tau}$ is decreasing with respect to $t$ so that we see that there exists a limit of $v(t)$ as $tarrow\infty$ . If we can show that $v(t)arrow 0$ a.e. on $\Omega$ ae $tarrow\infty$ , then $1-v(t)=1-e^{-\int_{0}^{t}u(\tau)d\tau}arrow 1$ as $tarrow\infty$ for a.e. $x\in\Omega$ . Since $z$ is the ratio of the volume of consumed calcium hydroxide to the volume of the total calcium hydroxide, $z=1$ a.e. on $\Omega$ implies that calcium hydroxide is fully consumed at almost everywhere in the concrete. Accordingly, finally, we see that the concrete is carbonated at almost everywhere.
In future, we try to consider the system consisting of (1.1) and a parabolic equation which represents a mathematical model of moisture transport handled in [1,3] as a mathematical model of concrete carbonation phenomenon.