## 3 Verification of Proposed Model  
### Carbonation Depth of Cracked Concrete  
Experimental results from [PERSON] et al. [5] were used to verify the proposed model. [PERSON] et al. [5] conducted an experimental study on the carbonation of concrete with different crack widths and mixing proportions. The concrete mixing proportions are shown in Table 1. Concrete with high water  
Figure 1: Flowchart of proposed model. CH: calcium hydroxide; \(RCH_{CE}\): mass of CH from the hydration of 1-unit mass cement; \(C_{0}\): cement content; \(\alpha\): degree of cement hydration; CSH: calcium silica hydrate; \(f_{S,C}\): silica weight fraction in cement; \(\varepsilon\): porosity of concrete; \(W\): water content in concrete; \(\rho_{W}\): density of water; \(\Delta c_{C}\): porosity reduction due to the carbonation of concrete; \(dx_{C}\): increase of carbonation depth in time increment \(dt\); \(CR\): carbonation rate parameter; D\({}_{\text{eq}}\): equivalent diffusion coefficient; \(g(t)\): durability failure criteria; \(CV\): cover depth of concrete; \(p_{f}\): probability of carbonation durability failure; \(n(g(t)<0)\): number of carbonation durability failures out of a total \(N\) trials. to cement ratio represented low strength concrete. The concrete specimens were cured under water for 28 days at 20 \({}^{\circ}\)C. After curing, the specimens were put into a chamber at 65% RH for two weeks. Next, the concrete specimens were put into the CO\({}_{2}\) chamber for accelerated carbonation testing (CO\({}_{2}\) concentration was 10%, RH = 65%, temperature was about 25 \({}^{\circ}\)C). Splitting tests were used to induce 0.05 to 0.25 mm cracks into the concrete specimens. By using the kinetic cement hydration model [9], the degree of cement hydration, CH content, CSH content and porosity of concrete were calculated and are shown in Figure 2. With decreases in the water to cement ratios, the available depositing space of the hydration products and capillary water concentration decreased and consequently, the degree of hydration was reduced (shown in Figure 2a). Concrete with lower water-to-cement ratios had higher CH content (shown in Figure 2b) and CSH content (shown in Figure 2c) and lower porosity (shown in Figure 2d). Carbonation is significantly influenced by concrete relative humidity. The carbonation of concrete tended to be the highest for environmental relative humidity of 50-70% [9]. When the relative humidity \begin{table}
\begin{tabular}{c c c c c} \hline W/C & **Water (kg/m3)** & **Cement (kg/m3)** & **Sand (kg/m3)** & **Coarse Aggregate (kg/m3)** \\ \hline
0.45 & 191 & 424 & 668 & 1058 \\
0.55 & 184 & 335 & 762 & 1058 \\
0.65 & 182 & 280 & 829 & 1041 \\ \hline \end{tabular}
\end{table} 
Table 1: Mixing proportions of concrete. Figure 2: Hydration related properties of concrete. (**a**) degree of hydration; (**b**) CH content; (**c**) CSH content; (**d**) porosity of concrete. was less than 50%, there was insufficient moisture for the progression of the carbonation reaction. While the relative humidity was higher than 70%, the diffusion of CO\({}_{2}\) was hindered due to the increase in the saturation degree of the concrete pores. On the other hand, the rate of cement hydration was also significantly influenced by relative humidity. Cement hydration virtually ceased at about 80% RH for concrete with various water to binder ratios and mixing proportions [9]. In the study by [PERSON] et al. [5], after water curing, the concrete was exposed to 65% relative humidity. The relative humidity in the surface skin of concrete approximately equaled that of the environmental relative humidity [2; 9]. When the relative humidity in the surface skin concrete was less than 80%, the hydration of cement in the surface skin concrete stopped [9]. Hence, further hydration in the carbonation zone of concrete was marginal and could be ignored. The calculation results from the hydration model can be used as input parameters for evaluating the concrete carbonation depth.