## 3 Comparison between Circular and Slab Diffusion Models  
### Chloride Concentration Estimation  
The sound and reliable estimation of chloride ion concentration on the surface of steel bars is significant. Here, this concentration, labeled \(C\), is compared for estimates from the slab diffusion and the circular diffusion model. The 28-day diffusion coefficient \(D_{28}\) is set at 4.35 cm\({}^{2}\)/a, with an age factor \(m\) of 0.53, and the surface chloride concentration \(C_{0}\) is assumed at 2% of the binder weight. The effect of the shape on the chloride concentration on the surface of rebar during diffusion process is analyzed, as shown in Figures 4 and 5. Figures 4 and 5 highlight that for an RC circular section, the slab diffusion model underestimates the chloride concentration. As the radius increasing, the difference between the circular diffusion and slab diffusion models deceases gradually, implying that the effect of the circular cross-section diminishes slowly. Figure 4: The chloride concentration versus concrete cover thickness at a diffusion time of 100 years. Figure 5: The chloride concentration versus diffusion time at a concrete cover thickness of 5 cm. Figure 3: The influence of time-variant diffusion coefficients on chloride concentration distribution. For an RC circuar column, \(C_{\text{cir}}/C_{\text{slab}}\) is introduced to measure the error on the work of using the slab diffusion model. The impact of the model parameters on \(C_{\text{cir}}/C_{\text{slab}}\) is disscused according to the sensitivity analysis methodology. Diffusion time, radius of the RC circuar column, and concrete cover thickness are labeled as \(t\), \(R\), and \(x\), respectively. Obviously, according to Equations (11) and (12), it can be found that \(C_{\text{cir}}/C_{\text{slab}}\) has no relation with \(C_{0}\). The change interval of other model parameters are listed in Table 2. During this analysis, one model parameter varies, with the others kept fixed. The final result is presented in Figure 6. It can be seen in Figure 6 that both higher \(x\) and smaller \(R\) will lead to the increase in \(C_{\text{cir}}/C_{\text{slab}}\). Nevertheless, \(m\), \(D_{28}\), and \(t\) have a very slight influence on this value. Distinctly, both \(R\) and \(x\) are more sensitivities to \(C_{\text{cir}}/C_{\text{slab}}\) than the other model parameters. On the basis of the observation from the sensitivity analysis, \(C_{\text{cir}}/C_{\text{slab}}\) is rewritten as a function using the radius of column and the concrete cover thickness. Assuming that \(D_{28}\), \(m\), \(t\), and \(R\) are fixed at the mean value, the \(C_{\text{cir}}/C_{\text{slab}}\) variation against \(x\) is plotted in Figure 7. The regression fitting result indicates that \(C_{\text{cir}}/C_{\text{slab}}\) can also be modeled using a linear function. Hence, \(C_{\text{cir}}/C_{\text{slab}}\) is expressed as follows:  
\[\frac{C_{\text{cir}}}{C_{\text{slab}}}(R,x)=k(R)\times x+b(R) \tag{13}\]  
where the coefficents \(k\) and \(b\) are functions with \(R\) only. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Mean Value** & **Lower Limitation** & **Upper Limitation** \\ \hline \(D_{28}\) & 3.435 cm\({}^{2}\)/a & 2.51 cm\({}^{2}\)/a & 4.36 cm\({}^{2}\)/a \\ \(m\) & 0.53 & 0.37 & 0.69 \\ \(R\) & 50 cm & 20 cm & 80 cm \\ \(x\) & 4 cm & 0 cm & 8 cm \\ \(t\) & 50a & 0a & 100a \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter value range of sensitivity analysis. Figure 6: Sensitivity analysis of \(C_{\text{cir}}/C_{\text{slab}}\) with respect to the model parameters. Figure 7: Plot of \(C_{\text{cir}}/C_{\text{slab}}\) versus cover thickness. Considering a series of \(R\), the values of \(k\) and \(b\) corresponding to each \(R\) are obtained by adopting the same regression fitting analysis with Figure 7. The final results are presented in Figures 8 and 9. In Figures 8 and 9, the specific forms of \(k(R)\) and \(b(R)\) are expressed after the regression fitting analysis again.