## 3 Mathematical Model
Hence, its identification can be reduced to the determination of positioning the second and third approximation knots, \(C_{\text{f(1)}}\) and \(C_{\text{f(2)}}\), and the values of the \(C_{\text{b}}\) function at 3 knots: \(C_{\text{b}}\Big{(}C_{\text{f(1)}}\Big{)}=f_{(0,1)}\Big{(}C_{\text{f(1)}} \Big{)}=f_{(1,2)}\Big{(}C_{\text{f(1)}}\Big{)}\), \(C_{\text{b}}\Big{(}C_{\text{f(2)}}\Big{)}=f_{(1,2)}\Big{(}C_{\text{f(2)}} \Big{)}=f_{(2,3)}\Big{(}C_{\text{f(2)}}\Big{)}\), and \(C_{\text{b}}\Big{(}C_{\text{f(3)}}\Big{)}=f_{(2,3)}\Big{(}C_{\text{f(3)}} \Big{)}\). Based on these, it is obviously possible to calculate the slopes \(a_{(i)}\) and constant terms \(b_{(i)}\) from Equation (22). The first and fourth approximation knots were assumed according to Equation (21) as known, i.e., at the abscissas \(C_{\text{f(0)}}=0\) and \(C_{\text{f(3)}}=C_{\text{fe}}\). In accordance with the experimental conditions, it was calculated for a 3% NaCl solution that \(C_{\text{fe}}=523.3\ \text{mol}/\text{m}^{3}\). The variability of the isotherm according to Equation (21) for \(n=3\) with the data necessary for its identification is illustrated generally in Figure 3. The last and possibly the middle segments of the isotherm, for the highest concentrations close to \(C_{\text{fe}}\) and in accordance with the assumptions described earlier, are able to effectively capture the \"skin effect\" and estimate the thickness of the \"skin\" of the concrete when plotting on their basis the variability of the apparent diffusion coefficient \(D_{\text{app}}\) along the sample depth. Figure 3: Illustration of chloride binding isotherm according to Equation (21) for \(n=3\). Isotherm parameters known according to the experimental conditions: \(C_{\text{f(0)}}\) and \(C_{\text{f(3)}}\). Isotherm parameters to be determined from the inverse problem: \(C_{\text{f(1)}}\), \(C_{\text{f(2)}}\), \(C_{\text{b}}\Big{(}C_{\text{f(1)}}\Big{)}\), \(C_{\text{b}}\Big{(}C_{\text{f(2)}}\Big{)}\), and \(C_{\text{b}}\Big{(}C_{\text{f(3)}}\Big{)}\). The identification of material parameters is proposed to be carried out by minimizing the following objective function:  
\[F(\mathbf{y})=\sum_{i=1}^{s}\left(\overline{C}_{\text{t model }(i)}(\mathbf{y})- \overline{C}_{\text{texp}(i)}\right)^{2} \tag{25}\]  
where:  
\[\mathbf{y}=\left[y_{i}\right]_{i=1,\ldots,s}, \tag{26}\]  
\[\left[D_{\text{eff(n=3)}},C_{\text{f(1)}},C_{\text{f(2)}},C_{\text{b}}\Big{(}C _{\text{f(1)}}\Big{)},C_{\text{b}}\Big{(}C_{\text{f(2)}}\Big{)},C_{\text{b}} \Big{(}C_{\text{f(3)}}\Big{)}\right]=\text{argmin }_{\mathbf{y}\in T}(F), \tag{27}\]  
\[\overline{C}_{\text{t model}(i)}=\frac{1}{\Delta I}\int_{\mathbb{x}_{i}- \frac{\Delta I}{2}}^{x_{i}+\frac{\Delta I}{2}}C_{\text{t}}(x)\ dx, \tag{28}\]  
\(\overline{C}_{\text{texp}(i)}\)--mean total chloride concentration (mol/m\({}^{3}\)) determined experimentally on the \(i\)-th sampling section (based on Table 4), \(\overline{C}_{\text{t model}(i)}\)--mean total chloride concentration (mol/m\({}^{3}\)) determined for the \(i\)-th sampling section on the basis of the adopted diffusion model in the initial-boundary problem corresponding to the experimental conditions, \(\mathbf{y}\)--vector of variables, \(D_{\text{eff(m=3)}}\)--effective chloride diffusion coefficient (m\({}^{2}\)/s) when assuming \(n=3\) in Equation (21), \(\Delta\)--length (m) of sampling section, \(x_{i}\)--coordinate at the middle of the \(i\)-th sampling section (m), \(s\)--number of sampling sections, \(T\)--domain of the objective function \(F\). In order to apply the presented mathematical model to estimate the unknown materials parameters, the knowledge of the open porosity of concrete, \(\varepsilon\), is also required, and it is assumed that \(\varepsilon\) is known based on the experimental measurement (Table 3). \(T\) is the domain of searching the solution of the problem, and the selection of its rationally justified boundary is a key issue due to the fact that the minimum of \(F\) describes physically acceptable results.