## 3 Mathematical Model
The most simple mathematical model should be chosen for the description of the phenomenon, bearing in mind the limitation of the number of identified parameters to the necessary minimum and obtaining the best possible conditioning of the coefficient inverse problem, but with the possibility of a reasonable quantification of the effect. Due to the first assumption, the surface layers must not be removed from the samples before the chloride diffusion process was started, and the experiment should be conducted in the isothermal and saturated conditions on concrete samples of at least 90 days to avoid excessive hydration process impact on the process. At the same time, the experiment should last long enough to capture changes in the concentration distribution as much as possible, but also as short as possible to limit the impact of hydration of cement relics in concrete skeleton on changes in the diffusion coefficient (e.g., [1]). Hence, the authors decided to use the research of the diffusion process lasting 180 days from its start which was described in Section 2. Moreover, since the binding effect strongly depends on the cement composition [32], the data for the samples with different composition depending on the used type of binder were the same selected. Due to the second assumption in the article, it is proposed to use the mathematical model described above in which the first-degree non-uniform spline function will be used to approximate the relationship (5) in the form:  
\[C_{b}=\left\{ \begin{array}{cccc}f_{(0,1)}(C_{f})=a_{(1)}C_{f}+b_{(1)}&\text{ for}&C_{f}\in\left[C_{f(0)}\,,\,C_{f(1)}\right]\\ f_{(1,2)}(C_{f})=a_{(2)}C_{f}+b_{(2)}&\text{for}&C_{f}\in\left[C_{f(1)}\,,\,C_{ f(2)}\right]\\ \vdots\\ f_{(i-1,i)}(C_{f})=a_{(i)}C_{f}+b_{(i)}&\text{for}&C_{f}\in\left[C_{f(i-1)}\,, \,C_{f(i)}\right]\\ \vdots\\ f_{(n-1,n)}(C_{f})=a_{(n)}C_{f}+b_{(n)}&\text{for}&C_{f}\in\left[C_{f(n-1)}\,, \,C_{f(n)}\right]\end{array} \right. \tag{21}\]  
where:  
\[a_{(i)}=\frac{C_{b}\Big{(}C_{f(i)}\Big{)}-C_{b}\Big{(}C_{f(i-1)}\Big{)}}{C_{ f(i)}-C_{f(i-1)}},\,b_{(i)}=\frac{C_{f(i)}\cdot C_{b}\Big{(}C_{f(i-1)} \Big{)}-C_{f(i-1)}\cdot C_{b}\Big{(}C_{f(i)}\Big{)}}{C_{f(i)}-C_{f(i-1)}}, \tag{22}\]  
\(i=1,2,\ldots,n\); \(C_{f(0)}=0<C_{f(1)}<C_{f(2)}<\ldots<C_{f(i)}<\ldots<C_{f(n)}=C_{fe}\)--free chloride concentrations which are the approximation knots for the function \(C_{b}\); \(a_{(i)}\), \(b_{(i)}\)--the slope and constant term of the function \(f_{(i-1,i)}\), respectively. Of course, \(a_{(i)}\) and \(b_{(i)}\) must be selected in such a way as to maintain the continuity of the isotherm, i.e.,  
\[f_{(i-1,i)}\Big{(}C_{\text{f(i)}}\Big{)}=f_{(i,i+1)}\Big{(}C_{\text{f(i)}}\Big{)} \ \ \text{for}\ \ i=1,2,\ldots,n-1; \tag{23}\]  
and that the function \(C_{\text{b}}\) is non-decreasing and is zero for the zero concentration of free chlorides, i.e.,  
\[a_{(i)}\geq 0\ \ \text{for}\ \ i=1,2,\ldots,n;\ \text{and}\ b_{(1)}=0. \tag{24}\]  
The authors decided to express \(C_{\text{b}}\) in the form of dependence (21) due to its relative ease, typical for splines, in approximating functions with a complex course. In particular, during the conducted experiments of 6-month diffusive saturation of concrete samples with chlorides, the authors expected for some of the concretes significant changes in their apparent diffusion coefficient \(D_{\text{app}}\) for the concentrations close to \(C_{\text{fe}}\) because, in such samples, they occur only at the outer surface where the \"skin effect\" may influence the process in parallel. Capturing this type of changes using isotherms according to Equation (20) is not efficiently possible. In the further part of article, concerning the analysis of the experiment results, the form of the \(C_{\text{b}}\) function was used according to Equation (21) for \(n=3\).