## 3 Mathematical Model  
The mathematical model was formulated taking into account the basic objectives of the research (Section 1) and the conditions of the experiment (Section 2). It was assumed that the transport of chloride ions in concrete under fully saturated conditions at constant temperature can be described by the following system of two local mass balances--respectively for the free chloride ions in the pore liquid and chlorides bound by the concrete skeleton (without distinction to chemically or physically bound chlorides):  
\[\varepsilon\;M_{\text{CI}}\frac{\partial C_{\text{f}}}{\partial t}=-\varepsilon \;\
abla\Delta\mathbf{j}_{\text{f}}-(1-\varepsilon)\;S, \tag{3}\]  
\[M_{\text{CI}}\frac{\partial C_{\text{b}}}{\partial t}=S \tag{4}\]  
where: \(C_{\text{f}}\)--concentration of free chlorides (mol/m\({}^{3}\)), \(C_{\text{b}}\)--concentration of bound chlorides (mol/m\({}^{3}\)), \(\mathbf{j}_{\text{f}}\)--vector of free chloride mass flux density (kg/(s\(\cdot\)m\({}^{2}\))) in the pore liquid, \(S\)--density of the mass source of bound chlorides (kg/(s\(\cdot\)m\({}^{3}\))) in the concrete skeleton, \(\
abla\)--nabla operator (1/m), \(t\)--time (s). The form of mass balance (4) for the bound chlorides corresponds to the situation in which it is assumed that the adsorption of chlorides by the skeleton is so fast compared to the diffusion process that it can be considered immediate. If the function of the chloride binding isotherm is known:  
\[C_{\text{b}}=f(C_{\text{f}}), \tag{5}\]  
and in addition, we can assume the isotropy of the material, the isothermality of the process, and take into account the diffusion of free chlorides using [PERSON]'s law with an effective diffusion coefficient independent of the chloride concentration, i.e.,  
\[\mathbf{j}_{\text{f}}=-D_{\text{eff}}\;M_{\text{CI}}\;\
abla C_{\text{f}} \tag{6}\]  
Figure 2: Average concentrations of total chlorides \(\overline{C}_{\text{t exp}}\) in the successive 2 mm thick sampling sections of concrete C1–C5 samples (based on the values from Table 4). Coordinate \(x=0\) relates to the specimen external surface exposed to the chloride solution. where: \(D_{\text{eff}}\)--effective constant diffusion coefficient in (m\({}^{2}\)/s), then we can obtain the following differential equation based on the mass balances (3) and (4):  
\[\frac{\partial C_{\text{f}}}{\partial t}=\frac{\varepsilon\,D_{\text{eff}}}{ \varepsilon+(1-\varepsilon)\frac{\partial C_{\text{b}}}{\partial C_{\text{f}} }}\
abla^{2}C_{\text{f}}. \tag{7}\]  
Equation (7) can also be presented as:  
\[\frac{\partial C_{\text{f}}}{\partial t}=D_{\text{app}}\
abla^{2}C_{\text{f}} \tag{8}\]  
where:  
\[D_{\text{app}}(C_{\text{f}})=\frac{\varepsilon\,D_{\text{eff}}}{\varepsilon+(1 -\varepsilon)\frac{\partial C_{\text{b}}}{\partial C_{\text{f}}}}\, \tag{9}\]  
is the apparent chloride diffusion coefficient in (m\({}^{2}\)/s), generally dependent on the concentration of free chlorides and taking into account the process of binding them in the skeleton. Equation (8) in the case of one-dimensional diffusion transport will simplify to the form:  
\[\frac{\partial C_{\text{f}}}{\partial t}=D_{\text{app}}\frac{\partial^{2}C_{ \text{f}}}{\partial x^{2}} \tag{10}\]  
where: \(x\)--spatial variable (m).