## 3 Thermodynamic Modelling of Cement Hydration
### OPC Clinker Rate Equations  
The dissolution of the four main clinker phases was described with the empirical rate equations proposed by [24] and modified by [PERSON] and coworkers [7], using Equations (3)-(5) for any specified time step. In this paper, the empirical expressions and the constants used in their approach can be found in Table 3, described by [PERSON] et al. [22]. A variable time step was used so that, here, only 47 steps were required to predict the dissolution of the four clinker phases for 1000 days of hydration.  
\[R_{t}=\frac{K}{N}(1-\alpha_{t})(-ln(1-\alpha_{t}))^{(1-N)}\cdot\frac{A}{A_{0} }\cdot exp\bigg{[}\frac{E_{a}^{m}}{R}\bigg{(}\frac{1}{T}-\frac{1}{T_{0}}\bigg{)} \bigg{]}\cdot\bigg{(}\frac{RH-0.55}{0.45}\bigg{)}^{4}\cdot f\Big{(}\frac{w}{c} \bigg{)} \tag{3}\]  
\[R_{t}=\frac{K(1-\alpha_{t})^{\frac{2}{3}}}{1-(1-\alpha_{t})^{\frac{1}{3}}} \cdot exp\bigg{[}\frac{E_{a}^{m}}{R}\bigg{(}\frac{1}{T}-\frac{1}{T_{0}}\bigg{)} \bigg{]}\cdot\bigg{(}\frac{RH-0.55}{0.45}\bigg{)}^{4}\cdot f\Big{(}\frac{w}{c} \bigg{)} \tag{4}\]  
\[R_{t}=K(1-\alpha_{t})^{N}\cdot exp\bigg{[}\frac{E_{a}^{m}}{R}\bigg{(}\frac{1}{ T}-\frac{1}{T_{0}}\bigg{)}\bigg{]}\cdot\bigg{(}\frac{RH-0.55}{0.45}\bigg{)}^{4} \cdot f\Big{(}\frac{w}{c}\bigg{)} \tag{5}\]  
\[f\Big{(}\frac{w}{c}\Big{)}=(1+3.333\cdot(\mathrm{H}\cdot w/c-\alpha_{\mathrm{ tot}}))^{4};\ \mathrm{for}\ \alpha_{\mathrm{tot}}\ >\ \mathrm{H}\cdot w/c \tag{6}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{3}{c}{**Oxide Proportions (g/100 g Cement)**} & \multicolumn{3}{c}{**Phase Compositions (g/100 g Cement)**} \\ \hline \multicolumn{3}{c}{**OPC**} & \multicolumn{2}{c}{**Limestone**} & \multicolumn{2}{c}{**OPC**} & \multicolumn{2}{c}{**Limestone**} \\ \hline SiO\({}_{2}\) & 20.2 & SiO & 0.8 & C\({}_{3}\)S & 66.5 & C\({}_{3}\)S & 64.60 \\ \hline Al\({}_{2}\)O\({}_{3}\) & 4.9 & Al\({}_{2}\)O\({}_{3}\) & 0.3 & C\({}_{2}\)S & 10.30 & C\({}_{2}\)S & 9.30 \\ \hline Fe\({}_{2}\)O\({}_{3}\) & 3.2 & Fe\({}_{2}\)O\({}_{3}\) & 0.3 & C\({}_{3}\)A & 7.50 & C\({}_{3}\)A & 7.40 \\ \hline CaO & 63.9 & CaO & 55 & C\({}_{4}\)AF & 8.50 & C\({}_{4}\)AF & 7.80 \\ \hline MgO & 1.8 & MgO & 1.8 & CaO\_free & 0.93 & CaO-free & 0.89 \\ \hline Na\({}_{2}\)O & 0.42 & Na\({}_{2}\)O & \textless{}0.01 & Calcite & 0.60 & Calcite & 4.60 \\ \hline K\({}_{2}\)O & 0.78 & K\({}_{2}\)O & \textless{}0.01 & Gypsum & 3.10 & Gypsum & 3.00 \\ \hline CaO-free & 0.93 & CaO-free & \textless{}0.01 & Periclase & 0.90 & Periclase & 0.90 \\ \hline CO\({}_{2}\) & 0.26 & CO\({}_{2}\) & 42.5 & K\({}_{2}\)SO\({}_{4}\) & 1.30 & K\({}_{2}\)SO\({}_{4}\) & 1.30 \\ \hline SO\({}_{3}\) & 2.29 & SO\({}_{3}\) & 0.05 & Na\({}_{2}\)SO\({}_{4}\) & 0.21 & Na\({}_{2}\)SO\({}_{4}\) & 0.20 \\ \hline Periclase & 0.9 & Periclase & 0.9 & & & Alkali & \\ \hline \multicolumn{3}{c}{Blaine fineness (m\({}^{2}\)/kg)} & \multicolumn{2}{c}{K\({}_{2}\)O} & 0.05 & K\({}_{2}\)O & 0.05 \\ \hline \multicolumn{3}{c}{413} & \multicolumn{2}{c}{429} & Na\({}_{2}\)O & 0.33 & Na\({}_{2}\)O & 0.31 \\ \hline \multicolumn{3}{c}{Ignition loss} & \multicolumn{2}{c}{MgO} & 0.94 & MgO & 0.87 \\ \hline \multicolumn{3}{c}{0.37} & \multicolumn{2}{c}{43.4} & SO\({}_{3}\) & 0.11 & SO\({}_{3}\) & 0.11 \\ \hline \end{tabular}
\end{table}
Table 3: Cement oxide proportions (taken from [20]).