## 4 Results of Kinetic Modeling
### Carbonation Kinetics Comparative Analysis
#### 4.2.2 Diffusion Control Region (Apparent Model)  
Using the methodology described in Section 3.4.2, an apparent model was used to determine the kinetic parameters of the diffusion control region. Data fitting plots are shown in Figures 8 and 9 and apparent model parameters are recorded in Table 6.  
Figure 8: Apparent model: (**a**) 1/X(\(t\)) versus time; (**b**) data fitting for diffusion control region. Figure 7: Intrinsic reaction rate model: (**a**) Data fitting with model; (**b**) Arrhenius plot. Figure 9: Apparent model: (**a**) Arrhenius plot for diffusion region; (**b**) conversion predicted using apparent model compared to original data. In order to validate the model, the obtained values of \(X_{u}\) and \(k\) were used to calculate \(X(t)\) (Equation (23)) and plotted against the isothermal data for the diffusion region. Figure 9b shows a good agreement between experimental data and the model. The kinetic parameters were obtained by performing data fitting on the Arrhenius plot (Figure 9a). An activation energy of 59.95 kJ mol\({}^{-1}\) was calculated for this region. Kinetic parameters for both the kinetic and diffusion control regions are presented in Table 7 and compared with selected literature results. [PERSON] states that \(k\) can be regarded as the intrinsic chemical reaction rate constant \(k_{\mathrm{s}}\)[37], as well as the diffusion constant \(D\), although the different units mean that it cannot be directly compared with reaction rate constants for studies such as those using pore models [17]. For both the chemical reaction and diffusion control regions, the reaction rates and values of \(X_{u}\) (see Table 6) tend to increase with increasing temperature, which is consistent with reaction kinetics increasing with higher temperatures (as described by Arrhenius relationships; i.e., Equation (4)). The reaction rates for P-CaO are seen to be slightly slower than those in the literature. The activation energies of P-CaO are also slightly lower than expected for both the kinetic control region (literature values typically fall between 20 and 32 kJ mol\({}^{-1}\)) and the diffusion region (literature values are typically between 80 and 200 kJ mol\({}^{-1}\)) [30]. The variation in the activation energies has been attributed to the differences in morphology and texture of the CO\({}_{2}\) sorbent [46; 56]. While it has been suggested that morphology of CaO does not \begin{table}
\begin{tabular}{c c c c} \hline \hline
**T (\({}^{\circ}\)C)** & **k (min\({}^{-1}\))** & **b (min)** & \(X_{u}\) \\ \hline
650 & 0.098 & 8.80 & 0.863 \\
700 & 0.158 & 5.54 & 0.875 \\
750 & 0.210 & 4.16 & 0.872 \\ \hline \end{tabular}
\end{table} 
Table 6: Carbonation reaction: Apparent model parameters. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & **Kinetic Control Region** & **[PERSON] [37]** & **[PERSON] [37]** & **[PERSON] et al.** \\
**Reference** & **This Work** & **[PERSON] et al. [56]** & **Data: [PERSON] and** & **Data: [PERSON] and [PERSON] [39]** & [17] \\ \hline \multirow{3}{*}{**Material**} & \multirow{3}{*}{P-CaO} & Strassburg limestone & \multirow{3}{*}{Calcined limestone} & \multirow{3}{*}{Calcony precipitated CaCO} & \multirow{3}{*}{Katowice limestone} \\  & & & & & \\ \cline{3-5}  & & Intrinsic reaction rate model & & & \\ \cline{3-5}  & & & model & & \\ \cline{3-5} T (\({}^{\circ}\)C) & k **(min\({}^{-1}\))** & & & & \\ \cline{3-5} \(650\) & 0.24 & 0.97 & 0.925 \({}^{*}\) & 0.858 & – \\
700 & 0.27 & 0.81 & – & – & – \\
750 & 0.30 & 0.67 & – & – & – \\ \(S\) **(m\({}^{2}\) g\({}^{-1}\))** & 13.8 & 29 & 15.6 & 12.8 & 12.9 \({}^{**}\) \\ \(E_{a}\) **(dJ mol\({}^{-1}\))** & 17.45 & 24.0 & 72.2 & 72.7 & 19.2 \\ \(k_{u}\) **(m\({}^{2}\) kmol\({}^{-1}\) s\({}^{-1}\))** & \(3.34\times 10^{-6}\) & \(4.68\times 10^{-6}\) & \(6.0\times 10^{-7}\) & – & \(5.29\times 10^{-6}\) \\ \hline \hline \end{tabular}  
* Calculated at 655 \({}^{\circ}\)C [37]; ** calculated from specific surface area (m\({}^{2}\) m\({}^{-3}\)) assuming \(\rho\) (non-porous CaCO) of 2710 kg m\({}^{-3}\).  
\end{table}