# Green Production of Glycerol Ketals with a Clay-Based Heterogeneous Acid Catalyst
## 3 Results and Discussion
### Glycerol Ketalization Kinetic Studies
#### 3.3.4 Kinetic Model Fitting  
Considering the previous results, it is evident that a kinetic model with a direct reaction (ketalization) and a reverse reaction (hydrolysis) is needed. The results of the reuse of all catalysts reported in Section 3.1, at any contact time, suggest a deactivation process for systems 1 and 3. In the literature, for solketal and other ketals, kinetic models of the type Potential, Langmuir-Hinshelwood-Hougen-Watson (LHHW) and Eley-Rideal (ER) are proposed and fitted to the results [17; 40; 41]. In this case, and for all systems, we have proposed and fitted a potential model considering that reactions are almost elemental (that is, the closing and opening reactions of the ketals are supposed to be very fast in comparison to reactions needing the collision of reagent molecules). This is in agreement with all kinetic models in literature, as far as we know, as, even in the hyperbolic models, the numerator always present terms for the direct and the reverse reactions that are first-order with respect to ketones and alcohols, on one side, and to ketals and water, on the other, with no term accounting for the presence of the intermediate. Considering the deactivation of the catalyst during the chemical process, and given the results in the previous section, a first-order kinetic equation with remaining activity at infinitum time (partial deactivation) is proposed and coupled to the previous kinetic equation proposed for the ketalization and hydrolysis reactions. Therefore, the proposed kinetic model is:  
\[r_{1}\ =\ k_{1}\cdot C_{cataly}\cdot C_{ketone}\cdot C_{glycerol} \tag{5}\]  
\[r_{2}\ =\ k_{2}\cdot C_{cataly}\cdot C_{ketal}\cdot C_{water} \tag{6}\]  
\[C_{cataly}\ =\ \beta+(1-\beta)\cdot e^{-k_{A}\cdot t} \tag{7}\]  
where the first rate equation corresponds to the ketalization (direct) reaction and the second one to the hydrolysis. Equation (6) reflects in an integral mode the evolution of the catalyst activity because, although it is a first-order process, this process only affects a part of the activity present at zero time,  
Figure 15: Evolution of the initial ketalization rates for all systems in subsequent reutilization cycles (System 1: acetone; System 2: butanone or methylethylketone (MEK); System 3: cyclohexanone). Reaction conditions: 25 \({}^{\circ}\)C, 50 g/L catalyst AC (system1); 40 \({}^{\circ}\)C, 200 g/L catalyst AC (system 2); 40 \({}^{\circ}\)C, 50 g/L catalyst AC (system 3); ketone/glycerol molar ratio = 6 (all systems). which is (1-\(\beta\)) if total activity is 1 and remaining activity at infinitum time is \(\beta\). Constants k\({}_{1}\) and k\({}_{2}\) are second-order constants, while k\({}_{d}\) is a first-order constant. As indicated at the beginning of this section devoted to results and their discussion, the proposed kinetic model was fitted, for each system, to each run, to runs where one of the variables was changed and to all runs at the same time. Results of this last fitting for all systems are collected in Tables 3 and 4 and in Figures 16-18. Table 3 collects statistical parameters related to the goodness of the fit of the model to all data for each system. In all cases, the values of SSR and RMSE are really low, given the high number of data (N), indicating a close fit of the model to all data (this can be even better observed in the figures). Again, very high values of Fisher F at 95% confidence, much higher than the threshold values at that level of confidence (typically in the range 20-100) show that the model is adequate, so the hypotheses behind it seem acceptable, and the fit of the model to data for any of the systems is correct. Finally, the percentage of variation explained (VE) is near 100%, so, again the predicted temporal evolution rate for the ketal is very similar to the experimental one. On the other side, Table 4 displays the kinetic parameters of the model for all the systems.