## 2 Materials and Methods
### Methods
#### 2.2.6 Statistical Methods  
Kinetic data retrieved at diverse catalyst concentrations, reagent molar ratios and temperature values are employed to propose, fit and validate a potential kinetic model based on the assumption of an elemental bimolecular reaction between glycerol and ethylene carbonate. Fitting of the kinetic model to diverse datasets was performed using a combination of a numerical integration of the ODE of the model with a variable step Euler method and a non-linear regression algorithm named NL2 SOLV, an evolved form of the Levenberg-Marquardt gradient method. These algorithms were implemented in the Aspen Custom Modeler v 12.1 software. The kinetic model was first fit to data at 80 \({}^{\circ}\)C, ethylene carbonate to glycerol molar ratio (M) equal to 2 and catalyst concentration (\(\mathrm{C_{cat}}\)) from 2 to 6% \(w/w\) of glycerol (9-27 g\(\cdot\)L\({}^{-1}\)) to retrieve a first value for the kinetic constant k and another for the \(\mathrm{K_{cat}}\) parameter. This parameter value was fixed for the rest of the fitting process, serving as a validating tool. A second fitting stage included datasets at 100 \({}^{\circ}\)C at a fixed \(\mathrm{C_{cat}}\) (2% \(w/w\)) to have a second value for k and an approximate value of the activation energy \(\mathrm{E_{a}}\) using the Arrhenius equation (Equation (2)). Finally, for the kinetic modelling with fresh catalyst, a final fitting and validation stage was the multi-temperature fitting of the model to retrieve the optimal value for the activation energy and the preexponential Arrhenius term. This approach allowed for the obtention of a kinetic model valid for the wide experimental reality studied. For the stability studies, a more complex kinetic model was considered as deactivation was present and could be modelled by an additional first-order kinetic equation, but a model fitting was simpler: the model was fitted to one experiment at a time to collect the kinetic parameters and analyze their evolution with the reuse cycles. \[k=exp\left(k_{0}-\frac{E_{a}}{R}\frac{1}{T}\right) \tag{2}\]  
where \(E_{a}\) is the activation energy, \(R\) is the ideal gas constant (also known as the universal gas constant, or molar gas constant), its exact value is 8.31 \(\mathrm{J}\cdot\mathrm{K}^{-1}\cdot\mathrm{mol}^{-1}\), \(T\) is the temperature in K, \(k\) is the rate constant of the reaction, the proportionality constant \(k_{0}\) is the pre-exponentialfactor, or the frequency factor that takes into consideration the frequency at which reactive molecules collide and the probability that a collision will lead to a successful collision for reaction. Apart from the usual physical considerations (positive values for the kinetic constants, positive value for the activation energy within the 10-300 kJ\(\cdot\)mol\({}^{-1}\) interval), several statistical criteria were analyzed to determine if each fitting was correct and, in the end, if the kinetic model was adequate, either in with fresh or with reused catalysts. Statistical analysis was based on the standard error for the kinetic parameters of each model and the goodness-of-fit parameters based on the least squared method. In fact, all goodness-of-fit statistical criteria applied were based on minimizing the sum of variances or squared residuals (RSE) as indicated in Equation (3), where \(\chi_{\rm e}\) stands for the experimental values of glycerol conversions, while \(\chi_{\rm c}\) are the conversion values calculated with the kinetic model. A derivate parameter is the standard error of estimate (SEE), a parameter that is calculated by using Equation (4). [PERSON]'s F value (F) is estimated through Equation (5) and should be higher than a threshold value for the given number of data and kinetic parameters (for our case, they are in the range 20-30). A fine fitting means a low value of RSS and, in consequence, for RSE, while the value of [PERSON]'s F should be high and, in any case, higher than the aforementioned threshold values at 95% confidence.