# Biodiesel and Biolubricant Production from Waste Cooking Oil: Transesterification Reactor Modeling
## 3 Results and Discussion
### Kinetic Study  
Batch tests were carried out in the laboratory to assess the apparent rate constant of the transesterification reaction leading to the biolubricant product. Also, the effects of the pressure, temperature, and catalyst concentration on this constant were studied. According to Equation (14), a plot of the right-hand-side term against time should yield a straight line, which slope is the apparent rate constant. As an example, Figure 8 shows the plot for the data for one of the runs carried out. The apparent kinetic constant (k\({}^{\prime}\)) is dependent on the pressure, temperature, and catalyst concentration. Regarding the pressure, according to the results included in Figure 3a, a working pressure of 510 mmHg has the same effect on the reaction rate of the process like those obtained at lower working pressures (which would imply higher energy costs). The effects of the temperature (T) and catalyst concentration (C\({}_{\text{cat}}\)) on the kinetic constant are represented by Equation (15) as follows:  
\[k^{\prime}=k_{0}\times C_{cat}^{n}\times e^{(\frac{-E_{0}}{R\cdot T})} \tag{15}\]  
In this case, the pre-exponential factor (\(k_{0}\)), n, and the activation energy (E\({}_{\text{a}}\)) are unknown. To study the effect of the temperature on the kinetic constant, runs were performed in the 80-140 \({}^{\circ}\)C range, with the catalyst concentration fixed at 0.3% \(w/w\). Table 3 shows the apparent rate constants (k\({}^{\prime}\)) obtained from these runs. \begin{table}
\begin{tabular}{c c c} \hline \hline
**T\({}_{r}\)\({}^{\circ}\)C** & **T, K** & \(k^{\prime}\) \\ \hline
80 & 353 & 0.0141 \\
100 & 373 & 0.0176 \\
120 & 393 & 0.0245 \\
140 & 413 & 0.0339 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Apparent kinetic constants (k\({}^{\prime}\)) for biolubricant production at different temperatures. Figure 8: Determination of the apparent rate constant of the transesterification reaction for biolubricant production. Reaction conditions: initial FAME/TMP molar ratio = 3; T = 120 \({}^{\circ}\)C; stirring rate = 350 rpm; working pressure = 210 mmHg; catalyst concentration = 1% \(w/w\). To obtain the activation energy, Equation (16) was applied as follows:  
\[\ln\ k^{\prime}=\ln(k_{0}\times 0.3^{n})-E_{a}\times\frac{1}{R\times T} \tag{16}\]  
Figure 9 shows the linear regression of the experimental data according to Equation (16), resulting in an activation energy value of 17,858 J\(\cdot\)mol\({}^{-1}\). To calculate k\({}_{0}\) and n, a series of runs at a constant temperature (80 \({}^{\circ}\)C) was carried out. Thus, the apparent rate constant can be expressed by Equation (17) as follows:  
\[k^{\prime}=k_{0}\times C_{cat}^{n}\times e^{(\frac{-E_{a}}{R\cdot(80+23)})} \tag{17}\]  
Table 4 shows the apparent rate constant values (k\({}^{\prime}\)) at 80 \({}^{\circ}\)C and for different catalyst concentrations in the 0.3-0.9% \(w/w\) range. If a logarithmic approach is applied to Equation (17), Equation (18) is obtained as follows:  
\[\ln\ k_{2}^{\prime}=n\times\ln\ C_{cat}+\ln\left(k_{0}\times e^{(\frac{-E_{a}} {R\times(80+23)})}\right) \tag{18}\]  
Thus, Figure 10 shows a linear regression according to Equation (18), yielding k\({}_{0}\) = 51.8 M\({}^{-1}\cdot\)min\({}^{-1}\) and n = 1.8. \begin{table}
\begin{tabular}{c c} \hline \hline
**Catalyst Concentration, \% \(w/w\)** & \(k^{\prime}2\) \\ \hline
0.3 & 0.0141 \\
0.5 & 0.0237 \\
0.7 & 0.0468 \\
0.9 & 0.1096 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Apparent kinetic constants (k\({}^{\prime}2\)) for biolubricant production at different temperatures. Figure 9: Dependence of the apparent constant (k\({}^{\prime}\)) on the reaction temperature according to the Arrhenius law. The parameters required in Equation (16) were obtained, thus allowing for the estimation of the rate constant at different temperatures and for catalyst concentrations. Although this constant has been calculated for a pressure of 510 mmHg, the effect of the pressure on the reaction rate was negligible in the range 210-510 mmHg.