# Biodiesel and Biolubricant Production from Waste Cooking Oil: Transesterification Reactor Modeling
## 3 Results and Discussion
### Industrial Equipment, Reactor Design, and Economic Feasibility Study
#### 3.5.3 Reactor Design
If a cylindrical reactor is considered, its inner diameter and height can be calculated to minimize the amount of material used for its construction, according to Equations (21) and (22) as follows:  
\[S=\left(2\pi\times R\times h\right)+\left(2\pi\times R^{2}\right) \tag{21}\]  
\[V=\pi\times R^{2}\times h \tag{22}\]  
If both equations are represented, Equation (23) is obtained, where the surface is a function of R as follows:  
\[S(R)=\frac{2\times V}{R}+\left(2\pi\times R^{2}\right) \tag{23}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Reagent** & **Molecular Weight, g\(\cdot\)mol\({}^{-1}\)** & **Density, kg\(\cdot\)m\({}^{-3}\)** & **Mass Flow, kg\(\cdot\)h\({}^{-1}\)** & **Inlet Mass, kg** & **Inlet Volume, m\({}^{3}\)** \\ \hline FAMEs & 284.52 & 880 & 198.37 & 1586.96 & 1.8 \\ TMP & 134.17 & 1080 & 22.33 & 178.66 & 0.17 \\ Sodium methoxide & 54.03 & 970 & 5.95 & 47.62 & 0.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Inlet flows for the second transesterification. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Reagent** & **Molecular Weight, g\(\cdot\)mol\({}^{-1}\)** & **Density, kg\(\cdot\)m\({}^{-3}\)** & **Mass Flow, kg\(\cdot\)h\({}^{-1}\)** & **Inlet Mass, kg** & **Inlet Volume, m\({}^{3}\)** \\ \hline WCO & 900 & 920 & 209.2 & 1673.80 & 1.82 \\ Methanol & 32.04 & 792 & 44.68 & 357.44 & 0.45 \\ Sodium methoxide & 54.03 & 970 & 6.28 & 50.2 & 0.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Inlet flows for the first transesterification. By representing the surface area as a function of the radius, a line is analogous to a parabola. This function, at its lowest point, presents a slope that equals 0; that is, the derivative of the surface, when R is minimized, equals 0, as observed in Equation (24). \[S^{\prime}(R_{min})=0=4\pi\times R-\frac{2\times V}{R^{2}} \tag{24}\]  
Once the radius is obtained, the inner diameter and height can be calculated, as well as the surface area of the reactor, as shown in Table 11. The wall thickness was considered according to previous studies [37]. Concerning the size of the stirring system, a six-blade disk stirrer was selected, following Equations (25) and (26), for the design of the blades as follows:  
\[\frac{D_{a}}{D_{t}}=\frac{1}{3};\;\frac{H}{D_{t}}=1;\;\frac{J}{D_{t}}=\frac{1}{ 12} \tag{25}\]  
\[\frac{E}{D_{t}}=\frac{1}{3};\;\frac{W}{D_{a}}=\frac{1}{5};\;\frac{L}{D_{a}}= \frac{1}{4} \tag{26}\]  
where \(\rm{D_{a}}\) is the diameter of the stirrer, \(\rm{D_{t}}\) is the inner diameter of the reactor, H is the height of the reaction medium, J is the width of the deflector, E is the distance from the stirrer and the bottom of the reactor, W is the width of the blades, and L is the length of the blades. By applying the previous equations, the results shown in Table 12 were obtained as follows:  
Regarding the vacuum pump, it should be noted that the requirements of this equipment were previously established (510 mmHg and 40 min of working time), resulting in 5.64 L-min\({}^{-1}\) of methanol that are removed from the reaction medium in the case of the second transesterification when a pump with a power of 2.2 kW is selected for this purpose.