Enhancement of Biodiesel Production via Ultrasound Technology: A Mathematical Study

Biodiesel is one of the alternative renewable energy sources that has received a lot of attention since it is clean, green energy. Different sources can be used for the production of biodiesel, but the most appropriate and economical method relies on the transesterification of methanol with the nonedible vegetable oil from the fruit of the Jatropha curcas plant. Molar ratio, vessel diameter, catalyst concentration, and ultrasound all have a substantial influence on the synthesis of biodiesel by the transesterification process. Among these factors, the diameter of the vessel and the ultrasonic effect through mass transfer limitations have a significant impact on successful reaction completion. In this research work, we have developed a mathematical model to analyze the three-step transesterification process and side saponification reaction in the presence of a potassium hydroxide catalyst. The model considers the influence of mixing intensity variations, including ultrasound, on the mass transfer in different phases. The mass transfer rate is calculated using the modified Dittus-Boelter correlation. An optimal control approach through the minimum principle by Pontryagin is applied to maximize the production of biodiesel at minimal cost. The novelty of this research, which we have derived from our analytical as well as numerical results, considering industrial processes, is that more than 97% biodiesel yield conversion is to be obtained at 50 kHz ultrasound frequency, a 6:1 methanol-to-Jatropha-oil molar ratio, and 1 m of vessel diameter within 50 min using optimal control theory.