Unveiling the generalization of the derivative order with a novel application of the fractional order model to green soybean oil extraction

Abstract

Green solvents have represented suitable options for substituting nonrenewable substances applied to vegetable oil extraction processes. The general power-law equations used for this modeling present poor results. For better modeling, the generalization of the order of the equation is required. Scientific literature and laboratory data were collected in this study. The fractional model was derived from the standard power law first-order model using the Caputo fractional derivative and the Mittag-Leffler function. The fractional model (FOKM) and first-order model (FOM) were compared after data reconciliation using various statistical metrics and tests. The FOKM achieved the best statistical performance with Radj2 over 0.9841 compared to 0.9043 for FOM. The AIC for FOKM was higher in almost all cases, and Fisher's exact test showed that FOM is not statistically similar to FOKM. Both models presented the residues' normality by the Shapiro-Wilk normality test; however, a higher correlation between the parameters was found for the FOKM. It must be emphasized that the fractional model produced statistically significant improvements when fitting the data, showing the relevance of generalized order equations. This novel application of FOKM in green soybean oil extraction may help other researchers understand this phenomenon and provide a better representation of the process for industrial scale-up and optimization.