Fig. 8: Relationship between the absolute temperature and the pseudo second-order initial extraction rate h for the extraction of oil from Irvingia gabonensis kernel at particles size of 0.5mm.
Obviously, the initial extraction rate, C1, increased with temperature, so did the constant that is related to maximum extraction yield, C2. Similarly, the initial extraction rate, h, extraction capacity, Cs, and the second-order rate constant, k, all increased with temperature. These findings were in line with the findings of Rakotondramasy et al. [42] for the solid–liquid extraction of protopine from Fumariaofficinalis L. It is important to note that when the temperature was kept at 308K, for a particle size of 0.5 mm, the initial extraction rate C1 for hyperbolic model was 6.21 min-1. This value was slightly higher than 5.15gL-1 min-1 obtained for the initial extraction rate in pseudo second-order model. On the other hand, the constant related to maximum extraction yield C2, for hyperbolic model, at the same constant temperature of 308 K and 0.5 mm particle size was 0.085 min-1, while the second-order rate constant, k was 0.0011 Lg-1min-1. However, the extraction capacity, Cs, at the same temperature and particle size was 68.97gL-1. This value was very close to pseudo second-order model calculated oil yield, Ct, 68.52 wt. %. This is an indication of the fitting of the second-order model for the extraction of oil from IGK.
Activation energy determination
The linearized Arrhenius equation [Equation (14)], was used to determine the relationship between k and T, the k0 and Ea. This was done by plotting In k versus 1000/T for pseudo second-order kinetic model (Fig. 9). The plot shows that the rate constant increases with the increases in temperature. However, a modified form of Arrhenius equation [Equation (25)] was used to determine the relationship between C2 and T, the k0 and Ea for hyperbolic kinetic model. Like the pseudo second-order model, this was carried out by plotting In C2 against 1000/T (Fig. 10).
\(\text{In\ }C_{2}\ =In\ k_{0}\ +\ \left(-\ \frac{E_{a}}{R}\ \bullet\ \frac{1}{T}\right)\)(25)
Using Equations 14 and 25, activation energies were calculated from the slopes and the values of temperature independent factors calculated from the intercept for the pseudo second-order (Fig. 9) and hyperbolic (Fig. 10) models, respectively. The relationships for the activation energy of extractions at 328K and 2.5 mm particle size, modeled using pseudo second-order and hyperbolic models, are given by Equations (26) and (27), respectively.
\(k\ =\ k_{0}\text{\ exp\ }\left(\frac{-11900}{8.314\ \times 328}\right)\)(26)
\(C_{2}\ =\ k_{0}\text{\ exp}\left(\frac{-18790}{8.314\ \times 328}\right)\)(27)
Their respective R2 values for pseudo second-order and hyperbolic models were 0.9582 and 0.9453.
In both models, the activation energies were positive, which is an indication that the extraction of oil from IGK is an endothermic process. In the case of pseudo second-order models, the activation energies for average particles size of 0.5, 1.0, 1.5, 2.0 and 2.5 mm were 5.49, 5.57, 5.79, 6.11, and 11.90 kJ/mol, respectively. Similarly, for hyperbolic model, the activation energies at these average particle sizes were 7.03, 11.03, 12.52, 16.63 and 18.79 kJ/mol, respectively. These results show that irrespective of the kinetic model used, the rate constants were dependent on the temperature, and they increased with increase in temperature. Also, the rate constants for the models were more temperature sensitive for larger particles size than for the smaller ones. This is manifested in the higher values of the activation energies obtained for larger average particles sizes. This observation is in close agreement to that obtained by Bucic-Kojic et al. [43] for the extraction of polyphenols from grape seeds. Thus, the influence of temperature on the extraction rate constant was more pronounce in larger particles size, than in the smaller ones. Finally, it could be seen that the activation energy values obtained for hyperbolic model at different particles size diameters, were higher than those obtained for pseudo second-order model. This could be attributed to the higher values of the rate constant obtained for hyperbolic model, compared to those obtained for pseudo second-order model [18].
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