The seeds of Moringa oleifera were harvested at a facility located at (23° 7′ 0″ N, 82° 23′ 0″ W) in Havana, Cuba. They were globular, three-winged seeds and covered with a thick blackish seed coat. The weight of 100 seeds of Moringa oleifera was found to be 23.90 g ± 0.38. The seed contained 72.74% ± 1.26 kernels and 27.26% ± 1.21 husks. According to literature reports (Salaheldeen et al., 2014; Díaz et al., 2017), the contents and properties of Moringa seeds depend on species and environmental conditions. After removing the husk, the kernels were milled and dried.

The size-reduced particles were analyzed using the Cuban standard (NC, 2008). A representative sample of 14 g of milled kernels was taken, according to the standard report. The average diameter of the particle conglomerate (Dp) in m was determined from Equation 1 (Rosabal and Garcel, 2006).

Where Δxi is the retained fraction in the bottom sieve of each pair and Dpi the average size of the particles retained in the bottom sieve of each pair.

The milled kernels were dried at 55 °C in a stove (model DHG 9146A, Shanghai Huitai Instrument Manufacturing CO., LTD, Shanghai, China). Removing water during this operation made the oil extraction more efficient, since drying favored the breakdown of the water-oil emulsions, guaranteeing the oil easily dissolved in the organic solvent. The moisture determination was carried out by weighing the difference using an analytical balance, model SARTORIUS BS 124S. The drying curve was obtained to define the operation time. Weighing was carried out every half hour until reaching constant weight.

A lab scale Soxhlet apparatus fitted with a 1-L round-bottom flask and a condenser was used to extract the oil from the kernels. About 10 g of dried sample were used for each extraction. Hexane (purity > 98%, specific gravity: 0.659, refractive index: 1.375) was added as solvent according to a solvent-kernels ratio (6:1) (mass: volume ratio) in accordance with previous studies (Díaz et al., 2017; Tabio et al., 2018). An evaporation process was applied to the mixture of oil and solvent in order to remove one from the other. The process was developed in a rotary evaporator (IKA-WERK HB 4 basic, Germany). The extracted oil yield was expressed in mass percentage, which is defined as the weight of the oil extracted over the weight of the sample.

The extraction kinetics of the Moringa oil were studied in the Soxhlet using hexane at 69 °C (boiling temperature) as solvent. During the experimental runs, the change in time of the extracted oil percentage was obtained. The measurements were carried out in triplicate.

The calculation method to apply the solution to Fick´s diffusion second law, using natural logarithms was followed by a simple regression analysis of the solute’s adimensional concentration with respect to time.

Oilseeds have a cellular structure. Therefore, the internal diffusion within the solid structure is the determining mechanism of the extraction rate. Therefore, this type of phenomenon is represented by Fick’s diffusion second law for a non-steady state (Varzakas et al., 2005). To obtain the effective diffusivity coefficient, the kinetics of the extractive process must be experimentally determined and the aforementioned law must be used. Fick’s diffusion second law for non-steady state was used to assess whether the extraction of Moringa oleifera oil using Soxhlet was controlled by internal resistance (diffusive transport). To mathematically model the diffusion according to this law, the following conditions must be taken into account:

The effective diffusivity of the solid is constant throughout the sphere and the differential equation that describes the diffusion process in spherical coordinates can be obtained according to Hinnes et al. (1987) (Equation 2).

Where ∂ c ∂ t is the cumulative term, ∂ 2 c ∂ r 2 + 2 ∂ c r ∂ r the diffusive term, r the sphere radius in m, t is the time of diffusion in s and Def are the effective diffusivity coefficients of solute in m²/s.

To model the mass transfer, an initial and boundary conditions are required:

For t = 0 c = c₀ (c₀: initial solute concentration (g/mL))

For t > 0 c = 0 at r = a, a: Radius on the surface of the sphere

The differential equation mathematics solution can be determined from Equation 3:

Because triglycerides have different molecular structures and masses, it is better to measure the amount of oil with respect to the solid mass. Therefore, the concentration ratio c c 0 is transformed into the relationship between the remaining oil mass (q) and the initial mass of oil (q₀), obtaining the q q 0 ratio.

For longer times, series converge and, consequently, for most practical calculations, a few points are sufficient. Indeed, all the series points, except the first one, are negligible. The solution of Fick’s diffusion second law is reduced to Equation 4, where the amount of solute remaining in the solid at time t with respect to the initial amount is:

Where q q 0 is the relationship between the remaining and initial quantity of oil, q the amount of remaining oil in g at time t, q₀ the initial amount of oil in g and π is the mathematical constant 3.1418.

Equation 4 was linearized by applying natural logarithms to obtain Equation 5. The fit models were tested for fit quality. For simple regression analysis, the Statgraphics Centurion statistical program, version XV was used. The slope of the line was used to estimate the effective diffusion coefficient. An initial oil content (c₀) of 45% was considered since it is the maximum reported by Abdulkareem et al. (2011); Ayerza (2011) and Efeovbokhan et al. (2015) for Moringa oleifera. The equivalent radius of the sphere was taken as the average radius of the particle conglomerate obtained by the sieving operation.

The sieving analysis served to obtain the distribution of the present size of particles in the Moringa oleifera seeds kernels after the milling process (Table 1 and Figure 1).

A varied distribution of particle sizes can be observed in Figure 1. The largest amount of milled seeds kernels (56.09%) began to cluster in smaller diameter sieves (between 1.0·10^-3 and 0.5·10^-3 m). The average diameter of the particle conglomerate (determined by Equation 1) was 0.602·10^-3 m. This result corroborates the fact that with this milling method it was possible to further decrease the particle size and therefore the oil extraction process was expected to be favored.

It has been reported (Abdulkareem et al., 2011) that particle size can directly affect the rate and yield of the extraction process. Generally, at smaller particle sizes the contact area between the solvent and the solid increases, which caused the oil transfer rate to increase (Abdulkareem et al., 2011).

Abdulkareen et al. (2011), recommend a small size range (between 0.7·10^-3 and 0.5·10^-3 m) so that each particle required approximately the same extraction time. The results obtained in the present work are in agreement with this aspect, the average diameter of the milled seeds kernels being included in the aforementioned interval. The diffusion coefficient for internal mass transport depends of the solid structure. To reduce the particle size of oilseeds the cellular wall is broken, thus increasing the access to the lipid content. It means that oil could be extracted with a higher extraction rate.

The initial moisture content of Moringa oleifera seeds kernels was 4.61%. This result was found to be lower than that of Moringa oleifera from Pakistan (5.70%) and Yucatan (5.84%) (Anwar and Bhanger, 2003; Ortiz et al., 2012). However, the above-mentioned moisture reports are related to the whole seed (husk and kernel). For this reason, the husk moisture content of Moringa oleifera from Cuba should be taken into account to make a precise comparison. Recently a husk moisture content of 6.25% was reported for Moringa oleifera grown in Cuba (Pfeil et al., 2020). Therefore, a seed moisture content of 5.05% could be obtained by mass balance using this last value.

The drying curve serves to determine the moisture content of the kernels at any instant in time. The results are shown in Figure 2.

The moisture content of Moringa oleifera seed kernels decreased from 4.61 to 0.22% during the drying process as can be seen in Figure 2. From the aforementioned, a drying time of two hours was established because the moisture decreased to percentages lower than 0.3%.

The results for kinetic extraction are shown in Figure 3.

An oil extraction behavior over time is observed in Figure 3, which indicated, as expected, that oil yield is time dependent. The amount of extracted oil is practically constant for extraction times of six hours and longer, which was demonstrated with the asymptotic behavior of the curve. The high rate of extraction observed at the early stages may be due to the high solubility of the oil in the freshly charged solvent and the high concentration of the oil at the solid surface. The freshly charged solvents (lean oil) created a positive gradient or the needed driving force that resulted in a higher mass transfer rate of oil in to the extracting solvents. After the optimum point had been reached, the extracting solvents (oil rich) gave rise to lower driving force or slower extraction rates (Efeovbokhan et al., 2015). The experimental kinetics of the extractive process were required to determine the effective diffusivity coefficient.

It should be expected that internal mass transport will be the controlling phase during the extraction process of Moringa oleifera oil using Soxhlet apparatus because of the cellular structure of abovementioned oilseed. This phenomenon was verified through Fick’s diffusion second law in a non-steady state, taking into account the kinetics of the extraction process as shown in Figure 3. Fick’s mathematic solution was linearized by applying the natural logarithms in Equation 5.

The established initial oil content of Moringa oleifera kernels and sample mass were 45% and 10 g, respectively. The remaining oil content was calculated as a mass fraction per unit of initial oil mass ( q q 0 ). Table 2 shows the experimental remaining oil masses determined from kinetics extraction with hexane.

A linear fit was corroborated for the experimental data when ln (q / q₀) is plotted against time (Figure 4).

The model obtained is represented by Equation (6):

The tests of model fit were verified to validate the quality. The analysis of variance (ANOVA) of the model and the statistical significance of its parameters are presented in Tables 3 and 4.

Since the P-value in the ANOVA table is less than 0.05, there is a statistically significant relationship between ln (q/q₀) and time at the 95.0% confidence level. The R-Squared statistic indicates that the model as fitted explains 97.9506% of the variability in ln (q/q₀). The correlation coefficient equals -0.9897, indicating a relatively strong relationship between the variables.

The adjusted R² statistic, which is more suitable for comparing models with different numbers of independent variables, was 97.65%. The high value of the determination coefficient R² justified a good correlation between the parameter and for this reason, the model is suitable for a satisfactory representation of the real process.

The Durbin-Watson (DW) statistic tests the residuals to determine if there is any significant correlation based on the order in which they occur in the data. Since the P-value (0.0560) is greater than 5.0%, there is no indication of serial autocorrelation in the residuals at the 5.0% significance level. The residuals, which are the difference between the observed values and the predicted values, should lie on a straight line in the normal probability plot.

Therefore, these three suppositions imply that the suggested model (Equation 6) is adequate and can be used to describe the oil extraction process of Moringa oleifera in the Soxhlet extractor, responding to the diffusive phenomenon according to Fick’s diffusion second law in a non-steady state.

Three oil extraction experiments were developed according to the description in the materials and methods section. For any extraction process, it is possible to obtain a high quantity of the product in the first hours, with a time from which the effectiveness of the extraction is very low. The difference between the overall maximum at 12 h and the value at 6 h, as obtained from the respective seed samples, extracting solvent and times was only nominal (see Figure 3). That is the main reason for fixing six hours as the extraction time for the experiments. This would not affect the oil extraction percentage and will be related to the energy consumption during the operation with the Soxhlet apparatus. This means that if the cost of solvent extraction is considered, the optimum conditions must be used. The nominal oil yield after this optimum point bears a negative cost on the overall extraction cost (Efeovbokhan et al., 2015).

The experimental extraction oil percentages of Moringa oleifera from Cuba were 40.12, 40.56 and 42.48%. These results were higher than those reported for Moringa oleifera Wild (NWFP, Pakistan) 34.80% and Moringa oleifera variety Mbololo (Kenya) 35.70%. However, the extracted oil content was comparable to Moringa oleifera Sindh (Pakistan) 40.39% (Anwar and Rashid, 2007). The differences found might be attributed to the diversity of natural habitats and agroclimatic constraints (Efeovbokhan et al., 2015).

In addition, the extraction percentage obtained from the phenomenological model (41.09%) was compared to the experimental results. The relative error was lower than 5% in all cases, which validated the degree of accuracy of the phenomenological model to predict the extraction percentage of Moringa oleifera oil in Soxhlet.

The effective diffusivity (Def) was obtained from the slope of the fitted model (Equation 6). The equivalent sphere radius was 0.301·10^-3 m as calculated from the average diameter of the particle conglomerate (0.602·10^-3 m). The effective diffusivity for Moringa oleifera oil through the kernels was obtained for the first time. The value of 0.685·10^-12 m²/s, fully matched with reports of effective diffusion coefficient (between 10^-12 to 10^-10 m²/s) for other solids (Doulia and Gekas, 2001; Cacace and Mazza, 2003; Varzakas et al., 2005). Diffusion within a solid structure is a more complex phenomenon than inside liquids and gases. For this reason, lower diffusion coefficient and slower mass transfer are to be expected.