Extraction modeling, kinetics, and thermodynamics of solvent extraction of Irvingia gabonensis kernel oil, for possible industrial application

The effects of the temperature, time, and particle size, in addition to the kinetics and thermodynamics parameters, of Irvingia gabonensis kernel oil (IGKO) yield were investigated. The highest oil yield of 68.80% (by weight) was obtained at 55°C, 150 minutes, and 0.5 mm. The evaluated physicochemical properties of IGKO indicated that viscosity, acidity, dielectric strength, flash, and pour points were 19.37 mm2 s−1, 5.18 mg KOH g−1, 25.83 kV, 285°C, and 17°C, respectively, suggesting its feasibility as transformer fluid upon further treatments. Between the pseudo‐second order and hyperbolic kinetic models studied, the former best describes the experimental data. ΔH, ΔS, and ΔG values of IGKO extraction at 0.5 mm and 328 K were, 251.81 kJ mol−1, 1.08 kJ mol−1, and −105.49 kJ mol−1, respectively, indicating the endothermic, irreversible, and spontaneous nature of the process. The kinetic model equations that describe the process were successfully developed for both models based on the process parameters.

Some of such oil seeds and nuts include but are not limited to Irvingia gabonensis (IG), soya bean, palm trees, Jatropha curcas, groundnut, Terminalia catappa L., and so on.
I gabonensis otherwise known as wild bush mango or "Ogbono" in south eastern part of Nigeria is a member of the Simarubaceae family. 4 It is an economic tree with its origin traced to most tropical forest of West and Central Africa. 5 In West Africa, I gabonensis is seen as the most important tree being encouraged for domestication. 6,7 Thus, it has attracted the attention of the World Agroforestry Centre (formerly the International Centre for Research in Agroforestry, ICRAF), together with its partners, thereby making it their choice tree in their agroforestry tree domestication programme. 8 Seasonally (between April to July), I gabonensis tree produces lots of edible fruits which are often not consumed, since the kernel component has greater value. 5 Nevertheless, there is greater utilization of the kernel, hence, it is a common practice to split the fruit into two using cutlass, in other to remove the split cotyledon (kernel) with knife, after which the flashy mesocarp is discarded, while the kernel is used for number of purposes. 9 Over the years, researches on I gabonensis kernel (IGK) majorly have been on its nutritional and medicinal applications, as well as the used of the milled kernels as condiment in soup as thickener, due to its rich fat and protein content. [10][11][12] Medicinally, it is used in body weight reduction of obese persons. 13 Little attention has been paid on the industrial applications. However, its kernels have local industrial application, such as its use in the production of local soap (due its high oil content). 14 Previously, researches have shown that IGK exhibits very high oil content which ranges between 60% and 69.76%. As such, provides desired raw material (oil) for the industrial utilization. 11,[15][16][17] Nevertheless, few researches have been conducted on the possible application of I gabonensis kernels oil (IGKO) industrially for biodiesel production. 15 Therefore, there is need to extend the utilization of IGKO in the production of transformer oil (TO), since to the best knowledge of the authors, no published work have been recorded in this direction. 18 Since the growing concern associated with the environmental impact, cost of power provision (TO), fluctuation in global crude oil price, as well as its nonrenewable nature, justifies the essence of the present study. Therefore, there is need to develop "environmentally acceptable" transformer fluid sourced from vegetable oils like IGKO.
Prior to the use of vegetable oil like IGKO for industrial applications, there is need for the oil to be extracted from the seeds/kernels. In other to achieve this goal, the choice of extraction method becomes very important. 19 Several extraction methods exist. Some of these methods are solvent extraction, sonication-assisted extraction; microwave-assisted extraction, supercritical fluid extraction, accelerated solvent extraction, and so on. 20 However, solvent extraction method using Soxhlet extractor was adopted in this study because of the simplicity, high oil yield, and oil quality associated with the method. 20,21 Solvent extraction method has been utilized severally for extraction of oil from fruits, seeds and nuts. Some of these fruits, seeds, and nuts are, Hazelnut (Corylus avellana L.), 22 Maclura pomifera (Rafin.) Schneider seed, 23 Prunus armeniaca L., 24 Sacha inchi (Plukenetia volubilis) seeds, 25 coconut waste, 26 T catappa L. kernel, 27 Colocynthis Vulgaris Shrad. 28 Similarly, I gabonensis is not left out, as solvent extraction methods have been utilized to extract oil from it. 11,15 In solvent extraction, it is important to note that the knowledge of the kinetic of oil extraction is of paramount importance because it assists in the determination of the highest oil yield within the studied time intervals, hence, the need to carry out extensive study on the kinetics of oil extraction from I gabonensis seed kernels. 29 Previously, researchers have carried out studies on the kinetics of oil extraction from seeds, and nuts. For instance, oil extraction kinetics have been applied to the extraction of oil from Colocynthis Vulgaris Shrad, 28 J curcas, 30 sunflower seeds, 31,32 fluted pumpkin seed, 33 coconut waste, 26 Neem seed (Azadirachta indica A. Juss) 34 and Prunus persica, 35 and from these studies it has been established that the ease of oil extraction from different seed/nuts varies. 36 Hence, there is need to study of the kinetics of oil extraction from IGKs, and to the best of knowledge of the authors', there has not been any published work in that regard.
It is worthy of note that during oil extraction process, the extraction rate (the rate at which equilibrium is attained) is influenced by factors such as, solute and solvent diffusion capacity, size, shape, internal structure of seeds particles (matrix), and the dissolution rate of the solvent on the oil soluble substances (solute). 21 In other words, the kinetics of IGKO extraction consists of the release of oil from porous or cellular matrices, into the solvent through the process of mass transfer mechanisms. This oil (solute) linked to the solid matrix of the kernel particles by either physical or chemical forces must be transported to the solvent phase by dissolution process. 37 For this to occur, three important steps have to be taken into consideration: (1) solvent penetration into the seed matrix (tissue), (2) intercellular miscella formation, and (3) extracted oil diffusion into the exterior miscella. 29 In other words, mathematical modeling of oil extraction kinetics from seeds and nuts is an activity of great importance. This is due to its economic benefits to industries. In the light of this and other benefits, it is necessary to develop models for extraction process based on the process parameters. In order to achieve this, the estimated process parameters, needs to be used in the development of the model. Such model must consider the phase behavior, state of equilibrium, solubility, diffusion, and dissolution of the process. 37,38 Several models have been used by researchers in the study of oil extraction kinetics process for oil seeds like, olive cake, 39 sunflower, 40,41 rapeseed (canola). 29 Although extraction kinetics has been extensively studied by many researchers, there is limited or no studies, available in the literature on oil extraction kinetics, thermodynamics and modeling of IGKO extraction, hence, the justification for the present study. Therefore, the objectives of this study were to study the influence of process parameters of temperature, time, and particle size on IGK oil yield, as well as to fit the obtained experimental data into two closely related extraction kinetic models (hyperbolic and pseudo-second order), so as to determine the model that best fit the experimental kinetic data. Also, the kinetic models of the extraction processes under different process parameters were established for predicting the extraction processes. In addition, the coefficient of determination (R 2 ) and for statistical error analysis functions (root mean square [RMS], the average relative error [ARE%], and the SE of estimation [SEE]), were used to study the fitting of the extraction kinetic models, to the experimentally obtained kinetics data. Furthermore, Arrhenius equation was used to evaluate the effect of extraction temperature on the kinetic models. The thermodynamic parameters of oil extraction from IGKs were also evaluated. Also, the physicochemical characterization of the IGKO was carried with the aim of evaluating its potential as base fluid for TO production. Finally, Fourier transform infrared spectroscopy (FTIR) was afterward used to ascertain the functional groups present in the IGKO.

Sample collection and preparation
IGKs were procured from Nkwo-Agu market, Umuaga in Udi Local Government Area, Enugu State, Nigeria. They were oven dried at temperature of 60 • C for 12 hours. Thereafter, the dried samples were milled using manual grinder. After which, sieved with different sieve sizes to obtain five different average particle sizes (0.5, 1.0, 1.5, 2.0, and 2.5 mm). The ground samples were sealed and stored until they were ready for use.

Solvent extraction experiment using Soxhlet extractor
Fifteen grams of dried milled IGK powder of a particular particle size were packed in a thimble of the Soxhlet extractor. The extractor was then filled with 150 mL of n-hexane. The experiments were performed at five different temperatures (35,40,45,50, and 55 • C) and at five different extraction times (30,60,90,120, and 150 minutes) for each particular average particle size (0.5, 1.0, 1.5, 2.0, and 2.5 mm). The extraction temperature was measured using an electronic thermometer (±0.1 • C, Hanna HI-9063), while the time was measured using a stop watch. The oil yield was calculated using AOAC method no. 920.85 42 using automatic Soxhlet apparatus (Soxtec 2050 FOSS, Denmark) in line with manufacturer manual guidelines. After each extraction process, the solvent was removed at 60 • C using rotary evaporator (model N-1000S-W, EYELA, Tokyo, Japan). The extraction done under every set of conditions was performed three times and the average value recorded. The oil yield of IGK was calculated using Equation (1). (1)

Kinetics
The analysis and design of extraction processes needs relevant kinetic data since it is the most important information to be used to understand the extraction process. In order to obtain these kinetic data, experiments were carried out at temperatures (35,40,45,50, and 55 • C) and at extraction times (30,60,90, 120, and 150 minutes) for each particular particle size (0.5, 1.0, 1.5, 2.0, and 2.5 mm) as stated earlier. Thereafter, the obtained experimental kinetics data were fitted to hyperbolic and pseudo-second order models.

Hyperbolic model
Hyperbolic model has been applied in food engineering science as Peleg's model in Equation (2).
where K 1 is the model rate constant, K 2 is the model capacity constant, c is the concentration of solute in the extraction solvent at any time (g L −1 ), and c 0 is the initial concentration of the solute in the sample particle (g m −3 ) and is usually equal to zero. Therefore, the extraction curves (% oil yield vs time) exhibit similar sharp as the sorption curves (moisture content vs time) proposed by Peleg. Hence, the feasibility of using the same mathematical model proposed by Peleg 43 to describe the kinetics of oil extraction from IGKs.
However, in this case, the extraction rate at the very beginning C 1 (min −1 ) and constant related to maximum extraction yield C 2 (min −1 ) were taken into consideration. 19 Thus, the hyperbolic model used to describe the oil extraction from IGKs is expressed as in Equation (3).
Recently, Equation (3) has been used to model the extraction of oil from T catappa L. kernel, 19 extraction of protopine from Fumaria officinalis L., 44 as well as in the extraction of total polyphenols from grapes. 45 Equation (3) results from a second-order rate law as could be seen in Equation (4). As such, it is important to state that Peleg's model 43 and pseudo-second order integrated rate law, Equation (4), are both hyperbolic Equations. 46 where Ck 2 t and C s k in Equation (4) are equivalent to C 1 and C 2 , respectively in Equation (3). k is the second-order extraction rate constant while C t and C s are the concentrations of oil in the solution at any time t and at saturation, respectively (g L −1 ). From Equation (3), it is important to state that the extraction is first-order at the very onset, and drops to zero-order in the latter phase of the extraction process. As such, when C 2 t ≪ 1, Equation (3) reduces to Equation (5).
And when t → s, the equilibrium is reached (y i = y e ), so Hence, C 1 is a constant that is related to the rate of extraction at the beginning, while the ratio C 1 /C 2 , the Peleg capacity constant which is related to the maximum of extraction yield, is the equilibrium concentration of the extracted oil.

Pseudo-second order model
The second-order rate law had been used over the years to model solvent extraction of a number of substances from plants, leaves, seeds, and nuts. 47,48 Extraction kinetic models that are based on a second-order rate law are usually used in both conventional and non-conventional extractions. 47,48 It provides a suitable illustration of solid-liquid extraction process, as such it was applied to describe the kinetics of oil extraction from IGKs. For a second-order rate law, the rate of dissolution of the oil contained in the solid to solution can be described by Equation (8).
where: dC s dt = the extraction rate (g L −1 min −1 ) k = the second-order extraction rate constant (L g −1 min −1 ) C s = the extraction capacity (concentration of oil at saturation in g L −1 ) C t =ȳ = the concentration of oil in the solution at any time (g L −1 ), t (min) By taking the initial and boundary condition t = 0 to t = t and C t = 0 to C t = C t , the integrated rate law for pseudo-second order extraction was obtained as Equation (9).
This can be further linearized in the form of Equation (11).
Thus, as t approaches 0, the initial extraction rate, h, is written as in Equation (12).
When Equation (10) is rearranged, the concentration of oil at any time can be obtained, Equation (13).
The initial extraction rate, h, the extraction capacity, C s and the pseudo-second order extraction, k, can be calculated experimentally by plotting t/C t vs t where by C s and k are determined from the intercept and slope of the linear plot, respectively.

Temperature effects
Arrhenius equation was used in evaluating the effect of temperature of extraction on the kinetic models. It was used to describe the relationship between extraction rate constant (k) and temperature (T). Equation (14) shows the Arrhenius equation.
Equation (15) can also be re-written in the form shown in Equation (16). When this is done, the unit of E a is written as (KJ mol −1 ).
where k 0 is the pre-exponential factor for extraction rate constant (L g −1 min −1 ), E a represents the activation energy of extraction (J mol −1 ). R is the ideal gas constant (8.314 J mol −1 K −1 ), T is the temperature of extraction in Kelvin (K). The pre-exponential factor, k 0 and the activation energy, E a can be determined using the natural logarithm of Equation (15). The plot of Ln (k) against 1000/T was used to calculate k 0 and E a .

Thermodynamic parameters
The thermodynamic parameters enthalpy change (ΔH) and entropy change (ΔS), for the extraction of oil from IGKs were calculated using Van't Hoff Equation (17).
Equation (17) can be re-written to include the Gibbs free energy change in the form of Equation (18).
The Gibbs free energy change was calculated using Equation (19).
where k is equilibrium constant, Y T is the yield of oil at temperature T, Y u is the percentage of the un-extracted oil, m L is amount of IGK in liquid at equilibrium temperature T, m s is amount of IGK in solid at equilibrium temperature T, R is gas constant (8.314 J (mol K) −1 ), while ΔH (kJ mol −1 ), ΔS (kJ mol −1 ), and ΔG (kJ mol −1 ) are enthalpy, entropy, and Gibbs free energy, respectively.

Statistical analysis
The degree at which the models studied statistically represent the data obtained experimentally were by the evaluation of correlation coefficient (R 2 ) using Equation (21), RMS, 19 ARE%, 49 and the SEE. 49 The error functions were computed using the expressions in Equations (22), (23), and (24) for RMS, ARE, and SEE, respectively.
where N is the number of experimental data points.ȳ cal andȳ exp are the calculated and experimental values, respectively, in Equation (22). Similarly, x and y are experimental and calculated values, respectively in Equations (23) and (24).  50 On the other hand, the viscosity and dielectric strength (DS) were measured according to ASTM D445, 51 and IEC 60156, 52 standard methods, respectively. The oil samples were tested three times and the average value taken.

FTIR analysis
The FTIR analysis of the IG oil sample was carried out using BUCK Scientific Infrared Spectrophotometer Model 530.

Effect of temperature
Temperature of extraction is one of the most essential parameter in extraction process. This is because of the very high sensitivity of the chemical constituent of plants, seeds, nuts, kernels, and leaves to heat. Solute solubility, as well as the diffusion coefficient, increases with the increase in the extraction temperature, as such, influencing the extraction process. 53 The effect of temperature on extraction rate of oil from IGK has been studied over temperatures range of 35 • C-55 • C, keeping the particle size constant at particle sizes (0.5, 1.0, 1.5, 2.0, and 2.5 mm), in each case ( Figure 1A-E). It could be observed that increasing the temperature from 35 • C to 55 • C, resulted in increase of the oil extraction yield, irrespective of the solute particle size. This could be attributed to the increase in diffusion of oil and decrease in its viscosity as the temperatures increased. 26,54 Similarly, the mass transfer coefficient of the process also increased with temperature thereby affecting diffusion. 26 Figure 1A-E shows the extraction of oil from IGK using Soxhlet extractor operated at maximum time of 150 minutes. It was seen that the rate of extraction was fast at the beginning of the process, and gradually reduces. This was due to the dissolution of free oil from the surface of the IGK when exposed to the fresh solvent, thereby leading to quick oil extraction, hence, a resultant rapid increase in the rate of extraction. Thus, there was easy solubility of the oil in the solvent, leading to fast extraction of the oil. 26 This is in agreement with the findings of Sulaiman et al 26 Figure 1A-E shows the effects of temperature on the oil yield of IGK at particle sizes of 0.5, 1.0, 1.5, 2.0, and 2.5 mm, respectively. Similar to the effect of time, oil yield increased with increase in temperature from 35 • C to 55 • C. Thus, increasing the temperature from 35 • C to 55 • C favors the extraction yield. This was also due to the increase in the diffusivity of the IGK oil and decrease in solvent viscosity at increased temperature. 26 From the plots in Figure 1A-E, it is evident that the extraction process was very fast in the beginning, between 30 and 90 minutes. Afterward, it gradually slowed down between 90 and 150 minutes. This phenomenon was due to internal diffusion. In this present study, like the previous works in the literature, the oil yield of IGK increased with temperature and time. The highest oil yield of 68.8% was obtained at 55 • C, 150 minutes, and 0.5 mm particle size.

Effect of particle size
Extraction rate increases as the particle size decreases. 26,56 Figure 2A,B shows the effect of particle size on the extraction of oil from IGK using hexane as solvent. The particle sizes considered in this study were 0.5, 1.0, 1.5, 2.0, and 2.5 mm. It could be seen that the oil yields obtained at 35 • C and 150 minutes, during extractions using these particles sizes were, 60.08%, 42.01%, 39.70%, 37.18%, and 35.4%, respectively. Hence, highest oil yield was obtained with the smallest particle size of 0.5 mm, and the least with the largest particle size of 2.5 mm. These findings were in line with the works of Sulaiman et al 26 and Huang et al, 57 for the extractions of solid coconut waste oil and Baizhu, respectively. The additional oil extracted from smaller particle size was attributed to the larger interfacial area of the solid present in them. Also, the solvent requires minimal distance to penetrate the solid particles in other to extract oil from it. In other words, larger interfacial area contributes to increase in pore diffusion between the solute (solid) and the solvent. On the other hand, larger particles have limited contact surface area, which causes more resistance to solvent penetration and oil diffusion. Hence, smaller quantity of oil would be transported from the inside of the larger particles to the surrounding solution. 26,55

Kinetic parameters
The values of the kinetic parameters and the individual error estimates for the nonlinear kinetics of hyperbolic and pseudo-second order models are presented in Table 1. It could be observed that majority of the kinetic parameters of C 1 and C s for hyperbolic and pseudo-second order models, respectively, increased with the increase in temperature. This could be attributed to the reason behind the increase in the oil yield with temperature rise. 19,44 Similarly, the kinetic parameters of C 2 and K also increased with increase in temperature as observed in Table 1. This increase in oil yield with temperature was as a result of the thermodynamic effect of oil solubilization inside the solid seed particles. 34 The     The principles for the determination of the model that best fit the experimental data for the two nonlinear models studied were R 2 , RMS, ARE % , and SEE. Conventionally, the higher the value of the R 2 and the lower the values of the error estimates (RMS, ARE%, and SEE), the better the model to fit the experimental data. 19 , for hyperbolic and pseudo-second order models, respectively. From these values, the average RMS values for both models were all less than ±5%, while those of SEE and ARE were all greater than 5%. Therefore, on the basis of the pseudo-second order model's values for RMS, ARE % , and SEE, which were lower than those of hyperbolic model, in addition to its higher R 2 value, pseudo-second order model, gave better fit to the experimental kinetic data when compared to the hyperbolic model. As such, pseudo-second order model was chosen as a better model for oil extraction from IGK. Similar result was reported by Agu et al 58 for the modeling of oil extraction from C vugaris Shrad seed using five different kinetic models.
Furthermore, it could be seen in Table 1 that the experimental and models' calculated oil yields for both hyperbolic and pseudo-second order models, were relatively close. For instance, the highest calculated models' oil yields for hyperbolic and pseudo-second order, were 68.63% and 68.88%, respectively. These values were obtained at 55 • (328 K), 0.5 mm particle size, and 150 minutes. As seen in Table 1, these values were very close to the 68.80% obtained experimentally. This is therefore, an indication that both models fit the extraction of oil from IGK using n-hexane as solvent. In other words, variation in particle size, extraction time and temperature, influenced the oil yield significantly, as evident form the obtained results. 19,26,59 Figure 3 shows the increase in IGK oil yield during the extraction of oil from the ground IGK using n-hexane at different particle sizes and extraction temperature. The extraction curve exhibits the shape of a typical Soxhlet and batch extraction of substances from plant materials as could be seen in previous works. 26,59 From the curve, it was observed that at a particular temperature, the IGK oil yield increased rapidly at the onset of the extraction, and gradually slows in the later stages. This initial extraction stage was characterized by an exceeding fast extraction rate. This fast extraction rate was due to the exposure of the milled IGK particles to fresh solvent which makes the solubilization of the free oil on the surface of the IGK very easy, as such, oil was quickly extracted. 26 However, during the later extraction stages, the oil diffused from the interior of the IGK particles and dissolve in the solvent. The oil yield of IGK was found to increase with rise in extraction temperature. This was attributed to better oil solubility at higher temperatures as could be seen in Figure 3A-E. 26 From Figure 3A-E, it is seen that at a particular temperature, higher oil yields were obtained at lower particle size due to bigger interfacial area of the kernel particles. 57

Temperature dependence and temperature effects
Similar to the effect of time, oil yield increased with increase in temperature from 35 • C to 55 • C as could be seen in Figure 1A-E, which was earlier presented. Therefore, increasing the temperature from 35 • C to 55 • C favors the extraction yield. This is because of the ease of penetration of the IGK matrix by the already energized n-hexane solvent molecules, 60 hence, substantiating the temperature dependence of oil extraction process. Since increase in temperature enhances softening of IGK, thus, improves the mass transfer coefficient of extraction, leading to improved extraction oil yield. 60,61 In other words, it is important to state at this point that rate of extraction at the very beginning, C 1 , and the constant related to maximum extraction yield, C 2 , were determined at different temperatures. They were dependent on temperature as seen in Figures 4 and 5, respectively. Similarly, the extraction capacity, C s , the second-order extraction rate constant, k, and the initial extraction rate, h, were also determined at different temperatures. They were also dependent on temperature as evident in Figures 6, 7, and 8, respectively.
Obviously, the initial extraction rate, C 1 , increased with temperature; so did the constant that is related to maximum extraction yield, C 2 . Similarly, the initial extraction rate, h, extraction capacity, C s , and the second-order rate constant, k, It is important to note that when the temperature was kept at 308 K, for particle size of 0.5 mm, the initial extraction rate C 1 for hyperbolic model was 6.21 minutes −1 . This value was slightly higher than 5.15 g L −1 minute −1 obtained for the initial extraction rate in pseudo-second order model. On the other hand, the constant related to maximum extraction yield C 2 , for hyperbolic model, at the same constant temperature of 308 K and 0.5 mm particle size was 0.085 minute −1 ; while the second-order rate constant, k was 0.0011 L g −1 minute −1 . However, the extraction capacity, C s , at the same temperature and particle size was 68.97 g L −1 . This value was very close to pseudo-second order model calculated oil yield, C t , 68.52 g L −1 . 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 k 0 and E a . This was done by plotting In k vs 1000/T for pseudo-second order kinetic model (Figure 9). The plot shows that the rate constant increases with the increases in temperature. However, a modified form of Arrhenius equation

F I G U R E 9 Arrhenius plots for the extraction of oil from
Irvingia gabonensis kernel at 0.5, 1.0, 1.5, 2.0 and 2.5 mm particle sizes for pseudo-second order kinetic model

F I G U R E 10 Arrhenius plots for the extraction of oil from
Irvingia gabonensis kernel at 0.5, 1.0, 1.5, 2.0, and 2.5 mm particle sizes for hyperbolic kinetic model  (25)) was used to determine the relationship between C 2 and T, the k 0 and E a for hyperbolic kinetic model. Like the pseudo-second order model, this was carried out by plotting In C 2 against 1000/T ( Figure 10).
Using Equations (14) and (25), activation energies were calculated from the slopes, and the values of temperature independent factors were calculated from the intercept, for the pseudo-second order ( Figure 9) and hyperbolic ( Figure 10) models, respectively. The relationships for the activation energy of extractions at 328 K and 2.5 mm particle size, modeled using pseudo-second order and hyperbolic models, are given by Equations (26) and (27), respectively.
Their respective R 2 values for pseudo-second order and hyperbolic models were 0.9582 and 0.9453. In both models, the activation energies were positive, 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 −1 , respectively; while the activation energies for hyperbolic model were 7.03, 11.03, 12.52, 16.63, and 18.79 kJ mol −1 , 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. They also show that 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 45 for the extraction of polyphenols from grape seeds. Thus, the influence of temperature on the extraction rate constant was more pronounced in larger particles size, than in the smaller ones. Finally, it was observed 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. 19

Modeling
From Equations (7) and (12), the initial extraction rates h and C 1 for pseudo-second order and hyperbolic models, respectively, could be modeled by plotting In h vs 1/T and In C 1 vs 1/T for the respective models. Figures 11 and 12 show their respective plots. From the plots, the relationships (28) and (29) were established at temperature of 328 K and 2.5 mm particle size. The 2.5 mm particle size was chosen because the least oil yield was obtained with it, as such, the successful development of a model with it, would ensure enhanced high oil recovery. This is due to the fact that temperature influence on the rate constant during extraction was more prominent in larger particles size, when compared to the smaller ones. In other words, the rate constants for the models were more temperature sensitive for larger particles size than for the smaller ones, as manifested in the higher values of the activation energies obtained for larger average particles sizes. 45 It could be seen that their initial extraction rates where close, although pseudo-second order model had the higher value than that of hyperbolic. This also reflects in the values obtained in Equations (28) and (29).
For the pseudo-second order model, Figure 13 shows the relationship that exists between extraction capacity, C s and temperature at particles size of 2.5 mm and the plot lead to Equation (30).

F I G U R E 11 Relationship between the initial extraction rate,
In h and temperature, for oil extraction from Irvingia gabonensis kernel at 2.5 mm particles size, using pseudo-second order model F I G U R E 12 Relationship between the initial extraction rate, In C 1 and temperature, for oil extraction from Irvingia gabonensis kernel at 2.5 mm particles size, using hyperbolic model

mm
By the combination of Equations (13), (28), and (30), the equation that describes the development of C t vs time and temperature model for pseudo-second order model can be given as Equation (31).
This equation shows the model for the evaluation of oil yield during solvent extraction of oil from IGK, for different temperature at any given time, using pseudo-second order model. This equation simply explains that the longer the time of extraction and the higher the extraction temperature are, the higher would be the concentration C t . Also, Figure 14 shows the relationship between the constant related to maximum extraction yield C 2 (min −1 ) and temperature at 2.5 mm particle size. The plot gave rise to Equation (32).
The combination of Equations (3), (29), and (32), the equation that describes the development ofȳ vs time and temperature model, for hyperbolic model can be written as Equation (33). This Equation (33) shows the model for the evaluation of oil yield during solvent extraction of oil from IGK, for different temperature at any given time, using hyperbolic model.
The models represented by Equations (31) and (33) for pseudo-second order and hyperbolic models, respectively, were compared with the experimental data. Figure 15 shows the comparison between the experimental and the models calculated IGK oil yield for different particle sizes and time, at temperature of 55 • C. From the plots ( Figure 15 and Table 1), good fit between the experimental and the calculated models' data was obtained for both pseudo-second order and hyperbolic models. This is an indication of the validity of the relationships.
Furthermore, Table 1 shows the results of the IGK oil yields obtained at different temperatures and particle size diameter at 150 minutes, and compared with the calculated models' oil yields values. The compared results of the experimental and calculated models' oil yields data indicate good agreement of the models with the experimental data, as evident from the low error analysis values (see Table 1).

Thermodynamic parameters
The values of the equilibrium constant and other thermodynamic parameters of IGK oil extraction are presented in Table 2, while the plots of In K vs 1/T for different particles sizes of 0.5, 1.0, 1.5, 2.0, and 2.5 mm, which were used in the determination of thermodynamics parameters (ΔH, ΔS, and ΔG) values are shown in Figure 16. For the thermodynamics of IGK oil extraction, the enthalpy values for the process were in the ranges of 251.81-569.28 kJ mol −1 , for the various particle sizes considered. The enthalpy values for the IGK oil extraction in the present study, were within the ranges (266.18-435.87 kJ mol −1 ) and (182.81-598.74 kJ mol −1 ) for C vugaris Shrad seed and T catappa kernel oil extractions, reported by Agu et al 58 and Menkiti et al, 60 respectively. However, the enthalpy result in this work was higher than (4-13.5 kJ mol −1 ) reported by Meziane and Kadi, 62 for olive cake oil. This difference in the enthalpy values was due to seeds morphology, as morphology often affects oil extraction. 58 The positive values of the enthalpy change are an indication that IGK oil extraction process was endothermic. 60 Similarly, as could be seen in Table 2, the entropy change of the process was also positive. The entropy values for IGK oil extraction ranged between 1.08 and 1.99 kJ mol −1 , with larger IGK particle sizes having higher entropy change values. The implication of this positive entropy values is that the process was irreversible in nature. 26,62 For the Gibbs free energy change values of the process, the entire values were negative. Hence, this is an indication of the feasibility and spontaneous nature of the process. From Table 2, it could be seen that the ΔG values were highly negative and in the range of −43.22 to −105.49 kJ mol −1 . This relatively high negative value(s) of ΔG is an indication that the extraction process was highly spontaneous. 26,56

Physicochemical properties of IGKO
The physicochemical characteristics of IGKO are shown in Table 3. In terms of IGK oil yield, it was found to be 68.80% (by weight) (see Table 3). This value was higher than oil yield values reported for C vulgaris Shrad, 28 cottonseed, 63 and soybean, 64 hence, an indication of its economic benefit and possible industrial application of IGKO. As a result, IGKO  could constitute an alternative source of oil for industrial application due to its relatively high oil yield. Also, Ekpe et al, 5 Matos et al, 12 and Zoué et al, 11 reported 67.33%, 73.83% (in mass), and 69.76% (in mass), respectively, for IGKO. These values were higher than that obtained in this work. The variation in the IGKO yield in this work when compared to those in the literature could be linked to the extraction methods and conditions, in addition to the type of solvent used. 60 Also, this difference in IGKO yield could also be attributed to factors like, geographical location, seed variety, and period of harvest. 60,65 From Table 3, it could be seen that the viscosity and acidity of IGKO were 19.37 mm 2 S −1 and 5.18 mg KOH g −1 , respectively. Hence, the value of viscosity in this work was found to be lower than 45 mm 2 S −1 for IGKO, but higher than 3.2 mm 2 S −1 for the IGK oil biodiesel, as reported by Bello et al. 15 For the acid value, the IGKO acid value (5.18 mg KOH g −1 ) in this work was found to be lower than 9.40 mg KOH g −1 , reported by Etong et al, 66 but higher than 4.67 mg KOH g −1 and 1.2 mg KOH g −1 , reported by Zoué et al 11 and Bello et al, 15 respectively. As already stated, the difference in the viscosity and acidity of IGKO in this work when compared to those reported elsewhere could be attributed to the breed of IGK used. 58,60 This difference in viscosities of IGKO in this work and those in the literatures could be due to differences in the extraction temperatures, since temperature significantly affects viscosity. 60 As seen in Table 3, the IV of the IGKO (98.75 g/I 2 /100 g oil) in this work was higher than the 32.43 g/I 2 /100 g oil and 4.17 g/I 2 /100 g oil reported by Zoué et al 11 and Yusuf et al, 67 respectively. The high IVs of the oil are an indication of the high level of unsaturation nature of the oils. As evident in Table 3, the density and moisture content of IGKO were 900 g cm −3 and 3.75 mg kg −1 , respectively. This density was lower than 930 g cm −3 , reported by Bello et al, 15 while the moisture content was higher than 0.023 reported by Matos et al. 12 The difference in the moisture content could be attributed to the initial moisture content of the IGK sample prior to the extraction process, as well as the method of extraction used. 68 Furthermore, the pour and flash points values of IGKO were 17 • C and 285 • C, respectively (see Table 3). These values were lower than 28 • C and 300 • C, respectively, reported by Bello et al 15 15 The high flash point of IGKO in this study is an indication of the safety handling nature of the oil; hence, it can easily be stored at room temperature. 69 The DS value of IGKO was 25.83 kV (Table 3). This value (25.83 kV) was found to be lower than those of soybean oil (39 kV) 70 and T catappa kernel oil (30.61 kV), 60 but slightly higher than that of palm kernel oil (25 kV). 69 Although this value is lower than the minimum requirement of 40-60 kV for conventional mineral TO, it is important to note that the DS value of IGKO can be improved with further purification and transesterification. 71

FTIR analysis of IGKO
The result in Figure 17 was analyzed and compared with known signature of identified materials in the FTIR library. 72 The peak center for IGKO sample ( Figure 17) at 900.7966 cm −1 is characteristics of P-F stretching, indicating the presence of phosphorus compounds. The peak at 1047.5 cm −1 is a characteristic of C-O stretching, indicating the presence of alcohol and phenol, which are oxygen-containing compounds, while the peak at 1369.536 cm −1 , is a characteristic of aromatic nitro compound NO 2 stretching, indicating the presence of nitrogen-containing compounds. The peaks centers at 1473.258 cm −1 and 1664.323 cm −1 , are characteristics of C=C stretching (indicating the presence of aromatic compounds) and nitrite N-O stretching (indicating the presence of nitrogen-containing compounds), respectively. In a similar way, the peak at 2020.284 cm −1 is characteristics of combination N-H stretching and combination O-H stretching, indicating the presence of organic compounds. The peaks at 2353.372 cm −1 and 2659.711 cm −1 are characteristics of phosphorus acid/ester P-H stretching and phosphorus acid/ester O-H stretching, respectively; that indicate presence of phosphorus compounds. Also, the peaks centered at 3002.457 cm −1 and 3289.143 cm −1 are characteristic of O-H stretching, indicating the presence of carboxylic acids, which are oxygen-containing compounds and water. Finally, the peak at 3772.403 cm −1 is beyond the infrared band of 3700 wavenumber (cm −1 ) for organic compounds as such, could not be identified.

CONCLUSION
In this work, it has been established that the process parameters (temperature, time, and particles size), influenced the IGK oil yield. This is due to the fact that increases in temperature and time resulted in the increase in the IGK oil yield, while smaller particle size gave higher IGK oil yield. The highest oil yield of 68.80% was obtained at 55 • C, 150 minutes, and 0.5 mm particle size. The physicochemical properties of the IGK oil indicated its potential for use as transformer fluid upon further treatment/modification, since no article in the literature have reported the DS of IGKO and its other TO properties. Thus, results of the authors' work on chemical modification of IGKO for possible use as transformer fluid will be published in no distant time. Of the two kinetic models studied, pseudo-second order gave better fitting to the experimental data than hyperbolic model; while the successful modeling of IGKO extraction using the aforementioned models, justifies the importance of this study. In addition, the activation energies determined using Arrhenius equation and modified form of Arrhenius equation for pseudo-second order and hyperbolic models, respectively; were all positive, an indication that oil extraction from IGK is an endothermic process. The obtained results indicate that irrespective of the model used, the rate constant k and the constant related to maximum extraction yield C 2 , for pseudo-second order and hyperbolic models, respectively, were temperature dependent; as they increased with temperature increase. Also, the constants k and C 2 for pseudo-second order and hyperbolic models, respectively, 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. Kinetic models equations were successfully developed to describe the IGK oil extraction processes under the different process parameters (temperature, time, and particle size) for both models. Finally, the ΔG, ΔS, and ΔH values obtained at the different particles sizes during the extraction indicated that the process was spontaneous, irreversible, and endothermic.