The source of triglycerides was chosen considering its composition, which should include relatively, high percentage of monounsaturated fatty acids (C16: 1, C18: 1), low proportion of polyunsaturated acids (C18: 2, C18: 3) and adequate content of saturated fatty acids (C16: 0, C18: 0) (Betiku and Adepoju, 2013). Because virgin soybean oil is one of the most abundant oils in Mexico, and its composition is a feasible candidate for this process (Table 1), it was selected for the transesterification experiments.

Methanol (CH₃OH) was chosen as the reactant for the transesterification, and the inorganic agent was sodium hydroxide (NaOH). These two compounds form sodium methoxide, following an inorganic reaction, prior to be added to the triglycerides source. An important factor in the transesterification process is the initial mixing of the methoxide and the triglyceride (TG) phases (Noureddini and Zhu, 1997). To avoid formation of two liquid layers, the triglyceride and the methoxide solution were preheated prior to mixing.

Mechanical agitation was continuously applied to maintain uniform concentration of the reactants. According to Darnoko and Cheryan (2000), between 120 and 600 rpm there is not limitation to mass transfer; therefore, agitation rate has been set at 250 rpm to avoid foams formation.

Theoretically, three molecules of methanol are required by one molecule of triglyceride to produce three molecules of FAME and one molecule of glycerol. As common practice, excess of alcohol is used in the production of FAME to ensure that the oil or fats will be completely converted to esters. The yield to the FAME is increased proportionally to the excess of alcohol to a maximum point; however, increasing the amount of alcohol beyond this value will not improve the yield, but it will improve the cost of alcohol recovery (Leung and Guo, 2006). Additionally, the molar ratio is associated with the type of catalyst used, the most used ratio in alkaline medium is 6:1 (Zhang et al., 2003). Regarding the reaction time, an excessive increase in the reaction time will result in a significant reduction in the yield of the product since some FAME will form soaps (Eevera et al., 2009). The temperature clearly influences performance; higher temperature can decrease the viscosity of the oil resulting in an increase in the reaction ratio and reduced reaction time. Simultaneously, if the temperature increases beyond the optimum, the yield will decrease, because at higher temperature the triglyceride saponification reaction accelerates. Finally, the yield to FAME increases proportionally to the amount of alkali supplied up to 1.5wt%; if more alkali is supplied, then saponification reactions are favored (Leung et al., 2006). One measure of the quality of the FAME produced is reflected by the density, which is compared with the EN14214 standard according to Bulla-Pereira (2014).

Samples of transesterification products were obtained following a 2⁴⁻¹ fractional factorial design of experiments, with five replicas at the central point to estimate standard deviation (Ogunnaike, 2009) (Table 2). Part of the methanol necessary as reactant was converted into sodium methoxide, measured as (CH₃OH/TG) ratio, and the rest was blended within the oil, measured as (methanol in situ, vol%). The former one is expected to form sodium methoxide using the sodium hydroxide recovered after esters formation. The pH was monitored as response variable, using a potentiometer; in addition, the yield to FAME, the (FAME/G) ratio and the final density of FAME were followed (Table 2).

In order to minimize the measurement error and the bias (Ogunnaike, 2009), experimental runs were randomized using the Minitab 17^TM statistical program; 13 experiments and duplicates were performed following the 2⁴⁻¹ design of experiments (see Appendix for the order of experiments). The kinetic constants were calculated following the mass balances up to the time the equilibrium is reached, as it is explained in Estimation of the Kinetics Rates’ Constants subsection.

The transesterification reactions of triglycerides follow a sequence of complex inorganic-organic mechanisms. This section describes the reactions involved, considering them ionic ones and with formation of an activated complex intermediate. All the samples were prepared in batch reactors (flasks) at the conditions proposed in the design of experiments.

The reaction pathways described for the inorganic formation of sodium methoxide (Eq. 1) and the sequential reaction steps of the triglyceride transesterification (Eqs. 2a, 4b), were modeled by classic mass balances for batch reactors (Eq. 5), developing nine kinetic Eqs. 6–14. d [ C ] d t = − f C ( C → ; k → ) (5) d [ H + ] d t = d [ C H 3 O − ] d t = ( k i n ′ − k i n ) ( [ C H 3 O − ] [ N a + ] [ H + ] [ O H − ] ) −                                       k 1 [ T G ] [ C H 3 O − ] [ N a + ] [ H + ] [ O H − ] + k 1 ′ [ D G ∗ ] [ E 1 ] −                                       k 3 [ D G ] [ C H 3 O − ] [ N a + ] [ H + ] [ O H − ] + k 3 ′ [ M G ∗ ] [ E 2 ] −                                       k 5 [ M G ] [ C H 3 O − ] [ N a + ] [ H + ] [ O H − ] + k 5 ′ [ G ∗ ] [ E 3 ] (6) d [ D G ∗ ] d t = k 1 [ T G ] [ C H 3 O − ] [ N a + ] [ H + ] [ O H − ] − k 1 ′ [ D G ∗ ] [ E 1 ] − k 2 [ D G ∗ ] (7) d [ M G ∗ ] d t = k 3 [ D G ] [ C H 3 O − ] [ N a + ] [ H + ] [ O H − ] − k 3 ′ [ M G ∗ ] [ E 2 ] − k 4 [ M G ∗ ] (8) d [ G ∗ ] d t = k 5 [ M G ] [ C H 3 O − ] [ N a + ] [ H + ] [ O H − ] − k 5 ′ [ G ∗ ] [ E 3 ] − k 6 [ G ∗ ] (9) d [ G ] d t = k 6 [ G ∗ ] (10) d E 2 d t = d [ D G ∗ ] d t = k 1 [ D G ∗ ] [ N a + ] [ H + ] [ O H − ] − k 1 ′ [ E 1 ] [ D G ∗ ] (11) d E 1 d t = d [ M G ∗ ] d t = k 3 [ M G ∗ ] [ N a + ] [ H + ] [ O H − ] − k 3 ′ [ E 2 ] [ M G ∗ ] (12) d E 3 d t = d [ G ∗ ] d t = k 5 [ G ∗ ] [ N a + ] [ H + ] [ O H − ] − k 5 ′ [ E 3 ] [ G ∗ ] (13) d [ N a + ] d t = d [ O H − ] d t = ( k i n ′ − k i n ) ( [ C H 3 O − ] [ N a + ] [ H + ] [ O H − ] ) − k 1 [ T G ] [ C H 3 O − ] [ N a + ] [ H + ] [ O H − ] + k 1 ′ [ D G ∗ ] [ E 1 ] + k 2 [ D G ∗ ] − k 3 [ D G ] [ C H 3 O − ] [ N a + ] [ H + ] [ O H − ] − k 3 ′ [ M G ∗ ] [ E 2 ] + k 4 [ M G ∗ ] − k 5 [ M G ] [ C H 3 O − ] [ N a + ] [ H + ] [ O H − ] + k 5 ′ [ G ∗ ] [ E 3 ] + k 6 [ G ∗ ]   (14) Because of instantaneous lifetime of the transition compounds (DG*, MG*, G*), their concentrations, are assumed to be pseudo-stationary. Thus, the instantaneous concentrations of these compounds (Eqs. 15–17) are obtained from their balances (Eqs. 7–9) in terms of the balance of hydrogen ions (Eq. 6). [ D G ∗ ] = k 1 [ T G ] 1 ( k i n ′ − k i n ) k 1 ′ [ E 1 ] d [ H + ] d t (15) [ M G ∗ ] = k 3 [ D G ] 1 ( k i n ′ − k i n ) k 3 ′ [ E 2 ] d [ H + ] d t (16) [ G ∗ ] = k 5 [ M G ] 1 ( k i n ′ − k i n ) k 5 ′ [ E 3 ] d [ H + ] d t (17)

Substituting the concentrations of the intermedium compounds, kinetic expressions in terms of the change of pH along the reaction time are obtained (Eqs. 18–20). d [ T G ] d t = k 1 [ T G ] 1 ( k i n ′ − k i n ) d [ H + ] d t − k 1 ′ [ D G ∗ ] [ E 1 ] (18) d [ D G ] d t = k 3 [ D G ] 1 ( k i n ′ − k i n ) d [ H + ] d t − k 3 ′ [ M G ∗ ] [ E 2 ] (19) d [ M G ] d t = k 5 [ M G ] 1 ( k i n ′ − k i n ) d [ H + ] d t − k 5 ′ [ G ∗ ] [ E 3 ] (20)

This model (Eqs. 6, 18–20) is used to estimate the kinetic constants of the proposed pathways of transesterification of virgin soybean oil (TG) with sodium methoxide; the latter one formed by the reaction of methanol and sodium hydroxide. It has to be noticed that, taking the point of view from Process Engineering, the mass balances necessary to be included are only the ones independent by stoichiometry. Checking the reactions, it is possible to identify one independent inorganic reaction (Eq. 6), and three independent organic ones. Therefore, after simplification of the mass balance equations for the organic reactions, we need only three of them, which are in Eqs. 18–20.

The design of experiments described in Selection of Reactants and Operating Conditions Section was run in random order (Appendix Table A1) obtaining the results in Table 3. The ANOVA (Appendix Table A2) strongly suggests that there is influence of the factors on the measured responses (p-value < 0.00001, for all factors-response combinations); therefore, deeper analysis was performed.

In order to follow the behavior of the sodium hydroxide, the pH of the first wash of the esters blend was measured (Figure 1); in all the experiments its values were observed between 10 and 12. This value can be used to calculate the concentration of pOH⁻ ions, which is consequence of the amount of alkaline species absorbed by the FAME. Besides, this amount of species is a reply to two experimental factors, mainly, methanol in situ and initial amount of supply of sodium hydroxide to prepare the sodium methoxide extra situ.

The statistical analysis of this information was carried out using a Pareto diagram (Figure 2), which shows the effect of the different combinations of factors in the pH response. As it can be seen, in addition to NaOH, the combination with the temperature at which transesterification is carried out and the methanol in situ (directly fed to the reaction), influence the amount of wash water necessary to neutralize the FAME. This track was particularly useful, since for this design of experiments it was proposed to add excess of methanol in situ and, meanwhile, decrease the amount of methoxide prepared extra situ; so that the proposed reaction mechanism could be verified; here a chain reaction has been carried out.

By measuring the (FAME/G) product ratio, it was found that its value (6/1) agrees with the literature (Mumatz et al., 2017); therefore, in this range of operating conditions good FAME production is achieved (Figure 3). Following the present design of experiments, the highest yield was obtained in experiment 9, which is the maximum point.

The next response measured was the final density of FAME, which is affected by excess of methanol fed, and by the amount of NaOH (Figure 4). This behavior was expected since methanol is fed in situ without reacting completely, and there is not a removal stage as the distillation proposed in the literature (Zhang et al., 2003); therefore, some excess of methanol could be present within the mixture of methyl esters.

After this analysis, it is noted that the temperature interval used was too short to exhibit any significant effect. Therefore, it was considered that the results obtained do not correspond to different temperatures, but are duplicate experiments at the same average temperature of 48°C. With this, the ability to estimate activation energies is lost, but precision in mass balances is increased.

Results from the 12 experiments described previously were introduced to the mass balances (22–35), here molar reaction advances ( ε _k ) were evaluated for each chemical entity involved in the process. The aim of this section is to estimate the amounts of reaction, in terms of moles, preserving the convergence in mass balances.

Taking as example the experiment with the highest yield to FAME (Exp. 9, Figure 3), solving mass balances in Table 4, and using the global conversion of the triglyceride, ξ _TG= 0.98, the first reaction advance is calculated (Eq. 36). By stoichiometry, the five advances related to esterification reactions are obtained (Eq. 37). These advances are used to solve mol balances (26–35). ε 1 = T G 0 ∗ ξ T G = 0.0500     m o l / b a t c h ∗ 0.98 = 0.0490     m o l / b a t c h (36) ε 1 = ε 2 = ε 3 = ε 4 = ε 5 = ε 6 = 0.0490 m o l / b a t c h (37)

Progress of sodium methoxide reaction can be followed by measuring pH (Eq. 38) and using the definition of measure of acidity (Eq. 39); this equivalence allows to determinate (OH⁻) concentration. p H = − l o g [ H + ] (38) p H + p O H = 14 (39)

Experimental value of the pH, measured prior to the washing step, was 10.08. Applying (Eq. 39), the value pOH = 3.92 is calculated (Eq. 40) and converted into concentration (Eq. 41). Taking molecular weight of sodium hydroxide, (MW_NaOH= 40 g/mol), and molar fraction of hydroxide in the molecule, (χOH= 17/40 = 0.425), it is straightforward to calculate sodium hydroxide concentration in the end products (Eq. 42). Once this value is obtained, the balance of sodium hydroxide can be solved from (24), and the advance of the inorganic reaction is calculated (Eq. 43). p O H = 14 – 9.27 = 4.73 (40) [ O H − ] = 10 − 4.73 = 1.86 x 10   − 5 M (41) [ N a O H ] o u t p u t = 1.86 x 10 − 5 M / 0.425 = 4.37 x 10 − 5 M (42) ε i n = 6.5 x 10 − 3   m o l / b a t c h + ε 2 + ε 4 + ε 6 − 4.37 x 10 − 5   m o l / b a t c h = 0.1564   m o l / b a t c h (43)

Then, the mass balance for sodium methoxide (30) is solved and thus the value of balance of water (25) is obtained (Eq. 44); global results are shown in (Table 5, 22a-35). C H 3 O − N a + = H 2 O = ε i n − ε 1 − ε 3 − ε 5 = 0.0064   m o l   /   b a t c h (44)

After the advances of reaction were calculated, the theoretical value obtained for the mass balance for glycerol was compared with the experimental weight of the heavier phase, which mainly consists of glycerol. Relative convergence was higher than 99%, excepting in case of the experiment 11; this one was discarded.

Once these data are known, the problem of the theoretical project focuses on estimating the value of the kinetic rates’ constants for this operating region. According to Leal et al. (1991) and the expression of Gibbs free energy (Eq. 45a), the value for the reference temperature of 25°C is obtained. Using the actual value of free energy at the reference point (Eq. 45b), the equilibrium constant for the reaction of sodium methoxide can be estimated (Eq. 46). Production of sodium methoxide is carried out at approximately 48°C (321.15 K); this temperature was substituted in (Eq. 46) to obtain the value of the equilibrium constant of the inorganic reaction (Eq. 47). Δ G 0 = Δ H 0 − T Δ S 0 (45a) Δ G 0 = 4780   J m o l − ( 298   K ) ( − 0.011   J m o l ⋅ K ) = 4783.278   J m o l (45b) K i n = k i n k i n ′ = exp ( − Δ G 0 R g   T ) (46) K i n = exp ( − 4783.278   J m o l ( 8.314   J m o l ⋅ K )   ( 321.15   K ) ) =   [ C H 3 O − ] [ N a + ] ⋅ [ H + ] [ O H − ] [ C H 3 O − ] [ H + ] ⋅ [ N a + ] [ O H − ]   = 0.1667 (47)

In order to continue with the estimation of kinetic constants for the simulation of the FAME production process, and based on the reaction schemes presented above, minimization using the Powell’s method (included in Matlab^TM) of the objective function (Eqs. 48a,b) was performed under the only constraint: Kinetic rate constants exhibit positive values (Eq. 48c). min { ∑ i = 1 n Exp [ ( y FAMEexp − y FAMEpred ) i 2 + ( ( FAME G ) exp − ( FAME G ) pred ) i 2 ] } (48a) ↔ min [ ∑ i = 1 n Exp [ Ψ FAME + Ψ Ratio ] ] (48b)

Subjected to: k 1 , … , k 6 ≥ 0 (48c) Here Ψ F A M E and Ψ R a t i o are the quadratic errors of estimation of yield to FAME and the ratio of the yield to products, respectively.

The estimation of the kinetic rate constants depends on precision of measurement of the yields to FAME and the ratio (Figure 2) of products. Applying the usual methodology of error propagation (Eq. 49), in order to estimate confidence intervals (α ≥ 95%), the propagated errors of the six kinetic rate constants were calculated (Table 4). Δ k m = | ( ∂ k ∂ Ψ F A M E ) m | Δ Ψ F A M E + | ( ∂ k ∂ Ψ R a t i o ) m | Δ Ψ R a t i o   (49)

The mean values of the kinetic rate constants (Table 5) were used to simulate the evolution of triglyceride consumption and FAME production for all experiments; an example is given for the best experiment (# 9, Figure 5).

As consequence of the good conversion of TG, DG and MG, production of FAME (Figure 5) shows continuous growing trend; for this experiment calculation of the yield to FAME (Eq. 50) agrees with the results observed for TG conversion and with data reported in literature (Vicente et al., 2004; Marchetti et al., 2007). y FAME = ∑ esters T G 0  *  100   ⇒ y FAME = 0.0494219 0.0500000 * 100 = 98.84  wt . % (50)

In fact, for this experiment conversion of TG follows first order trend (Figure 5); also, conversion is practically complete. The same is true for DG and MG, which is confirmed by the yield to G (Eq. 51). G pred G exp = 0.01847 0.02189 = 0.8438 (51)

The kinetic constants estimated (Table 4) were, also, used to simulate the evolution of sodium methoxide (Figure 6). It is interesting to see that formation of sodium hydroxide (inorganic reaction in 1) is taking place along the whole experimental time; therefore, sodium hydroxide is playing the role of reactant rather than acting as catalyst. Moreover, it is not necessary to “prepare” a batch of the basic solution prior to addition to the triglyceride, because the inorganic reaction is able to take place simultaneously to the transesterification one, as soon as sodium hydroxide is recovered and there is methanol available in the reacting mixture (see reaction mechanism, Eqs. 2–4).

Finally, it was found that there is inverse correlation between the yield to FAME and the amount of water necessary for the purification of the FAME; the impact of each factor on the yield to FAME (Eq. 52), shows that the amount of sodium hydroxide added at the beginning is the only significant one. H 2 O w a s h ( m l ) = 462.7 − 9.167 ∗ T e m p e r a t u r e + 33.89 ∗ M e t h a n o l + 53.57 ∗ N a O H − 24.25 ∗ M e t h a n o l e x c − 0.5556 ∗ T e m p e r a t u r e ∗ M e t h a n o l + 0.5000 ∗ T e m p e r a t u r e ∗ M e t h a n o l e x c − 22.50 ∗ C t P t (52)