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Kinetic study of the transesterification reaction by artificial neural networks and parametric particle swarm optimization
Fuel 267 (2020) 117221
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Full Length Article
Kinetic study of the transesterification reaction by artificial neural networks and parametric particle swarm optimization
T
Diego Galvana, Hágata Cremascoa, Ana Carolina Gomes Mantovanib, Evandro Bonac, ⁎ Mário Killnera, Dionisio Borsatoa, a
Chemistry Department, State University of Londrina (UEL), 86.057 970 Londrina, PR, Brazil Physics Department, State University of Londrina (UEL), 86.057 970 Londrina, PR, Brazil c Programa de Pós-Graduação em Tecnologia de Alimentos (PPGTA), Universidade Tecnológica Federal do Paraná (CM), Câmpus Campo Mourão, 87.301 899 Campo Mourão, Paraná, Brazil b
GRAPHICAL ABSTRACT
ARTICLE INFO
ABSTRACT
Keywords: PSO Self-organizing maps (SOM) Runge-Kutta Rate constants Soybean oil Methanolysis
In this work, the mechanism of soybean oil transesterification reaction was investigated by Particle Swarm Optimization (PSO), and the reaction progress was monitored by Artificial Neural Networks (ANN) using SelfOrganizing Maps (SOM). The PSO was used to adjust the values of the reaction rate constants (kn) in the four proposed mechanisms/models approaches. The mean error values obtained by the PSO method showed a good agreement between the experimental and simulated data for all evaluated approaches, ranging from 4.3 to 8.5%. The results suggest that under the evaluated conditions reverse reactions may be disregarded, following a secondorder kinetic model for the three stepwise reactions. The SOM proved to be an efficient tool for exploring data obtained during the transesterification process, allowed to organize the data in the form of clusters presenting in the weight maps the relationship between the monitored variables during the progress of the transesterification
⁎
Corresponding author. E-mail address: [email protected] (D. Borsato).
https://doi.org/10.1016/j.fuel.2020.117221 Received 8 November 2019; Received in revised form 24 January 2020; Accepted 26 January 2020 0016-2361/ © 2020 Elsevier Ltd. All rights reserved.
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reaction. The PSO and ANN type SOM has been shown efficient tools for investigating the kinetics of the transesterification reaction, and they offer good perspectives for further kinetic studies of other types of industrial processes.
applied in pattern recognition and classification. The ANNs are mathematical models whose architecture was inspired by biological neural networks and consist of simple processing units that store empirical knowledge through a learning process [25]. The Self Organizing Maps (SOM) is a type of ANN in which the intrinsic relationships between samples or variables in a dataset are represented using an unsupervised learning method [26]. These sets of techniques can be applied in a wide range of areas, including food process control [27,28] and fuels [29,30]. In this work, four different kinetic models and mechanisms approach for the soybean oil transesterification reaction was evaluated. The optimization of the kinetic parameters for each case was obtained by the PSO method. The monitoring of the reaction progress was also performed using the SOM-type of ANN, based on experimental and modeled results.
1. Introduction The demand for fossil fuels in recent decades has been increasing, causing resource scarcity and negative environmental impacts due to overuse. Renewable and less polluting substitute fuels have been increasingly researched. Among these alternatives, biodiesel is gaining huge space [1]. The industrial production of biodiesel is well known worldwide and has a well-developed technology, making the process reasonably inexpensive when compared to other methods of renewable energy production. In addition, there are several alternatives to raw materials that can minimize the costs of production [1,2]. Although the process is well known, the kinetic study of the transesterification reaction is still necessary due to its wide industrial use. Regardless of the type of process, operating conditions or the raw material used, the kinetic mechanism is basically the same. However, the set of kinetic and thermodynamic parameters that describe this mechanism may differ for each case [2]. These parameters are of fundamental importance in defining reaction rates for chemical reactors simulations, design, and development, as well as process control [3,4]. In the transesterification reaction one mol of triglyceride and three mols of short-chain alcohol, in the presence of a catalyst, react forming three mols of ester and one mol of glycerol. This reaction is a set of three consecutive and reversible steps, in which the triglyceride reacts with alcohol, forming the diglyceride, being converted to monoglyceride and finally glycerol and three ester molecules. Each ester molecule is formed in each step of the reaction [2]. It should be noted that in this reaction excess alcohol is commonly used to shift the reaction equilibrium towards product formation [3,5]. Some studies proposed for the reaction mechanism and the kinetic models applied for the transesterification reaction carried out under similar conditions to those obtained in the present work. The vast majority of studies indicate that the transesterification reaction mechanism occurs through three consecutive reversible reactions following second-order kinetics [2–12,14,15,17,18,20–22]. Furthermore, some authors evaluated the same mechanism including the fourth and second-order derivation reaction [6,7,15,21], or the saponification reactions [9], making the kinetic model more complex. Although less frequently, other studies use kinetic models of first-order [3,13,19], while other authors have evaluated irreversibility [3,8,16,19]. For the kinetic study of reactions, it is necessary to adjust the rate constants. Therefore, it is necessary to use an optimization method. Analysis of experimental data for determination/optimization of chemical reaction kinetic parameters can be obtained by numerical iterative methods, linear and differential equations [5]. Over the years, several optimization methods have been developed. Some of these methods used in the transesterification reaction. Among these methods, Particle Swarm Optimization (PSO) stands out. The PSO method was originally developed by Kennedy & Eberhart [23], inspired by the collective behavior of animals. In PSO the set of candidate solutions to the optimization problem is defined as a swarm of particles that can flow through the parameter space, defining trajectories. These trajectories are driven by their own and their neighbors' best performances [24]. This algorithm is a powerful and stochastic scalable computation method that can be used to find the global optimum in a complex search space. It is an easy-to-implement, more efficient and better-performing method that converges faster than other scalable methods, requiring less computational memory [5]. Another tool that has been gaining ground is the Artificial Neural Networks (ANN), which are nonlinear multivariate methods that can be
2. Experimental 2.1. Transesterification reaction All transesterification reactions were carried out by mixing 500 g of soybean oil (Soya, Brazil), 140 mL of methanol (1:6 M ratio of oil:methanol) and 0.50% or 0.75%w/w sodium hydroxide (Fmaia, Brazil) as catalyst, previously dissolved in methanol (Anidrol, Brazil). The reaction was conducted with mechanical stirring maintained at 150 rpm for 62 min and keeping the laboratory temperature controlled at 20 ± 2 °C. Reactions for the two different catalyst concentrations were performed in triplicate. 2.2. Sampling of transesterification reactions During the development of the reaction, were acquired 16 aliquots of 7.0 mL of the reaction medium for each reaction, in the times of 3, 7, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58 and 62 min for analysis in the Proton Nuclear Magnetic Resonance Analysis – 1H NMR (NMR 400 MHz, Bruker). The 1H NMR analysis and the operating conditions were used according to Galvan et al. [2]. 2.3. Determination of conversion rates The molar percentage conversion rates can be obtained by equations of triglycerides, diglycerides, monoglycerides, fatty acid methyl esters, glycerol and methanol compounds present in the reaction medium during the transesterification reactions. This equation was adapted from the work of Nieva-Echevarría et al. [31] and was published in Galvan et al. [2]. 2.4. Kinetic mathematical model In this study, different mechanisms/models were evaluated. Therefore, the overall and three-step of forward and reverse reactions (Eqs. (1)–(4)), were used to model the reversible and irreversible transesterification reaction. The kinetic models were evaluated without and with shunt mechanism [7]. Stepwise reactions:
TG + MeOH
2
k1 k2
DG + FAME
(1)
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DG + MeOH
k3 k4
MG + MeOH
MG + FAME
k5 k6
transfer and the chemical reaction control the overall kinetics of the process kinetics. Finally, the reaction approached the equilibrium state [4]. For this kinetic study, the assumptions are introduced following Noureddini and Zhu [7], Darnoko and Cheryan [8], Vicente et al. [11] and Stamenković et al. [4].
(2)
GL + FAME
(3)
(i) The initial stage of mass transfer control was negligible. In this sense, the chemical reaction stage was supposed to control the reaction rate; (ii) the proportion of free fatty acid in the oil was negligible, so the free fatty acid neutralization can be ignored; (iii) the saponification reaction is negligible at low temperatures. Since the saponification reactions that consume the catalyst were supposed to be negligible, the catalyst concentration remains almost constant; (iv) the values of the DG and MG molar conversions are represented by the sum of the diglycerides and monoglycerides isomers, respectively.
Overall reaction:
TG + 3MeOH
k7 k8
GL + 3FAME
(4)
The reaction steps are in equations from (1)–(4), where k1–8 are rate constants (k1, k3, k5, and k7 are rate constants for forwarding reactions; k2, k4, k6, and k8 are rate constants for reverse reactions), triglyceride (TG), diglyceride (DG), monoglyceride (MG), glycerol (GL), methanol (MeOH) and fatty acid methyl esters (FAME). A system of differential equations based on a kinetic model was presented by Noureddini & Zhu [7] and Silva et al. [18]. The differential kinetic rate law equations obtained from Eqs. (1)–(4) are given in the differential equations from (5)–(10). This system of equations has different functions (ODE, ordinary differential equation) to solve the differential equation system that can be resolved using numerical methods in the Matlab software (MathWorks, USA).
TG = t
k1.cTG .cMeOH +k2.cDG .cFAME
DG = k1.cTG .cMeOH t
k2.cDG .cFAME
MG = k3. cDG .cMeOH t
GL = k5.cMG .cMeOH t
FAME = k1.cTG .cMeOH t cMG .cMeOH MeOH = t
FAME t
k4.cMG .cFAME
3 3 k 7.cTG .cMeOH + k8.cGL .cFAME
k3.cDG .cMeOH
(6)
k5.cMG .cMeOH +k6.cGL .cFAME
(7)
3 k8.cGL .cFAME
k2.cDG .cFAME +k3. cDG .cMeOH 3 k6.cGL .cFAME +k 7.cTG .cMeOH
The Particle Swarm algorithm [24,32] was applied to optimize the reaction rate constants kn (n ∊ {1,2,…,8}) for transesterification reactions, based on the resolution of the ODEs. The values of kn were optimized by the PSO method by minimizing the objective function. The PSO algorithm was executed in a Matlab routine. The PSO routine was executed with some restrictions applied to the values of rate constants k1-8, all the lower (lb) and upper bounds (ub) applied to reactions were 1 × 10−11 L mol−1 min−1 for lb to 1.0 L mol−1 min−1 for ub. These values of the bounds applied to the variables were defined based on preliminary tests. The PSO routine was executed with a swarm of 30 particles, and maximum tolerated error of 1 × 10−3 for the convergence of the calculations between the value of the objective function computed, in a closed-loop, until the function could not achieve any better results for 20 subsequent interactions.
(5)
k 4.cMG .cFAME
3 k6.cGL .cFAME +k 7.cTG .cMeOH
2.5. Particle Swarm optimization (PSO)
(8)
k4.cMG .cFAME +k5. 3 k8.cGL .cFAME
(9)
2.6. Artificial Neural Networks (ANN) The Kohonen Self-Organizing Map (SOM) algorithm begins by initializing the first grid with random synaptic weights and no organization applied to the map. Then, three key processes take place: competition, cooperation and synaptic adaptation [27]. SOM-type ANN routine developed was used according to the algorithm described in Haykin [25] and was applied using the Matlab software. The molar percentage conversion rate values of the numerical and experimental data of the compounds present in the reaction medium, for the two catalyst concentrations, at each reaction time, were tabulated and presented to the neural network. The SOM was applied to numerical and experimental data isolated. The grid topology consists of a 12 × 12 with 7,000 training epochs to ensure a convergence of the
(10)
where c is the molar concentrations of triglyceride (TG), diglyceride (DG), monoglyceride (MG), glycerol (GL), methanol (MeOH), and fatty acid methyl esters (FAME), respectively, in the reaction mixture, expressed in mol L−1. Kinetic rate constants (kn) expressions of the reaction are expressed in L mol−1 min−1. Table 1 shows four proposed mechanisms/models that were evaluated in this study. In equations from (5) to (10), the reaction mechanism with the shunt reaction k7 and k8 were included (study 1), without the shunt-reaction scheme (study 2), k7 and k8 are set equal to zero; and for the overall reaction (study 4), from k1 to k6 are set equal to zero. For the irreversible reaction (study 3), k2, k4, and k6 are rate constants for the reverse reactions are set equal to zero. In this work, the program was run using the Particle Swarm algorithm to estimate and optimized the kinetic rate constants with a Runge-Kutta method as the numerical integrator to do the numerical solution of the differential equation system that represents each proposed mechanism. In modeling was selected ODEs solver function named ode45 which implemented nonstiff solutions of Runge-Kutta 4, 5 order method. The computed concentration values versus reaction time for each time for the NMR analysis were used as inputs in the Matlab PSO routine, from which optimized kinetic rate constants were automatically obtained. Was considered a relative tolerance error of 1 × 10−4 in the modeling. The transesterification reaction occurs via the initial heterogeneous regime, followed by the pseudo-homogenous regime, where the mass
Table 1 Mechanisms/models approach evaluated. Rate constants
k1 k2 k3 k4 k5 k6 k7 k8
(TG → DG) (TG ← DG) (DG → MG) (DG ← MG) (MG → GL) (MG ← GL) (TG → GL) (TG ← GL)
Mechanism/model approach Study 1
Study 2
Study 3
Study 4
✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
✓ ✓ ✓ ✓ ✓ ✓ × ×
✓ × ✓ × ✓ × × ×
× × × × × × ✓ ✓
✓ kn included; × not included. 3
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average quantization error. The initial neighborhood relationship was 1.0, with an initial learning rate of 0.1, decaying exponentially with the training epochs to 1.35 × 10−3.
[14,18,22]. Increasing the catalyst concentration for all proposed models resulted in an increase of the order of magnitude of the rate constants values, indicating that the applied kinetic models are adequate to the experimental data. Comparing the formation and reverse reactions for each proposed study, it is noted that the values of formation rate constants increase in the order of k1 to k3 to k5 at both catalyst concentrations. That is, the reaction of triglyceride to diglyceride is slower than that of diglyceride to monoglyceride, which is slower than that of monoglyceride to glycerol. This same behavior for k2, k4, and k6, which represent the reverse reactions, is not observed in some of the evaluated models. Low k1 values were also reported by Vicente et al. [11] and [12]. This fact implicates that at low temperatures, the reaction of the triglyceride forming the diglyceride is the slowest, and therefore this reaction step is the rate-determining step. The mass transfer that occurs during oil and alcohol miscibility also contributes to low reaction rates at this early stage. When comparing the kinetic models with (study 1) and without the shunt-reaction scheme (study 2) it is possible to notice that the values of rate constants were similar for both models (Table 2), indicating that the inclusion of the shunt mechanism does not cause significant changes in the values of k1 to k6. Furthermore, the values of k7 and k8 showed orders of magnitude lower than k1 to k6. Therefore, it is not necessary to include a shunt mechanism to describe transesterification kinetics. Similar behaviors were obtained by Darnoko and Cheryan [8] and Narváez et al. [15] when the shunt mechanism was included in the modeling, but Freedman et al. [6] obtained better adjustments of experimental data when the combination of second-order consecutive and fourth-order shunt reaction kinetic model was applied. The behavior of an irreversible versus a reversible model was evaluated in studies 2 and 3 since the excess of methanol shifts the reaction equilibrium ensuring higher yields of methyl esters. This practice is commonly used in the biodiesel industries. Comparing the two reactions, it is possible to observe that the formation rate constant values remained practically unchanged for both reversible and irreversible reactions (Table 2). The values of the reverse rate constants (study 2) indicate that the reaction is an irreversible process in this condition since they presented orders of magnitude lower than the formation ones, i.e. these reactions are not favorable. According to Vicente et al. [11], the constant k6 that represents the reaction of glycerol with methyl ester to form monoglycerides and methanol is not favored and is considered irreversible. This fact is due to the immiscibility of methyl esters and glycerol, which involves a great mass transfer resistance. Figs. 1 and 2 show a comparison between the average and simulated experimental conversion rates along with the reaction progress for studies 3 and 4 proposed. These models were chosen for demonstration purposes since one represents the three irreversible stepwise reactions without the shunt mechanism, and the other represents the reversible
2.7. Statistical test The theoretical and experimental data related to triglyceride, diglyceride, monoglyceride, glycerol, methanol and fatty acid methyl esters were compared through the percentage error to assess the quality of the adjustment [33]. N
%Error = 100 i=1
|Cexp
Ccalc |
Ccalc
1 N
(11)
where: C¯calc = estimated mean concentration by numerical solution; C¯ exp = average experimental concentration; and N = number of analyzed samples. 3. Results and discussion The determination of molar conversion rates of the main species present in the reaction medium over time was obtained by 1H NMR and allowed the reaction monitoring using catalyst concentrations of 0.50 and 0.75% w/w. The values of the conversion rates of the species at each time for both catalyst concentrations were used as input data in the PSO routine, so it was possible to optimize the rate constant values according to each proposed mechanism. Table 2 presents the values of the optimized rate constants within the constraint limits assigned to the variables, for the four proposed kinetic models, that were evaluated by the PSO algorithm. For the first study, the reaction mechanism with the shunt reaction was evaluated by mathematical modeling, including k7 and k8. In this model, besides the three consecutive reversible reactions, a shunt reaction in which three methanol molecules react simultaneously with a triglyceride molecule was included [6,15]. The second study also included the three reversible reaction steps, but without the shunt-reaction scheme. In the third study, only the three irreversible steps of the reaction without the shunt mechanism were inserted into the modeling. The irreversibility of the reaction was evaluated due to the amount of excess methanol used in the reactions to shift the equilibrium towards product formation. Under these conditions, the reverse rate constants are so small compared to the direct ones that they can be neglected. In the fourth study, only the global reversible reaction was modeled, considering the reactant species and reaction products, without the inclusion of intermediates. According to Table 2, all values of the formation rate constants were higher than the reversals, for all mechanisms and concentrations evaluated, so that the equilibrium tends to the formation of the products. This behavior is in agreement with other works found in the literature Table 2 Rate constants optimized by the particle swarm algorithm. Study
Catalyst
Rate constants (L mol−1·min−1)a k1(TG ⟶ DG)
1 2 3 4
0.75% 0.50% 0.75% 0.50% 0.75% 0.50% 0.75% 0.50%
1.72 6.97 1.88 6.99 1.54 6.99 – –
× × × × × ×
10−2 10−3 10−2 10−3 10−2 10−3
k2(TG ← DG) 6.18 3.59 1.27 7.53 – – – –
× × × ×
10−6 10−6 10−2 10−6
k3(DG ⟶ MG) 6.35 2.65 6.98 2.67 6.96 2.64 – –
× × × × × ×
10−2 10−2 10−2 10−2 10−2 10−2
k4(DG ← MG) 6.32 1.18 4.44 7.15 – – – –
× × × ×
10−2 10−5 10−2 10−6
k5(MG ⟶ GL) 9.17 6.69 2.46 6.64 1.30 6.44 – –
× × × × × ×
10−1 10−2 10−1 10−2 10−1 10−2
k6(MG ← GL) 8.18 1.32 1.92 4.39 – – – –
× × × ×
10−2 10−8 10−2 10−6
k7(TG ⟶ GL) 3.33 8.14 – – – – 1.05 3.62
× 10−5 × 10−8
× 10−3 × 10−4
k8(TG ← GL) 5.11 1.10 – – – – 2.94 1.03
× 10−7 × 10−8
× 10−10 × 10−11
a Restriction applied to the variables rate constants k1-8, all the lower (lb) and upper bounds (ub) applied to reactions were 1.0 × 10−11 L mol−1 min−1 for lb to 1.0 L mol−1 min−1 for ub. Where k1–8 are rate constants (k1, k3, k5, and k7 are rate constants for forwarding reactions; k2, k4, k6, and k8 are rate constants for reverse reactions).
4
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Fig. 1. Kinetic modeling curves (– simulated) and mean experimental points (● experimental) for the composition of reaction mixture without the shunt mechanism (study 3: three-step reactions to model the irreversible transesterification reaction).
overall reaction according to previous discussions. However, the same interpretation can be made for other studies. In general, the kinetic profiles (Figs. 1 and 2) did not change between the evaluated studies. However, the kinetic profile changed abruptly when the catalyst concentration was changed. The increase in the catalyst concentration from 0.50 to 0.75% showed that the reagent consumption rates and the product formation occurred abruptly, reaching equilibrium in approximately 58 min, while in the 0.50% reaction was not possible to visualize the reaction equilibrium during the 62 min evaluated. According to Figs. 1 and 2, there is an increase in the concentration of methyl esters together with an increase in glycerol concentration of 3:1, once one glycerol molecule is released every three methanol molecules, and one triglyceride molecule is consumed, in accordance with equation (4). In addition, one may also notice an increase in intermediate concentrations in the first few minutes of reaction, reaching a peak, which then slowly decays until finally remain constant (Fig. 1). Table 3 shows the error values obtained for each species in each modeling as well as the absolute error. According to Kadi et al. [5], the lower the parameter error values, the greater the accuracy for the reactor sizing. This factor directly affects the investment costs, maintenance, and energy spent during the reaction.
Table 3 Errors obtained for each study evaluated. Study
1 2 3 4
Cat.
0.75% 0.50% 0.75% 0.50% 0.75% 0.50% 0.75% 0.50%
Error (%)a
Abs. Errorb
TG
DG
MG
GL
FAME
MeOH
5.31 7.60 5.15 6.90 6.34 6.97 3.40 9.65
1.84 3.87 1.85 3.44 2.49 3.51 – –
2.56 2.13 1.78 2.02 2.63 1.99 – –
2.86 5.19 2.61 3.74 6.05 3.84 3.53 5.76
12.39 16.45 14.29 16.29 16.67 15.89 10.60 17.40
12.95 15.37 11.31 14.21 16.90 14.42 8.53 16.49
0.063 0.084 0.062 0.078 0.085 0.078 0.043 0.082
a
Eq. (11). Sum of absolute errors. Where: triglyceride (TG), diglyceride (DG), monoglyceride (MG), glycerol (GL), methanol (MeOH) and fatty acid methyl esters (FAME). b
According to Table 3, the mean error values indicate that all studies had similar values, ranging from 4.3 to 8.5%. The errors of methyl esters and methanol were the highest values, but these errors can be considered acceptable when it comes to kinetic modeling.
Fig. 2. Kinetic modeling curves (– simulated) and mean experimental points (● experimental) for the composition of the reaction mixture (study 4: overall reactions to model the reversible transesterification reaction). 5
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Fig. 3. Weight maps overlaid by topological maps for molar conversion rates of mono-, di- and triglycerides. Samples are coded as a function of reaction time and catalyst concentration. Position 1 and Position 2 indicate the position of the winning neuron.
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Fig. 4. Weight maps overlaid by topological maps for the molar conversion rates of methyl esters, glycerol, and methanol. Samples are coded as a function of reaction time and catalyst concentration. Position 1 and Position 2 indicate the position of the winning neuron.
7
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Similar error values are reported in the literature when it comes to modeling the transesterification reaction under analogous conditions. This result shows the robustness of the PSO method for optimization and the convergence of the rate constant values, which reflects a perfect agreement of the numerical results with the experimental ones. According to Kadi and cols. [5], the methods used in the kinetics of transesterification reaction, like the linear equation and iterative methods, depending on the initial solution and convergence towards non-optimum solutions is highly probable. Kadi et al. [5] reported estimated mean errors between numerical and experimental results of 0.05 and 1% when using the PSO to determine the kinetic parameters in transesterification of the rapeseed oil. Bashiri & Pourbeiram [3] using the Monte Carlo simulation for the study of the transesterification kinetics of the soybean oil, reported average errors of up to 3%. Vicente et al. [11] e [12] when evaluating the transesterification kinetics of soybean oil and Brassica carinata reported error values between 3.3 and 61%. By modeling transesterification kinetics of the soybean oil by Finite Elements associated the Simplex Optimization, Galvan and coworkers [2] obtained an average error of 10.2% and percentage errors for each species ranging from 3.53% to 16.96%. Stamenkovic et al. [4], obtained a TG error of up to 4.6% when studying the transesterification reaction of sunflower oil. The progress of the transesterification reaction was also evaluated by SOM-type of ANN, which is based on less conventional statistical principles, without the need for in-depth knowledge in statistics and multivariate analysis. Experimental and modeled conversion rate values, for each species present in the reaction medium over time intervals, were used as input variables for the SOM network. The modeled data presented to the network were obtained from study 2, but the same interpretation can be made for the other studies. The experimental and modeled datasets contained 204 points (6 columns/species, for 34 rows/times) each. After the training stage, a topological map (sample distribution) was generated and for each species, the respective weight map was evaluated. In the topological map, each sample is associated with the respective winning neuron, i.e. the one that best represents the sample in the SOM. The SOM network classifies the input data as clusters that can be formed by one or more neurons. The definition of groups is characterized by the presence of empty neurons between clusters [27]. Figs. 3 and 4 illustrate the weight maps, overlaid by the topological map for the sample segmentation. In weight maps, the observed values for the input variables are indicated by the color scale. In general, topological maps indicated the formation of clusters for the initial, intermediate and final reaction times. It is also possible to observe that the times of the samples with 0.50% and 0.75% of catalyst had different winning neurons. According to Figs. 3 and 4, it can be observed that the SOM segmented the samples as a function of the transesterification reaction time almost identically for the simulated and experimental samples. Analyzing the weight maps the interpretation of the transesterification reaction kinetics is easily understood and corroborates the modeled results. Weight maps adequately describe the kinetic mechanism of the transesterification reaction, where the triglyceride molecules first react with methanol to form diglycerides which are converted to monoglycerides and finally to glycerol and methyl esters. Moreover, in the weight maps for intermediates, it is possible to see that there is an increase in the reaction rates in the initial times, reaching a maximum value, followed by a decrease.
disregarded, following a three-step kinetic model without the inclusion of the shunt mechanism. The application of SOM in combination with the evolution graphs of the species present in the reaction medium proved to be an efficient tool for the exploration of data obtained during the transesterification process. The SOM was able to organize the data obtained in clusters and presented in the weight maps the relationship between the monitored variables during the transesterification reaction progress. The PSO algorithms and the SOM-type ANN were very important to achieve the best conditions of the kinetic parameters and to evaluate the monitoring of the transesterification reaction with minimal computational work since many variables were evaluated. Declaration of Competing Interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement Diego Galvan: Conceptualization, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing - review & editing. Hágata Cremasco: Conceptualization, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing - review & editing. Ana Carolina Gomes Mantovani: Methodology, Writing - original draft, Writing - review & editing. Evandro Bona: Methodology, Software, Supervision, Writing - review & editing. Mário Killner: Methodology, Software, Validation, Data curation, Project administration. Dionisio Borsato: Funding acquisition, Methodology, Project administration, Software, Supervision, Writing - review & editing. Acknowledgments The State University of Londrina (UEL) for the technical support and CAPES for granting the scholarship. The authors also thank the Spectroscopy Laboratory (LABSPEC) – UEL. Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) for funding and fellowships awarded to D. Galvan (CAPES process Nr. 88881.131632/ 2016-01). References [1] Gebremariam SN, Marchetti JM. Economics of biodiesel production. Energy Convers Manage 2018;168:74–84. [2] Galvan D, Chendynski LT, Mantovani AC, Quadri M, Killner M, Cremasco H, et al. Mathematical modeling of the transesterification reaction by finite elements: optimization of kinetic parameters using the simplex sequential method. J Braz Chem Soc 2020;31:313–9. [3] Bashiri H, Pourbeiram N. Biodiesel production through transesterification of soybean oil: a kinetic Monte Carlo study. J Mol Liq 2016;223:10–5. [4] Stamenković OS, Todorović ZB, Lazić ML, Veljković VB, Skala DU. Kinetics of sunflower oil methanolysis at low temperatures. Bioresour Technol 2008;99:1131–40. [5] Kadi MA, Akkouche N, Awad S, Loubar K, Tazerout M. Kinetic study of transesterification using particle swarm optimization method. Heliyon 2019;5:e02146. [6] Freedman B, Butterfield RO, Pryde EH. Transesterification kinetics of soybean oil 1. J Am Oil Chemists’ Soc 1986;63:1375–80. [7] Noureddini H, Zhu D. Kinetics of transesterification of soybean oil. J Am Oil Chemists’ Soc 1997;74:1457–63. [8] Darnoko D, Cheryan M. Kinetics of palm oil transesterification in a batch reactor. J Am Oil Chemists’ Soc 2000;77:1263–7. [9] Komers K, Skopal F, Stloukal R, Machek J. Kinetics and mechanism of the KOH—catalyzed methanolysis of rapeseed oil for biodiesel production. Eur J Lipid Sci Technol 2002;104:728–37. [10] Leevijit T, Wisutmethangoon W, Prateepchaikul G, Tongurai C, Allen M. A second order kinetics of palm oil transesterification. Sustain Energy Environ 2004:277–81. [11] Vicente G, Martínez M, Aracil J, Esteban A. Kinetics of sunflower oil methanolysis. Ind Eng Chem Res 2005;44:5447–54. [12] Vicente G, Martinez M, Aracil J. Kinetics of brassica c arinata oil methanolysis. Energy Fuels 2006;20:1722–6.
4. Conclusion Different approaches to kinetic models and mechanisms for the transesterification reaction were investigated by PSO. All studies showed good agreement between experimental and simulated data, with errors similar to those reported in the literature. The kinetic results suggest that under the evaluated conditions, reverse reactions may be 8
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D. Galvan, et al. [13] Wenzel B, Tait M, Módenes A, Kroumov A. Modelling chemical kinetics of soybean oil transesterification process for biodiesel production: an analysis of molar ratio between alcohol and soybean oil temperature changes on the process conversion rate. Bioautomation 2006;5:13–22. [14] Bambase ME, Nakamura N, Tanaka J, Matsumura M. Kinetics of hydroxide-catalyzed methanolysis of crude sunflower oil for the production of fuel-grade methyl esters. J Chem Technol Biotechnol 2007;82:273–80. [15] Narváez PC, Rincón SM, Sánchez FJ. Kinetics of palm oil methanolysis. J Am Oil Chemists’ Soc 2007;84:971–7. [16] Chea KY, Choo YM, Ma AN, Basiron Y. Production Technology of Palm Diesel, Procedings of the 1998 PORIM Internatinal Biofuel and Lubricant Conference n.d. [17] Lima DR de. Produção de esteres etilicos (biodiesel) a partir da transesterização basica de oleo residual. University of Campinas, 2008. [18] da Silva N de L, Rivera EC, Batistella CB, Maciel Filho R, Maciel MRW. Biodiesel production from vegetable oils: Operational strategies for large scale systems. Computer Aided Chemical Engineering, vol. 25, Elsevier; 2008, p. 1101–6. [19] Ramezani K, Rowshanzamir S, Eikani MH. Castor oil transesterification reaction: a kinetic study and optimization of parameters. Energy 2010;35:4142–8. [20] Issariyakul T, Dalai AK. Comparative kinetics of transesterification for biodiesel production from palm oil and mustard oil. Can J Chem Eng 2012;90:342–50. [21] Janajreh I, ElSamad T, AlJaberi A, Diouri M. Transesterification of waste cooking oil: kinetic study and reactive flow analysis. Energy Proc 2015;75:547–53. [22] Okullo AA, Temu AK. Modelling the kinetics of Jatropha oil transesterification. Energy Power Eng 2015;7:135–43. [23] Eberhart R, Kennedy J. Particle swarm optimization. Proceedings of the IEEE international conference on neural networks, vol. 4, Citeseer; 1995, p. 1942–8. [24] Marini F, Walczak B. Particle swarm optimization (PSO). A tutorial. Chemometrics
Intell Lab Syst 2015;149:153–65. [25] Haykin S. Redes neurais: princípios e prática. Bookman Editora 2007. [26] Peng D, Bi Y, Ren X, Yang G, Sun S, Wang X. Detection and quantification of adulteration of sesame oils with vegetable oils using gas chromatography and multivariate data analysis. Food Chem 2015;188:415–21. [27] Cremasco H, Borsato D, Angilelli KG, Galão OF, Bona E, Valle ME. Application of self-organising maps towards segmentation of soybean samples by determination of inorganic compounds content. J Sci Food Agric 2016;96:306–10. [28] Cremasco H, Galvan D, Angilelli KG, Borsato D, de Oliveira AG. Influence of film coefficient during multicomponent diffusion–KCl/NaCl in biosolid for static and agitated system using 3D computational simulation. Food Sci Technol 2019;39:173–81. [29] Kimura M, Savada FY, Tashima DLM, Romagnoli ÉS, Chendynski LT, Silva LRC, et al. Application of the self-organizing map in the classification of natural antioxidants in commercial biodiesel. Biofuels 2018:1–6. [30] Walkoff AR, Antunes SRM, Arrúa MEP, Silva LRC, Borsato D, Rodrigues PRP. Selforganizing maps neural networks applied to the classification of ethanol samples according to the region of commercialization. Orbital: the electronic. J Chem 2017;9:248–55. [31] Nieva-Echevarría B, Goicoechea E, Manzanos MJ, Guillén MD. A method based on 1 H NMR spectral data useful to evaluate the hydrolysis level in complex lipid mixtures. Food Res Int 2014;66:379–87. [32] Mezura-Montes E, Coello CAC. Constraint-handling in nature-inspired numerical optimization: past, present and future. Swarm Evol Comput 2011;1:173–94. [33] Bona E, da Silva RSSF, Borsato D, Silva LHM, de Souza Fidelis DA. Multicomponent diffusion modeling and simulation in prato cheese salting using brine at rest: the finite element method approach. J Food Eng 2007;79:771–8.
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Full Length Article
Kinetic study of the transesterification reaction by artificial neural networks and parametric particle swarm optimization
T
Diego Galvana, Hágata Cremascoa, Ana Carolina Gomes Mantovanib, Evandro Bonac, ⁎ Mário Killnera, Dionisio Borsatoa, a
Chemistry Department, State University of Londrina (UEL), 86.057 970 Londrina, PR, Brazil Physics Department, State University of Londrina (UEL), 86.057 970 Londrina, PR, Brazil c Programa de Pós-Graduação em Tecnologia de Alimentos (PPGTA), Universidade Tecnológica Federal do Paraná (CM), Câmpus Campo Mourão, 87.301 899 Campo Mourão, Paraná, Brazil b
GRAPHICAL ABSTRACT
ARTICLE INFO
ABSTRACT
Keywords: PSO Self-organizing maps (SOM) Runge-Kutta Rate constants Soybean oil Methanolysis
In this work, the mechanism of soybean oil transesterification reaction was investigated by Particle Swarm Optimization (PSO), and the reaction progress was monitored by Artificial Neural Networks (ANN) using SelfOrganizing Maps (SOM). The PSO was used to adjust the values of the reaction rate constants (kn) in the four proposed mechanisms/models approaches. The mean error values obtained by the PSO method showed a good agreement between the experimental and simulated data for all evaluated approaches, ranging from 4.3 to 8.5%. The results suggest that under the evaluated conditions reverse reactions may be disregarded, following a secondorder kinetic model for the three stepwise reactions. The SOM proved to be an efficient tool for exploring data obtained during the transesterification process, allowed to organize the data in the form of clusters presenting in the weight maps the relationship between the monitored variables during the progress of the transesterification
⁎
Corresponding author. E-mail address: [email protected] (D. Borsato).
https://doi.org/10.1016/j.fuel.2020.117221 Received 8 November 2019; Received in revised form 24 January 2020; Accepted 26 January 2020 0016-2361/ © 2020 Elsevier Ltd. All rights reserved.
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reaction. The PSO and ANN type SOM has been shown efficient tools for investigating the kinetics of the transesterification reaction, and they offer good perspectives for further kinetic studies of other types of industrial processes.
applied in pattern recognition and classification. The ANNs are mathematical models whose architecture was inspired by biological neural networks and consist of simple processing units that store empirical knowledge through a learning process [25]. The Self Organizing Maps (SOM) is a type of ANN in which the intrinsic relationships between samples or variables in a dataset are represented using an unsupervised learning method [26]. These sets of techniques can be applied in a wide range of areas, including food process control [27,28] and fuels [29,30]. In this work, four different kinetic models and mechanisms approach for the soybean oil transesterification reaction was evaluated. The optimization of the kinetic parameters for each case was obtained by the PSO method. The monitoring of the reaction progress was also performed using the SOM-type of ANN, based on experimental and modeled results.
1. Introduction The demand for fossil fuels in recent decades has been increasing, causing resource scarcity and negative environmental impacts due to overuse. Renewable and less polluting substitute fuels have been increasingly researched. Among these alternatives, biodiesel is gaining huge space [1]. The industrial production of biodiesel is well known worldwide and has a well-developed technology, making the process reasonably inexpensive when compared to other methods of renewable energy production. In addition, there are several alternatives to raw materials that can minimize the costs of production [1,2]. Although the process is well known, the kinetic study of the transesterification reaction is still necessary due to its wide industrial use. Regardless of the type of process, operating conditions or the raw material used, the kinetic mechanism is basically the same. However, the set of kinetic and thermodynamic parameters that describe this mechanism may differ for each case [2]. These parameters are of fundamental importance in defining reaction rates for chemical reactors simulations, design, and development, as well as process control [3,4]. In the transesterification reaction one mol of triglyceride and three mols of short-chain alcohol, in the presence of a catalyst, react forming three mols of ester and one mol of glycerol. This reaction is a set of three consecutive and reversible steps, in which the triglyceride reacts with alcohol, forming the diglyceride, being converted to monoglyceride and finally glycerol and three ester molecules. Each ester molecule is formed in each step of the reaction [2]. It should be noted that in this reaction excess alcohol is commonly used to shift the reaction equilibrium towards product formation [3,5]. Some studies proposed for the reaction mechanism and the kinetic models applied for the transesterification reaction carried out under similar conditions to those obtained in the present work. The vast majority of studies indicate that the transesterification reaction mechanism occurs through three consecutive reversible reactions following second-order kinetics [2–12,14,15,17,18,20–22]. Furthermore, some authors evaluated the same mechanism including the fourth and second-order derivation reaction [6,7,15,21], or the saponification reactions [9], making the kinetic model more complex. Although less frequently, other studies use kinetic models of first-order [3,13,19], while other authors have evaluated irreversibility [3,8,16,19]. For the kinetic study of reactions, it is necessary to adjust the rate constants. Therefore, it is necessary to use an optimization method. Analysis of experimental data for determination/optimization of chemical reaction kinetic parameters can be obtained by numerical iterative methods, linear and differential equations [5]. Over the years, several optimization methods have been developed. Some of these methods used in the transesterification reaction. Among these methods, Particle Swarm Optimization (PSO) stands out. The PSO method was originally developed by Kennedy & Eberhart [23], inspired by the collective behavior of animals. In PSO the set of candidate solutions to the optimization problem is defined as a swarm of particles that can flow through the parameter space, defining trajectories. These trajectories are driven by their own and their neighbors' best performances [24]. This algorithm is a powerful and stochastic scalable computation method that can be used to find the global optimum in a complex search space. It is an easy-to-implement, more efficient and better-performing method that converges faster than other scalable methods, requiring less computational memory [5]. Another tool that has been gaining ground is the Artificial Neural Networks (ANN), which are nonlinear multivariate methods that can be
2. Experimental 2.1. Transesterification reaction All transesterification reactions were carried out by mixing 500 g of soybean oil (Soya, Brazil), 140 mL of methanol (1:6 M ratio of oil:methanol) and 0.50% or 0.75%w/w sodium hydroxide (Fmaia, Brazil) as catalyst, previously dissolved in methanol (Anidrol, Brazil). The reaction was conducted with mechanical stirring maintained at 150 rpm for 62 min and keeping the laboratory temperature controlled at 20 ± 2 °C. Reactions for the two different catalyst concentrations were performed in triplicate. 2.2. Sampling of transesterification reactions During the development of the reaction, were acquired 16 aliquots of 7.0 mL of the reaction medium for each reaction, in the times of 3, 7, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58 and 62 min for analysis in the Proton Nuclear Magnetic Resonance Analysis – 1H NMR (NMR 400 MHz, Bruker). The 1H NMR analysis and the operating conditions were used according to Galvan et al. [2]. 2.3. Determination of conversion rates The molar percentage conversion rates can be obtained by equations of triglycerides, diglycerides, monoglycerides, fatty acid methyl esters, glycerol and methanol compounds present in the reaction medium during the transesterification reactions. This equation was adapted from the work of Nieva-Echevarría et al. [31] and was published in Galvan et al. [2]. 2.4. Kinetic mathematical model In this study, different mechanisms/models were evaluated. Therefore, the overall and three-step of forward and reverse reactions (Eqs. (1)–(4)), were used to model the reversible and irreversible transesterification reaction. The kinetic models were evaluated without and with shunt mechanism [7]. Stepwise reactions:
TG + MeOH
2
k1 k2
DG + FAME
(1)
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DG + MeOH
k3 k4
MG + MeOH
MG + FAME
k5 k6
transfer and the chemical reaction control the overall kinetics of the process kinetics. Finally, the reaction approached the equilibrium state [4]. For this kinetic study, the assumptions are introduced following Noureddini and Zhu [7], Darnoko and Cheryan [8], Vicente et al. [11] and Stamenković et al. [4].
(2)
GL + FAME
(3)
(i) The initial stage of mass transfer control was negligible. In this sense, the chemical reaction stage was supposed to control the reaction rate; (ii) the proportion of free fatty acid in the oil was negligible, so the free fatty acid neutralization can be ignored; (iii) the saponification reaction is negligible at low temperatures. Since the saponification reactions that consume the catalyst were supposed to be negligible, the catalyst concentration remains almost constant; (iv) the values of the DG and MG molar conversions are represented by the sum of the diglycerides and monoglycerides isomers, respectively.
Overall reaction:
TG + 3MeOH
k7 k8
GL + 3FAME
(4)
The reaction steps are in equations from (1)–(4), where k1–8 are rate constants (k1, k3, k5, and k7 are rate constants for forwarding reactions; k2, k4, k6, and k8 are rate constants for reverse reactions), triglyceride (TG), diglyceride (DG), monoglyceride (MG), glycerol (GL), methanol (MeOH) and fatty acid methyl esters (FAME). A system of differential equations based on a kinetic model was presented by Noureddini & Zhu [7] and Silva et al. [18]. The differential kinetic rate law equations obtained from Eqs. (1)–(4) are given in the differential equations from (5)–(10). This system of equations has different functions (ODE, ordinary differential equation) to solve the differential equation system that can be resolved using numerical methods in the Matlab software (MathWorks, USA).
TG = t
k1.cTG .cMeOH +k2.cDG .cFAME
DG = k1.cTG .cMeOH t
k2.cDG .cFAME
MG = k3. cDG .cMeOH t
GL = k5.cMG .cMeOH t
FAME = k1.cTG .cMeOH t cMG .cMeOH MeOH = t
FAME t
k4.cMG .cFAME
3 3 k 7.cTG .cMeOH + k8.cGL .cFAME
k3.cDG .cMeOH
(6)
k5.cMG .cMeOH +k6.cGL .cFAME
(7)
3 k8.cGL .cFAME
k2.cDG .cFAME +k3. cDG .cMeOH 3 k6.cGL .cFAME +k 7.cTG .cMeOH
The Particle Swarm algorithm [24,32] was applied to optimize the reaction rate constants kn (n ∊ {1,2,…,8}) for transesterification reactions, based on the resolution of the ODEs. The values of kn were optimized by the PSO method by minimizing the objective function. The PSO algorithm was executed in a Matlab routine. The PSO routine was executed with some restrictions applied to the values of rate constants k1-8, all the lower (lb) and upper bounds (ub) applied to reactions were 1 × 10−11 L mol−1 min−1 for lb to 1.0 L mol−1 min−1 for ub. These values of the bounds applied to the variables were defined based on preliminary tests. The PSO routine was executed with a swarm of 30 particles, and maximum tolerated error of 1 × 10−3 for the convergence of the calculations between the value of the objective function computed, in a closed-loop, until the function could not achieve any better results for 20 subsequent interactions.
(5)
k 4.cMG .cFAME
3 k6.cGL .cFAME +k 7.cTG .cMeOH
2.5. Particle Swarm optimization (PSO)
(8)
k4.cMG .cFAME +k5. 3 k8.cGL .cFAME
(9)
2.6. Artificial Neural Networks (ANN) The Kohonen Self-Organizing Map (SOM) algorithm begins by initializing the first grid with random synaptic weights and no organization applied to the map. Then, three key processes take place: competition, cooperation and synaptic adaptation [27]. SOM-type ANN routine developed was used according to the algorithm described in Haykin [25] and was applied using the Matlab software. The molar percentage conversion rate values of the numerical and experimental data of the compounds present in the reaction medium, for the two catalyst concentrations, at each reaction time, were tabulated and presented to the neural network. The SOM was applied to numerical and experimental data isolated. The grid topology consists of a 12 × 12 with 7,000 training epochs to ensure a convergence of the
(10)
where c is the molar concentrations of triglyceride (TG), diglyceride (DG), monoglyceride (MG), glycerol (GL), methanol (MeOH), and fatty acid methyl esters (FAME), respectively, in the reaction mixture, expressed in mol L−1. Kinetic rate constants (kn) expressions of the reaction are expressed in L mol−1 min−1. Table 1 shows four proposed mechanisms/models that were evaluated in this study. In equations from (5) to (10), the reaction mechanism with the shunt reaction k7 and k8 were included (study 1), without the shunt-reaction scheme (study 2), k7 and k8 are set equal to zero; and for the overall reaction (study 4), from k1 to k6 are set equal to zero. For the irreversible reaction (study 3), k2, k4, and k6 are rate constants for the reverse reactions are set equal to zero. In this work, the program was run using the Particle Swarm algorithm to estimate and optimized the kinetic rate constants with a Runge-Kutta method as the numerical integrator to do the numerical solution of the differential equation system that represents each proposed mechanism. In modeling was selected ODEs solver function named ode45 which implemented nonstiff solutions of Runge-Kutta 4, 5 order method. The computed concentration values versus reaction time for each time for the NMR analysis were used as inputs in the Matlab PSO routine, from which optimized kinetic rate constants were automatically obtained. Was considered a relative tolerance error of 1 × 10−4 in the modeling. The transesterification reaction occurs via the initial heterogeneous regime, followed by the pseudo-homogenous regime, where the mass
Table 1 Mechanisms/models approach evaluated. Rate constants
k1 k2 k3 k4 k5 k6 k7 k8
(TG → DG) (TG ← DG) (DG → MG) (DG ← MG) (MG → GL) (MG ← GL) (TG → GL) (TG ← GL)
Mechanism/model approach Study 1
Study 2
Study 3
Study 4
✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
✓ ✓ ✓ ✓ ✓ ✓ × ×
✓ × ✓ × ✓ × × ×
× × × × × × ✓ ✓
✓ kn included; × not included. 3
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average quantization error. The initial neighborhood relationship was 1.0, with an initial learning rate of 0.1, decaying exponentially with the training epochs to 1.35 × 10−3.
[14,18,22]. Increasing the catalyst concentration for all proposed models resulted in an increase of the order of magnitude of the rate constants values, indicating that the applied kinetic models are adequate to the experimental data. Comparing the formation and reverse reactions for each proposed study, it is noted that the values of formation rate constants increase in the order of k1 to k3 to k5 at both catalyst concentrations. That is, the reaction of triglyceride to diglyceride is slower than that of diglyceride to monoglyceride, which is slower than that of monoglyceride to glycerol. This same behavior for k2, k4, and k6, which represent the reverse reactions, is not observed in some of the evaluated models. Low k1 values were also reported by Vicente et al. [11] and [12]. This fact implicates that at low temperatures, the reaction of the triglyceride forming the diglyceride is the slowest, and therefore this reaction step is the rate-determining step. The mass transfer that occurs during oil and alcohol miscibility also contributes to low reaction rates at this early stage. When comparing the kinetic models with (study 1) and without the shunt-reaction scheme (study 2) it is possible to notice that the values of rate constants were similar for both models (Table 2), indicating that the inclusion of the shunt mechanism does not cause significant changes in the values of k1 to k6. Furthermore, the values of k7 and k8 showed orders of magnitude lower than k1 to k6. Therefore, it is not necessary to include a shunt mechanism to describe transesterification kinetics. Similar behaviors were obtained by Darnoko and Cheryan [8] and Narváez et al. [15] when the shunt mechanism was included in the modeling, but Freedman et al. [6] obtained better adjustments of experimental data when the combination of second-order consecutive and fourth-order shunt reaction kinetic model was applied. The behavior of an irreversible versus a reversible model was evaluated in studies 2 and 3 since the excess of methanol shifts the reaction equilibrium ensuring higher yields of methyl esters. This practice is commonly used in the biodiesel industries. Comparing the two reactions, it is possible to observe that the formation rate constant values remained practically unchanged for both reversible and irreversible reactions (Table 2). The values of the reverse rate constants (study 2) indicate that the reaction is an irreversible process in this condition since they presented orders of magnitude lower than the formation ones, i.e. these reactions are not favorable. According to Vicente et al. [11], the constant k6 that represents the reaction of glycerol with methyl ester to form monoglycerides and methanol is not favored and is considered irreversible. This fact is due to the immiscibility of methyl esters and glycerol, which involves a great mass transfer resistance. Figs. 1 and 2 show a comparison between the average and simulated experimental conversion rates along with the reaction progress for studies 3 and 4 proposed. These models were chosen for demonstration purposes since one represents the three irreversible stepwise reactions without the shunt mechanism, and the other represents the reversible
2.7. Statistical test The theoretical and experimental data related to triglyceride, diglyceride, monoglyceride, glycerol, methanol and fatty acid methyl esters were compared through the percentage error to assess the quality of the adjustment [33]. N
%Error = 100 i=1
|Cexp
Ccalc |
Ccalc
1 N
(11)
where: C¯calc = estimated mean concentration by numerical solution; C¯ exp = average experimental concentration; and N = number of analyzed samples. 3. Results and discussion The determination of molar conversion rates of the main species present in the reaction medium over time was obtained by 1H NMR and allowed the reaction monitoring using catalyst concentrations of 0.50 and 0.75% w/w. The values of the conversion rates of the species at each time for both catalyst concentrations were used as input data in the PSO routine, so it was possible to optimize the rate constant values according to each proposed mechanism. Table 2 presents the values of the optimized rate constants within the constraint limits assigned to the variables, for the four proposed kinetic models, that were evaluated by the PSO algorithm. For the first study, the reaction mechanism with the shunt reaction was evaluated by mathematical modeling, including k7 and k8. In this model, besides the three consecutive reversible reactions, a shunt reaction in which three methanol molecules react simultaneously with a triglyceride molecule was included [6,15]. The second study also included the three reversible reaction steps, but without the shunt-reaction scheme. In the third study, only the three irreversible steps of the reaction without the shunt mechanism were inserted into the modeling. The irreversibility of the reaction was evaluated due to the amount of excess methanol used in the reactions to shift the equilibrium towards product formation. Under these conditions, the reverse rate constants are so small compared to the direct ones that they can be neglected. In the fourth study, only the global reversible reaction was modeled, considering the reactant species and reaction products, without the inclusion of intermediates. According to Table 2, all values of the formation rate constants were higher than the reversals, for all mechanisms and concentrations evaluated, so that the equilibrium tends to the formation of the products. This behavior is in agreement with other works found in the literature Table 2 Rate constants optimized by the particle swarm algorithm. Study
Catalyst
Rate constants (L mol−1·min−1)a k1(TG ⟶ DG)
1 2 3 4
0.75% 0.50% 0.75% 0.50% 0.75% 0.50% 0.75% 0.50%
1.72 6.97 1.88 6.99 1.54 6.99 – –
× × × × × ×
10−2 10−3 10−2 10−3 10−2 10−3
k2(TG ← DG) 6.18 3.59 1.27 7.53 – – – –
× × × ×
10−6 10−6 10−2 10−6
k3(DG ⟶ MG) 6.35 2.65 6.98 2.67 6.96 2.64 – –
× × × × × ×
10−2 10−2 10−2 10−2 10−2 10−2
k4(DG ← MG) 6.32 1.18 4.44 7.15 – – – –
× × × ×
10−2 10−5 10−2 10−6
k5(MG ⟶ GL) 9.17 6.69 2.46 6.64 1.30 6.44 – –
× × × × × ×
10−1 10−2 10−1 10−2 10−1 10−2
k6(MG ← GL) 8.18 1.32 1.92 4.39 – – – –
× × × ×
10−2 10−8 10−2 10−6
k7(TG ⟶ GL) 3.33 8.14 – – – – 1.05 3.62
× 10−5 × 10−8
× 10−3 × 10−4
k8(TG ← GL) 5.11 1.10 – – – – 2.94 1.03
× 10−7 × 10−8
× 10−10 × 10−11
a Restriction applied to the variables rate constants k1-8, all the lower (lb) and upper bounds (ub) applied to reactions were 1.0 × 10−11 L mol−1 min−1 for lb to 1.0 L mol−1 min−1 for ub. Where k1–8 are rate constants (k1, k3, k5, and k7 are rate constants for forwarding reactions; k2, k4, k6, and k8 are rate constants for reverse reactions).
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Fig. 1. Kinetic modeling curves (– simulated) and mean experimental points (● experimental) for the composition of reaction mixture without the shunt mechanism (study 3: three-step reactions to model the irreversible transesterification reaction).
overall reaction according to previous discussions. However, the same interpretation can be made for other studies. In general, the kinetic profiles (Figs. 1 and 2) did not change between the evaluated studies. However, the kinetic profile changed abruptly when the catalyst concentration was changed. The increase in the catalyst concentration from 0.50 to 0.75% showed that the reagent consumption rates and the product formation occurred abruptly, reaching equilibrium in approximately 58 min, while in the 0.50% reaction was not possible to visualize the reaction equilibrium during the 62 min evaluated. According to Figs. 1 and 2, there is an increase in the concentration of methyl esters together with an increase in glycerol concentration of 3:1, once one glycerol molecule is released every three methanol molecules, and one triglyceride molecule is consumed, in accordance with equation (4). In addition, one may also notice an increase in intermediate concentrations in the first few minutes of reaction, reaching a peak, which then slowly decays until finally remain constant (Fig. 1). Table 3 shows the error values obtained for each species in each modeling as well as the absolute error. According to Kadi et al. [5], the lower the parameter error values, the greater the accuracy for the reactor sizing. This factor directly affects the investment costs, maintenance, and energy spent during the reaction.
Table 3 Errors obtained for each study evaluated. Study
1 2 3 4
Cat.
0.75% 0.50% 0.75% 0.50% 0.75% 0.50% 0.75% 0.50%
Error (%)a
Abs. Errorb
TG
DG
MG
GL
FAME
MeOH
5.31 7.60 5.15 6.90 6.34 6.97 3.40 9.65
1.84 3.87 1.85 3.44 2.49 3.51 – –
2.56 2.13 1.78 2.02 2.63 1.99 – –
2.86 5.19 2.61 3.74 6.05 3.84 3.53 5.76
12.39 16.45 14.29 16.29 16.67 15.89 10.60 17.40
12.95 15.37 11.31 14.21 16.90 14.42 8.53 16.49
0.063 0.084 0.062 0.078 0.085 0.078 0.043 0.082
a
Eq. (11). Sum of absolute errors. Where: triglyceride (TG), diglyceride (DG), monoglyceride (MG), glycerol (GL), methanol (MeOH) and fatty acid methyl esters (FAME). b
According to Table 3, the mean error values indicate that all studies had similar values, ranging from 4.3 to 8.5%. The errors of methyl esters and methanol were the highest values, but these errors can be considered acceptable when it comes to kinetic modeling.
Fig. 2. Kinetic modeling curves (– simulated) and mean experimental points (● experimental) for the composition of the reaction mixture (study 4: overall reactions to model the reversible transesterification reaction). 5
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Fig. 3. Weight maps overlaid by topological maps for molar conversion rates of mono-, di- and triglycerides. Samples are coded as a function of reaction time and catalyst concentration. Position 1 and Position 2 indicate the position of the winning neuron.
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Fig. 4. Weight maps overlaid by topological maps for the molar conversion rates of methyl esters, glycerol, and methanol. Samples are coded as a function of reaction time and catalyst concentration. Position 1 and Position 2 indicate the position of the winning neuron.
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Similar error values are reported in the literature when it comes to modeling the transesterification reaction under analogous conditions. This result shows the robustness of the PSO method for optimization and the convergence of the rate constant values, which reflects a perfect agreement of the numerical results with the experimental ones. According to Kadi and cols. [5], the methods used in the kinetics of transesterification reaction, like the linear equation and iterative methods, depending on the initial solution and convergence towards non-optimum solutions is highly probable. Kadi et al. [5] reported estimated mean errors between numerical and experimental results of 0.05 and 1% when using the PSO to determine the kinetic parameters in transesterification of the rapeseed oil. Bashiri & Pourbeiram [3] using the Monte Carlo simulation for the study of the transesterification kinetics of the soybean oil, reported average errors of up to 3%. Vicente et al. [11] e [12] when evaluating the transesterification kinetics of soybean oil and Brassica carinata reported error values between 3.3 and 61%. By modeling transesterification kinetics of the soybean oil by Finite Elements associated the Simplex Optimization, Galvan and coworkers [2] obtained an average error of 10.2% and percentage errors for each species ranging from 3.53% to 16.96%. Stamenkovic et al. [4], obtained a TG error of up to 4.6% when studying the transesterification reaction of sunflower oil. The progress of the transesterification reaction was also evaluated by SOM-type of ANN, which is based on less conventional statistical principles, without the need for in-depth knowledge in statistics and multivariate analysis. Experimental and modeled conversion rate values, for each species present in the reaction medium over time intervals, were used as input variables for the SOM network. The modeled data presented to the network were obtained from study 2, but the same interpretation can be made for the other studies. The experimental and modeled datasets contained 204 points (6 columns/species, for 34 rows/times) each. After the training stage, a topological map (sample distribution) was generated and for each species, the respective weight map was evaluated. In the topological map, each sample is associated with the respective winning neuron, i.e. the one that best represents the sample in the SOM. The SOM network classifies the input data as clusters that can be formed by one or more neurons. The definition of groups is characterized by the presence of empty neurons between clusters [27]. Figs. 3 and 4 illustrate the weight maps, overlaid by the topological map for the sample segmentation. In weight maps, the observed values for the input variables are indicated by the color scale. In general, topological maps indicated the formation of clusters for the initial, intermediate and final reaction times. It is also possible to observe that the times of the samples with 0.50% and 0.75% of catalyst had different winning neurons. According to Figs. 3 and 4, it can be observed that the SOM segmented the samples as a function of the transesterification reaction time almost identically for the simulated and experimental samples. Analyzing the weight maps the interpretation of the transesterification reaction kinetics is easily understood and corroborates the modeled results. Weight maps adequately describe the kinetic mechanism of the transesterification reaction, where the triglyceride molecules first react with methanol to form diglycerides which are converted to monoglycerides and finally to glycerol and methyl esters. Moreover, in the weight maps for intermediates, it is possible to see that there is an increase in the reaction rates in the initial times, reaching a maximum value, followed by a decrease.
disregarded, following a three-step kinetic model without the inclusion of the shunt mechanism. The application of SOM in combination with the evolution graphs of the species present in the reaction medium proved to be an efficient tool for the exploration of data obtained during the transesterification process. The SOM was able to organize the data obtained in clusters and presented in the weight maps the relationship between the monitored variables during the transesterification reaction progress. The PSO algorithms and the SOM-type ANN were very important to achieve the best conditions of the kinetic parameters and to evaluate the monitoring of the transesterification reaction with minimal computational work since many variables were evaluated. Declaration of Competing Interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement Diego Galvan: Conceptualization, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing - review & editing. Hágata Cremasco: Conceptualization, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing - review & editing. Ana Carolina Gomes Mantovani: Methodology, Writing - original draft, Writing - review & editing. Evandro Bona: Methodology, Software, Supervision, Writing - review & editing. Mário Killner: Methodology, Software, Validation, Data curation, Project administration. Dionisio Borsato: Funding acquisition, Methodology, Project administration, Software, Supervision, Writing - review & editing. Acknowledgments The State University of Londrina (UEL) for the technical support and CAPES for granting the scholarship. The authors also thank the Spectroscopy Laboratory (LABSPEC) – UEL. Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) for funding and fellowships awarded to D. Galvan (CAPES process Nr. 88881.131632/ 2016-01). References [1] Gebremariam SN, Marchetti JM. Economics of biodiesel production. Energy Convers Manage 2018;168:74–84. [2] Galvan D, Chendynski LT, Mantovani AC, Quadri M, Killner M, Cremasco H, et al. Mathematical modeling of the transesterification reaction by finite elements: optimization of kinetic parameters using the simplex sequential method. J Braz Chem Soc 2020;31:313–9. [3] Bashiri H, Pourbeiram N. Biodiesel production through transesterification of soybean oil: a kinetic Monte Carlo study. J Mol Liq 2016;223:10–5. [4] Stamenković OS, Todorović ZB, Lazić ML, Veljković VB, Skala DU. Kinetics of sunflower oil methanolysis at low temperatures. Bioresour Technol 2008;99:1131–40. [5] Kadi MA, Akkouche N, Awad S, Loubar K, Tazerout M. Kinetic study of transesterification using particle swarm optimization method. Heliyon 2019;5:e02146. [6] Freedman B, Butterfield RO, Pryde EH. Transesterification kinetics of soybean oil 1. J Am Oil Chemists’ Soc 1986;63:1375–80. [7] Noureddini H, Zhu D. Kinetics of transesterification of soybean oil. J Am Oil Chemists’ Soc 1997;74:1457–63. [8] Darnoko D, Cheryan M. Kinetics of palm oil transesterification in a batch reactor. J Am Oil Chemists’ Soc 2000;77:1263–7. [9] Komers K, Skopal F, Stloukal R, Machek J. Kinetics and mechanism of the KOH—catalyzed methanolysis of rapeseed oil for biodiesel production. Eur J Lipid Sci Technol 2002;104:728–37. [10] Leevijit T, Wisutmethangoon W, Prateepchaikul G, Tongurai C, Allen M. A second order kinetics of palm oil transesterification. Sustain Energy Environ 2004:277–81. [11] Vicente G, Martínez M, Aracil J, Esteban A. Kinetics of sunflower oil methanolysis. Ind Eng Chem Res 2005;44:5447–54. [12] Vicente G, Martinez M, Aracil J. Kinetics of brassica c arinata oil methanolysis. Energy Fuels 2006;20:1722–6.
4. Conclusion Different approaches to kinetic models and mechanisms for the transesterification reaction were investigated by PSO. All studies showed good agreement between experimental and simulated data, with errors similar to those reported in the literature. The kinetic results suggest that under the evaluated conditions, reverse reactions may be 8
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