Mechanistic Modeling of Palmitic Acid Esterification via

Ind. Eng. Chem. Res. 2011, 50, 1177–1186

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Mechanistic Modeling of Palmitic Acid Esterification via Heterogeneous Catalysis Nuttapol Lerkkasemsan,† Nourredine Abdoulmoumine,‡ Luke Achenie,*,† and Foster Agblevor‡ Department of Chemical Engineering and Department of Biological Systems Engineering, Virginia Polytechnic Institute and State UniVersity, Blacksburg, Virginia 24061, United States

Biodiesel has emerged as a promising renewable and clean energy alternative to petrodiesel. While biodiesel has traditionally been prepared through homogeneous base catalysis, heterogeneous acid catalysis has been investigated recently due to its ability to convert cheaper but high “free fatty acid” content oils such as waste vegetable oil while decreasing production cost. In this work, the esterification of free fatty acid over sulfated zirconia and activated acidic alumina was studied experimentally in a batch reactor. The models of the reaction over the catalysts were developed in two parts. First, a kinetic study was performed using a deterministic model to develop a suitable kinetic expression, and the related parameters were estimated. Second, a stochastic model was developed as a pseudo-verification of the nature of the reaction at the molecular level. The esterification of palmitic acid obeyed the Eley-Rideal mechanism in which palmitic acid and methanol are adsorbed on the surface for SO4/ZrO2-550 °C and AcAl2O3, respectively. The coefficients of determination of the deterministic model were 0.98, 0.99, and 0.99 for SO4/ZrO2-550 °C at 40, 60, and 80 °C, respectively, and 0.99, 0.96, and 0.98 for AcAl2O3 at the same temperature. The deterministic and stochastic models were in good agreement when the number of molecules in the system was large enough. In general for a system consisting of few molecules, the stochastic model is suggested, whereas for a large system, the deterministic continuum model is suggested. If computational effort is not a bottleneck, then the stochastic model could be used in all systems. 1. Introduction Scientific inquiry into the production of biofuels as an alternative source of energy has increased significantly in recent years. This trend was prompted by environmental concerns and the growing evidence of the depletion of natural crude oil reservoirs. Biodiesel has emerged as a renewable biofuel for the replacement of petroleum-based diesel or petrodiesel. Its appeal is due to the fact that it has virtually all of the desirable properties of petroleum-based diesel while being biodegradable and most importantly, carbon neutral.1 Biodiesel could be produced by the transesterification of triacylglycerols through base catalysis (see Figure 1) or esterification of free fatty acids and triacylglycerols by acid catalysis as in Figure 2. However, virtually all industrial biodiesel plants use KOH or NaOH bases to convert refined edible vegetable oils with methanol to its equivalent fatty acid methyl esters (FAME) due to the faster reaction rate of homogeneous base catalysis. In addition, in homogeneous base catalysis, feedstock selection is key to the success and economic feasibility of biodiesel production. Both water and free fatty contents are crucial components in the reaction, and their concentrations must be minimized to prevent side reactions such as saponification and ester hydrolysis. Indeed, in the presence of water, ester hydrolysis of fatty acids on the triacylglycerol backbone could take place, leading to the formation of additional free fatty acids. Free fatty acids could then react with the base catalyst to form fatty acid soaps through saponification. According to Lotero et al., the free fatty acid contents in the oil should be no more than 0.5 wt % beyond which the downstream separation process is hindered by excessive soap formation due to the saponification side reaction.2 Therefore, while homogeneous base catalysis achieves high biodiesel yields, it limits the choice of feedstock * To whom correspondence should be addressed. E-mail: [email protected]. † Department of Chemical Engineering. ‡ Department of Biological Systems Engineering.

to refined oils due to its compliance with the criteria described above. This limitation hinders the competitiveness of biodiesel against petrodiesel. It is estimated that over 80% of the production cost of biodiesel is associated with the cost of refined vegetable oil which is already comparable to the selling price of petrodiesel.3 Consequently, to compete on a large scale, lower cost feedstock and simpler processing schema should be used for biodiesel production. This objective could be achieved by using cheaper and high free fatty acid content oil such as waste vegetable oil and animal fat necessitating a solution for handling the free fatty acids. Among homogeneous processes proposed for biodiesel production from high free fatty acid content oils, a two-stage process where free fatty acids are first converted by acid catalysis via esterification followed by base catalysis has been employed to reduce the free fatty acid content to an acceptable value.4 This approach unfortunately increases the chemical consumption since the acid catalyst used in the first stage must be neutralized before the second stage resulting in increased chemical usage. While heterogeneous base catalysis could also

Figure 1. Transesterification of triacylglerol to produce three fatty acid alkyl esters by base catalysis.

Figure 2. Esterification of a free fatty acid to an alkyl fatty acid ester by acid catalysis.

10.1021/ie100891n  2011 American Chemical Society Published on Web 12/15/2010

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be employed, such a catalyst suffers from the same limitations as its homogeneous counterpart with low esterification activity. Heterogeneous acid catalysts have the most merit because they could simultaneously catalyze the conversion of both free fatty acids and triacylglycerols while reducing the production cost due to catalyst reusability and decrease in separation cost. Berrios et al. reported the kinetic study of free fatty acids in sunflower oil with sulfuric acid as the homogeneous catalyst and modeled the reaction with pseudohomogeneous first order for the forward reaction and second order for the reverse reaction.5 Similarly, Tesser et al. described the esterification of oleic acid over an ion-exchange sulfonic resin by a pseudosecond order kinetic model.6 Meunier and co-workers investigated the esterification of palmitic acid in sunflower oil over a silicasupported Nafion resin and compared its activity with sulfated zirconium oxide.7 These authors showed that the silica-supported Nafion resin performed better than the zirconium oxide. H3PW12O40 heteropolyacid was used as homogeneous catalyst for the ethanolysis of various fatty acids with yields similar to those obtained through the traditional sulfuric acid homogeneous catalysis.8 SnCl2 was successfully used for the esterification of oleic acid in soybean oil using ethanol, and the catalyst was found to yield comparable results with homogeneous H2SO4 catalyzed reactions.9 Despite the recent surge in biodiesel research, there are few investigations on free fatty acid esterification and even fewer investigations of the reaction mechanism over heterogeneous catalysts. Several models (including Langmuir-Hinshelwood and Eley-Rideal models) were used to study the kinetics of palmitic acid esterification on sulfated zirconium oxide and activated acidic alumina catalysts; the models that best fit the experimental data are reported in this article. The model selection criteria were root-mean-square deviation, standard deviation, coefficient of determination, and experimental intuition. Furthermore, the Gillespie model was used to for further verification of the model. The rest of the manuscript is organized as follows. The experimental effort is described in section 2. This is followed by the modeling strategy. Finally we provide and discuss the results. 2. Experiments Details of the experimental protocol will be published elsewhere. Therefore, in this paper, we will provide only a brief summary. Materials. Zirconium oxide and activated acidic alumina were obtained from Alfa Aeasar (Alfa Aesar, Ward Hill, MA). Palmitic acid was purchased from Sigma Aldrich (St. Louis, MO). Methanol, n-hexane, and acetonitrile were all purchased from Fischer Scientific (Fair Lawn, NJ) as analytical grade solvents and used as received. Catalyst Preparation and Characterization. Zirconium oxide was ground to pass through a 20 mesh and subsequently sulfated according the procedure described in procedure described elsewhere.10 After sulfation by incipient impregnation, the catalyst was activated at 550 °C for 6 h and this material was labeled as SO4/ZrO2-550 °C. The activated acidic alumina catalyst was used as received without sulfation but was calcined at 550 °C for 3 h before use and was labeled as AcAl2O3. The sulfated zirconium oxide catalyst was characterized by XPS, SEM-EDS, and BET. The acidic alumina catalyst was characterized by SEM-EDS for chemical composition, and the surface area was provided by the supplier. Palmitic Acid Esterification. The reaction was carried out in a water bath maintained at the appropriate temperature. In

the reaction flask, 2.56 g (0.01 mol) of palmitic acid and 32.04 g (1 mol) of methanol were added and placed in the water bath first to reach the desired temperature before adding the appropriate amount of catalyst. The free fatty acid to methanol content was increased to a 1:100 molar ratio to ensure complete immersion of palmitic acid in the reaction mixture. Furthermore, the higher methanol content prevented palmitic acid precipitation during sample collection, improving the accuracy of the procedure. 3. Modeling Deterministic Kinetic Modeling. SO4/ZrO2-550 °C and AcAl2O3 were used to perform the kinetic experiments at 40, 60, and 80 °C at 10% (w/w) and 100% (w/w). Several models (including Langmuir-Hinshelwood and Eley-Rideal models) were tested and the ones that best fit the experimental data are reported in this article. The total active site on SO4/ZrO2-550 °C was 2.05 wt %. Owing to the lack of actual experimental data, we employed the total active site which was measured via chemisorption of ethylene on alumina; this was reported to be 35% mole.11 It was assumed that an Eley-Rideal mechanism occurred with palmitic acid on the catalyst surface and methanol in the bulk fluid with palmitic acid as the rate-determining step. For AcAl2O3 however, the reaction occurred between methanol on the catalyst surface and palmitic acid in the bulk fluid with methanol adsorption as the rate-determining step. SO4/ZrO2 at 550 °C. The reaction mechanisms of palmitic acid esterification on sulfated zirconia were hypothesized to follow the Eley-Rideal mechanism. The reaction consisted of three elementary reactions as described below. KAB

A + S S AS KS

AS + M S ES + W KDB

ES S E + S Here A is palmitic acid, S is an active site on the surface, M is methanol, W is water, E is palmitic methyl ester, AS and ES are adsorbed intermediates. The kinetic model was built based on following assumptions: 1. The rate of the noncatalyzed reactions could be neglected compared to the catalyzed ones. 2. The diffusion rate of product and reactant over the catalyst surface could be neglected. 3. There were no differences in the activity and accessibility of sites on catalyst surface. 4. The surface reaction was the rate limiting step. 5. The adsorption and desorption of reactants and products were fast and were therefore at equilibrium. 6. There was no palmitic methyl ester and water before the reaction. Based on these assumptions, the following kinetic rate law was derived:

(

rate )

( ))

CWCE KDBKS CE 1 + KABCA + KDB

K1 KABCACM -

where KS, KAM, and KD represent the equilibrium constants of surface reaction, adsorption of palmitic acid, and desorption of palmitic methyl ester, K1 ) CTks,ks is the reverse reaction rate constant.


The concentration of the reactants and products can be expressed as CA ) CA0(1 - X),

CM ) CA0(100 - X)

CE ) CA0X,

CW ) CA0X

Therefore, the equation can be expressed as

rate )

(

K1 KABCA0 (1 - X)(100 - X) 2

1 + KABCA0(1 - X) +

( )) CA02X2

KDBKS CA0X KDB

AcAl2O3 Kinetic Model. Similarly, the reaction mechanism of palmitic acid esterification on AcAl2O3 was hypothesized to follow the Eley-Rideal mechanism. It also consisted of three elementary reactions as KAM

M + S S MS KS

M·S + A S W·S + E KDW

M·S S W + S where M is methanol, S is an active site, A is palmitic acid, W is water, E is palmitic acid ester, and MS and WS are surface intermediates. The kinetic model was built based on the same assumptions for SO4/ZrO2. The following kinetic rate law can be derived:

(

CWCE KDWKSKAMCA rate ) CWCE CE 1+ + KDWKSCA KDW K1 CM -

)

CE ) CA0X,

From the definition of the propensity function

CW ) CA0X

Therefore, the equation can be expressed as

rate )

(

K1 CA0(100 - X) 1+

KDWKSKAM

CA0X2 KDWKS(1 - X)

+

∑ p(τ, j|x, t) dτ)

) p0(τ|x, t) × (1 -

CM ) CA0(100 - X)

CA0X2

In the stochastic approach, it is assumed that each reaction proceeds independently and randomly and can occur with a certain probability associated with the thermodynamic properties of the reacting molecules. The stochastic chemical kinetics describes interactions involving a discrete number of molecules. The stochastic algorithm is appropriate when the number of molecules in a system is small. In a given initial number of molecules, there are many possible time evolutions, each of which has their own probability. The summation of all the probabilities has to be one. There is a link between deterministic and stochastic simulation. The reaction rate constants of the deterministic simulation can be interpreted in terms of probabilities in the stochastic simulation. The differential equations in the deterministic simulation are similar to the stochastic approach. The nature of chemical reactions is stochastic.13 Each reaction is a discrete event that takes place with a given probability. An aspect of this simulation is that it can determine whether a proposed reaction mechanism is consistent with observed result. The stochastic algorithm we employed is as follows.14 1. P0(τ|x,t) as the probability that no reaction occurs in the time interval [t, t + τ). 2. p(τ,j|x,t) is the probability that the next reaction will be the jth reaction and occur during the time interval [t + τ, t + τ + dτ). 3. X(t) ) x is the number of molecules of a given species. 4. aj(X(t)) is a propensity function of the jth reaction. 5. Vj is a state-change vector of jth reaction. 6. cj is a reaction rate constant of jth reaction. For a small enough time step, the assumption is that what happens over [t, t + τ) is independent of what happens over[t + τ, t + τ + dτ). p0(τ + dτ|x, t) ) p0(τ|x, t) ∪ p0(τ|x, t + τ) dτ ) p0(τ|x, t) × p0(τ|x, t + τ) dτ

Here KS, KAM, KD represent the equilibrium constants of surface reaction, adsorption of palmitic acid, and desorption of palmitic acid methyl ester, and K1 ) CTkAM, kAM is the forward rate constant of adsorption of methanol. The concentration of the reactants and products can be expressed as CA ) CA0(1 - X),

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)

CA0X KDW

Stochastic Simulation. Computer simulations can be vital in investigating a chemical reaction. The simulation can aid in the exploration of the complex dynamics of reaction with ease. There are two approaches to consider in computer simulation, namely deterministic and stochastic.12 The deterministic approach uses a set of differential equations to explain the time dependence of the concentrations in the chemical system.

M

∑ a (x) dτ)

p0(τ + dτ|x, t) ) p0(τ|x, t)(1 -

k

k)1

From the proof in ref 15 p0(τ, j|x, t) )

aj(x) (a (x) e-asum(x)τ) asum(x) sum

Here, (aj(x))/(asum(x)) is the next reaction index which corresponds to a discrete random variable that controls the chance of picking the jth reaction while asum(x) e-asum(x)τ is the time until the next reaction and is the density function for a continuous random variable with an exponential distribution. Assuming good mixing and negligible surface diffusion, the resulting algorithm can be summarized in the following pseudocode: 1. Evaluate {ak(X(t))}kM) 1 and M

asum(X(t)): )

∑ a (X(t)) k

k)1

2. Draw two independent uniform (0,1) random numbers, ξ1 and ξ2. 3. Set j to be the smallest integer satisfying

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∑ a (X(t)) > ξ a

1 sum(X(t))

k

k)1

4. Set

()

1 ξ2 τ) asum(X(t)) ln

5. Set X(t + τ) ) X(t) + Vj 6. Return to step 1. SO4/ZrO2-550 °C Stochastic Simulation. There are six subchemical reactions that occur during the esterification of palmitic acid over SO4/ZrO2-550 °C. The solution contains seven species and X ) Xn is the number of molecules of the nth species. The six subreactions are illustrated below: C1

X1 + X2 98 X3

[] [] [] [] [] []

-1 1 0 -1 1 0 1 -1 -1 V1 ) 0 V2 ) 0 V3 ) -1 0 0 1 0 0 1 0 0 0

j

(1)

0 0 0 0 1 -1 1 0 0 V4 ) 1 V5 ) 0 V6 ) 0 -1 -1 1 -1 0 0 0 1 -1

AcAl2O3 Stochastic Simulation. The reactions are expressed as below: C1

X1 + X2 98 X3

(7)

a1 ) C1X1X2

(7a)

C2

a1 ) C1X1X2 C2

X3 98 X1 + X2

(1a)

X3 98 X1 + X2

(8)

a2 ) C2X2

(8a)

(2)

C3

X3 + X4 98 X5 + X6

(9)

(2a)

a3 ) C3X3X4

(9a)

X3 + X4 98 X5 + X6

(3)

X5 + X6 98 X3 + X4

(10)

a3 ) C3X3X4

(3a)

a4 ) C4X5X6

(10a)

a2 ) C 2 X2 C3

C4

C5

C4

X5 + X6 98 X3 + X4 a4 ) C4X5X6

X5 98 X7 + X2

(11)

a5 ) C5X5

(11a)

(4)

(4a)

X2 + X7 98 X5

(12)

X5 98 X7 + X2

(5)

a6 ) C6X2X7

(12a)

a5 ) C 5 X5

(5a)

C5

C6

X2 + X7 98 X5

(6)

C6

where X1 is for methanol, X2 is for the vacant sites of the catalyst, (dm3)/(gcat min) is for methanol adsorbed on the catalyst, X is for palmitic acid, X5 is for palmitic acid methyl ester/methanol intermediate adsorbed on the catalyst, X6 is for palmitic acid methyl ester and finally X7 is for water. 4. Results and Discussion

a6 ) C6X2X7

(6a)

Here X1 is for palmitic acid, X2 is for the vacant sites on the catalyst, X3 is for the palmitic acid attached to the catalyst, X4 is for the methanol, X5 is for palmitic acid methyl ester attached to the catalyst, X6 is for water, and X7 is for the palmitic acid methyl ester. In step 5, the state vectors (Vj) of reactions 1-6 are

4.1. Characterization of Catalysts. The catalysts used in this study were characterized to determine the elemental chemical composition and the surface area, both important characteristics to know for future reference. The surface areas of both catalysts are shown in Table 1. The elemental composition analysis of SO4/ZrO2-550 °C and AcAl2O3 was carried out by SEM-EDS and XPS as shown in Table 2 and Table 3. While both techniques are adequate for elemental analysis, XPS

Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011 Table 1. BET Surface of the Catalysts SO4/ZrO2-550 °C

AcAl2O3

90.9

150

Surface area (m2/g)

Table 2. Elemental Composition of SO4/ZrO2-550°C

Zr O

SEM-EDS

XPS

74.31 ( 9.39 20.99 ( 8.81

15.07 ( 3.27 48.86 ( 3.79

C S

SEM-EDS

XPS

3.66 ( 1.55 1.04 ( 0.25

34.01 ( 4.07 2.05 ( 0.29

Table 3. Elemental Composition of AcAl2O3 by SEM-EDS Al O C

40.15 ( 0.64 54.10 ( 0.85 5.75 ( 0.55

is more surface sensitive with penetration depth of 1-10 nm. SEM-EDS scans deeper with a penetration depth of 0.3-5 µm, therefore providing a more accurate overall elemental composition of the catalysts. 4.2. Deterministic Kinetic Modeling. The esterification of palmitic acid with methanol on SO4/ZrO2-550 °C and AcAl2O3 follows the Eley-Rideal mechanism. For SO4/ZrO2-550 °C, the reaction of adsorbed palmitic acid with methanol in bulk fluid is the rate determining step for SO4/ZrO2-550 °C (Figure 3). In contrast, the adsorption of methanol on an active site on the catalyst was the rate-limiting step in AcAl2O3 (Figure 4). The disparity between the adsorbed species is attributed to the nature of the acidic sites on the catalysts. Sulfated zirconium

Table 4. Kinetic Parameters of Palmitic Acid Esterification on SO4/ZrO2-550°C

temp (°C)

K1 × 10-4 (g (dm3)2/ 3 (gcat min2))

KAB × 10-2 (dm3/mol)

KDB (mol/dm3)

KS × 10-3 (dm3/ (gcat min))

40 60 80

3.78 16.12 51.10

59.75 26.74 13.38

2.01 2.01 9.08

4.21 19.38 92.41

Table 5. Kinetic Parameters of Palmitic Acid Esterification on AcAl2O3

temp (°C)

K1 × 10-4 (g (dm3)2/ 3 (gcat min2))

KAM (dm3/mol)

KDB (mol/dm3)

KS × 10-1 (dm3/ (gcat min))

40 60 80

1.36 0.91 1.50

2.01 0.14 0.08

2.17 2.01 2.01

0.02 0.21 1.14

oxide has been reported to have Bro¨nsted and Lewis acid. When the reaction was carried over pure zirconium oxide, the esterification yield was minimal therefore it was assumed that the catalyst activity was mostly due to the Bro¨nsted acid sites generated by the sulfation. Activated acidic alumina however only has Lewis acid sites. The adsorption of palmitic acid is believed to be a nucleophilic interaction between the Bro¨nsted acid site and the carbonyl of the carboxylic moiety similar to homogeneous acid catalyzed esterification. On the activated acidic alumina however, Lewis acid sites are predominant leading to the chemisoption of methanol. Therefore, Bro¨nsted acid behaves differently from Lewis acid in the esterification of free fatty acid. The estimated kinetic parameters are reported in Table 4 and Table 5 for SO4/ZrO2-550 °C and AcAl2O3, respectively. The coefficients of determination of the deterministic model were 0.98, 0.99, and 0.99 for SO4/ZrO2-550 °C at 40, 60, and 80 °C, respectively, and 0.99, 0.96, and 0.98 for AcAl2O3 at the same temperature. To evaluate the efficiency of the catalysts, the activation energies were determined. The reaction rate constant is related to the reaction temperature through the Arrhenius equation. Therefore, the overall reaction activation energy can be calculated from equation below using the reaction rate constant:

(

KS ) A exp Figure 3. Esterification of palmitic acid on zirconia sulfate. Fitting of Eley-Rideal model to experimental data for palmitic acid esterification over SO4/ZrO2-550 °C at 40, 60, and 80 °C.

Figure 4. Esterification of palmitic acid on alumina. Fitting of Eley-Rideal model to experimental data for palmitic acid esterification over SO4/ZrO2550 °C at 40, 60, and 80 °C.

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∆E RT

)

Both the frequency factor A and the activation energy Ea were obtained by regressing ln(KS) versus 1/T. The activation energy of palmitic acid esterification was 70.81 kJ/mol and 93.71 KJ/ mol for SO4/ZrO2-550 °C and AcAl2O3, respectively. From the activation energy of both catalysts, SO4/ZrO2-550 °C seems to be a better catalyst in the esterification of palmitic acid which is a free fatty acid. However, this catalyst requires sulfation and temperature activation before use. On the other hand, AcAl2O3 does not require sulfation, though 3 hours of heating at 550 °C is required in order remove moisture and atmospheric carbon from the surface. Therefore, it is more convenient for industrial use where easy-to-prepare catalysts are needed, although the operation cost of AcAl2O3 may be higher because of the higher activation energy requiring higher temperature. The esterification mechanisms of palmitic acid which occur over both SO4/ZrO2-550 °C and AcAl2O3 are depicted in Figure 5 and Figure 6, respectively. 4.3. Predicted Conversion Using Deterministic Model. To validate the models, we used the models to predict the percent conversion at 210 and 240 min at all temperatures for all catalysts. We then compared the predicted values with actual experimental data.

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Figure 8. Prediction of esterification of palmitic acid on alumina. Table 6. Reaction Parameters Determined by Stochastic Model on SO4/ZrO2-550 °C

Figure 5. Palmitic acid esterification over SO4/ZrO2-550 °C.

temp k1 ((dm3/ k2 (dm3 k3 ((dm3/ k4 (dm3/ k5 ((mol/ k6 (mol/ (°C) mol)2) /mol) (gcat min))2) (gcat min)) dm3)2) dm3) 40 60 80

0.30 0.80 1.14

0.5 3 8.5

1.28 6.55 6.92

304.04 337.84 74.88

4.02 1 4.54

2 0.5 0.5

Table 7. Reaction Parameters Determined by Stochastic Model on AcAl2O3 temp (°C) 40 60 80

Figure 6. Palmitic acid esterification over AcAl2O3.

Figure 7. Prediction of esterification of palmitic acid over zirconia sulfate.

As can be seen in Figure 7 and Figure 8, the deterministic models used to predict the conversion at time greater than 180 min showed good accuracy. The values of the deterministic models are very close to the experimental data. Therefore, these rate law models can be used to predict the experimental data at

k1 × 10-2 ((dm3/ k2 × 10-3 k3 ((dm3/ k4 (dm3/ k5 ((mol/ k6 (mol/ mol)2) (dm3/mol) (gcat min))2) (gcat min)) dm3)2) dm3) 1.89 1.27 2.08

0.94 9.10 271.59

9.70 6.23 10.82

500.00 300.00 95.00

1.19 16.08 20.10

0.55 8.00 10.00

different temperature between 40 and 80 °C and at longer reaction time. 4.4. Results for Stochastic Kinetic Modeling. The stochastic simulation is used as a tool for observing the nature of the reaction which occurs over the surface of the catalysts. In systems such as some biological processes, the reaction can be accurately simulated using a stochastic simulation. The major difference at the molecular level between deterministic and stochastic is that the deterministic model makes a continuum assumption in the concentrations, whereas the stochastic model investigates the intrinsic nature of the reaction and therefore considers the number of molecules that are reacting. In a chemical reaction each individual reaction is an event that occurs with a certain probability. From a practical point of view, the deterministic model is easier to set up and faster in terms of calculation compared to the stochastic model. When dealing with a small number of molecules, the deterministic model is not appropriate. To observe the reactions which occur over the surface of the catalyst, a small number of molecules in the system will be observed. There is a link between deterministic and stochastic simulation. The rate constant (Cj) for stochastic simulation are estimated from kinetic parameters (kj) (Tables 6 and 7) in the deterministic simulation as C1 )

k1 , nAvol

C2 ) k2,

C3 )

C4 ) In addition

k3 , nAvol

k4 , nAvol

C5 ) k5,

C6 )

k6 nAvol


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Figure 9. Stochastic simulation vs Eley-Rideal model on SO4/ZrO2-550 °C at 40 °C: (a) stochastic simulation on SO4/ZrO2-550 °C; (b) compare conversion from stochastic model vs deterministic model on SO4/ZrO2-550 °C.


KAM )

k1 , k2

KS )

k3 k4

and kDB )

k5 k6

Palmitic Acid Esterification over SO4/ZrO2-550 °C. The reaction parameters of palmitic acid esterification over SO4/ ZrO2-550 °C are determined as in Table 6. Also, the results of the stochastic simulations for SO4/ZrO2-550 °C at different temperatures are shown in Figures 9-11. As the number of molecules in the system increased, the stochastic simulation algorithms generated smooth graphs that are comparable to those produced in the deterministic model. This was also observed as the temperature increased due to increased frequency of collision and therefore higher likelihood

of reaction. The number of molecules employed were 1800, 1200, and 600 for both SO4/ZrO2 and AcAl2O3 at temperatures of 40, 60, and 80 °C, respectively. The stochastic simulation acts as a pseudovalidation of the Eley-Rideal mechanism. The deterministic behavior represents the average of all these possible stochastic evolutions. Thus we note that the deterministic model and stochastic simulation are very close as illustrated in Figure 9 through 14 for both catalysts. Consequently, the deterministic result should be enough to describe the mechanism of the system. However, for other more complex catalyst such as enzyme, the stochastic simulation could be essential in investigating the reaction mechanisms.

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Figure 12. Stochastic simulation vs Eley-Rideal model on AcAl2O3 at 40 °C: (a) stochastic simulation on AcAl2O3; (b) compare conversion from stochastic model vs deterministic model on AcAl2O3.

Palmitic Acid Esterification over AcAl2O3. The reaction parameters of palmitic acid esterification over AcAl2O3 were determined as in Table 7. In addition, the results of stochastic simulations for alumina at different temperatures are shown in Figures 12-14. The rate constant (Cj) can be calculate from (kj) as C1 )

k1 , C ) k2, nAvol′ 2

k3 , nAvol′ k4 , C5 ) k5, C4 ) nAvol′ C3 )

C6 )

k6 nAvol

Computational Time for Stochastic Simulations. The computational time (in CPU seconds) for palmitic acid esterification over both systems (SO4/ZrO2-550 °C and AcAl2O3) are given in Table 8. The runs were conducted on a Intel Core2 Duo P8400 2.27 GHz computer running the VISTA operating system. We note that the CPU time increases from 40 to 60 °C and then drops afterwardsit is not clear why this is the case. However we conjecture that this probably has to do with the balance between two driving forces: (a) increasing collision rate and therefore the greater the probability that a reaction occurs within a very short time interval and (b) the faster depletion of reactants at higher temperatures.


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Figure 13. Stochastic simulation vs Eley-Rideal model on AcAl2O3 at 60 °C: (a) stochastic simulation on AcAl2O3; (b) compare conversion from stochastic model vs deterministic model on AcAl2O3.

Figure 14. Stochastic simulation vs Eley-Rideal model on AcAl2O3 at 80 °C: (a) stochastic simulation on AcAl2O3; (b) compare conversion from stochastic model vs deterministic model on AcAl2O3. Table 8. CPU Time for Stochastic Simulations Temperature [)] (C)

CPU time for zirconia [)] (s)

CPU time for alumina [)] (s)

40 60 80

24.2347 173.6447 0.6084

1.3728 8.7205 1.7784

5. Conclusion We employed both deterministic and stochastic simulation in this study. The stochastic simulation generated smooth graphs (which are a characteristic of the deterministic simulation) as the number of molecules in the system increased. This was also observed as the temperature increased due to increased fre-

quency of collision and therefore higher likelihood of reaction. These results help us to answer the question, what is the critical number of molecules required for the stochastic simulation to track the deterministic simulation? This would give a rough guide as to when to employ which method. Since in general the deterministic simulation is faster than the stochastic simulation, there would be a gain in speed if the deterministic simulation is adequate. However if speed is of no consequence, then the safer route is to simply employ stochastic simulation. The second goal was to use the stochastic simulation as a pseudovalidation of the deterministic simulation. The results suggest that the Eley-Rideal mechanism is appropriate to explain the reaction mechanism which occurs in

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the esterification of palmitic acid over both SO4/ZrO2-550 °C and AcAl2O3. The reaction of adsorbed palmitic acid with methanol in bulk fluid is the rate determining step for SO4/ ZrO2-550 °C. On the other hand, the adsorption of methanol on an active site on the catalyst was the rate-limiting step in AcAl2O3. This difference in adsorbed species is attributed the nature of the acid sites: SO4/ZrO2-550 °C has mostly Bro¨nsted acid sites, whereas AcAl2O3 has Lewis acid sites. Furthermore, the deterministic model and the stochastic simulation are in good agreement. It is therefore sufficient to use the deterministic model for the future kinetic investigation of free fatty esterification over heterogeneous catalysts. Furthermore, we showed that these models can be used to predict the conversion for different times. The activation energy of SO4/ZrO2-550 °C is 70.81 kJ/mol while that of AcAl2O3 is 93.70 kJ/mol indicating that SO4/ZrO2-550 °C is better in the esterification for free fatty acids. Both SO4/ZrO2-550 °C and AcAl2O3 are therefore suitable catalysts for the conversion of high free fatty acid containing oils such as waste cooking oil, a cheaper alternative feedstock, for the production of renewable diesel through an environmentally friendly process. Finally, this investigation of the esterification of free fatty acids will not only help in reactor design but could also serve as a model for further kinetic studies of transesterification of vegetable oil to biodiesel. Moreover, the combination of deterministic modeling and stochastic simulation can be used as a model to study more complex catalyst such as enzyme catalyst. Acknowledgment The authors duly acknowledge the Institute for Critical Technology and Applied Science at Virginia Polytechnic Institute and State University for funding this project. Nomenclature A ) palmitic acid E ) palmitic acid methyl ester M ) methanol S ) catalyst active site W ) water CT ) total catalyst active sites AS ) palmitic acid adsorbed on an active site ES ) intermediate palmitic acid methyl ester on the catalyst surface CA ) concentration of palmitic acid, M CB ) concentration of methanol, M CE ) concentration of palmitic acid methyl ester, M CW ) concentration of water, M CA0 ) initial concentration of palmitic acid, M KAB ) adsorption parameter of palmitic acid, dm3/mol KDB ) desorption parameter of palmitic acid, mol/dm3

KAM ) adsorption parameter of methanol, dm3/mol KS ) equilibrium rate constant, KS ) k-S/kS k-s ) kinetic constants of the forward reaction, dm3/(gcat min) ks ) kinetic constants of the reverse reaction, dm3/(gcat min) X ) conversion of palmitic acid

Literature Cited (1) Kulkarni, M. G.; Gopinath, R.; Meher, L. C.; Dalai, A. K. Solid acid catalyzed biodiesel production by simultaneous esterification and transesterification. Green Chem. 2006, 8 (12), 1056–1062. (2) Lotero, E.; Liu, Y.; Lopez, D. E.; Suwannakarn, K.; Bruce, D. A.; Goodwin, J. G. Synthesis of biodiesel via acid catalysis. Ind. Eng. Chem. Res. 2005, 44 (14), 5353–5363. (3) (a) Apostolakou, A. A.; Kookos, I. K.; Marazioti, C.; Angelopoulos, K. C. Techno-economic analysis of a biodiesel production process from vegetable oils. Fuel Process. Technol. 2009, 90 (7-8), 1023–1031. (b) Di Serio, M.; Tesser, R.; Pengmei, L.; Santacesaria, E. Heterogeneous catalysts for biodiesel production. Energy Fuels 2007, 22 (1), 207–217. (c) Encinar, J. M.; Gonzalez, J. F.; Rodriguez-Reinares, A. Biodiesel from used frying oil. Variables affecting the yields and characteristics of the biodiesel. Ind. Eng. Chem. Res. 2005, 44 (15), 5491–5499. (4) Canacki, M.; Gerpen, J. V. Biodiesel production from oils and fats with high free fatty acids. Trans. Am. Soc. Agric. Eng. 2001, 44, 1429– 1436. (5) Berrios, M.; Siles, J.; Martı´n, M. A.; Martı´n, A. A kinetic study of the esterification of free fatty acids (FFA) in sunflower oil. Fuel 2007, 86 (15), 2383–2388. (6) Tesser, R.; Casale, L.; Verde, D.; Di Serio, M.; Santacesaria, E. Kinetics of free fatty acids esterification: Batch and loop reactor modeling. Chem. Eng. J. 2009, 154 (1-3), 25–33. (7) Ni, J.; Meunier, F. C. Esterification of free fatty acids in sunflower oil over solid acid catalysts using batch and fixed bed-reactors. Appl. Catal. A: Gen. 2007, 333 (1), 122–130. (8) Cardoso, A.; Augusti, R.; Da Silva, M. Investigation on the esterification of fatty acids catalyzed by the H3PW12O40 heteropolyacid. J. Am. Oil Chem. Soc. 2008, 85 (6), 555–560. (9) Cardoso, A.; Neves, S.; da Silva, M. Esterification of oleic acid for biodiesel production catalyzed by SnCl2: A kinetic investigation. Energies 2008, 1 (2), 79–92. (10) Arata, K. Preparation of solid superacid catalysts. Sekiyu Gakkaishi 1996, 39 (3), 185–193. (11) Amenomiya, Y.; Cvetanovic, R. J. Active sites of alumina and silica-alumina as observed by temperature programmed desorption. J. Catal. 1970, 18 (3), 329–337. (12) Mira, J.; Fenandez, C. G.; Urreaga, J. M. Two examples of deterministic versus stochastic modeling of chemical reactions. J. Chem. Educ. 2003, 80 (12), 1488–1493. (13) de Levie, R. Stochastics, the basis of chemical dynamics. J. Chem. Educ. 2000, 77 (6), 771–774. (14) Gillespie, D. T. Stochastic simulation of chemical kinetics. Annu. ReV. Phys. Chem. 2007, 58, 35–55. (15) Higham, D. J. Modeling and simulating chemical reactions. SIAM ReView 2008, 50 (2), 347–368.

ReceiVed for reView April 15, 2010 ReVised manuscript receiVed October 19, 2010 Accepted November 24, 2010 IE100891N

Mechanistic Modeling of Palmitic Acid Esterification via

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