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 PolytechnicInstitute and State UniVersity, Blacksburg, Virginia 24061, United States

Biodiesel has emerged as a promising renewable and clean energy alternative to petrodiesel. While biodieselhas traditionally been prepared through homogeneous base catalysis, heterogeneous acid catalysis has beeninvestigated recently due to its ability to convert cheaper but high “free fatty acid” content oils such as wastevegetable oil while decreasing production cost. In this work, the esterification of free fatty acid over sulfatedzirconia and activated acidic alumina was studied experimentally in a batch reactor. The models of the reactionover the catalysts were developed in two parts. First, a kinetic study was performed using a deterministicmodel to develop a suitable kinetic expression, and the related parameters were estimated. Second, a stochasticmodel was developed as a pseudo-verification of the nature of the reaction at the molecular level. Theesterification of palmitic acid obeyed the Eley-Rideal mechanism in which palmitic acid and methanol areadsorbed on the surface for SO4/ZrO2-550 °C and AcAl2O3, respectively. The coefficients of determinationof 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 werein good agreement when the number of molecules in the system was large enough. In general for a systemconsisting of few molecules, the stochastic model is suggested, whereas for a large system, the deterministiccontinuum model is suggested. If computational effort is not a bottleneck, then the stochastic model could beused in all systems.

1. Introduction

Scientific inquiry into the production of biofuels as analternative source of energy has increased significantly in recentyears. This trend was prompted by environmental concerns andthe growing evidence of the depletion of natural crude oilreservoirs. Biodiesel has emerged as a renewable biofuel forthe replacement of petroleum-based diesel or petrodiesel. Itsappeal is due to the fact that it has virtually all of the desirableproperties of petroleum-based diesel while being biodegradableand most importantly, carbon neutral.1

Biodiesel could be produced by the transesterification oftriacylglycerols through base catalysis (see Figure 1) or esteri-fication of free fatty acids and triacylglycerols by acid catalysisas in Figure 2. However, virtually all industrial biodiesel plantsuse KOH or NaOH bases to convert refined edible vegetableoils with methanol to its equivalent fatty acid methyl esters(FAME) due to the faster reaction rate of homogeneous basecatalysis. In addition, in homogeneous base catalysis, feedstockselection is key to the success and economic feasibility ofbiodiesel production. Both water and free fatty contents arecrucial components in the reaction, and their concentrations mustbe minimized to prevent side reactions such as saponificationand ester hydrolysis. Indeed, in the presence of water, esterhydrolysis of fatty acids on the triacylglycerol backbone couldtake place, leading to the formation of additional free fatty acids.Free fatty acids could then react with the base catalyst to formfatty acid soaps through saponification. According to Lotero etal., the free fatty acid contents in the oil should be no morethan 0.5 wt % beyond which the downstream separation processis hindered by excessive soap formation due to the saponificationside reaction.2 Therefore, while homogeneous base catalysisachieves high biodiesel yields, it limits the choice of feedstock

to refined oils due to its compliance with the criteria describedabove. This limitation hinders the competitiveness of biodieselagainst petrodiesel.

It is estimated that over 80% of the production cost ofbiodiesel is associated with the cost of refined vegetable oilwhich is already comparable to the selling price of petrodiesel.3

Consequently, to compete on a large scale, lower cost feedstockand simpler processing schema should be used for biodieselproduction. This objective could be achieved by using cheaperand high free fatty acid content oil such as waste vegetable oiland animal fat necessitating a solution for handling the free fattyacids.

Among homogeneous processes proposed for biodieselproduction from high free fatty acid content oils, a two-stageprocess where free fatty acids are first converted by acidcatalysis via esterification followed by base catalysis has beenemployed to reduce the free fatty acid content to an acceptablevalue.4 This approach unfortunately increases the chemicalconsumption since the acid catalyst used in the first stage mustbe neutralized before the second stage resulting in increasedchemical usage. While heterogeneous base catalysis could also

* To whom correspondence should be addressed. E-mail:achenie@vt.edu.

† Department of Chemical Engineering.‡ Department of Biological Systems Engineering.

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

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

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

10.1021/ie100891n 2011 American Chemical SocietyPublished on Web 12/15/2010

Dow

nloa

ded

by I

ND

IAN

IN

ST O

F T

EC

HN

OL

OG

Y B

OM

BA

Y o

n Se

ptem

ber

4, 2

015

| http

://pu

bs.a

cs.o

rg

Pub

licat

ion

Dat

e (W

eb):

Dec

embe

r 15

, 201

0 | d

oi: 1

0.10

21/ie

1008

91n

be employed, such a catalyst suffers from the same limitationsas its homogeneous counterpart with low esterification activity.Heterogeneous acid catalysts have the most merit because theycould simultaneously catalyze the conversion of both free fattyacids and triacylglycerols while reducing the production costdue to catalyst reusability and decrease in separation cost.

Berrios et al. reported the kinetic study of free fatty acids insunflower oil with sulfuric acid as the homogeneous catalystand modeled the reaction with pseudohomogeneous first orderfor the forward reaction and second order for the reverse reac-tion.5 Similarly, Tesser et al. described the esterification of oleicacid over an ion-exchange sulfonic resin by a pseudosecondorder kinetic model.6 Meunier and co-workers investigated theesterification of palmitic acid in sunflower oil over a silica-supported Nafion resin and compared its activity with sulfatedzirconium oxide.7 These authors showed that the silica-supportedNafion resin performed better than the zirconium oxide.H3PW12O40 heteropolyacid was used as homogeneous catalystfor the ethanolysis of various fatty acids with yields similar tothose obtained through the traditional sulfuric acid homogeneouscatalysis.8 SnCl2 was successfully used for the esterification ofoleic acid in soybean oil using ethanol, and the catalyst wasfound to yield comparable results with homogeneous H2SO4

catalyzed reactions.9 Despite the recent surge in biodieselresearch, there are few investigations on free fatty acid esteri-fication and even fewer investigations of the reaction mechanismover heterogeneous catalysts.

Several models (including Langmuir-Hinshelwood andEley-Rideal models) were used to study the kinetics of palmiticacid esterification on sulfated zirconium oxide and activatedacidic alumina catalysts; the models that best fit the experimentaldata are reported in this article. The model selection criteriawere root-mean-square deviation, standard deviation, coefficientof determination, and experimental intuition. Furthermore, theGillespie model was used to for further verification of the model.The rest of the manuscript is organized as follows. Theexperimental effort is described in section 2. This is followedby the modeling strategy. Finally we provide and discuss theresults.

2. Experiments

Details of the experimental protocol will be publishedelsewhere. Therefore, in this paper, we will provide only a briefsummary.

Materials. Zirconium oxide and activated acidic aluminawere 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 purchasedfrom Fischer Scientific (Fair Lawn, NJ) as analytical gradesolvents and used as received.

Catalyst Preparation and Characterization. Zirconiumoxide was ground to pass through a 20 mesh and subsequentlysulfated according the procedure described in procedure de-scribed elsewhere.10 After sulfation by incipient impregnation,the catalyst was activated at 550 °C for 6 h and this materialwas labeled as SO4/ZrO2-550 °C. The activated acidic aluminacatalyst was used as received without sulfation but was calcinedat 550 °C for 3 h before use and was labeled as AcAl2O3. Thesulfated zirconium oxide catalyst was characterized by XPS,SEM-EDS, and BET. The acidic alumina catalyst was charac-terized by SEM-EDS for chemical composition, and the surfacearea was provided by the supplier.

Palmitic Acid Esterification. The reaction was carried outin a water bath maintained at the appropriate temperature. In

the reaction flask, 2.56 g (0.01 mol) of palmitic acid and 32.04g (1 mol) of methanol were added and placed in the water bathfirst to reach the desired temperature before adding the ap-propriate amount of catalyst. The free fatty acid to methanolcontent was increased to a 1:100 molar ratio to ensure completeimmersion of palmitic acid in the reaction mixture. Furthermore,the higher methanol content prevented palmitic acid precipitationduring sample collection, improving the accuracy of theprocedure.

3. Modeling

Deterministic Kinetic Modeling. SO4/ZrO2-550 °C andAcAl2O3 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 arereported in this article. The total active site on SO4/ZrO2-550°C was 2.05 wt %. Owing to the lack of actual experimentaldata, we employed the total active site which was measuredvia chemisorption of ethylene on alumina; this was reported tobe 35% mole.11 It was assumed that an Eley-Rideal mechanismoccurred with palmitic acid on the catalyst surface and methanolin the bulk fluid with palmitic acid as the rate-determining step.For AcAl2O3 however, the reaction occurred between methanolon the catalyst surface and palmitic acid in the bulk fluid withmethanol adsorption as the rate-determining step.

SO4/ZrO2 at 550 °C. The reaction mechanisms of palmiticacid esterification on sulfated zirconia were hypothesized tofollow the Eley-Rideal mechanism. The reaction consisted ofthree elementary reactions as described below.

Here A is palmitic acid, S is an active site on the surface, Mis methanol, W is water, E is palmitic methyl ester, AS and ESare adsorbed intermediates. The kinetic model was built basedon following assumptions:

1. The rate of the noncatalyzed reactions could be neglectedcompared to the catalyzed ones.

2. The diffusion rate of product and reactant over the catalystsurface could be neglected.

3. There were no differences in the activity and accessibilityof 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 wasderived:

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

A + S SKAB

AS

AS + MSKS

ES + W

ES SKDB

E + S

rate )K1(KABCACM - ( CWCE

KDBKS))

1 + KABCA +CE

KDB

1178 Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011

Dow

nloa

ded

by I

ND

IAN

IN

ST O

F T

EC

HN

OL

OG

Y B

OM

BA

Y o

n Se

ptem

ber

4, 2

015

| http

://pu

bs.a

cs.o

rg

Pub

licat

ion

Dat

e (W

eb):

Dec

embe

r 15

, 201

0 | d

oi: 1

0.10

21/ie

1008

91n

The concentration of the reactants and products can beexpressed as

Therefore, the equation can be expressed as

AcAl2O3 Kinetic Model. Similarly, the reaction mechanismof palmitic acid esterification on AcAl2O3 was hypothesized tofollow the Eley-Rideal mechanism. It also consisted of threeelementary reactions as

where M is methanol, S is an active site, A is palmitic acid, Wis water, E is palmitic acid ester, and MS and WS are surfaceintermediates. The kinetic model was built based on the sameassumptions for SO4/ZrO2. The following kinetic rate law canbe derived:

Here KS, KAM, KD represent the equilibrium constants of surfacereaction, adsorption of palmitic acid, and desorption of palmiticacid methyl ester, and K1 ) CTkAM, kAM is the forward rateconstant of adsorption of methanol.

The concentration of the reactants and products can beexpressed as

Therefore, the equation can be expressed as

Stochastic Simulation. Computer simulations can be vitalin investigating a chemical reaction. The simulation can aid inthe 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 ap-proach uses a set of differential equations to explain the timedependence of the concentrations in the chemical system.

In the stochastic approach, it is assumed that each reactionproceeds independently and randomly and can occur with acertain probability associated with the thermodynamic propertiesof the reacting molecules. The stochastic chemical kineticsdescribes interactions involving a discrete number of molecules.The stochastic algorithm is appropriate when the number ofmolecules in a system is small. In a given initial number ofmolecules, there are many possible time evolutions, eachof which has their own probability. The summation of all theprobabilities has to be one.

There is a link between deterministic and stochastic simula-tion. The reaction rate constants of the deterministic simulationcan be interpreted in terms of probabilities in the stochasticsimulation. The differential equations in the deterministicsimulation are similar to the stochastic approach. The nature ofchemical reactions is stochastic.13 Each reaction is a discreteevent that takes place with a given probability. An aspect ofthis simulation is that it can determine whether a proposedreaction mechanism is consistent with observed result. Thestochastic algorithm we employed is as follows.14

1. P0(τ|x,t) as the probability that no reaction occurs in thetime interval [t, t + τ).

2. p(τ,j|x,t) is the probability that the next reaction will bethe 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).

From the definition of the propensity function

From the proof in ref 15

Here, (aj(x))/(asum(x)) is the next reaction index which corre-sponds to a discrete random variable that controls the chanceof picking the jth reaction while asum(x) e-asum(x)τ is the time untilthe next reaction and is the density function for a continuousrandom variable with an exponential distribution.

Assuming good mixing and negligible surface diffusion, theresulting algorithm can be summarized in the followingpseudocode:

1. Evaluate {ak(X(t))}k ) 1M and

2. Draw two independent uniform (0,1) random numbers, �1

and �2.3. Set j to be the smallest integer satisfying

CA ) CA0(1 - X), CM ) CA0

(100 - X)

CE ) CA0X, CW ) CA0

X

rate )K1(KABCA0

2(1 - X)(100 - X) - (CA0

2X2

KDBKS))

1 + KABCA0(1 - X) +

CA0X

KDB

M + S SKAM

MS

M ·S + ASKS

W ·S + E

M ·S SKDW

W + S

rate )K1(CM -

CWCE

KDWKSKAMCA)

1 +CWCE

KDWKSCA+

CE

KDW

CA ) CA0(1 - X), CM ) CA0

(100 - X)

CE ) CA0X, CW ) CA0

X

rate )K1(CA0

(100 - X) -CA0

X2

KDWKSKAM)

1 +CA0

X2

KDWKS(1 - X)+

CA0X

KDW

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

) p0(τ|x, t) × (1 - ∑ p(τ, j|x, t) dτ)

p0(τ + dτ|x, t) ) p0(τ|x, t)(1 - ∑k)1

M

ak(x) dτ)

p0(τ, j|x, t) )aj(x)

asum(x)(asum(x) e-asum(x)τ)

asum(X(t)): ) ∑k)1

M

ak(X(t))

Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011 1179

Dow

nloa

ded

by I

ND

IAN

IN

ST O

F T

EC

HN

OL

OG

Y B

OM

BA

Y o

n Se

ptem

ber

4, 2

015

| http

://pu

bs.a

cs.o

rg

Pub

licat

ion

Dat

e (W

eb):

Dec

embe

r 15

, 201

0 | d

oi: 1

0.10

21/ie

1008

91n

4. Set

5. Set

6. Return to step 1.SO4/ZrO2-550 °C Stochastic Simulation. There are six

subchemical reactions that occur during the esterification ofpalmitic acid over SO4/ZrO2-550 °C. The solution contains sevenspecies and X ) Xn is the number of molecules of the nthspecies. The six subreactions are illustrated below:

Here X1 is for palmitic acid, X2 is for the vacant sites on thecatalyst, X3 is for the palmitic acid attached to the catalyst, X4

is for the methanol, X5 is for palmitic acid methyl ester attachedto the catalyst, X6 is for water, and X7 is for the palmitic acidmethyl ester.

In step 5, the state vectors (Vj) of reactions 1-6 are

AcAl2O3 Stochastic Simulation. The reactions are expressedas below:

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 isfor palmitic acid, X5 is for palmitic acid methyl ester/methanolintermediate adsorbed on the catalyst, X6 is for palmitic acidmethyl ester and finally X7 is for water.

4. Results and Discussion

4.1. Characterization of Catalysts. The catalysts used inthis study were characterized to determine the elementalchemical composition and the surface area, both importantcharacteristics to know for future reference. The surface areasof both catalysts are shown in Table 1. The elemental composi-tion analysis of SO4/ZrO2-550 °C and AcAl2O3 was carriedout by SEM-EDS and XPS as shown in Table 2 and Table 3.While both techniques are adequate for elemental analysis, XPS

∑k)1

j

ak(X(t)) > �1asum(X(t))

τ )ln( 1

�2)

asum(X(t))

X(t + τ) ) X(t) + Vj

X1 + X298C1

X3 (1)

a1 ) C1X1X2 (1a)

X398C2

X1 + X2 (2)

a2 ) C2X2 (2a)

X3 + X498C3

X5 + X6 (3)

a3 ) C3X3X4 (3a)

X5 + X698C4

X3 + X4 (4)

a4 ) C4X5X6 (4a)

X598C5

X7 + X2 (5)

a5 ) C5X5 (5a)

X2 + X798C6

X5 (6)

a6 ) C6X2X7 (6a)

V1 ) [-1-110000

]V2 ) [11-10000

]V3 ) [00-1-1110

]V4 ) [0

011-1-10

]V5 ) [0100-101

]V6 ) [0-10010-1

]X1 + X298

C1X3 (7)

a1 ) C1X1X2 (7a)

X398C2

X1 + X2 (8)

a2 ) C2X2 (8a)

X3 + X498C3

X5 + X6 (9)

a3 ) C3X3X4 (9a)

X5 + X698C4

X3 + X4 (10)

a4 ) C4X5X6 (10a)

X598C5

X7 + X2 (11)

a5 ) C5X5 (11a)

X2 + X798C6

X5 (12)

a6 ) C6X2X7 (12a)

1180 Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011

Dow

nloa

ded

by I

ND

IAN

IN

ST O

F T

EC

HN

OL

OG

Y B

OM

BA

Y o

n Se

ptem

ber

4, 2

015

| http

://pu

bs.a

cs.o

rg

Pub

licat

ion

Dat

e (W

eb):

Dec

embe

r 15

, 201

0 | d

oi: 1

0.10

21/ie

1008

91n

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 composi-tion of the catalysts.

4.2. Deterministic Kinetic Modeling. The esterification ofpalmitic acid with methanol on SO4/ZrO2-550 °C and AcAl2O3

follows the Eley-Rideal mechanism. For SO4/ZrO2-550 °C, thereaction of adsorbed palmitic acid with methanol in bulk fluidis the rate determining step for SO4/ZrO2-550 °C (Figure 3). Incontrast, the adsorption of methanol on an active site on thecatalyst was the rate-limiting step in AcAl2O3 (Figure 4).

The disparity between the adsorbed species is attributed tothe nature of the acidic sites on the catalysts. Sulfated zirconium

oxide has been reported to have Bronsted and Lewis acid. Whenthe reaction was carried over pure zirconium oxide, theesterification yield was minimal therefore it was assumed thatthe catalyst activity was mostly due to the Bronsted acid sitesgenerated by the sulfation. Activated acidic alumina howeveronly has Lewis acid sites. The adsorption of palmitic acid isbelieved to be a nucleophilic interaction between the Bronstedacid site and the carbonyl of the carboxylic moiety similar tohomogeneous acid catalyzed esterification. On the activatedacidic alumina however, Lewis acid sites are predominantleading to the chemisoption of methanol. Therefore, Bronstedacid behaves differently from Lewis acid in the esterificationof free fatty acid. The estimated kinetic parameters are reportedin Table 4 and Table 5 for SO4/ZrO2-550 °C and AcAl2O3,respectively. The coefficients of determination of the determin-istic model were 0.98, 0.99, and 0.99 for SO4/ZrO2-550 °C at40, 60, and 80 °C, respectively, and 0.99, 0.96, and 0.98 forAcAl2O3 at the same temperature.

To evaluate the efficiency of the catalysts, the activationenergies were determined. The reaction rate constant is relatedto the reaction temperature through the Arrhenius equation.Therefore, the overall reaction activation energy can be calcu-lated from equation below using the reaction rate constant:

Both the frequency factor A and the activation energy Ea wereobtained by regressing ln(KS) versus 1/T. The activation energyof palmitic acid esterification was 70.81 kJ/mol and 93.71 KJ/mol for SO4/ZrO2-550 °C and AcAl2O3, respectively. From theactivation energy of both catalysts, SO4/ZrO2-550 °C seems tobe a better catalyst in the esterification of palmitic acid whichis a free fatty acid. However, this catalyst requires sulfationand temperature activation before use. On the other hand,AcAl2O3 does not require sulfation, though 3 hours of heatingat 550 °C is required in order remove moisture and atmosphericcarbon from the surface. Therefore, it is more convenient forindustrial use where easy-to-prepare catalysts are needed,although the operation cost of AcAl2O3 may be higher becauseof the higher activation energy requiring higher temperature.The esterification mechanisms of palmitic acid which occur overboth SO4/ZrO2-550 °C and AcAl2O3 are depicted in Figure 5and Figure 6, respectively.

4.3. Predicted Conversion Using Deterministic Model. Tovalidate the models, we used the models to predict the percentconversion at 210 and 240 min at all temperatures for allcatalysts. We then compared the predicted values with actualexperimental data.

Table 1. BET Surface of the Catalysts

SO4/ZrO2-550 °C AcAl2O3

Surface area (m2/g) 90.9 150

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

SEM-EDS XPS SEM-EDS XPS

Zr 74.31 ( 9.39 15.07 ( 3.27 C 3.66 ( 1.55 34.01 ( 4.07O 20.99 ( 8.81 48.86 ( 3.79 S 1.04 ( 0.25 2.05 ( 0.29

Table 3. Elemental Composition of AcAl2O3 by SEM-EDS

Al 40.15 ( 0.64O 54.10 ( 0.85C 5.75 ( 0.55

Figure 3. Esterification of palmitic acid on zirconia sulfate. Fitting ofEley-Rideal model to experimental data for palmitic acid esterification overSO4/ZrO2-550 °C at 40, 60, and 80 °C.

Figure 4. Esterification of palmitic acid on alumina. Fitting of Eley-Ridealmodel to experimental data for palmitic acid esterification over SO4/ZrO2-550 °C at 40, 60, and 80 °C.

Table 4. Kinetic Parameters of Palmitic Acid Esterification onSO4/ZrO2-550°C

temp (°C)

K1 × 10-4

(g (dm3)2/(gcat

3 min2))KAB × 10-2

(dm3/mol)KDB

(mol/dm3)KS × 10-3 (dm3/

(gcat min))

40 3.78 59.75 2.01 4.2160 16.12 26.74 2.01 19.3880 51.10 13.38 9.08 92.41

Table 5. Kinetic Parameters of Palmitic Acid Esterification onAcAl2O3

temp (°C)

K1 × 10-4

(g (dm3)2/(gcat

3 min2))KAM

(dm3/mol)KDB

(mol/dm3)KS × 10-1 (dm3/

(gcat min))

40 1.36 2.01 2.17 0.0260 0.91 0.14 2.01 0.2180 1.50 0.08 2.01 1.14

KS ) A exp(-∆ERT )

Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011 1181

Dow

nloa

ded

by I

ND

IAN

IN

ST O

F T

EC

HN

OL

OG

Y B

OM

BA

Y o

n Se

ptem

ber

4, 2

015

| http

://pu

bs.a

cs.o

rg

Pub

licat

ion

Dat

e (W

eb):

Dec

embe

r 15

, 201

0 | d

oi: 1

0.10

21/ie

1008

91n

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

different temperature between 40 and 80 °C and at longerreaction time.

4.4. Results for Stochastic Kinetic Modeling. The stochasticsimulation is used as a tool for observing the nature of thereaction which occurs over the surface of the catalysts. Insystems such as some biological processes, the reaction can beaccurately simulated using a stochastic simulation. The majordifference at the molecular level between deterministic andstochastic is that the deterministic model makes a continuumassumption in the concentrations, whereas the stochastic modelinvestigates the intrinsic nature of the reaction and thereforeconsiders the number of molecules that are reacting. In achemical reaction each individual reaction is an event that occurswith a certain probability. From a practical point of view, thedeterministic model is easier to set up and faster in terms ofcalculation compared to the stochastic model. When dealing witha small number of molecules, the deterministic model is notappropriate.

To observe the reactions which occur over the surface of thecatalyst, a small number of molecules in the system will beobserved. There is a link between deterministic and stochasticsimulation. The rate constant (Cj) for stochastic simulation areestimated from kinetic parameters (kj) (Tables 6 and 7) in thedeterministic simulation as

In addition

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

Figure 6. Palmitic acid esterification over AcAl2O3.

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

Figure 8. Prediction of esterification of palmitic acid on alumina.

Table 6. Reaction Parameters Determined by Stochastic Model onSO4/ZrO2-550 °C

temp(°C)

k1 ((dm3/mol)2)

k2 (dm3

/mol)k3 ((dm3/

(gcat min))2)k4 (dm3/

(gcat min))k5 ((mol/

dm3)2)k6 (mol/

dm3)

40 0.30 0.5 1.28 304.04 4.02 260 0.80 3 6.55 337.84 1 0.580 1.14 8.5 6.92 74.88 4.54 0.5

Table 7. Reaction Parameters Determined by Stochastic Model onAcAl2O3

temp(°C)

k1 × 10-2

((dm3/mol)2)

k2 × 10-3

(dm3/mol)k3 ((dm3/

(gcat min))2)k4 (dm3/

(gcat min))k5 ((mol/

dm3)2)k6 (mol/

dm3)

40 1.89 0.94 9.70 500.00 1.19 0.5560 1.27 9.10 6.23 300.00 16.08 8.0080 2.08 271.59 10.82 95.00 20.10 10.00

C1 )k1

nAvol, C2 ) k2, C3 )

k3

nAvol,

C4 )k4

nAvol, C5 ) k5, C6 )

k6

nAvol

1182 Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011

Dow

nloa

ded

by I

ND

IAN

IN

ST O

F T

EC

HN

OL

OG

Y B

OM

BA

Y o

n Se

ptem

ber

4, 2

015

| http

://pu

bs.a

cs.o

rg

Pub

licat

ion

Dat

e (W

eb):

Dec

embe

r 15

, 201

0 | d

oi: 1

0.10

21/ie

1008

91n

Palmitic Acid Esterification over SO4/ZrO2-550 °C. Thereaction parameters of palmitic acid esterification over SO4/ZrO2-550 °C are determined as in Table 6. Also, the results ofthe stochastic simulations for SO4/ZrO2-550 °C at differenttemperatures are shown in Figures 9-11.

As the number of molecules in the system increased, thestochastic simulation algorithms generated smooth graphs thatare comparable to those produced in the deterministic model.This was also observed as the temperature increased due toincreased 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 temperaturesof 40, 60, and 80 °C, respectively. The stochastic simulationacts as a pseudovalidation of the Eley-Rideal mechanism. Thedeterministic behavior represents the average of all thesepossible stochastic evolutions. Thus we note that the determin-istic model and stochastic simulation are very close as illustratedin Figure 9 through 14 for both catalysts. Consequently, thedeterministic result should be enough to describe the mechanismof the system. However, for other more complex catalyst suchas enzyme, the stochastic simulation could be essential ininvestigating the reaction mechanisms.

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 conversionfrom stochastic model vs deterministic model on SO4/ZrO2-550 °C.

Figure 10. Stochastic simulation vs Eley-Rideal model on SO4/ZrO2-550 °C at 60 °C: (a) stochastic simulation on SO4/ZrO2-550 °C; (b) compare conversionfrom stochastic model vs deterministic model on SO4/ZrO2-550 °C.

KAM )k1

k2, KS )

k3

k4and kDB )

k5

k6

Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011 1183

Dow

nloa

ded

by I

ND

IAN

IN

ST O

F T

EC

HN

OL

OG

Y B

OM

BA

Y o

n Se

ptem

ber

4, 2

015

| http

://pu

bs.a

cs.o

rg

Pub

licat

ion

Dat

e (W

eb):

Dec

embe

r 15

, 201

0 | d

oi: 1

0.10

21/ie

1008

91n

Palmitic Acid Esterification over AcAl2O3. The reactionparameters of palmitic acid esterification over AcAl2O3 weredetermined as in Table 7. In addition, the results of stochasticsimulations for alumina at different temperatures are shown inFigures 12-14.

The rate constant (Cj) can be calculate from (kj) as

Computational Time for Stochastic Simulations. Thecomputational time (in CPU seconds) for palmitic acid esteri-fication over both systems (SO4/ZrO2-550 °C and AcAl2O3) aregiven in Table 8. The runs were conducted on a Intel Core2Duo P8400 2.27 GHz computer running the VISTA operatingsystem. We note that the CPU time increases from 40 to 60 °Cand then drops afterwardsit is not clear why this is the case.However we conjecture that this probably has to do with thebalance between two driving forces: (a) increasing collision rateand therefore the greater the probability that a reaction occurswithin a very short time interval and (b) the faster depletion ofreactants at higher temperatures.

Figure 11. Stochastic simulation vs Eley-Rideal model on SO4/ZrO2-550 °C at 80 °C: (a) stochastic simulation on SO4/ZrO2-550 °C; (b) compare conversionfrom stochastic model vs deterministic model on SO4/ZrO2-550 °C.

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

C1 )k1

nAvol′, C2 ) k2, C3 )k3

nAvol′,

C4 )k4

nAvol′, C5 ) k5, C6 )k6

nAvol

1184 Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011

Dow

nloa

ded

by I

ND

IAN

IN

ST O

F T

EC

HN

OL

OG

Y B

OM

BA

Y o

n Se

ptem

ber

4, 2

015

| http

://pu

bs.a

cs.o

rg

Pub

licat

ion

Dat

e (W

eb):

Dec

embe

r 15

, 201

0 | d

oi: 1

0.10

21/ie

1008

91n

5. Conclusion

We employed both deterministic and stochastic simulationin this study. The stochastic simulation generated smooth graphs(which are a characteristic of the deterministic simulation) asthe number of molecules in the system increased. This was alsoobserved 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 criticalnumber of molecules required for the stochastic simulation totrack the deterministic simulation? This would give a roughguide as to when to employ which method. Since in generalthe deterministic simulation is faster than the stochastic simula-tion, there would be a gain in speed if the deterministicsimulation 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 apseudovalidation of the deterministic simulation.

The results suggest that the Eley-Rideal mechanism isappropriate to explain the reaction mechanism which occurs in

Figure 13. Stochastic simulation vs Eley-Rideal model on AcAl2O3 at 60 °C: (a) stochastic simulation on AcAl2O3; (b) compare conversion from stochasticmodel 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 stochasticmodel 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 24.2347 1.372860 173.6447 8.720580 0.6084 1.7784

Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011 1185

Dow

nloa

ded

by I

ND

IAN

IN

ST O

F T

EC

HN

OL

OG

Y B

OM

BA

Y o

n Se

ptem

ber

4, 2

015

| http

://pu

bs.a

cs.o

rg

Pub

licat

ion

Dat

e (W

eb):

Dec

embe

r 15

, 201

0 | d

oi: 1

0.10

21/ie

1008

91n

the esterification of palmitic acid over both SO4/ZrO2-550 °Cand AcAl2O3. The reaction of adsorbed palmitic acid withmethanol in bulk fluid is the rate determining step for SO4/ZrO2-550 °C. On the other hand, the adsorption of methanolon an active site on the catalyst was the rate-limiting step inAcAl2O3. This difference in adsorbed species is attributed thenature of the acid sites: SO4/ZrO2-550 °C has mostly Bronstedacid sites, whereas AcAl2O3 has Lewis acid sites. Furthermore,the deterministic model and the stochastic simulation are in goodagreement. It is therefore sufficient to use the deterministicmodel for the future kinetic investigation of free fatty esterifi-cation over heterogeneous catalysts. Furthermore, we showedthat these models can be used to predict the conversion fordifferent times. The activation energy of SO4/ZrO2-550 °C is70.81 kJ/mol while that of AcAl2O3 is 93.70 kJ/mol indicatingthat SO4/ZrO2-550 °C is better in the esterification for free fattyacids. Both SO4/ZrO2-550 °C and AcAl2O3 are therefore suitablecatalysts for the conversion of high free fatty acid containingoils such as waste cooking oil, a cheaper alternative feedstock,for the production of renewable diesel through an environmen-tally friendly process. Finally, this investigation of the esteri-fication of free fatty acids will not only help in reactor designbut could also serve as a model for further kinetic studies oftransesterification of vegetable oil to biodiesel. Moreover, thecombination of deterministic modeling and stochastic simulationcan be used as a model to study more complex catalyst such asenzyme catalyst.

Acknowledgment

The authors duly acknowledge the Institute for CriticalTechnology and Applied Science at Virginia PolytechnicInstitute and State University for funding this project.

Nomenclature

A ) palmitic acidE ) palmitic acid methyl esterM ) methanolS ) catalyst active siteW ) waterCT ) total catalyst active sitesAS ) palmitic acid adsorbed on an active siteES ) intermediate palmitic acid methyl ester on the catalyst surfaceCA ) concentration of palmitic acid, MCB ) concentration of methanol, MCE ) concentration of palmitic acid methyl ester, MCW ) concentration of water, MCA0 ) initial concentration of palmitic acid, MKAB ) adsorption parameter of palmitic acid, dm3/molKDB ) desorption parameter of palmitic acid, mol/dm3

KAM ) adsorption parameter of methanol, dm3/molKS ) 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. Solidacid catalyzed biodiesel production by simultaneous esterification andtransesterification. 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 fromvegetable oils. Fuel Process. Technol. 2009, 90 (7-8), 1023–1031. (b) DiSerio, M.; Tesser, R.; Pengmei, L.; Santacesaria, E. Heterogeneous catalystsfor biodiesel production. Energy Fuels 2007, 22 (1), 207–217. (c) Encinar,J. M.; Gonzalez, J. F.; Rodriguez-Reinares, A. Biodiesel from used fryingoil. 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 fatswith high free fatty acids. Trans. Am. Soc. Agric. Eng. 2001, 44, 1429–1436.

(5) Berrios, M.; Siles, J.; Martın, M. A.; Martın, A. A kinetic study ofthe 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 sunfloweroil 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 theesterification 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 forbiodiesel production catalyzed by SnCl2: A kinetic investigation. Energies2008, 1 (2), 79–92.

(10) Arata, K. Preparation of solid superacid catalysts. Sekiyu Gakkaishi1996, 39 (3), 185–193.

(11) Amenomiya, Y.; Cvetanovic, R. J. Active sites of alumina andsilica-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 ofdeterministic 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. SIAMReView 2008, 50 (2), 347–368.

ReceiVed for reView April 15, 2010ReVised manuscript receiVed October 19, 2010

Accepted November 24, 2010

IE100891N

1186 Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011

Dow

nloa

ded

by I

ND

IAN

IN

ST O

F T

EC

HN

OL

OG

Y B

OM

BA

Y o

n Se

ptem

ber

4, 2

015

| http

://pu

bs.a

cs.o

rg

Pub

licat

ion

Dat

e (W

eb):

Dec

embe

r 15

, 201

0 | d

oi: 1

0.10

21/ie

1008

91n