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Fluid Phase Equilibria
j o ur nal homep age: www.elsev ier .com/ locate / f lu id
hermodynamic simulation of transesterification reaction by Gibbs energyinimization
aison M. Yancy-Caballero, Reginaldo Guirardello ∗
chool of Chemical Engineering, University of Campinas (UNICAMP), Av. Albert Einstein 500, 13083-852 Campinas, SP, Brazil
r t i c l e i n f o
rticle history:eceived 6 September 2012eceived in revised form1 December 2012ccepted 18 December 2012vailable online 25 December 2012
a b s t r a c t
In this study, simultaneous chemical and phase equilibrium calculations were carried out by the methodof direct Gibbs energy minimization to perform a thermodynamic analysis of transesterification reactionof soybean oil with both ethanol and methanol in order to improve the processes for producing biodiesel.The CONOPT solver was used to solve the problem as a nonlinear programming model in the GAMS®
23.2.1 software. In addition, the UNIFAC model was employed to describe the liquid phase non-idealities.A strategy of balance of radicals is proposed to satisfy the requirement of conservation of number of moles
eywords:iodieselibbs energy minimizationhemical and phase equilibriumon-linear programming
in the specific case of transesterification reaction, in order to take into account the degrees of freedomof reacting system without explicitly writing all individual reactions. The results showed that the use ofoptimization techniques associated with the GAMS software are useful and efficient tools to calculatethe chemical and phase equilibrium by minimizing of the Gibbs energy, provided that different initialguesses are used. Furthermore, the computational times spent in the calculations were quite small.
AMS
. Introduction
Nowadays there has been a growing interest in finding alter-ative sources of clean and safe energy that can fully or partiallyeplace fossil fuels, due to global environmental concerns, the oilrice rise and the possibility of depletion of crude oil reserves inhe future. Biodiesel production processes from vegetable oils wereeveloped, for this reason, in order to reduce dependence on crudeil and enable sustainable development [1–4]. The importance ofiodiesel as an alternative substitute to mineral diesel implies that
t is a non-toxic and biodegradable fuel, and it is also produced fromenewable sources. Moreover, biodiesel has a higher cetane numbernd a more favorable combustion emission profile than petroleum-ased diesel (such as reducing levels of carbon monoxide) [5,6].
Transesterification is the most used process to obtain biodiesel.t is a reaction between a triacylglycerol (fat or oil) and an alco-ol that produces fatty acid alkyl esters (biodiesel) and glycerol asub-product. The transesterification of vegetable oils is catalyzed
y acid/base catalysts to increase the reaction rate and its yield.n excess of alcohol is required to shift the equilibrium of the reac-
ion [7,8]. According to the United States Department of Agriculture
∗ Corresponding author at: School of Chemical Engineering, University of Cam-inas (UNICAMP), P.O. Box 6066, Av. Albert Einstein 500, 13083-852 Campinas, SP,razil. Tel.: +55 19 3521 3955; fax: +55 19 3521 3965.
(USDA), biodiesel is currently produced from soybean oil in manycountries, e.g. the United States, Argentina and Brazil. Methanolis the most common used alcohol because it is cheaper than othertypes of alcohol, among others reasons; nevertheless, ethanol is thepredominant alcohol in Brazil due to its low cost and accessibilityin this country, so there is great interest in biodiesel productionfrom this alcohol there [7,9–11]. The generated glycerol is sep-arated from the oil phase when the equilibrium conditions areachieved. Two liquid phases co-exist in equilibrium at the final sys-tem: one of them is rich in glycerol and the other one is rich infatty acid methyl/ethyl esters. The unreacted alcohol is distributedbetween these two liquid phases in proportions depending on theinteraction characteristic energies of each compound [12,13].
So the prediction of chemical and phase equilibrium has aremarkable application in many industrial processes, such asbiodiesel production and more specifically at the separationstage. For this reason, several optimization techniques have beenproposed in many studies [14–22] to solve the problem of simulta-neous chemical and phase equilibrium calculation by direct Gibbsenergy minimization. Hence the chemical and phase problem issolved in one step with this method, since it is not necessary toperform a phase stability analysis [15,23]. A non-linear program-ming approach has been used frequently by many researchers[15,16,18] to accomplish this. Nonetheless if non-convex thermo-
dynamic models are used, the problem may have multiple localoptima; this makes its solution difficult, so that a reliable initialguess is necessary to solve the problem. Despite of this, to over-come the difficulties coming from local minima, some researchers
ami number of atoms of element m in component iAi parameter for saturation pressure in DIPPR correla-
tion of component iBi parameter for saturation pressure in DIPPR correla-
tion of component iCi parameter for saturation pressure in DIPPR correla-
tion of component iCpi ideal-gas heat capacity for component iDi parameter for saturation pressure in DIPPR correla-
tion of component iDAG diacylglycerolsEi parameter for saturation pressure in DIPPR correla-
tion of component ifij fugacity of component i in the mixture in phase jf ◦i
fugacity of pure component i in the reference stateat system temperature and reference pressure
G Gibbs energy of the systemGi partial molar Gibbs energy of component igli number of radicals RlCO in component iHi partial molar enthalpy of component iMAG monoacylglycerolsnij number of moles of component i in phase jn0
iinitial number of moles for component i
NC number of components or species in the systemNE number of elements in the systemNL number of organic fatty acid radicalsNP number of potential phases in the systemNR number of independent reactions in the systemP absolute pressure of systemPsat
ivapor–liquid saturation pressure of component i
pi number of radicals RO in component iqi number of radicals CH2O-CHO-CH2O in component
iR universal gas constantT temperatureTAG triacylglycerolsxi molar fraction in liquid phase of component iyi molar fraction in gas phase of component i
Greek letters�ij chemical potential of component i in phase j�◦
ichemical potential of reference at system tempera-ture and reference pressure of pure component i
�i fugacity coefficient of component i in vapor phase� ij activity coefficient of pure component i in liquid
phase j�ik stoichiometric coefficient of the component i in the
reaction k�k extent of the reaction k�g◦
fstandard-state Gibbs energy of formation
�h◦f
standard-state enthalpy of formation
SuperscriptsV vapor phaseL liquid phase
Subscriptsi component in the mixturej phase in the system
l organic fatty acid radicalm number of different types of atoms in the systemk independent reaction in the system
d Phase Equilibria 341 (2013) 12– 22 13
have extended the Gibbs energy formulation with non-convexthermodynamic models to a linear programming problem: Whiteet al. [14] considered ideal behavior of all studied systems to avoidlocal minima. Rossi et al. [19] developed a method to linearize theGibbs function; the main objective of their technique was to ensurethat global optimum was found when employed convex and non-convex thermodynamic models. They also used two approaches forthe problem of equilibrium: one gamma-phi approach and one phi-phi approach. The Wilson, NRTL and UNIQUAC models were usedto represent the liquid phase non-idealities in the former approach.In the latter, the Peng–Robinson equation of state and also thequadratic mixing rule proposed by Van der Waals were used torepresent the vapor and liquid phases.
Other researchers have taken into account stochastic methodsrather than deterministic methods to solve the problem of Gibbsenergy minimization. Zhu et al. [20], Zhu and Xu [24] applied theSimulated Annealing algorithm to calculate the phase equilibriumby Gibbs energy minimization when liquid phases are representedby NRTL and UNIQUAC models.
Nonlinear programming was employed in this survey to cal-culate the chemical and phase equilibrium by Gibbs energyminimization in conditions of constant pressure and temperature.The UNIFAC model was used for representing the liquid phasenon-idealities, and a possible formation of vapor phase with idealbehavior was admitted. The number of moles was considered as adecision variable while temperature, vapor pressure and the chem-ical potential of the pure component in the reference state wereconsidered parameters. An alternative strategy for satisfying thematerial balance was employed during the minimization process,since all of the components in transesterification reaction were con-sidered in the simulations, i.e. all esters and triacylglycerols presentin biodiesel and oil, respectively. So, a balance of radicals was usedin order to take into account the degrees of freedom of reactingsystem without explicitly writing all individual reactions.
2. Methodology
2.1. Thermodynamic model
The thermodynamic equilibrium condition for reactive multi-component systems, at constant pressure and temperature, and agiven initial composition can be obtained by direct minimizationof Gibbs energy of the system through the following expression:
G =NC∑i=1
NP∑j=1
nij ·(
�◦i + R · T · ln
fijf ◦i
)(1)
where NP is the number of phases, NC is the number of components,nij is the amount of moles of component i in j phase, and �◦
iis
the chemical potential of reference of pure component i and fij isfugacity for component i in phase j. The used reference state for theGibbs expression was ideal gas at 1 atm, so the fugacity of the purecomponent at the reference state (f ◦
i) is equal to unity.
Restrictions of material balance, either by stoichiometric formu-lation (Eq. (3)) or by non-stoichiometric formulation (Eq. (4)), andnon-negativity of number of moles must be observed according tothe following equations [23]:
Non-negativity of number of moles
nij ≥ 0, i = 1, . . . , NC; j = 1, . . . , NP (2)
Material balance by stoichiometric formulation
NP∑j=1
nij = n0i +
NR∑k=1
�ik · �k, i = 1, . . . , NC (3)
1 / Fluid Phase Equilibria 341 (2013) 12– 22
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4 D.M. Yancy-Caballero, R. Guirardello
Material balance by non-stoichiometric formulation
NC
i=1
⎡⎣ami ·
NP∑j=1
nij
⎤⎦ =
NC∑i=1
ami · n0i , m = 1, . . . , NE (4)
here n0i
is the initial number of moles of component i, NR andE are the number of reactions k and the number of elements m
n the system respectively, �ik is the stoichiometric coefficient ofomponent i in reaction k, �k is the extent of reaction k, and ami ishe number of atoms of element m in component i.
In order to improve the search for the global optimum, a math-matical manipulation of UNIFAC model was done, since this is aon-convex model and can present occurrence of multiple localinima. Thus, the UNIFAC model was rewritten as the difference
f two convex functions, where the concave portion is separable,s it was proposed by McDonald and Floudas [16]. Although theormulated problem is non-convex in general, and the CONOPTolver here implemented is a local solver that does not guaran-ee the global optimum, multiple initial guesses were used toolve the problem and obtain a feasible solution. It is importanto highlight that the three case studies (n-butyl acetate–water,ater–n-propanol–n-hexane and ethanol–benzene–water) pre-
ented by McDonald and Floudas [16], where they implementedhe UNIFAC model, were solved with the approach developed in thisork. The results obtained with these tests were the same as those
eported by McDonald and Floudas [16] as global solutions. Theseesearches guaranteed the global solutions by applying a branchnd bound algorithm that works partitioning the initial feasibleegion into subdomains in which the convex envelope of the Gibbsnergy is constructed so as to provide valid underestimators of thelobal solution. A sequence of nondecreasing lower bounds is thenenerated until �-global convergence is obtained.
.2. Parameter calculation
Some parameters as R, T and P remained constant through-ut the optimization process, and others parameters as Psat
iand
◦i
were calculated by means of known thermodynamic relations.hus, the chemical potential of the pure component in the referencetate was calculated by applying the two equations below [25]:
∂
∂T
(Gi
RT
)P
= − Hi
RT2(5)
∂Hi
∂T
)P
= Cpi (6)
here Gi is the partial molar Gibbs energy, Hi is the partial molarnthalpy and Cpi is the ideal-gas heat capacity of the component. The choice of the ideal gas as reference state is justified by thearge availability of thermodynamic data and prediction methodsor this state.
The parameters for calculating the reference chemical poten-ial of the pure component, e.g. the ideal-gas heat capacity, thetandard-state enthalpy and Gibbs energy of formation, were foundn the literature for some components as ethanol, methanol, glyc-rol and some esters [25–28]. These thermodynamic properties areot reported for the other components (monoacylglycerols, diacyl-lycerols and triacylglycerols), then they were predicted by group
ontribution methods, such as the Joback [29,30], Benson [29,31]nd Constantinou and Gani [29,32] methods.
The saturation pressure of the liquid Psati
was calculated as aunction of temperature through the equation from the Design
Fig. 1. Scheme of general transesterification reaction.
Institute for Physical Properties (DIPPR) Chemical Database 801 inDiadem Public v. 1.2 software [28].
Psat = Ai + Bi
T+ Ci · ln T + Di · TEi (7)
Nevertheless, the parameters of this correlation were not foundin DIPPR database for all components studied in this work, hence agroup contribution method developed by Ceriani and Meirelles [33]was utilized for predicting the vapor pressure, since this methodwas proposed specifically for fatty systems.
2.3. Proposed strategy for mass balance in the reactional system:radicals balance
The transesterification reaction can be written generally as isshown in Fig. 1:
where R1, R2, R3 are long-chain hydrocarbons deriving fromfatty acid chains, which can be equal or different, and R representsa carbon chain of alcohol. Transesterification reaction is commonlya sequence of three consecutive reversible reactions, where thetriacylglycerol is stepwise converted into diacylglycerol, monoa-cylglycerol and finally glycerol (Fig. 2a–c); 1 mol of fatty acid ester isproduced at each step of the reaction as shown in Fig. 2 [9,13]. Theseintermediate reactions should be considered for all oils (triacyl-glycerols, diacylglycerols and monoacylglycerols) of the reactionalsystem.
According to Smith and Missen [23] the stoichiometric for-mulation and the non-stoichiometric formulation are equivalents,provided that the number of independent reactions (NR) is equalto the difference between the number of components (NC) and thenumber of elements (NE):
NR = NC − NE (8)
If this condition is not satisfied, then the stoichiometric for-mulation is not equivalent to the non-stoichiometric formulation,since the degrees of freedom are not the same. In the specific caseof transesterification reaction of vegetable oils, many groups andcomponents are present in the reactional system, so that NR is lessthan NC − NE. The material balance given by non-stoichiometricformulation has more degrees of freedom than the material bal-ance given by stoichiometric formulation. Hence in this reactionthose material balances are not equivalents.
If the non-stoichiometric formulation is applied to the trans-esterification reaction considering all the triacylglycerols (TAG),diacylglycerols (DAG) and monoacylglycerols (MAG) present in thereaction, the model allows a radical R1CO to be transformed into adifferent radical R2CO, which does not occur under real transesteri-fication conditions.
One way to consider all possible intermediate reactions in thetransesterification is to perform a balance of radicals (or groups ofatoms), which remains unmodified in the reactions as it is shownbelow:
a Cx:y, x: number of carbons and y: number of double bonds.
here qi is the number of radicals CH2O-CHO-CH2O present inomponent i.
It is not necessary to balance the hydrogen radicals that arexchanged between alcohol and glycerol (it is different to do a bal-nce of all atoms H), because this is a consequence of the balancesentioned before (Eqs. (9), (10) and (11)), due to the degrees of
reedom. In this manner the balance of radicals satisfies the balancef atoms automatically, but the balance of atoms does not alwaysatisfy the balance of radicals. It is important to remark that theadicals balance was applied on the three consecutive reversibleeactions present in the transesterification, as it is shown in Fig. 2.
. Results and discussion
A thermodynamic analysis based on the Gibbs energy mini-ization was carried out for the transesterification reaction. Theain components considered in the simulation were: soybean oil,
thanol, methanol, methyl and ethyl esters and glycerol. Soybeanil is mainly formed by a TAGs mixture and some other minor com-ounds. However in this work, the full complexity of the soybeanil was only expressed in terms of TAG composition to simplify thealculations. Consequently, in order to estimate the TAG compo-ition of soybean oil a methodology proposed by Antoniosi-Filhot al. [34] was used. This is based on a statistical procedure thatonsiders lipase hydrolysis characteristics, where the fatty acidsre randomly distributed among the three positions of the glycerololecule. It is necessary to know the content of triacylglycerols
risaturated (triacylglycerols formed by esterification of glycerolnd three saturated fatty acids) to apply this method. This contentor soybean oil can be considered zero [35]. In this way, the TAGomposition was estimated from the fatty acids compositions giveny Rodrigues et al. [36], presented in Table 1. The TAG estimatedomposition is shown in Table 2.
In addition to TAG presented in Table 2, the possible formationsf DAG and MAG during the intermediate steps of the transesterifi-
ation reaction were taken into account. So, different combinationsere done for one and two substitutions for a hydrogen atom of the
lcohol in each TAG considered, with the purpose of determininghe possible formation of DAG and MAG in the reaction. Then each
a Only considered TAG with compositions greater than 5%.b x:y, x: number of carbons and y: number of double bonds. For abbreviations of
AG see Table 1.
LiHLn – –
a H: substitution in the glycerol chain. For abbreviations see Table 1.
TAG was split into 1,2- and 1,3-DAG and each DAG was also split intoMAG according to the stoichiometric relations of transesterificationreaction (Fig. 2). This procedure was already applied successfullyby Ceriani and Meirelles [33], Rodrigues et al. [36], Gonc alves andMeirelles [37], Lanza et al. [38]. In this fashion, the more completestudy of the transesterification was performed by simulating thisreaction including all of the components reported in Table 3.
The standard-state enthalpy and the Gibbs energy of forma-tion of TAG, DAG and MAG were estimated by utilizing the Joback,Benson and Constantinou and Gani methods [29–32], while theideal-gas heat capacity was only calculated by the Joback methodbecause of the large number of considered components and thesimplicity of this method too. The estimated properties by the dif-ferent group contribution methods for all of the compounds areshown in Appendix A, together to thermodynamic data taken fromliterature. The structural and binary interaction parameters for theUNIFAC equation were obtained from Magnussen et al. [39]. The oilconversion was calculated by utilizing Eq. (12):
Oil conversion (%) = (n0oil − nf
oil)
n0oil
× 100 (12)
The GAMS® 23.2.1 software (General Algebraic Modeling Sys-tem) with the CONOPT2 solver executed in a Pentium III (512 MB,900 MHz) was employed for calculating the chemical and phaseequilibrium problem. CONOPT is an efficient large-scale solverdeveloped by ARKI Consulting & Development A/S, designed fornon-linear programming (NPL) problems and it is based on feasibledirection methods, specifically on the GRG (Generalized ReducedGradient) algorithm.
3.1. Validation of the proposed methodology
In order to validate the proposed methodology, a case studyinvolving transesterification reaction of soybean oil with differ-ent alcohols available in the literature was selected to comparethe results obtained from simulations. Then, the study of transes-terification reaction developed by Colucci et al. [40] was selected,these researchers studied the alkaline transesterification reactionfor producing biodiesel from oil soybean. It was carried out atatmospheric pressure, at different levels of temperature and oil-to-alcohol ratios, by utilizing an ultrasonic mixing technique andpotassium hydroxide as catalyst. They investigated the effects of
2-propanol, 1-butanol, ethanol and methanol on the transesteri-fication reaction with 1.5% KOH as catalyst and 1:6 molar ratio ofoil-to-alcohol at 60 ◦C. The reaction time to achieve the equilibriumconversion was about two hours. In the present work we carried
Table 4Comparison of calculated and experimental soybean oil equilibrium conversion at 60 ◦C, atmospheric pressure, 1:6 molar ratio of oil-to-alcohol and different alcohols types.
Alcohol Colucci et al. [40] This work
TAG (%w) DAG (%w) Oil conversion (%) TAG (%w) DAG (%w) Oil conversion (%)
Methanol 0.00 0.54 99.33 0.00 0.03 99.96
oToc
motdwftnutuwshcam
3s
pbasw
eddst
TEGc
ec
Ethanol 0.00 0.67 99.11
1-Butanol 1.90 3.27 92.02
2-Propanol 43.39 0.00 29.21
ut simulations at the same conditions studied by Colucci et al. [40].he results obtained from simulations by our proposed methodol-gy and the results obtained experimentally by Colucci et al. [40]an be seen in Table 4.
It is possible to verify the good ability of the Gibbs energy mini-ization methodology for predicting the equilibrium conversion
f the transesterification reaction. From Table 4, it can be seenhat the equilibrium conversions reproduced the experimentalata with good fidelity by employing the formulated methodologyhen ethanol and methanol were used. The uncertainties obtained
or these cases were 0.63% and 0.72% in absolute percentage. Onhe other hand, the prediction of the equilibrium conversion wasot good enough when 1-butanol and 2-propanol were used. Thencertainties for these cases were 6.3% and 54%, respectively. Theoo large absolute percentage error found when 2-propanol wassed might come from the fact that the experimental value likelyas not the value of the conversion in equilibrium, because the
econdary and tertiary alcohols react slower with the oil and, per-aps the reaction time was not sufficient to achieve the equilibriumonversion. The proposed methodology was also able to predict themount of DAG produced and TAG unreacted with a good agree-ent to the experimental data.
.2. Results for simulation of transesterification reaction ofoybean oil with ethanol and methanol
Results obtained in terms of equilibrium conversion by the pro-osed methodology for the transterification reaction of oil soybeany varying initial molar ratio between reactants (oil-to-alcohol),t atmospheric pressure and two conditions of temperature, arehown in Tables 5 and 6 when reactions with ethanol and methanolere simulated:
The results presented in Tables 5 and 6 are predictive only, nev-rtheless they permit to perform an analysis of the reaction to
etermine the best conditions of operation in the biodiesel pro-uction processes. In addition, since our predictions depend oneveral parameters estimated by group contribution methods, e.g.he standard-state enthalpy and Gibbs energy of formation, an
able 5quilibrium conversion of soybean oil transesterification with ethanol calculated byibbs energy minimization and UNIFAC model. Results are in function of the groupontribution method used for predicting thermodynamic properties.
a Group contribution method employed for predicting the thermodynamic prop-rties (standard-state enthalpy and Gibbs energy of formation) used in thealculations.
0.00 0.05 99.820.36 0.89 97.82
24.26 12.84 45.20
analysis of sensibility was carried out to check how a variation in thevalues of these parameters could modify the predicted results. InTables 5 and 6 are tabulated equilibrium conversions as a functionof group contribution method used for estimating the standard-state enthalpy and Gibbs energy of formation. The results obtainedfor each considered group contribution method, in terms of equi-librium conversion, allow us to verify that the results from oursimulations are fairly sensitive to these properties. This is becauseof the shown differences of a method to another when the molarratio oil-to-alcohol was 1:3; it was the only case without completeconversion at equilibrium. Due to those differences, it can be con-cluded that if values accurate to these thermodynamic propertiesare not known, the quality of the results will be compromised. How-ever, in the cases where molar ratios 1:9 and 1:6 were used, theequilibrium conversion was complete without taking care of thegroup contribution method utilized for predicting these thermo-dynamic properties.
In relation to the phases formed after the transesterificationreaction, it can be observed that the UNIFAC model was able topredict a biodiesel-rich phase with small amounts of glycerol anda part of the unreacted alcohol, and another phase which is rich inglycerol with the remaining amount of unreacted alcohol and verysmall amounts of biodiesel. It all can be appreciated in Appendix B.
The highest equilibrium conversions were gotten from simula-tions with feed molar ratios oil-to-alcohol of 1:6 and 1:9. This is inagreement with optima feed ratios reported in the literature, whichis of 1:6 for alkaline transesterification [13,41] and 1:9 for acid-catalyzed transesterification [42]. The reaction temperature shouldbe less than the boiling point of the used alcohol with the purpose ofensuring that the reaction will be carried out in liquid phase. In con-sequence the used temperature commonly in biodiesel productionprocesses by transesterification is between 35 and 60 ◦C depend-ing on the type of alcohol and catalyst employed [9,13,43–45].Nonetheless, the results obtained suggest that the use of lowertemperatures than those reported in the literature do not have a
significant effect on the maximal equilibrium conversion of trans-esterification reaction with ethanol or methanol. As an effect of this,it is possible to say that the transesterification reaction (with both
Table 6Equilibrium conversion of soybean oil transesterification with methanol calculatedby Gibbs energy minimization and UNIFAC model. Results are in function of thegroup contribution method used for predicting thermodynamic properties.
a Group contribution method employed for predicting the thermodynamic prop-erties (standard-state enthalpy and Gibbs energy of formation) used in thecalculations.
1 / Flui
mi4toesrd
clnttiep1
4
srcso
i
Appendix A. Thermodynamic properties employed in this
TT
8 D.M. Yancy-Caballero, R. Guirardello
ethanol and ethanol), with the equilibrium conversions obtainedn this work, can be carried out to temperatures between 30 ◦C and0 ◦C maintaining a high conversion of the reaction. Furthermore,he use of molar ratios greater of 1:6 does not have major effectn equilibrium conversion; in contrast they would complicate thester and glycerol recovery due to increasing of solubility. It can beeen that a high conversion was also obtained when a feed molaratio 1:3 was utilized, which could facilitate the separation stageuring the alcohol recovery process.
It is important to explain that the reaction kinetic was notonsidered in the finished simulations, so to achieve the equi-ibrium conversion of this work it is necessary to develop aew highly active catalysts able to reach high conversion inhe transesterification reaction. In this manner, it is valuableo study the mass transfer phenomena due to the differencesn the molecular weight of reactants (soybean oil and alcohol),specially when ethanol is used in the reaction. Finally, the com-utational times spent in all of the simulations were less than0 s.
. Conclusion
The Gibbs energy minimization method implemented in theoftware GAMS for calculating the phase and chemical equilib-ium was robust, fast and reliable for predicting the equilibriumonversion of the transesterification reaction of vegetable oils. Con-equently, it can also be applied to other types of reactional systems
f multicomponent mixtures.
There is an advantage when the problem of simultaneous chem-cal and phase equilibrium calculation by the direct Gibbs energy
a DIPPR® Database Diadem public v.1.2.0.b Lapuerta et al. [27].c Bucalá et al. [26].d Expression for ideal-gas heat capacity: Cpg
i[J mol−1 K−1] = Ai + Bi · T + Ci · T2 + Di · T3
d Phase Equilibria 341 (2013) 12– 22
minimization is formulated as a non-linear optimization probleminstead of using a strategy for finding roots: it avoids performing ananalysis of the problem in two steps, since it is not required a phase-stability analysis previously. The low computational time spent inall of the calculations is another advantage. Nevertheless, the useof non-linear programming techniques associated to GAMS to min-imize the Gibbs energy was only possible due to the manipulationof UNIFAC thermodynamic model and the use of multiple initialguesses, because of the non-convex nature of this model can causeproblems coming from the local minima. In relation to thermody-namic model employed to represent the liquid phase non-idealitiespresent in the reactional system of biodiesel, it is possible to saythat the UNIFAC model showed a good representation of the liq-uid phase behavior of systems constituted by ethanol, methanol,glycerol, ethyl and methyl esthers. Finally, radicals balance shownto be a valuable strategy for satisfying requirements of mass con-servation when many simultaneous reactions take place, since thisstrategy allows doing a material balance of reacting systems with-out explicitly writing all individual reactions.
Acknowledgement
This research work was supported by CAPES – Coordenac ão deAperfeic oamento de Pessoal de Ensino Superior – Brazil.
a Mixture of TAG, DAG (<1.0E−7) and MAG (<1.0E−7).
Table B2Results obtained from simulation of soybean oil transesterification with ethanol, using the predicted properties by Joback method. Results expressed in mass fraction.
a Mixture of TAG, DAG (<1.0E−7) and MAG (<1.0E−7).
Table B3Results obtained from simulation of soybean oil transesterification with ethanol, using the predicted properties by Constantinou and Gani method. Results expressed in massfraction.
Table B4Results obtained from simulation of soybean oil transesterification with methanol, using the predicted properties by Benson method. Results expressed in mass fraction.
a Mixture of TAG, DAG (<1.0E−7) and MAG (<1.0E−7).
Table B5Results obtained from simulation of soybean oil transesterification with methanol, using the predicted properties by Joback method. Results expressed in mass fraction.
a Mixture of TAG, DAG (<1.0E−7) and MAG (<1.0E−7).
Table B6Results obtained from simulation of soybean oil transesterification with methanol, using the predicted properties by Constantinou and Gani method. Results expressed inmass fraction.
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