Concentration-independent rate constant for biodiesel synthesisfrom homogeneous-catalytic esterification of free fatty acid

Saeikh Z. Hassan, Madhu Vinjamur n

Department of Chemical Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India

H I G H L I G H T S

� Rate constant dependency on reac-tion parameters is studied usingTaguchi approach.

� Second-order rate constant (SRC) ismodified for esterification of freefatty acids.

� Modified-SRC is reported as con-centration-independent rate con-stant (CIRC).

� CIRC is shown to be equivalent toBrønsted–Bjerrum relation.

� CIRC-based kinetics predictions arefound to be better than those basedon SRC.

G R A P H I C A L A B S T R A C T

(a) Taguchi analysis for parametric effects on concentration-independent rate constant (CIRC or k0).(b) Arrhenius plots for CIRC (using standard L9 orthogonal array design).

a r t i c l e i n f o

Article history:Received 8 August 2013Received in revised form26 October 2013Accepted 17 December 2013Available online 27 December 2013

Keywords:Concentration-independent rate constantTaguchi methodBiodieselEsterificationFree fatty acid

a b s t r a c t

Dependence of the second-order rate constant (SRC) on the reactants’ concentration, for esterification offatty acids, is found empirically in the published reports. Taguchi method is used to study dependencyof the rate constant on temperature, reactants’ concentration, and catalyst loading for the synthesis ofbiodiesel from sulfuric acid-catalyzed esterification of free fatty acid (FFA) with methanol. On the basisof general literature on dependency of rate constant on reactants’ concentration and the esterification-rate determining step, concentration term embedded in the units of SRC has been identified as catalystconcentration. Accordingly SRC has been modified to include the catalyst concentration in the rate law,which is consistent with the general mass-action rate law reported by Blum and Luus (1964) and Pekař(2011). Analysis of the modified rate constant showed it to be independent of the reactants’ concentra-tion. It is thus named as 0concentration independent rate constant (CIRC)0 . The predictions of kinetics ofthe esterification of FFA are better using CIRC than other commonly used rate forms. The modified rateconstant is shown to be equivalent to Brønsted–Bjerrum relation. Moreover, the modified rate constant ismore general than the reported ones for esterification of fatty acids and it predicts kinetics well over awide range of reaction parameters.

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1. Introduction

The cost of feedstock accounts for 60–80% of the total cost ofbiodiesel fuel (Haas et al., 2005; Zhang et al., 2003). Biodiesel

production can be profitable if it is made from low-cost feed-stocks, but such feed-stocks generally contain large amounts offree fatty acids (FFA) (Canakci and Gerpen, 2001; Diaz-Felix et al.,2009; Ghadge and Raheman, 2005). A systematic study of thereaction kinetics of biodiesel synthesis from esterification of FFA isconsidered here. Reaction rate for kinetic study can be modeled ask(T)[A]x[B]y for reactions represented by xAþyB-products. Thisrate model is established over a century and a half ago and it

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n Corresponding author. Tel.: þ91 22 25 76 7218; fax: þ91 22 25 72 6895.E-mail address: madhu@che.iitb.ac.in (M. Vinjamur).

Chemical Engineering Science 107 (2014) 290–301

follows the kinetic aspect of the law of mass action for anelementary reaction (Koudriavtsev et al., 2000). The exponents xand y are the stoichiometric coefficients for the reaction, and k(T)is the rate constant, which is generally a function of temperatureand independent of reactants’ concentrations.

The empirical equation of Arrhenius for rate constant is wellaccepted in the study of kinetics parameters from experimentaldata. Collision and transition state theories are also used fortheoretical calculation of rate constant (Holland and Anthony,1989). These theories give an insight into the molecular dynamicsof the reactions; especially transition state theory provides moredetails about the mechanistic approach at molecular level for therate constant estimation. However, these theories suffer frommany difficulties in estimation of practical values for rate con-stants (Holland and Anthony, 1989; Truhlar et al., 1996). Besidestemperature, the apparent rate constant can also depend on otherfactors such as catalyst type, catalyst loading, solvent, and reac-tants molar ratio (Fogler, 2006; Jong et al., 2009; Leyes andOthmer, 1945; Yadav and Mehta, 1994). The above theories andthe Arrhenius equation express the temperature dependency ofrate constant clearly, but not the influence of the other factors.

The linear increase of rate constant with the catalyst concentrationand molar ratio (MR) of alcohol to fatty acid are reported by Leyes andOthmer (1945), Ling and Geankoplis (1958), and Othmer and Rao(1950). Ling and Geankoplis (1958) reported a combined equation forthe rate constant of homogeneous-catalyzed esterification of oleic acidwith butanol as k¼93(0.401CCatalystþ0.000368)(0.0133MRþ0.042)(e16.76─7100/T) by relating the effects of catalyst concentration (CCatalyst),molar ratio (MR), and temperature. This equation only holds at 100 1Cand within the range of catalyst concentration and MR studied.Equivalent forms of the combined equation are also reported byLeyes and Othmer (1945) and Othmer and Rao (1950) for theesterification of fatty acids. However, Othmer and Rao (1950)mentioned that the combined equation obtained for the sulfuricacid-catalyzed esterification of oleic acid with butanol does not workfor MR410, catalyst loading beyond the range of 0.5–1.2 wt%, andtemperature other than 100 1C. Hence, such reported combinedrelations are specific to the reaction conditions under study.

The kinetics of complex reactions is studied by postulatingseveral elementary reaction steps where each of the elementaryreaction rate constants is strictly concentration independent.An overall (or apparent/effective/complex) rate constant can beobtained from the overall kinetics derived from the kinetics ofelementary steps. Recently, based on reactivity and activity coeffi-cients arguments, Jesudason (2008) concluded that the current formof elementary reaction rate constant is still incomplete. He stronglyargued that, due to historical conventions, rate constant of elemen-tary reactions is treated as strictly independent of reacting speciesconcentration. He showed from molecular dynamics simulation(without using any transition states) that even the rate constant ofelementary reaction depends on reacting species concentration incontrast to the current form of elementary rate constant.

Taguchi method is used in this study for a systematic study ofreaction kinetics for biodiesel synthesis by homogeneous-acid catalyticesterification of free fatty acid with methanol to understand the effectsof reaction parameters on rate constant. Taguchi method is a statisticalanalysis of a system and utilizes various standard orthogonal arrays forthe design of experiments (Bagachi, 1993; Phadke, 2008). This methodhas been mainly used for optimization of process variables usingsignal-to-noise ratio as the decision making tool. The method not onlyminimizes the number of experiments, but also exploits the variationof all variables simultaneously in order to understand the impact ofeach of them on the process and identify interaction among thevariables. The method is technically advanced and more systematic inapproach than the conventional method of “one factor at a time“(Czitrom, 1999).

Several researchers (Chongkhong et al., 2007; Gan et al., 2012;Khan et al., 2010) have used the Taguchi method for optimizationof reaction parameters (e.g., temperature, reactants proportion,reaction time, catalyst type, catalyst concentration, etc.) to reducethe FFA content in biodiesel production. Similarly, Taguchi meth-odology is applied for optimization of the reaction variables forobtaining the maximum yield of biodiesel from trans-esterification(Antolin et al., 2002; Wu and Leung, 2011; Yusup and Khan, 2010;Mahamuni and Adewuyi, 2010). In these kinds of studies, the yieldof biodiesel or fatty acid conversion is treated as a response factor.The application of Taguchi method to study the effects of reactionvariables on rate constant for biodiesel production from the low-cost feed-stocks has not been studied.

In the esterification reactions of fatty acids, the catalyst loadingterm is explicitly expressed in rate equation for heterogeneouscatalysis (Gangadwala et al., 2003; Izci et al., 2009; Tesser et al.,2005). In the homogeneous catalysis, however, there is yet noconsensus on the inclusion of catalyst loading in rate equation; insome reports catalyst loading appears in the rate equation (Bartet al., 1994; Boucher et al., 2008; Jong et al., 2009; Liu et al., 2006)and in some others it does not (Aafaqi et al., 2004; Berrios et al.,2007). This reported contradiction is also examined in this work.

Sulfuric acid-catalyzed esterification of oleic acid (mixed asfree fatty acid in refined sunflower oil) with methanol is carriedout as a model reaction for this study. Taguchi approach guidedmodification of the conventional rate constant to a concentration-independent rate constant (CIRC). Using CIRC and Taguchiorthogonal array design, a method is shown to predict kineticsof biodiesel synthesis at different levels of reaction parameters.

2. Experimental section

2.1. Materials

Methanol (GR grade, moistureo0.02%), water (HPLC grade),sulfuric acid (98% GR), pure oleic acid, oxalic acid dihydrate(GR grade), n-hexane (HPLC grade), and phenolphthalein indicatorare supplied by Merck India Ltd. Refined sunflower oil (max. 0.1 wt%FFA) is purchased from Liberty Oil Mills Ltd., Mumbai, India.Potassium hydroxide (extra pure AR) is obtained from SiscoResearch Laboratories Pvt. Ltd., Mumbai, India.

2.2. Taguchi design of experiments, experimental procedure, sampleanalysis, and data interpretation

The details of the design, procedure, and the analysis are givenelsewhere (Hassan and Vinjamur, 2013). In this study, the standardL9 (34) orthogonal array of experiments (see Table 1) is conductedfor sulfuric acid-catalyzed esterification of 50 wt% oleic acid(mixed as FFA in sunflower oil) with methanol. Each experimentin the L9 array is carried out three times and the average value ofconversion is taken for analysis. Uncertainty in the measuredconversion of the FFA is within72%.

3. Results and discussion

3.1. Rate model and initial reaction kinetics

Esterification reaction of oleic acid with methanol is: OleicAcidþMethanol2Methyl OleateþWater. For esterification reac-tions, several kinetics have been reported: reversible second-order (Aafaqi et al., 2004; Bart et al., 1994), second-order forforward reaction with no backward reaction (Liu et al., 2006;Smith, 1939), and pseudo first order when large quantities of

S.Z. Hassan, M. Vinjamur / Chemical Engineering Science 107 (2014) 290–301 291

alcohol are taken (Berrios et al., 2007; Kocsisova et al., 2005). Inour previous study (Hassan and Vinjamur, 2013), it is found thatthe collected kinetics data of esterification of oleic acid in sun-flower oil is best fitted to a reversible second-order reaction rate,and the equilibrium constants are found to be 1.53 (70.05), 1.67(70.04), and 1.91 (70.06) at 40 1C, 50 1C, and 65 1C, respectively.The rigorous thermodynamic approach with activity coefficients isprincipally the correct method for kinetics study. However, in ourprevious study (Hassan and Vinjamur, 2013), it is discussed indetail and concluded that the concentration-based approach isvalid for the kinetics and equilibrium analysis of homogeneous-catalytic esterification of fatty acids with alcohols. A similar conclu-sion is also drawn by Ronnback et al. (1997) and Keurentjes et al.(1994) by comparing both the activity-based and concentration-based approaches in their studies.

For the initial reaction kinetics, the fitted straight lines are notpassing through origin, as shown in Fig. 1a, which means that thereaction does not obey reversible second-order kinetics during theinitial period. Reverse hydrolysis of esters is considered negligible inthe initial reaction kinetics, thus irreversible second-order kineticmodel is used for the initial period of reaction as shown in Fig. 1b.Good fits to straight lines, passing through origin, are obtained for6–8min of the initial reaction kinetics. This irreversible kinetics in theinitial period is in agreement with sulfuric acid-catalyzed esterificationof fatty acids with methanol reported by Boucher et al. (2008) and Liuet al. (2006). Hence, irreversible second-order kinetics in initial periodfollowed by reversible second-order kinetics in later period describesthe esterification of free fatty acid with methanol.

3.2. Analysis of rate constants

The second-order rate constants (k or SRC) are calculated fromthe slopes of the fitted-lines to the kinetics data shown in ourprevious study (Hassan and Vinjamur, 2013). The SRCs as responsefactors and L9 array as design of experiments are taken as input toMINITAB 15 software for Taguchi analysis. The signal-to-noise ratio(SN ratio) is mainly used as the objective function in Taguchimethod. SN ratio indicates the sensitivity of signal to noise wheresignal is quality characteristic (or response factor) and noise isvariation (or deviation) in response factor. Larger-is-better SN ratiois applied to the rate constants because they should be as high aspossible to achieve maximum conversion in less time. Hence, theobjective function to be maximized is expressed as

SN ratio¼ �10 log 101n

∑n

i ¼ 1

1y2i

!ð1Þ

where y is the response factor (quality characteristic) and n is thenumber of observations. Results of larger-is-better SN ratio

analysis of SRCs are shown in the main effects plots (Fig. 2a),which describe the effect of each reaction parameter on SRCs.

An increase in SN ratio (larger-is-better) with a variable impliesthat the response factor increases with that variable. SN ratioanalysis of SRC in Fig. 2a shows appreciable increase in the SRCs(as a response factor) with increase in temperature from 40 to 65 1Cand catalyst loading from 0.5 to 2.0 wt%, whereas SRCs increasednegligibly as MR is raised from 3 to 9. The effects of reactionparameters on the SRCs are quantified by the analysis of variance(ANOVA) for the SN ratios (Table 2). The contributions of tempera-ture and catalyst loading to SRC are 65.89% and 26.98%, respectively,which are far higher than MR contribution (1.77%). As the effect ofMR is found to be negligible, the sum of squares of MR effect ispooled with error sum of squares to get improved values of F-ratio(pooled) and probability (P-value) at 95% confidence level. Thepooling procedure and the criteria to determine pooling are given indetail elsewhere (Phadke, 2008). It can be seen from Table 2 thatthe P-value for temperature and catalyst loading iso0.05, whichmeans that they are significant parameters at 95% confidence level.

A variation in MR changes reactants’ concentration, and fromthe above analysis, it can be deduced that the SRCs are nearlyindependent of reactants’ concentration because the effect of MRis negligible (1.77% contribution to SRC). According to the L9 array(Table 1), at each level of catalyst loading, experiments areconducted at different levels of MR and temperature. If Arrheniusplots (ln(k) vs. 1/T) are made for each of these levels of catalystloadings in the L9 array (Table 1), straight lines are expected topass through the data points because MR has negligible effect onSRC. Fig. 2b shows that the straight lines reasonably fit the data at

Table 1L9 orthogonal array of experiments for esterification of oleic acid (mixed 50 wt% asFFA in sunflower oil) with methanol.

Exp.no.

Catalyst loading(wt%)

Temperature(1C)

Molar ratio(MR)a

Empty(e)b

1 0.5 40 3 12 0.5 50 6 23 0.5 65 9 34 1.0 40 6 35 1.0 50 9 16 1.0 65 3 27 2.0 40 9 28 2.0 50 3 39 2.0 65 6 1

a Methanol-to-FFA mole ratio.b Empty means not assigned to any parameter and 1, 2, 3 are coded units.

Fig. 1. Rate kinetics for the initial period of reaction: (a) reversible second-orderkinetics and (b) irreversible second-order kinetics. Values of the reaction para-meters are given in legends with forward slash as catalyst loading (wt%),temperature (1C), and MR, respectively.

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0.5 and 1.0 wt% catalyst loading, but does not fit the data at 2 wt%.As in the case of Arrhenius plots in Fig. 2b, MR is not fixed.Therefore, MR could affect SRC due to which no linear fit is foundat 2 wt% catalyst loading. Effect of MR on rate constants ofesterification of oleic acid with alcohol is also reported by Lingand Geankoplis (1958) and Othmer and Rao (1950). This effectcontradicts the deduction that SRC is reactants’ concentration-independent.

Temperature influence on SRC is evident from Arrhenius expres-sion and catalyst affects SRC by lowering the activation energy ofreaction; but physical insight into the dependency of SRC on MR is notfound in the literature on esterification of fatty acids. SRC has aconcentration term in its unit in denominator, and the effect of MR onSRC exists possibly due to this embedded concentration term in SRC.Jesudason (2008) strongly argued that even the treatment of rateconstant of elementary reactions as strictly independent of reactants’concentrations is due to historical conventions. A general mass actionrate law reported by Blum and Luus (1964) and Pekař (2011) is given inEq. (2). The ϕ term in this equation is not present in the popularGuldberg–Waage kinetic mass action law:

r¼ kþ1ϕ Πm

i ¼ 1aνii �k�1ϕ Π

m

i ¼ 1aν0ii ð2Þ

where ϕ is some function of activities, ai is the activity of reactingspecies, and νi and ν0 i are the exponents which may be different fromthe stoichiometric coefficients. For thermodynamically non-ideal reac-tion, aAþbB2cC, the overall rate can be expressed using Eq. (3), asgiven elsewhere (Jesudason, 2008)

d½C�dt

¼ ψkþ1½A�αγαA½B�ηγηB�ψk�1½C�λγλC ð3Þ

where γ is the activity coefficient and ψ is the concentration dependentfactor. Boyd (1977) explained that ψ term is consistent with “thereciprocal of the activity coefficient of the activated complex” in

Eyring’s transition state theory. In Eyring’s model (Eyring and Eyring,1963; Butt, 2000), a reaction scheme, AþB2Activated Complex-Products, is postulated for the pseudo-elementary reaction. The reac-tion rate for the scheme is

r¼ k0ðTÞγAγBγ†

CACB ð4Þ

γ† is the activity coefficient of the activated complex. Furthermore, k0(T)is the temperature-dependent rate constant for thermodynamicallyideal systems which is described according to the Brønsted–Bjerrumrelation as

k ¼ k0ðTÞγAγBγ†

ð5Þ

Boudart (1968) assessed the non-ideality of electrolyte systemsand experimentally observed the dependence of rate constants onthe ionic strength of the liquid phase. A rate expression, purelyproportional to the product of activity of the reactants which doesnot account for γ† and its solvation by the liquid phase, is unable todescribe the experimental results. Eckert and Boudart (1963) alsoshowed that for a non-ideal gas phase reaction at high pressures,Eq. (6) is the correct rate expression for a homogeneous gas phasereaction where fugacity coefficient ϕ is used instead of γ

k¼ k0ðTÞϕAϕBZ

ϕ†ð6Þ

Z is the compressibility factor for the gas mixture. For 2HI-H2þ I2at high pressures, they obtained a pressure-independent rateconstant over a wide range of HI pressures.

Bronsted acids can catalyze the esterification of fatty acids byprotonation of the carbonyl oxygen to give an oxonium ion. Thision activates the nucleophilic attack of an alcohol to form atetrahedral intermediate. Subsequent dis-proportionation of theintermediate generates corresponding ester after loss of a proton(see Fig. 3) (Lilja et al., 2002; Liu et al., 2006; Yalcinyuva et al.

Fig. 2. Second-order rate constant (SRC): (a) SN ratio analysis and (b) Arrhenius plots.

Table 2Analysis of variance (ANOVA) for SRC (k).

Source DFa SSb MSc % Contribution F-ratio (pooled) P-value

Catalyst loading 2 60.998 30.499 26.98 7.56e 0.044 SignificantTemperature 2 148.964 74.482 65.89 18.47e 0.010 SignificantMolar ratio 2 4.006d 2.003 1.77Residual error 2 12.123 6.061 5.36Total 8 226.091 100Residual error (pooled) 4 16.128 4.032

a DF, degree of freedom.b SS, sum of squares.c MS, mean of sum of squares.d The SS of parameter added together with SS of residual error to form residual error SS (pooled).e Significant at 95% confidence level: F(0.05,2,4)¼6.94 (Bagachi, 1993).

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2008). Reaction of protonated fatty acid with alcohol is the ratedetermining step where the transition states (activated com-plex), corresponding to the protonated acid, form the tetrahedralintermediate (Aranda et al., 2008; Lilja et al., 2002; Bart et al.,1994).

However, Jesudason (2008) showed from the moleculardynamics simulation, without using any transition states, thateven the rate constant of elementary reaction depends on reac-tants’ concentration in contrast to the current form of elementaryrate constant. For reaction in solution, ψ in the form of the rateequation in Eq. (3) depends on the concentrations of all species inthe solution (Denbigh, 1955). According to Madon and Iglesia(2000) “… the number of activated complexes undergoing areactive vibration per unit volume, and not the thermodynamicactivity of the activated complexes doing so, determines thereaction rate.” In some other form of rate equation (Manes et al.,1950), shown in Eq. (7), φ is some complicated function inthe denominator and depends on all the species in the reactionmass:

r¼ kþ1∏mi ¼ 1½Ai�νiφ

�k�1∏mi ¼ 1½A�ν0ii

φð7Þ

[Ai] is the concentration of species Ai.At this point, a supposition can be made that the use of catalyst

concentration in place of γ† may help to obtain rate constants thatare independent of MR. Furthermore, the concentration of catalystis affected by MR as the volume of reaction mass increases withincrease in MR such that the amount of oil (containing 50% FFA)and catalyst loading is fixed. Since kinetics of acid-catalyzedmono-esterification can be studied with simple concentration-based rate approach (Hassan and Vinjamur, 2013; Keurentjes et al.,1994; Ronnback et al., 1997), activity coefficients of reactingspecies are taken as unity. Linear regression of our preliminarydata (not shown here) for rate constant relation with inverse oftemperature (Arrhenius relation) also suggested that the rateconstant can be obtained when the catalyst concentration is usedin the denominator of rate equation.

Therefore, SRC (k) is modified accordingly where volume andmoles embedded in its units are considered as volume of reactionmass (V) and moles of catalyst (NCatalyst), respectively. Thus,modified SRC can be written as

k¼ k0V

NCatalyst

� �¼ k0

ðcatalyst concentrationÞ ð8Þ

Fig. 3. Reaction mechanism of acid-catalyzed esterification of fatty acid with alcohol.

Fig. 4. Concentration-independent rate constant (CIRC): (a) SN ratio analysis and (b) Arrhenius plots.

Fig. 5. Interaction plots between temperature and catalyst loading for (a) SRC and(b) CIRC.

S.Z. Hassan, M. Vinjamur / Chemical Engineering Science 107 (2014) 290–301294

where k0 can be called as concentration-independent rate constant[(time)�1] referred to as CIRC henceforth. NCatalyst term in thedenominator of Eq. (8) suggests that increase of this termdecreases k (SRC). However, it is discussed in detail in Section3.4 (see Fig. 6) that CIRC is catalyst loading dependent and theratio of CIRC to NCatalyst indeed increases linearly with catalystloading. This linear increase implies that the rate of esterificationreaction increases with catalyst loading.

The CIRC values are calculated using Eq. (8). Similar to SRC,larger-is-better SN ratio analysis and ANOVA for SN ratio arecarried out on CIRC. The trends of SN ratio analysis of CIRC areshown in Fig. 4a. These trends are similar to those shown in Fig. 2afor SRC, which means CIRCs increase with temperature andcatalyst loading. But, no change in CIRC is found with increase in

MR from 3 to 9. The ANOVA table (Table 3) shows that the reactiontemperature and catalyst loading are significant parameters at 95%confidence level as P-valueo0.05, and their contribution to CIRC is22.24% and 75.65%, respectively; the contribution of MR to CIRC is0.30% only.

The residual error is 1.81% (Table 3) which is less than theresidual error 5.36% obtained in ANOVA for SRC (Table 2). Residualerror is the difference between total sum of squared deviationsand sum of squared deviations due to individual parameterseffects. L9 array with three parameters and no-interactionassumption is used in this study. Therefore, smaller residual error(o2%) represents more accuracy of effects of individual para-meters on CIRC and supports the validity of no-interactionassumption over the experimental range studied. The percentcontribution in Table 3 suggests that the catalyst loading influ-ences CIRC more than the temperature over the range of variablesstudied. Arrhenius plots of ln(k0) vs. 1/T are also made for each ofthe levels of catalyst loadings as shown in Fig. 4b. Arrhenius plotsfor CIRC (Fig. 4b) fit better to straight lines compared to theArrhenius plots for SRC (Fig. 2b); even at 2 wt% catalyst loadinggood fit to straight line is obtained.

Therefore, the following results of analysis of CIRC indicate thatCIRC is independent of MR unlike the SRC: no change in SN ratiowith increase in MR (Fig. 4a); 0.30% (close to zero) contribution byMR (Table 3); o2% residual error (Table 3); and straight lines fitsfor the Arrhenius plots (Fig. 4b).

3.3. Interactions of the parameters

The L9 orthogonal array is applied with an assumption of nointeractions among the parameters. This assumption is now requiredto be examined for validation of the previous deductions drawn fromSN ratio analysis, ANOVA, and Arrhenius plots for both SRC and CIRC. Ifthe rate constants are truly independent of MR, then only theinteraction between temperature and catalyst loading is required tobe understood. This interaction is examined with the help of interac-tion plots. When parameters are plotted on an interaction graph, acharacteristic of parallelism between any two parameters shows theabsence of interaction between them (Phadke, 2008). In L9 orthogonalarray, if variables in columns 1 (catalyst loading) and 2 (temperature)are independent of each other, then they would have no effect oncolumns 3 and 4. Otherwise, the effect of interaction between thevariables in columns 1 and 2 would be confounded with the effect ofvariables in columns 3 and 4. From the SN ratio analysis (Fig. 2a) andANOVA (Table 2) of SRC, effects on columns 3 and 4 are found to benegligible. Thus, it is concluded that SRC is nearly independent of MR(means reactants’ concentration independent). If this is true, thenthere should be no interaction between catalyst loading andtemperature.Fig. 6. Effect of catalyst loading on (a) CIRC and (b) k0/NCatalyst.

Table 3Analysis of variance (ANOVA) for CIRC (k0).

Source DFa SSb MSc % Contribution F-ratio (pooled) P-value

Catalyst loading 2 506.799 253.40 75.65 71.76e 0.001 SignificantTemperature 2 148.964 74.482 22.24 21.09e 0.008 SignificantMolar ratio 2 2.002d 1.001 0.30Residual error 2 12.123 6.061 1.81Total 8 669.888 100Residual error (pooled) 4 14.125 3.531

a DF, degree of freedom.b SS, sum of squares.c MS, mean of sum of squares.d The SS of parameter added together with SS of residual error to form residual error SS (pooled).e Significant at 95% confidence level: F(0.05,2,4)¼6.94 (Bagachi, 1993).

S.Z. Hassan, M. Vinjamur / Chemical Engineering Science 107 (2014) 290–301 295

Fig. 5a illustrates plot of interaction between temperature andcatalyst loading which is based on SN ratios obtained for SRC. Noparallel lines are obtained between any two levels of temperatureand catalyst loading. This means that the assumption of nointeraction in L9 orthogonal array study is invalid in the case ofanalysis of effects of three parameters on SRC. Therefore, thededuction “SRC is concentration-independent“ becomes invalid.

Fig. 5b illustrates the plot of interaction between temperatureand catalyst loading based on SN ratios obtained for CIRC. All thelines between 0.5, 1.0, and 2.0 wt% are almost parallel to eachother when temperature increases from 50 to 65 1C. Whentemperature is increased from 40 to 50 1C, parallel lines areobserved between 0.5 and 2.0 wt% catalyst loading. The only andslight deviation from parallel behavior at 1.0 wt% between 40 and50 1C is found. Parallelism between 0.5 and 2.0 wt% catalystloading satisfies the criteria of no interaction. Overall, a negligibleinteraction between temperature and catalyst loading isfound. Therefore, no-interaction assumption is valid for CIRC thussupports the deduction that CIRC is independent of MR.

3.4. Effects of catalyst loading and temperature on CIRC

It is found from Taguchi analysis that CIRC is independent ofMR, but depends on catalyst loading and temperature. One canargue that CIRC should not be affected by catalyst loading because

catalyst concentration term is separated (see Eq. (8)). Catalystconcentration is determined by catalyst loading and MR. At a fixedcatalyst loading, variation in MR affects the concentration ofcatalyst, but the total weight (or moles) of catalyst in the reactionmass remains constant. Change in weight (or moles) of catalystloading affects both catalyst concentration and CIRC, whereas onlychange in MR affects catalyst concentration and not CIRC. There-fore, mass (or moles) of catalyst, not concentration, affects CIRC.

Fig. 6a shows non-linear increase of CIRC with catalyst loading (inthe form of NCatalyst) at different temperatures. Increase in CIRC withtemperature rise at 0.5 wt% (0.002551mol/50 g OA) catalyst loading isnot appreciable. Also, at 1.0 wt% (0.005102mol/50 g OA) and 2.0 wt%(0.010204mol/50 g OA) catalyst loadings, increase in CIRC is smallwith temperature increase from 40 to 50 1C except at 65 1C. However,temperature dependency of CIRC followed the Arrhenius law as shownin Fig. 4b. Non-linear increase in CIRC with catalyst loading at differentreaction temperatures can be fitted to quadratic polynomials with acoefficient of determination (R2)40.99 as follows (see Fig. 6a):

k0 ¼ αN2CatalystþβNCatalyst ð9Þ

α and β are constants. Using Eqs. (8) and (9), SRC can be written as

k¼ ðαN2CatalystþβNCatalystÞðV=NCatalystÞ ¼ ðαNCatalystþβÞV ð10Þ

Change in NCatalyst at fixed MR means volume of reaction massremains constant, such that the amount of oil (containing 50% OA)

Table 4L27 (313) orthogonal array design of experiments (in coded unit)a.

Exp. no. CLb Tc 3 4 MRd 6 7 8 9 10 11 12 13

1 1 1 1 1 1 1 1 1 1 1 1 1 12 1 1 1 1 2 2 2 2 2 2 2 2 23 1 1 1 1 3 3 3 3 3 3 3 3 34 1 2 2 2 1 1 1 2 2 2 3 3 35 1 2 2 2 2 2 2 3 3 3 1 1 16 1 2 2 2 3 3 3 1 1 1 2 2 27 1 3 3 3 1 1 1 3 3 3 2 2 28 1 3 3 3 2 2 2 1 1 1 3 3 39 1 3 3 3 3 3 3 2 2 2 1 1 1

10 2 1 2 3 1 2 3 1 2 3 1 2 311 2 1 2 3 2 3 1 2 3 1 2 3 112 2 1 2 3 3 1 2 3 1 2 3 1 213 2 2 3 1 1 2 3 2 3 1 3 1 214 2 2 3 1 2 3 1 3 1 2 1 2 315 2 2 3 1 3 1 2 1 2 3 2 3 116 2 3 1 2 1 2 3 3 1 2 2 3 117 2 3 1 2 2 3 1 1 2 3 3 1 218 2 3 1 2 3 1 2 2 3 1 1 2 319 3 1 3 2 1 3 2 1 3 2 1 3 220 3 1 3 2 2 1 3 2 1 3 2 1 321 3 1 3 2 3 2 1 3 2 1 3 2 122 3 2 1 3 1 3 2 2 1 3 3 2 123 3 2 1 3 2 1 3 3 2 1 1 3 224 3 2 1 3 3 2 1 1 3 2 2 1 325 3 3 2 1 1 3 2 3 2 1 2 1 326 3 3 2 1 2 1 3 1 3 2 3 2 127 3 3 2 1 3 2 1 2 1 3 1 3 2

Reaction parameters Levels

1 2 3

Catalyst loadings (wt%) 0.5 1.0 2.0Temperature (K) 313 323 338Molar ratio (MR) 3 6 9

a Rows 1, 5, 9, 11, 15, 16 21, 22, and 26 with columns 1, 2, 5 and 6 are L9 experimentsbecause L9 is also a subset of L27. Columns 1, 2, and 5 are filled with catalyst loading,temperature, and MR, respectively, and the rest are empty columns. Exp. nos. 4, 7, 12, 13, 20,25, and 27 are validation runs which are other than nine experiments conducted using L9orthogonal array.

b CL, catalyst loading.c T, temperature.d MR, molar ratio.

S.Z. Hassan, M. Vinjamur / Chemical Engineering Science 107 (2014) 290–301296

is fixed, then Eq. (10) can be written as

k¼ α0NCatalystþβ0 ð11Þ

where α0 ¼ αV and β0 ¼ βV . Expression for rate constant in Eq. (11)is similar to the expression reported by Ling and Geankoplis (1958)for sulfuric acid-catalyzed esterification of oleic acid with butanolas

k¼ 0:401Cþ0:000368 ð12Þwhere C is the catalyst loading in wt% and Eq. (12) only holds at100 1C.

Eq. (8) tempts one to think that the SRC would decrease ascatalyst loading is increased, but this temptation is wrong. Theratio of CIRC to NCatalyst is plotted against catalyst loading (Fig. 6b)to understand the overall effect of catalyst loading on this ratio.Fig. 6b shows a linear increase in the ratio at different tempera-tures with an increase in catalyst loading. As per Eq. (8), this ratiois equivalent to SRC at fixed MR (as V remains constant). The linearbehavior of rate constant with catalyst loading is also reported byothers for esterification reactions (Fogler, 2006; Leyes and Othmer,1945; Ling and Geankoplis, 1958; Jong et al., 2009). Therefore, theoverall effect of CIRC to NCatalyst ratio would increase in thereaction rate with catalyst loading even though the catalystloading term appears in the denominator of the rate equation.

3.5. Prediction of kinetics

Three variables with three levels of each require a total of 33

experiments under full factorial design for complete analysis ofeffect of variables on response factor. The L27 (313) orthogonalarray is selected for full factorial design as shown in Table 4. Thecolumns 1, 2, and 5 are assigned to catalyst loading, temperatureand MR, respectively. The L9 orthogonal array is a subset of L27when columns 1, 2, 5, and 6 are combined with rows 1, 5, 9, 11, 15,16, 21, 22, and 26. For validation of kinetics predictions, exp. nos. 4,7, 12, 13, 20, 25, and 27 from L27 are carried out. Each experimentis carried out three times and average of the conversion of oleicacid is taken. Uncertainty in measured conversion of oleic acid iswithin 72.3%. In this section, comparison of predictions fromdifferent forms of rate equations is made.

3.5.1. CIRC-based kineticsOn the basis of the deduction “CIRC is reactant concentrations

independent“ and the orthogonal array design, a method can bedeveloped for predictions of reaction kinetics for all the experi-ments from full factorial design. Reaction kinetics is predictedusing reversible second-order rate equation as given in thefollowing equation:

�d½OA�dt

¼ k0V

NCatalyst

� �½CH3OH�½OA��

½H2O�½MeOA�Keq

� �ð13Þ

The above equation is obtained by replacing SRC with CIRC.Rate expression shown in Eq. (13) might mislead that increase incatalyst loading results decrease in reaction rate. However, it isdiscussed earlier in Section 3.4 (see Fig. 6) that CIRC is catalystloading dependent and the ratio of CIRC to NCatalyst increaseslinearly with catalyst loading. Therefore, the overall effect of CIRCto NCatalyst ratio would increase in the reaction rate with catalystloading. Integrating Eq. (13) in terms of conversion (X) of oleic acidgives

lnð1þMRþa2�2a1XÞð1þMR�a2�2a1XÞ

� �¼ k0

VNCatalyst

� �a2½OA�0tþ Intercept ð14Þ

where Intercept ¼ ln ½ð1þMRþa2Þ=ð1þMR�a2Þ� and rest of theterms are defined in our previous study (Hassan and Vinjamur,2013).

Values for Keq, CIRC (k0), and the Intercept are obtained from the L9experiments. Keq is a function of reaction temperature only; CIRCdepends on temperature, catalyst loading, and not on MR as discussedearlier; and Intercept depends on temperature and MR. Therefore, thevalues for Keq, CIRC, and the Intercept are tabulated accordingly inTable 5. Intercept values in Table 5 are taken from intersection of axisof ordinate by fitted-lines shown in our previous study (Hassan andVinjamur, 2013) instead of values which can be calculated fromIntercept formula shown in Eq. (14). The reason is initial reactionkinetics followed the irreversible second-order kinetics, and thus,straight lines for reversible second-order rate kinetics are not passingthrough origin as shown in Fig. 1a.

A systematic selection of Keq, CIRC, and the Intercept values canbe made for prediction of kinetics of L27 experiments. Forexample, the exp. no. 4 of L27 (Table 4) is carried out at levels 1,2, and 1 for catalyst loading, temperature, and MR, respectively.Using Table 5 the values for Keq, CIRC, and the Intercept are selectedas 1.67, 1.3070�10�05 min�1, and 2.7134, respectively, for theprediction of kinetics of exp. no. 4.

3.5.2. SRC-based kineticsOne can argue that if one starts with the elementary reaction

steps, an overall rate constant can be obtained which consists ofelementary rate constants only and then the overall rate constantdoes not depend on reactants’ concentration. A widely acceptedmechanism for acid-catalysis of esterification reactions is shown inFig. 3. The second reaction step (protonated oleic acid reacts withmethanol to form tetrahedral intermediate, which ultimatelyproduces protonated methyl oleate—MeOAþ and water) is con-sidered as the rate controlling step. Pre-equilibrium assumption isapplied for the first (protonation of carboxyl group of oleic acidwhich produces carbocations) and the last (dissociation of proto-nated methyl oleate into methyl oleate and proton) steps. The rateequation can be derived as given in the following equation:

�d½OA�dt

¼ k ½CH3OH�½OA��½H2O�½MeOA�

Keq

� �ð15Þ

Table 5Keq, CIRC, and Intercept values for predictions of kinetics.

Temperature Keq Catalyst loading Temperature CIRC (k0 , min�1) Temperature MR Intercept

1 1.53 1 1.17062E�05 1 2.74482 1.67 1 2 1.30701E�05 1 2 3.37073 1.91 3 2.30783E�05 3 3.7516

1 2.25564E�05 1 2.71342 2 3.79538E�05 2 2 3.1755

3 7.6276E�05 3 3.59271 7.01263E�05 1 2.6921

3 2 9.63631E�05 3 2 3.193 2.96719E�04 3 3.4763

S.Z. Hassan, M. Vinjamur / Chemical Engineering Science 107 (2014) 290–301 297

where k is the overall rate constant and consists of the rateconstant of the rate controlling step only, and Keq is a product ofK1, K2, and K3 (equilibrium constants of three elementary steps,respectively).

Thus, here in Eq. (15), k is a true second-order rate constant(means kinetic orders being equal to the molecularity of thereaction with respect to each of the reactants) obtained fromelementary reaction steps for esterification reactions. Also, thederived rate model in Eq. (15) is same as the rate kinetic modelobtained through kinetics data fitting shown in our previous study(Hassan and Vinjamur, 2013). So, SRC and the obtained overall rateconstant from mechanism are same. Therefore, validation experi-ments are also compared with the predictions when SRC is used inrate equation. Keq and the Intercept values in Table 5 are used as itis, and SRC values in place of CIRC are taken for predictions.

3.5.3. Catalyst concentration in numerator based kineticsIn some studies of homogenous-catalytic esterification of fatty

acid, catalyst concentration (CCatalyst ) is used in the numerator ofreversible second-order rate model (Bart et al., 1994; Jong et al.,2009; Keurentjes et al., 1994; Liu et al., 2006). The justification forthis rate form is that the rate of reaction increases with catalystconcentration. Therefore, this form of rate equation is also con-sidered for kinetics predictions with an assumption that the rateconstants involved are independent of reactants’ concentration.Similar to SRC-based predictions, Table 5 is used for predictionswith rate constant (k“) values calculated as

SRC ¼ ðk00ÞCCatalyst ¼ ðk00Þ NCatalyst

V

� �ð16Þ

3.5.4. Comparison of predicted kinetics:The CIRC-based predictions are compared with the validation

experiments in Fig. 7(panels a–c), and good agreement betweenexperiments and predictions are found. In the SRC-based predic-tions, Fig. 7(panels a–c) shows that predictions of exp. nos. 4, 7, 12,and 27 are in good agreement with experiments, but the predic-tion of exp. nos. 13, 20, and 25 are poor (Fig. 7(panels b and c)). Itmeans that the assumption of concentration-independent SRC isnot valid for all the experiments, even though SRC is found to betrue order rate constant from reaction mechanism. This confirmsthe earlier result of 0MR effect on SRC0 obtained from parametricinteraction-analysis and Arrhenius plots. Fig. 7(panels a–c) alsoshows that the predictions with the use of the catalyst concentra-tion in the numerator of rate equation are poorer than those usingCIRC and SRC. This implies that the k“ calculated from Eq. (16) isnot concentration independent.

Parity plots for all the three forms of kinetics predictions areshown in Fig. 8(panels a–c) against experimental results, and theroot mean square deviation (RMSD) of all the validation experi-ments are calculated using the following equation:

RMSD¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∑N

i fXiðexperimentÞ�XiðpredictedÞg2N

sð17Þ

Dashed line in parity plots is 73% error bounds for conversionfrom experiments. Fig. 8 shows that more points are within thiserror bounds in CIRC-based kinetics (Fig. 8a) than other forms ofrate kinetics. RMSD of all the validation experiments are smallerwhen predictions are based on CIRC as shown in the table (seeFig. 8). Therefore, it is clear that CIRC-based predictions are betterthan other commonly used rate forms. The high accuracy in CIRC-based predictions supports our earlier deduction that CIRC isindependent of reactants’ concentrations.

3.6. Effects of reaction parameters on kinetics

Effects of individual reaction parameters on reaction kineticsare discussed in this section. L27 array with three parameters canbe arranged in three series (catalyst series, temperature series, andMR series) for the study of effects of individual parameters onkinetics. A set of experiments are formed in each series by keepingtwo parameters constant and varying the series parameter. In plotsof each parameter series, dark dots are conversion obtained fromexperiments and solid lines are corresponding CIRC-based kineticspredictions. For all the parameters series, CIRC-based predictionsare in good agreement with experiments (see Fig. 9).

3.6.1. Effect of catalyst loadingFig. 9 shows the effect of catalyst loading on the conversion of

the fatty acid and in each set of experiments, the catalyst loading

Fig. 7. Comparison of validation experiments 4, 7, 12, 13, 20, 25, and 27 of L27 array(for details see Table 4) with kinetics predictions based on CIRC ( ), SRC ( ),and catalyst concentration in numerator ( ).

S.Z. Hassan, M. Vinjamur / Chemical Engineering Science 107 (2014) 290–301298

increases while temperature and MR are fixed. Fig. 9a is for the setof experiments L27/4, L27/13, and L27/22 which correspond to 0.5,1.0, and 2.0 wt% catalyst loading, respectively, at fixed temperatureof 50 1C and 3 MR. From the experiments in Fig. 9a, it is evidentthat the conversion of the fatty acid increases with the catalystloading. Similar results are also found for other set of experimentsin Fig. 9(panels b and c). At higher amount of catalyst, moreHþ� ions are present in the reaction mixture, which accelerate theesterification of fatty acid with methanol (Sendzikiene et al., 2004;Lucena et al., 2008). In every set of experiments shown in Fig. 9(panels a–c), CIRC-based predictions are in good agreement withexperiments. It means increase in predicted-conversion of fattyacid with the increase in catalyst loading is found even thoughNCatalyst is present in the denominator of the rate expression(Eq. (13)).

3.6.2. Effect of temperatureFig. 9(panels d–f) shows the effect of temperature on conver-

sion of fatty acid in the temperature series. Fig. 9d is for the set ofexperiments L27/1, L27/4, and L27/7 which correspond to 40, 50,and 65 1C, respectively, at fixed catalyst loading of 0.5 wt% and3 MR. From the experiments in Fig. 9d, increase in the conversionof the fatty acid with a temperature rise is observed. Similarresults are also found in other set of experiments in Fig. 9(panelse and f). The reaction is endothermic (ΔH¼ þ7:8270:02 kJ=mol);therefore, the conversion increases with temperature.

3.6.3. Effect of MREffect of MR on the conversion of fatty acid is shown in Fig. 9

(panels g–i). The set of experiments L27/25, L27/26, and L27/27 inFig. 9i corresponds to 3, 6, and 9MR, respectively, at fixed temperature

of 65 1C and 2 wt% catalyst. From the experiments in Fig. 9i, increase inthe conversion of the fatty acid with MR is observed. Similar trends arealso found in other set of experiments in Fig. 9(panels g and h). Thesetrends can be explained by Le Chatelier0s principle according to whichincrease in MR shifts the equilibrium of esterification reaction in theforward direction.

Therefore, CIRC-based reversible second-order rate kinetics isfound to be more accurate in the kinetic study of biodieselsynthesis from sulfuric acid-catalyzed esterification of FFA withmethanol than other commonly used rate forms. The method forkinetics predictions based on CIRC and Taguchi design shown inthis study could help to determine FFA content while carrying outpretreatment of low-cost feed-stocks to reduce FFA to desired levelprior to trans-esterification of oil. In this work, over the range ofreaction parameters studied, the maximum predicted conversionof FFA is 94.2% at 2 wt% catalyst, 65 1C, and 9 MR after 180 min ofreaction. Experiment showed that 97.54% conversion is achieved atthe same reaction conditions that means 50 wt% FFA is reduced to1.23 wt% which is in good agreement with CIRC-based predictedvalue of 2.89 wt% FFA.

4. Conclusions

Taguchi design of experiments using the standard L9 orthogonalarray is applied to study the effect of reaction parameters (molarratio of methanol-to-FFA, temperature, and catalyst loading) on theapparent rate constant of the esterification reaction of FFA withmethanol. The regressed second-order rate constant (k or SRC) didnot fit the Arrhenius plot of ln(k) vs. 1/T well. The misfit is due to thedependence of the SRC on the concentration of reactants. No physical

Fig. 8. Parity plots for conversion of oleic acid (X) from experiments with predicted values. Predictions are based on rate kinetics using (a) CIRC, (b) SRC and (c) CCatalyst innumerator. Dashed lines ( ) are 73% error bounds for X (experiments). RMSD for all the validation experiments are shown in the attached table.

S.Z. Hassan, M. Vinjamur / Chemical Engineering Science 107 (2014) 290–301 299

insight is found for this dependency in the literature for esterificationof fatty acids.

Based on the general literature on dependence of the rateconstant on the concentration of reactants and esterification-ratedetermining step, concentration term in the units of SRC has beenidentified as the concentration of the catalyst. Accordingly SRChas been modified to include the concentration of the catalyst inthe denominator of the rate law. This law is consistent with thegeneral mass-action rate law reported by Blum and Luus (1964)and Pekař (2011). The regressed values of the modified rate cons-tant fitted the Arrhenius plot well. Notably, this modified rateconstant is shown to be independent of the reactants’ concentra-tion and thus, named as 0concentration independent rate constant(CIRC)0. The kinetics of the esterification reaction are predictedbetter using CIRC than other forms (conventional rate constantand inclusion of the catalyst concentration term in the numerator).The modified rate constant is shown to be equivalent to Brønsted–Bjerrum relation. Moreover, the modified rate constant used in thisstudy is more general than the reported ones for esterification of

fatty acids and it predicts kinetics over a wide range of reactionparameters.

Acknowledgments

We thank Professor Prakash R. Apte, Electrical EngineeringDepartment, Indian Institute of Technology Bombay, for helpfuldiscussions and suggestions in the use of the Taguchi method forthis study.

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