Utilization of MATLAB to Simulate Kinetics of Transesterification of Palm Oil-Based Methyl Esters...

Chemical Product and ProcessModeling

Volume 5, Issue 1 2010 Article 13

Utilization of MATLAB to Simulate Kineticsof Transesterification of Palm Oil-Based

Methyl Esters with Trimethylolpropane forBiodegradable Synthetic Lubricant Synthesis

Hamidah Abd Hamid∗ Robiah Yunus†

Thomas S.Y. Choong‡

∗Universiti Putra Malaysia, meyda [email protected]†Universiti Putra Malaysia, [email protected]‡Universiti Putra Malaysia, [email protected]

Copyright c©2010 The Berkeley Electronic Press. All rights reserved.

Utilization of MATLAB to Simulate Kineticsof Transesterification of Palm Oil-Based

Methyl Esters with Trimethylolpropane forBiodegradable Synthetic Lubricant Synthesis∗

Hamidah Abd Hamid, Robiah Yunus, and Thomas S.Y. Choong

Abstract

This paper presents the simulation work on kinetic study for the transesterification reactionbetween palm oil-based methyl ester with trimethylolpropane to produce biodegradable lubricantbase oil. The reaction mechanism involves three stepwise reversible series-parallel elementaryreactions. New kinetic modeling approach is needed in this field due to the limitation of theearlier kinetic models developed for this reaction. The earlier kinetic study made use of manyassumptions in order to simplify the solutions. In this work, new rate equations were derived fromthe actual reaction mechanism, and MATLAB was used to obtain the simultaneous solutions of therate equations. The results obtained were compared with the experimental data used in the earlierstudy. The new simulation approach was found to describe the experimental values satisfactorily.

KEYWORDS: transesterification kinetics, biodegradable lubricant, palm oil methyl ester, MAT-LAB simulation

∗This work was supported by Department of Chemical and Environmental Engineering, UniversitiPutra Malaysia.

1. Introduction Vegetable oils such as canola or rapeseed, sunflower, soybean and palm oil are potential renewable resources for biodegradable lubricants to replace the conventional mineral oil-based lubricants due to the rising consumers’ awareness about the environment and new legislations introduced by several countries (Cecutti and Agius, 2008; Joseph et al., 2007; McCarthy et al., 2008; Schneider, 2006; Yunus, 2003). These vegetable oil-based lubricants were reported to have better lubrication properties than petroleum-based lubricants. However, they have some inherent performance limitations as lubricants such as poor oxidative stability, sensitivity to hydrolysis and low temperature fluidity (Wilson, 1998). These limitations could be overcome by using structural chemical modification as reported elsewhere (Erhan et al., 2008; Joseph et al., 2007; Kempermann and Murrenhoff, 1998; Yunus et al., 2003).

Since palm oil methyl ester (POME) plays an increasing important role as one of the starting materials in the oleochemical industry in Malaysia, it has been selected as the raw material in the synthesis of polyol esters as lubricant base stock. Trimethylolpropane is the polyol selected due to its branched structure that provides high thermal stability (Wilson, 1998). As a fundamental element in the commercial or pilot scale-up of the processing plan, an attempt to simulate the kinetic models using analytical solution has already been established (Yunus et al., 2004) to predict the yield of product as a function of the operating variables. Nevertheless, the models are only valid within the initial part of the process and not when the system is approaching equilibrium. In this approach, the assumptions of irreversible reactions for all reactions involved were used to simplify the rate equations and the solutions. However, to satisfy the assumption of irreversible reactions, high excess of raw material is needed and the side product must be continuously withdrawn from the system. Thus, the earlier kinetic models have inherent limitations and new solutions are needed. It is more convenient to simulate the reaction rates for complex reaction system by using a computer program with suitable numerical procedure to reduce time for the development of kinetic model (Brule et al., 2009; Levenspiel, 1999; Noureddini, 1998). A successful simulator for the kinetic models is essential to ensure high productivity and optimum cost. Several software tools could be useful to simulate the reaction kinetics (Hinsberg and Houle, 2009; The Mathworks Inc., 1999; Watts and Shampine, 2009). The attention was focused to MATLAB due to its strong presence in the simulation field and its user-friendly factors. The emphasis of this work is on the utilization of MATLAB as the mathematical simulator to determine the concentration-time profiles for various species involved in the transesterification reactions of POME with TMP in a batch

1

Abd Hamid et al.: Kinetic Simulation of Palm Oil-Based Lubricant Synthesis

Published by The Berkeley Electronic Press, 2010

reactor. The simulation results were then compared with some of the experimental data values obtained by Yunus (2003). 2. Methods 2.1 Kinetic modeling Reaction kinetics In order to obtain appropriate kinetic models for simulator implementation, the kinetics of transesterification reaction of POME with TMP were developed distinctively. The reaction involves three distinct steps of elementary reversible series-parallel reaction mechanism (Yunus et al., 2004) since there are three hydroxyl groups in TMP to be substituted with an ester group that is cleaved from POME. The reaction yields methanol (M) as a by-product as well as other intermediate products such as TMP monoesters (ME) and diesters (DE). The main product is TMP triesters (TE). The kinetic mechanism of the reaction to produce TMP esters is described as follows:

TMP + POME ME + M (1) ME + POME DE + M (2) DE + POME TE + M (3)

The overall reaction is given by:

TMP + 3POME TE + 3M (4) We assume that the density of each component is constant. From the

above mechanism, the decomposition of TMP to form ME with second order kinetics is expressed in terms of molar concentration as follows:

MME'1POMETMP1

TMP CCkCCkdt

dCrTMP −=−=− (5)

The previous experimental work used excess POME and continuous removal of methanol via vacuum (Yunus et al., 2004). When that assumptions were also applied to this kinetic model, the term k1CTMPCPOME is very much larger than the

k3 k3’

k1

k1’ k2

k2’

k0

k0’

2

Chemical Product and Process Modeling, Vol. 5 [2010], Iss. 1, Art. 13

http://www.bepress.com/cppm/vol5/iss1/13DOI: 10.2202/1934-2659.1458

one k1’ CMECM, i.e. k1CTMPCPOME >> k1’ CMECM, so that k1’ CMECM is neglected. Hence, the rate of formation of TMP is given by:

POMETMP1

TMP CCkdt

dC−=

(6) If we assume that the remaining two steps of the reaction mechanism are reversible and follow the second order kinetics, the rates of formation of ME, DE, TE, POME and methanol can be represented by the following equations:

MDE2POMEME2POMETMP1ME CCkCCkCCk

dtdC ′+−=

(7)

MTE3POMEDE3MDE2POMEME2DE CCkCCkCCkCCk

dtdC ′−+′−=

(8)

MTE3POMEDE3TE CCkCCk

dtdC ′−=

(9)

MTE3POMEDE3MDE2

POMEME2POMETMP1POME

CCkCCkCCk

CCkCCkdt

dC

′+−′+

−−=

(10)

MTE3POMEDE3MDE2

POMEME2POMETMP1M

CCkCCkCCk

CCkCCkdt

dC

′−+′−

+=

(11)

The molar concentration of species i, Ci is given by: TTii MρnC ⋅= (12) where ρT is density of the mixture, ni is mole of component i and MT is the total mass of the mixture. Weight fraction-based kinetics It is essential to transform the rate equations into the weight fraction model since the composition of each component involved was measured in terms of mass ratio of a species i.e., Xi that is expressed by:

3



T

iii M

MWnX ⋅= (13)

We redefined the rate equations by substituting equation 12 into equations 6 to 11. Based on equation 13, the rate of formation for TMP was derived in terms of weight fraction by multiplying with TTMP ρMW to yield:

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

POME

TPOMETMP1

TMP

MWXXk

dtdX ρ

(14) The same approach was used to transform the remaining rate equations in terms of weight fraction for ME, DE, POME and methanol. The resulting rate equations are as follows:

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅′+

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅⋅

=

MDE

TMEMDE2

POME

TPOMEME2

POMETMP

TMEPOMETMP1

ME

MWMW

MWXXk

MWXXk

MWMWMWXXk

dtdX

ρ

ρρ

(15)

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅′+⎟⎟

⎠

⎞⎜⎜⎝

⎛′−

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅⋅

=

TEM

TDEMTE3

M

TMDE2

POME

TPOMEDE3

POMEME

TDEPOMEME2

DE

MWMW

MWXXkMW

XXk

MWXXk

MWMWMWXXk

dtdX

ρρ

ρρ

(16)

⎟⎟⎠

⎞⎜⎜⎝

⎛′−⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

⋅=

M

TMTE3

POMEDE

TTEPOMEDE3

TE

MWXXk

MWMWMWXXk

dtdX ρρ

(17)

4



⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅′+

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅′+

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

MTE

TPOMEMTE3

MDE

TPOMEMDE2

DE

TPOMEDE3

ME

TPOMEME2

TMP

TPOMETMP1

POME

MWMWρMW

XXk

MWMWρMW

XXk

MWρ

XXk

MWρXXk

MWρXXk

dtdX

(18)

(19)

Equations 14 to 19 were used in the programming coding in MATLAB. Table 1 and 2 show the numerical values needed in order to simulate the kinetic models. Table 1: Dimensions for the simulation via MATLAB

Parameter Value ρT 820.76 g/L XPOME,0 0.9556 XTMP,0 0.0444 XME,0, XDE,0, XTE,0, XM,0 0 MWTMP 134 MWME 389 MWDE 644 MWTE 899 MWPOME 287 MWM 32

⎟⎟⎠

⎞⎜⎜⎝

⎛′+

⎟⎟⎠

⎞⎜⎜⎝

⎛′+⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅

−

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅

−

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅

−=

TE

TMTE3

DE

TMDE2

POMEDE

TMPOMEDE3

POMEME

TMPOMEME2

POMETMP

TMPOMETMP1

M

MWρXXk

MWρXXk

MWMWρMWXXk

MWMWρMWXXk

MWMWρMWXXk

dtdX

5



Table 2: Initial value of reaction rate constants of transesterification between POME and TMP at 100°C

Parameter Initial Value* k1 0.9 k2 0.7 k3 0.25 k2’ 0.1 k3’ 0.08

*Any values can be inserted as the input condition to simulate the kinetic model 2.2 Implementation of the kinetics models in MATLAB As mentioned in the introduction, the kinetic models derived using analytical solution that relied heavily on process assumptions such an irreversible process for all reactions are only valid at the early part of the reactions but not applicable when the system is approaching equilibrium (Yunus, 2003; Yunus et al., 2004). Hence, we implemented numerical procedure via MATLAB for the integration of the resulting differential equations from Section 2.2. Ordinary differential equation (ODE) solver in MATLAB There are several numbers of built-in ODE functions in MATLAB for solving numerically the ordinary differential equations (The Mathworks Inc., 1999). In our modeling, we selected ODE solver function named ode45 which implement nonstiff solutions of Runge-Kutta 4, 5 order method. The resulting kinetic equations in terms of weight fraction (equations 14-19) were implemented as the references in MATLAB by declaring the function namely wtfrackinetic: function dXdt=wtfrackinetic(t,X) %X(1)=xTMP, X(2)=xME, X(3)=xDE, X(4)=xTE, X(5)=xPOME, X(6)=xMethanol %mw1=mwTMP, mw2=mwME, mw3=mwDE, mw4=mwTE, mw5=mwPOME, mw6=mwMethanol %rho=density of the mixture #Declaration of reaction rate constants, density of mixture and each molecular weight global k1 k2 k2r k3 k3r rho mw1 mw2 mw3 mw4 mw5 mw6

6



#Ordinary differential equations for rates of transesterification reaction dXdt=[-k1*X(1)*X(5)*rho/mw5;k1*X(1)*X(5)*mw2*rho/mw1/mw5-k2*X(2)*X(5)*rho/mw5+k2r*X(3)*X(6)*rho*mw2/mw6/mw3;k2*X(2)*X(5)*rho*mw3/mw2/mw5-k3*X(3)*X(5)*rho/mw5+k3r*X(4)*X(6)*rho*mw3/mw4/mw6-k2r*X(3)*X(6)*rho/mw6;k3*X(3)*X(5)*rho*mw4/mw3/mw5-k3r*X(4)*X(6)*rho/mw6;-k1*X(1)*X(5)*rho/mw1-k2*X(2)*X(5)*rho/mw2-k3*X(3)*X(5)*rho/mw3+k3r*X(4)*X(6)*rho*mw5/mw4/mw6+k2r*X(3)*X(6)*rho*mw5/mw3/mw6;k1*X(1)*X(5)*rho*mw6/mw1/mw5+k2*X(2)*X(5)*rho*mw6/mw2/mw5+k3*X(3)*X(5)*rho*mw6/mw3/mw5-k3r*X(4)*X(6)*rho/mw4-k2r*X(3)*X(6)*rho/mw3]; XTMP, XME, XDE, XTE, XPOME and XM must be defined within the same matrix, and so by calling XTMP as X(1) XME as X(2), XDE as X(3), XTE as X(4), XPOME as X(5) and XM as X(6), they are listed as common matrix to matrix X. The parameters of the simulation including the initial conditions, the simulation time (we took the reaction period time as the simulation time), the values of rate constants and molecular weights were fixed and stored outside the function in global form, since the cost of the numerical integration depends critically on the expense of evaluating the ODE function (Brule et al., 2009). This function was sent to the solver selected (ode45) which was then be simulated and solved to determine the components concentration. The code that represents this operation is: [t,X]=ode45(‘wtfrackinetic’,tspan,X0); 2.3 Statistical analysis An analysis of variance (ANOVA) was performed for the statistical analysis of the experimental data by using PHStat add-in for Microsoft Excel. It was done to estimate whether there is a significant influence of temperature on kinetic parameters by setting a confidence level of 95% (p-value <0.05). The fit quality of the model was evaluated using the similar statistical tests as used by Vega-Gálvez et al. (2009): coefficient of determination (R²), sum of squares error (SSE), root mean square error (RMSE) and Chi-square (χ²) as shown in equations 20-23. The optimum criteria were determined by selecting the lowest values of SSE, RMSE and χ² that are closer to zero, and the highest values of R² (R² ≈ 1).

SSTSSER −= 12 , where ( )

2

1,exp,

1 ∑=

−=N

jiij XX

NSST (20)

7



( )2

1,,,exp,

1 ∑=

−=N

jijcalcij XX

NSSE (21)

( )21

1,exp,,,

1⎥⎦

⎤⎢⎣

⎡−= ∑

=

N

jijijcalc XX

NRMSE (22)

( )

zN

XXχ

N

j ijcalcij

−

−=∑ =1

2,,,exp,2 (23)

3. Results and discussion As the paper is only emphasized on the simulation practice via MATLAB for kinetics of transesterification between POME and TMP, only some of appropriate experimental data values were taken from Yunus (2003) and compared with the simulated product distribution values to confirm the suitability of the simulation kinetic models. We chose the data of synthesis of POME with TMP in vacuum condition with molar ratio of POME to TMP 10:1 and 0.9 wt% sodium methoxide catalyst based on the total mass at a molar ratio of 3.9:1, the maximum weight fraction of DE occurred after one minute (Yunus, 2003). This comparison will be discussed in Section 3.2. In Section 3.1 the discussion will focus on how the estimation of rate constants via MATLAB affects the distribution curves. The influences of temperature on the rate constants determined by means of ANOVA and statistical analysis of the proposed kinetic model will be discussed in Section 3.3 and 3.4. 3.1 Rate constants In this study, the approach was to develop a full set of transesterification kinetics (as indicated in equations 14 to 19) that were used in a function file created via MATLAB to simulate the product distribution curves at 70-110°C reaction temperatures. The determination of rate constants for the second and third steps of reaction mechanism was done by considering the location of maxima points of the intermediate products, ME and DE. Based on the assumption of an irreversible reaction for the first step of the mechanism, Figure 1 shows the distribution curves of TMP at 100°C using several values of rate constant k1 that is closely fitted to the experimental data.

Larger value of k1 would give better estimation of TMP and lower value of SSE but the SSE values for the second and third step of the transesterification

8



reaction increase when k1 was equal or larger than 0.97. Hence, the optimum value of k1 is 0.96 ± 0.005 at this temperature where similar outcome at other temperatures was also observed (see Table 3). It is observed that the limiting reactant TMP has reached zero concentration meaning that it was completely reacted after 3 minutes. This indicates that the formation of ME from TMP was very fast, and the reverse reaction of the first step shown in equation 1 was suppressed, which supports our hypothesis that the first step of the transesterification mechanism is an irreversible reaction.

Figure 1: Effects of k1 on simulated TMP during the transesterification between

palm oil methyl esters (POME) with trimethylolpropane (TMP) at 100°C.

Table 3: Average reaction rate constants at various temperatures in the synthesis

of palm oil TMP esters after optimum fitting.

Temperature (°C) Rate constants ((wt/wt)-1.(min)-1) k1 k2 k3 k2’ k3’

70 0.03 0.02568 0.00851 0 9 × 10-5

80 0.15 0.11420 0.03344 0 0.00714 90 0.31 0.17310 0.04368 0 0.00805 100 0.96 0.73842 0.24687 0 0.07758 110 0.23 0.12970 0.03432 0.01873 0.04007

9



The effect of rate constants k2 and k2’ on ME was also examined by the assumption of reversible reaction step as discussed earlier. The distribution curves of ME at 100°C and 110°C is shown in Figure 2. The weight fraction of ME is zero after approximately 10 minutes of reaction at 100°C, which implies that the formation of ME from DE was suppressed at this temperature. This result suggests the optimum range value of k2 of 0.73742-0.73942 where the ratio of k2/k1 would be between 0.7692 to 0.7702. The optimum rate constant k2’ is zero at 70 to 100°C except for 110°C where the optimum value of k2’ is 0.01873 ± 0.0005. Thus, the assumption of reversible reaction for the second step of the transesterification should be used at temperatures above or equal 110°C whereas the irreversible assumption could only be used for temperature 100°C and below. Table 3 shows the average rate constants determined by trial and error calculations via MATLAB. Figure 2: Effects of k2 and k2’ on simulated ME during the transesterification

between POME with TMP at (A) 100°C, k1=0.96, and (B) 110°C. See Figure 1 for abbreviations.

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The effects of rate constants k3 and k3’ were also studied in this paper. For the case of reversible reaction, the rate constants for the third step of transesterification mechanism from equation 2 are the function of the forward reaction rate constants k1 and k2 and also the reverse rate constant k2’. By using similar approach mentioned earlier, the effects of k3 and k3’ are depicted in Figure 3 at reaction temperature of 100°C after optimum fitting with the maximum points of experimental DE. The optimal values of rate constants k3 and k3’ at 100°C were determined to be 0.24687 and 0.07758 with the level of uncertainty of 0.00005 and 0.0001 respectively. An excellent estimation of DE distribution was achieved when the reversible reaction assumption was used rather than the use of irreversible reaction in the earlier study (Yunus et al., 2004). During the reaction period, the rate of DE formation from ME was higher than the rate of DE decomposition to TE (k2 > k3), which contributed to the accumulation of DE. The occurrence of the DE formation reversely from TE (indicated by nonzero value of k3’) was also being a factor to the more DE existence in the system. As shown by Yunus et al. (2004), the equal rate of forward reaction and reverse reaction would stabilize the concentration of DE when the third step of transesterification reaction finally reached equilibrium. Figure 3: Effects of k3 and k3’ on simulated diesters (DE) during the

transesterification between POME with TMP at 100°C. See Figure 1 for abbreviations.

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3.2 Comparison with experimental data Figure 4 shows the comparison between simulation values of products ME, DE and TE with the experimental data taken from Yunus (2003) at 100°C respectively. It is observed that the weight fractions of experimental DE and TE between 3 to 7 minutes did not match exactly with the simulated curves. This may be due to the fluctuation of temperature which took place during the reaction. Nevertheless, the simulated distributions of ME, DE and TE in general provide good agreement with the experimental data points. Figure 4: Product distribution curves for the transesterification of POME with

TMP. The experimental data are presented by the symbols, whereas the solid lines are simulated curves based on equations 14 -19. Molar ratio of POME /TMP was 10:1; 0.9 wt% sodium methoxide catalyst. See Figure 1 for abbreviations.

3.3 Influence of temperature on kinetic parameters From the ANOVA test performed to the experimental data and to all parameters used in the kinetic model, p-value <0.05 was obtained. This implies that there is a significant effect of temperature of the transesterification medium on these kinetic parameters and thus, the Arrhenius equation log10k = (-Ea/2.303R/T + C) could be applied. The data at 110°C were not included since the Arrhenius relationship in

12



this case is only valid for temperatures below 100°C. As tabulated in Table 4, the activation energies for k1, k2 and k3 in palm oil TMP ester synthesis are similar to those obtained in Yunus (2003), such as 33.9 kcal/mol for k1, 33.3 kcal/mol for k2 and 33.6 kcal/mol for k3. Linear regression analysis of the relationship gave determination coefficients of 0.982, 0.961 and 0.9328 for k1, k2 and k3, respectively. The activation energy for k3’ was higher compared to others (52.2 kcal/mol), which implies that the reverse reaction of TE to DE is more sensitive to temperature than the forward reaction. This might not be reliable since only 87.79% of the variation in temperature can be explained by the variability in k3’ value. For k2’, the relationship was not applicable because of the zero values obtained as presented in Table 3. Table 4: Activation energy for transesterification of TMP, ME and DE with palm

oil methyl esters.

Reaction Rate constant Activation energy, Ea (kcal/mol) R2

TMP ↔ ME k1 28.4 0.9820 ME ↔ DE k2 26.7 0.9610 DE ↔ TE k3 26.3 0.9328

k3’ 52.2 0.8779 3.4 Statistical analysis of the model Figure 5 shows the mean values of SSE, RMSE and χ² determined at five different temperatures. In general, the products distribution model proposed was fitted to the experimental data with values of SSE, RMSE and χ² close to zero. Hence, based on the results illustrated in Figure 5, the proposed model obtained best fit quality at reaction temperature at 100°C (SSE = 1.90 × 10-5; RMSE = 2.81 × 10-3; χ² = 1.24 × 10-4), followed by temperature at 110°C (SSE = 1.24 × 10-4; RMSE = 8.33 × 10-3; χ² = 6.21 ×10-4) and temperature at 80°C (SSE = 4.87 × 10-4; RMSE = 0.0171; χ² = 5.84 × 10-3). The model that gave lower fit quality among the selected temperatures were at 70°C (SSE = 5.45 × 10-4; RMSE = 0.0195; χ² = 6.53 × 10-3) and 90°C (SSE = 5.35 × 10-4; RMSE = 0.0184; χ² = 6.42 × 10-3), respectively. This implies that the temperature that is equal or more than 100°C encountered fewer errors. The other evidence that supports the proposed model is the coefficients of determination of the Arrhenius relationship for k1, k2 and k3 that were considerably high, as discussed in earlier section.

13



Figure 5: Statistical analysis of the kinetic model at five temperatures. Sum of

squares error (SSE), root mean square error (RMSE) and Chi-square (χ²).

4. Conclusion The kinetics of transesterification of POME with TMP was once modeled by analytical solution that relied heavily on the assumption of irreversible reaction. Due to the lack of accuracy for prediction of intermediate product distribution which is DE at its final equilibrium, the simulation of the reaction kinetics was done and compared with the experimental data values taken from Yunus (2003). An excellent fit of the curve lines to the experimental data points indicate that the proposed simulated model was valid at all stages of reaction. Moreover, the advantage of this approach is its simplicity in developing kinetic models without the need of an analytical transformation of rate equations and thus could be used to describe adequately the product distribution of palm oil TMP esters during the synthesis. The utilization of MATLAB was appropriate since it is a powerful tool to solve complicated equations from the kinetics study of complex reactions.

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Nomenclature C frequency factor Ci concentration of species i (mol/L) DE diesters i species i k0, k0’ rate constants for total reversible transesterification ((wt/wt)-1.(min)-1) k1, k2, k3 rate constants for reaction steps 1, 2, and 3 ((wt/wt)-1.(min)-1) k1’, k2’, k3’ rate constants for reverse reaction steps 1, 2, and 3 ((wt/wt)-1.(min)-1) M methanol MT total mass of mixture (g) ME monoesters MWi molecular weight of species i N number of response or experimental data ni number of mol of species i (mol) POME palm oil methyl esters R universal gas constant R² coefficient of determination RMSE root mean square error rTMP reaction rate for trimethylolpropane (mol/L.min) SSE sum of squares error SST total sum of squares T reaction temperature (K) t reaction time (min) TE triesters TMP trimethylolpropane esters

ijcalcX ,, calculated mass fraction of species i at j number of data (wt/wt)

ijX ,exp, experimental mass fraction of species i at j number of data (wt/wt) Xi,0 initial mass fraction for species i (wt/wt) Xi mass fraction of species i (wt/wt)

iX mean of mass fraction for species i (wt/wt) z number of constants Greek symbol ρT density of total mixture (g/L) χ² Chi-square

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Utilization of MATLAB to Simulate Kinetics of Transesterification of Palm Oil-Based Methyl Esters...

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