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Kinetic model for the increase of reaction order during polyesterification
Chemical Engineering and Processing 43 (2004) 1487–1493
Kinetic model for the increase of reaction order during polyesterification Tapio Salmi a,∗ , Erkki Paatero b , Per Nyholm a b
a Laboratory of Industrial Chemistry, Process Chemistry Centre, Åbo Akademi, Biskopsgatan 8, FIN-20500 Turku/Åbo, Finland Laboratory of Industrial Chemistry, Centre of Separation Technology, Lappeenranta University of Technology, FIN-53851 Lappeenranta, Finland
Received 29 March 2003; received in revised form 29 January 2004; accepted 29 January 2004 Available online 27 March 2004
Abstract A simple two-parameter model was proposed for the kinetics of polyesterification reactions. The kinetic model is based on the true reaction mechanism, i.e. on the shift of the ionic equilibria in the carboxylic acid (RCOOH) protolysis during the reaction, and on the bimolecular nucleophilic substitution of the protolysed acid with the alcohol (R OH). The model describes the increase of the reaction order with respect to the carboxylic acid from 1 to 2 as the esterification proceeds. The rate equation r = kcnCOOH cOH was used, where n is a function of the carboxylic acid concentration. The kinetic model was tested with the classical data of Flory obtained for diethylene glycol–adipic acid and the lauryl alcohol–adipic acid reactions. The model provided an excellent description of the data over the entire range of conversions of the carboxylic acids and it can be extended to new polyesterification systems. © 2004 Elsevier B.V. All rights reserved. Keywords: Carboxylic acid; Kinetic model; Polyesterification
1. Introduction It was shown in the classical work of Flory [1] that the reaction order of polyesterification reactions increases during the progress of the reaction. The esterification reaction of the carboxylic acid (RCOOH) and the alcohol (R OH), RCOOH + R OH = RCOOR + H2 O, is of first order with respect to the carboxylic acid in the beginning of the reaction and second order with respect to the acid in the end of the reaction. The change of the reaction order was demonstrated, for example, in the esterification of adipic acid and caproic acid with several alcohols, such as diethylene glycol and lauryl alcohol [1]. By using stoichiometric initial amounts of the reactants, the total reaction, was determined to 2 at lower acid conversions, but the order increased to 3 at conversions exceeding 70%. At the intermediate range of conversions, the rate data cannot be satisfactorily described by second or third order kinetic models. Abbreviations: A, ion pair, A = RC(OH)2 + RCOO− ; MRS, mean residual square; RCOOH, carboxylic acid; RCOOR, ester; R OH, alcohol; WSRS, weighted sum of residual squares ∗ Corresponding author. Fax: +358-2-215-4479. E-mail address: [email protected] (T. Salmi). 0255-2701/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2004.01.006
In later studies [2] it has been suggested that the order of the esterification reactions might be between 2 and 3, but rate equations having, for example, the reaction order of the carboxylic acid about 3/2 and that of the alcohol about 1 still fail to describe the kinetic data at high conversions, as demonstrated by Solomon [3]. According to Fang et al. [4], the increase of the reaction order is caused by the change of the dielectric constant and the viscosity of the liquid phase during the reaction. These authors derived a rate expression which well describes the experimental data of Flory over the entire conversion range. The drawback of their model is, however, that it contains three adjustable parameters in addition to three equilibrium constants. Therefore, the model of Fang et al. [4] has not been extensively used. The change of the reaction volume due to removed water is an important issue, which is of practical importance in the design and operation of polyesterification reactors. A simple approach is to assume instantaneous evaporation of water, but even more sophisticated calculations have been applied. In case that the amount of water liberated is measured during the course of experiment, in it can easily been taken into account in the determination of kinetic parameters [5]. In the review article of Fradet and Marechal [6], various rate
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equations are critically evaluated and simple equations for volume change are presented. It is evident that there is a need of a simple rate equation for engineering purposes, a rate equation, which is able to describe the polyesterification over wide ranges of conversions. In the present work, a new rate equation is developed, which predicts the increase of the reaction order in polyesterification. The rate equation is demonstrated with the esterification data of Flory [1]. We have selected the classical data of Flory, since these data are even nowadays presented in textbooks to illustrate polyesterification kinetics and the dilemma of a changing reaction order.
2. Mechanism and kinetics of esterification In this section we consider the polyesterification mechanism and kinetics in systems, from where the reaction product, water is continuously and instantaneously removed. Thus, the stirring of the reaction system is assumed to be efficient enough to suppress mass transfer limitations. The first step in esterification is the autoprotolysis of the carboxylic acid. In the absence of a catalyst the carboxylic acid itself acts a protolysing agent [4] +
RCOOH + RCOOH = RC(OH)2 + RCOO
−
(1)
In the beginning of the reaction, the cations and anions formed in reaction (1) co-exist as separate ions. As the polymerisation proceeds, and the reaction milieu is dominated by the polymers, the protolysis reaction gives predominantly an ion pair RCOOH + RCOOH = RC(OH)2 + RCOO− (A)
(2)
The protolysis reactions (1)–(2) are rapid; the rate determining step in the esterification is the nucleophilic attack of the alcohol to the cation (RC(OH)2 )+ or to the ion pair (A). The subsequent reaction steps are presumed to be rapid. Thus, all steps after (1) and (2) can be lumped together, and reactions (3) and (4) are obtained: RC(OH)2 + + R OH = RCOOR + H2 O( ↑ ) + H+
(3)
RC(OH)2 + RCOO− + R OH = RCOOR + RCOOH + H2 O( ↑ )
(4)
where the arrow denotes that the water formed is removed from the system by continuous bubbling of an inert gas (e.g. nitrogen) and application of a vacuum. In the beginning of the reaction, step (3) dominates, but as the reaction proceeds, the ion pairs (A) are formed, and the influence of step (4) on the overall kinetics increases. If the amount of water is small, reactions (3)–(4) can be treated as irreversible reactions, the rates of which are given by Eqs. (5) and (6), respectively
The total rate of esterification is r = r3 + r4
(7)
Since steps (1)–(2) are presumed to be rapid, the quasiequilibrium hypothesis is applied on them cRC(OH)2 cRCOO (8) K1 = 2 cRCOOH K2 =
cA 2 cRCOOH
(9)
In the absence of water, the concentrations of the cations and anions generated in step (1) are approximately equal, i.e. cRC(OH)2 = cRCOO , and the unknown concentrations cRC(OH)2 and cA are obtained from (8) and (9) cRC(OH)2 = K1 cRCOOH
1/2
(10)
2 cA = K2 cRCOOH
(11)
The expressions (10)–(11) are inserted in rate Eqs. (5) and (6), and the overall rate Eq. (7) finally becomes 1/2
r = (k3 K1
+ k4 K2 cRCOOH )cRCOOH c R OH
(12)
It should be kept in mind that the constants K1 and K2 are strictly speaking dependent on the mixture composition. Rate Eq. (12) can qualitatively explain the experimental ob1/2 servations: in the beginning of the reaction K1 is larger than K2 cRCOOH and the reaction is of first order with respect to the carboxylic acid, in the end of the reaction K2 cRCOOH 1/2 is larger than K1 and the reaction is second order with respect to the carboxylic acid. The main disadvantage of rate Eq. (12) is that constants K1 and K2 are functions of the liquid-phase properties. A description of the changes of K1 and K2 inevitably increases the number of parameters in the system. In the characterisation of industrial polymerisation processes, the basic data consists typically of the acid numbers, i.e. the measured conversions only. Therefore, our goal is to describe the esterification kinetics with a global model being as simple as possible. We thus apply a semi-empirical approach to rate Eq. (12), and approximate the term in the parenthesis by a power-law expression kcn−1 RCOOH The exponent (n) includes the effects from the shift of the ionic equilibria and the changes of the component activity coefficients. The rate Eq. (12) becomes r = kcnRCOOH c R OH
(13)
r3 = k3 cRC(OH)2 cR OH
(5)
where the exponent n is 1 in the beginning of the reaction and it approaches the limiting value of 2 as the equilibrium is attained. The basic task is to find a suitable empirical function for the exponent to provide a smooth increase of the reaction order. We assume that the increase of the reaction order is proportional to the reaction order itself and to the change of the carboxylic acid concentration:
r4 = k4 cA cR OH
(6)
dn = −anq dcRCOOH
(14)
T. Salmi et al. / Chemical Engineering and Processing 43 (2004) 1487–1493
where a is a proportionality coefficient and q, empirical exponent. Integration of Eq. (14) with the limits cRCOOH = c0 , n = 1 and cRCOOH = c∗ , n = 2, where c0 and c∗ denote the initial and equilibrium concentrations (i.e. concentration of the carboxylic group obtained after a long experiment in a closed system, where the equilibrium is attained), respectively, gives the value of parameter a: a=
ln 2 , c0 − c ∗
q=1
(q − 1)(1 − 21−q ) , a= c0 − c ∗
(15a)
(15b)
Integration of Eq. (14) from the initial values (n = 1, cRCOOH = c0 ) to general values (n, c) and insertion of the expressions for the constant a give the following equations for n: ∗
(16a) q = 1
(16b)
As the conversion of the acid is denoted by X, X = (c0 –c)/c0 , and the equilibrium concentration c∗ is assumed to be zero (water is evaporated from the system and thus a complete conversion of the carboxylic group is obtained), Eq. (16) is simplified to n = 2X
(17a)
(17b)
The behaviour of function (17), with different values of q, is illustrated in Fig. 1. The figure shows that the rapid increase of the reaction order towards the value 2 at high conversions can be obtained by selecting a sufficiently high value of q. For stoichiometric initial amounts of the carboxylic acid and the alcohol, the concentrations in Eq. (13) are always equal, cRCOOH = cR OH = c, and rate Eq. (13) is simplified to r = kcn+1
q = 1
n = 2(c0 −c)/(c0 −c ) , q = 1 1/(1−q) 1−q c0 − c , n = 1 − (1 − 2 ) c0 − c ∗
n = [1 − (1 − 21−q )X]1/(1−q)
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(18)
For a well-stirred (semi)batch reactor, from which the reaction product, water, is continuously and instantaneously removed, the mass balance equation of the non-volatile reactant (e.g. carboxylic acid, RCOOH = A) can be written as dnA = rA V (19) dt where nA is the amount of substance and V is the liquid volume. Inserting the concentration, nA = cA V and differentiating Eq. (19) gives after rearrangement dcA dV/dt = rA − cA (20) dt V To progress further, an assumption concerning the volume change is needed. By assuming that the reaction mixture contains the carboxylic acid, the alcohol and the ester and
Fig. 1. The dependence of the reaction order (n) on the conversion (X) of the carboxylic acid according to Eq. (17) for different values of the parameter (q).
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that the contributions of partial molar volumes are negligible, it can be shown that the volume is updated from the formula V = αnA + βn0A
(21)
The coefficients α and β are explained in Appendix A. The expressions nA = cA V and n0A = c0A V0 are inserted in Eq. (21) and the differentiation dV/dt is carried out. The update of the liquid volume is V βc0A = (22) V0 1 − αcA and the balance equation becomes dcA = rA (1 − αcA ) (23) dt Parameter α is thus a measure of the importance of the volume change. A rough estimate of it is obtained from the molar masses the reactants and the esterification product (see Appendix A). For the data which will considered in the sequel, the role of the correction term αcA is not important and it will be approximated to unity. The reaction stoichiometry gives rA = –r, the subscript (A) is omitted, and equation becomes dc = −r dt
(24)
A dimensionless concentration (y) is introduced, y = c/c0 , and a combination of Eqs. (18) and (24) give dy = −kcn0 yn+1 dt
(25)
The values of y and the conversion (X) are related by X = 1 − y. Integration of Eq. (25) yields y dy F(y) = − = kt (26) n n+1 1 c0 y By choosing a suitable value of q and carrying out the integration according to Eq. (26) numerically from y = y0 to the experimental points y(t), a plot of F(y) against t should give a straight line. Alternatively, the original differential Eq. (25) is solved numerically and non-linear regression analysis is applied directly to the y − t data.
3. Application of the rate equation on the data of P. Flory We tested the proposed rate equation with the classical data of Flory [1], on the kinetic results obtained in polyesterification of adipic acid with diethylene glycol at 166 ◦ C and in esterification of adipic acid with lauryl alcohol at 202 ◦ C. The data are presented in Tables I and II of the original reference [1], and the normalised concentrations of adipic acid versus the reaction time are shown in Figs. 2 and 3. The corresponding test plots for a third order reaction, 1/(1−X)2 = 1/y2 versus t were prepared by Flory [1], and these plots were straight lines only for conversions X > 0.8, depending slightly on the specific chemical cases under consideration. On the other hand, at the initial stage of the reaction the plots of a second order reaction. 1/(1 − X) versus
Fig. 2. Polyesterification kinetics of adipic acid with diethylene glycol at 166 ◦ C according to Flory [1]. The continuous lines represent model Eqs. (17) and (25).
T. Salmi et al. / Chemical Engineering and Processing 43 (2004) 1487–1493
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Fig. 3. Polyesterification kinetics of adipic acid with lauryl alcohol at 202 ◦ C according to Flory [1]. The continuous lines represent model Eqs. (17) and (25).
Fig. 4. The plots F(y) = kt according to Eq. (26) for the polyesterification of adipic acid with diethylene glycol. The data of Flory [1].
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t gave straight lines, but failed to describe the high conversion domain. Solomon [3] prepared the test plot for the reaction order 5/2—according to the proposal of Tang and Yao [2]—and even this rate equation was unable to describe the data at high conversions [6]. The plots F(y) versus t according to our model (21) for the diethylene glycol–adipic acid and lauryl alcohol–adipic acid reactions are presented in Figs. 4 and 5, respectively. As can be seen from the figures, straight lines are obtained for q-values of approximately 709 for the diethylene glycol–adipic acid reaction and for q-values of approximately 6–7 for the lauryl alcohol–adipic acid reaction. Furthermore, we estimated the rate constant k directly from the primary data, using non-linear regression analysis with Levenberg–Marquardt method and numerical solution of the differential Eq. (25) with Rosenbrock–Wanner method [7]. The value of q was either fixed a priori or it was estimated simultaneously with the rate constant. The results are summarised in Tables 1 and 2 for the diethylene glycol–adipic acid and the lauryl alcohol–adipic acid systems, respectively. The corresponding fits of the model to the data are shown in Figs. 2 and 3. The optimal value q is about 8 for the diethylene glycol–adipic acid reaction and about 6 for the lauryl alcohol–adipic acid system; de facto these integer values could very well be used to describe the kinetics of the actual systems. These values of q provide a satisfactory description of the rapid increase of the reaction order at conversions exceeding 70–80%. As can be seen from the statistics of the regression analysis (Tables 1
Table 1 Determination of the kinetic parameters of the polyesterification of adipic acid with ethylene glycol at 166 ◦ C by using the data of Flory [1] q 8.4 1 8 9
s (%) 3.4 – – –
k 0.027 0.035 0.027 0.027
s (%) 0.8 5.9 0.7 0.7
N 23 23 23 23
M 2 1 1 1
WSRS 0.31 0.22 0.34 0.36
MRS 10−3
× × 10−1 × 10−3 × 10−3
0.15 0.10 0.15 0.16
× × × ×
10−4 10−2 10−4 10−4
q = 8.4 gives the best fit whereas the other values are shown for comparison.
Table 2 Determination of the kinetic parameters of the esterification of adipic acid with lauryl alcohol at 166 ◦ C by using the data of Flory [1] q 6.4 1 6 7
s (%) 6.4 – – –
k 0.081 0.105 0.082 0.080
s (%) 1.7 6.0 1.3 1.3
N 14 14 14 14
M 2 1 1 1
WSRS 0.39 0.73 0.41 0.46
MRS 10−3
× × 10−2 × 10−3 × 10−3
0.32 0.56 0.32 0.35
× × × ×
10−4 10−2 10−4 10−4
q = 6.4 gives the best fit whereas the other values are shown for comparison.
and 2) and from the fits (Figs. 1 and 2) of the model, the rate equation proposed by us provides an excellent description of the esterification kinetics; the differences between the experimental and the calculated conversions were typically less than 0.5%, and the increase of the reaction order was predicted correctly.
Fig. 5. The plots F(y) = kt according to Eq. (26) for the polyesterification of adipic acid with lauryl alcohol. The data of Flory [1].
T. Salmi et al. / Chemical Engineering and Processing 43 (2004) 1487–1493
4. Conclusions A new empirical rate equation (Eqs. (13) and (16)) for engineering purposes was proposed for the esterification reactions to predict the increase of the reaction order during the progress of the reaction. The model was based on the esterification mechanism (1)–(4) comprising the shifts of the ionic equilibria (1)–(2) during the reaction. The new rate equation gave an excellent description of the classical esterification data of Flory (Figs. 2 and 3). The proposed approach has perspective in the prediction of polyesterification kinetics in laboratory and industrial scale. The effect of a non-stoichiometric initial ratio of the reactants as well as the volume contraction the reaction mixture should be investigated in future, by fitting new experimental data to the model. We believe that our approach can be used as a source of inspiration in the engineering modelling of polyesterification kinetics.
Appendix A. Nomenclature a c c0 c∗ F k M n n N q r
proportionality coefficient, Eq. (14) concentration of the carboxylic acid (RCOOH) (mol/l) initial concentration of the carboxylic acid (mol/l) equilibrium concentration of the carboxylic acid (mol/l) integral function, Eq. (26) (mol/l)−n rate constant (mol/l)−n number of parameters amount of substance (mol) exponent in rate Eq. (13) number of experimental observations exponent Eq. (14) rate (mol/(l min)
s t V X y
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standard deviation of a parameter (%) time (min) volume conversion of the carboxylic acid, X = 1 − c/c0 dimensionless concentration of the carboxylic acid, y = c/c0
Greek letters α volume correction parameter (= molar mass of reactants/densities of reactants − molar mass of product/density of product) (l/mol) β volume correction parameter, (= molar mass of product/density of product) (l/mol)
References [1] P. Flory, Kinetics of polyesterification: a study of the effects of molecular weight and viscosity on reaction rate, J. Am. Chem. Soc. 61 (1939) 3334–3340. [2] A.C. Tang, K.S. Yao, Mechanism of hydrogen ion catalysis in esterification. II. Studies on the kinetics of polyesterification reactions between dibasic acids and glycols, J. Polym. Sci. 35 (1959) 219–233. [3] D.H. Solomon, Polyesterification: Kinetics and Mechanism of Polymerisation, vol. 3, New York, Dekker, 1971. [4] Y.-R. Fang, C.-G. Lai, J.-L. Lu, M.-K. Chen, The kinetics and mechanism of polyesterification of binary acid and binary alcohol, Sci. Sin. 18 (1975) 72–87. [5] T. Salmi, E. Paatero, P. Nyholm, M. Still, K. Närhi, K. Immonen, Kinetic model for the main and side reactions in the polyesterification of carboxylic acids with diols, Chem. Eng. Sci. 49 (1994) 3601–3616. [6] A. Fradet, E. Marechal, 1982. Kinetics and Mechanisms of Polyesterifications I. Diols with Diacids, in Advances in Polymer Science, vol. 43, Springer-Verlag, Berlin, Heidelberg, pp. 51–142. [7] S. Vajda, P. Valkó, Reproche—Regression Program for Chemical Engineers, Manual, European Committee for Computers in Chemical Engineering Education, Budapest, 1985.
Kinetic model for the increase of reaction order during polyesterification Tapio Salmi a,∗ , Erkki Paatero b , Per Nyholm a b
a Laboratory of Industrial Chemistry, Process Chemistry Centre, Åbo Akademi, Biskopsgatan 8, FIN-20500 Turku/Åbo, Finland Laboratory of Industrial Chemistry, Centre of Separation Technology, Lappeenranta University of Technology, FIN-53851 Lappeenranta, Finland
Received 29 March 2003; received in revised form 29 January 2004; accepted 29 January 2004 Available online 27 March 2004
Abstract A simple two-parameter model was proposed for the kinetics of polyesterification reactions. The kinetic model is based on the true reaction mechanism, i.e. on the shift of the ionic equilibria in the carboxylic acid (RCOOH) protolysis during the reaction, and on the bimolecular nucleophilic substitution of the protolysed acid with the alcohol (R OH). The model describes the increase of the reaction order with respect to the carboxylic acid from 1 to 2 as the esterification proceeds. The rate equation r = kcnCOOH cOH was used, where n is a function of the carboxylic acid concentration. The kinetic model was tested with the classical data of Flory obtained for diethylene glycol–adipic acid and the lauryl alcohol–adipic acid reactions. The model provided an excellent description of the data over the entire range of conversions of the carboxylic acids and it can be extended to new polyesterification systems. © 2004 Elsevier B.V. All rights reserved. Keywords: Carboxylic acid; Kinetic model; Polyesterification
1. Introduction It was shown in the classical work of Flory [1] that the reaction order of polyesterification reactions increases during the progress of the reaction. The esterification reaction of the carboxylic acid (RCOOH) and the alcohol (R OH), RCOOH + R OH = RCOOR + H2 O, is of first order with respect to the carboxylic acid in the beginning of the reaction and second order with respect to the acid in the end of the reaction. The change of the reaction order was demonstrated, for example, in the esterification of adipic acid and caproic acid with several alcohols, such as diethylene glycol and lauryl alcohol [1]. By using stoichiometric initial amounts of the reactants, the total reaction, was determined to 2 at lower acid conversions, but the order increased to 3 at conversions exceeding 70%. At the intermediate range of conversions, the rate data cannot be satisfactorily described by second or third order kinetic models. Abbreviations: A, ion pair, A = RC(OH)2 + RCOO− ; MRS, mean residual square; RCOOH, carboxylic acid; RCOOR, ester; R OH, alcohol; WSRS, weighted sum of residual squares ∗ Corresponding author. Fax: +358-2-215-4479. E-mail address: [email protected] (T. Salmi). 0255-2701/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2004.01.006
In later studies [2] it has been suggested that the order of the esterification reactions might be between 2 and 3, but rate equations having, for example, the reaction order of the carboxylic acid about 3/2 and that of the alcohol about 1 still fail to describe the kinetic data at high conversions, as demonstrated by Solomon [3]. According to Fang et al. [4], the increase of the reaction order is caused by the change of the dielectric constant and the viscosity of the liquid phase during the reaction. These authors derived a rate expression which well describes the experimental data of Flory over the entire conversion range. The drawback of their model is, however, that it contains three adjustable parameters in addition to three equilibrium constants. Therefore, the model of Fang et al. [4] has not been extensively used. The change of the reaction volume due to removed water is an important issue, which is of practical importance in the design and operation of polyesterification reactors. A simple approach is to assume instantaneous evaporation of water, but even more sophisticated calculations have been applied. In case that the amount of water liberated is measured during the course of experiment, in it can easily been taken into account in the determination of kinetic parameters [5]. In the review article of Fradet and Marechal [6], various rate
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equations are critically evaluated and simple equations for volume change are presented. It is evident that there is a need of a simple rate equation for engineering purposes, a rate equation, which is able to describe the polyesterification over wide ranges of conversions. In the present work, a new rate equation is developed, which predicts the increase of the reaction order in polyesterification. The rate equation is demonstrated with the esterification data of Flory [1]. We have selected the classical data of Flory, since these data are even nowadays presented in textbooks to illustrate polyesterification kinetics and the dilemma of a changing reaction order.
2. Mechanism and kinetics of esterification In this section we consider the polyesterification mechanism and kinetics in systems, from where the reaction product, water is continuously and instantaneously removed. Thus, the stirring of the reaction system is assumed to be efficient enough to suppress mass transfer limitations. The first step in esterification is the autoprotolysis of the carboxylic acid. In the absence of a catalyst the carboxylic acid itself acts a protolysing agent [4] +
RCOOH + RCOOH = RC(OH)2 + RCOO
−
(1)
In the beginning of the reaction, the cations and anions formed in reaction (1) co-exist as separate ions. As the polymerisation proceeds, and the reaction milieu is dominated by the polymers, the protolysis reaction gives predominantly an ion pair RCOOH + RCOOH = RC(OH)2 + RCOO− (A)
(2)
The protolysis reactions (1)–(2) are rapid; the rate determining step in the esterification is the nucleophilic attack of the alcohol to the cation (RC(OH)2 )+ or to the ion pair (A). The subsequent reaction steps are presumed to be rapid. Thus, all steps after (1) and (2) can be lumped together, and reactions (3) and (4) are obtained: RC(OH)2 + + R OH = RCOOR + H2 O( ↑ ) + H+
(3)
RC(OH)2 + RCOO− + R OH = RCOOR + RCOOH + H2 O( ↑ )
(4)
where the arrow denotes that the water formed is removed from the system by continuous bubbling of an inert gas (e.g. nitrogen) and application of a vacuum. In the beginning of the reaction, step (3) dominates, but as the reaction proceeds, the ion pairs (A) are formed, and the influence of step (4) on the overall kinetics increases. If the amount of water is small, reactions (3)–(4) can be treated as irreversible reactions, the rates of which are given by Eqs. (5) and (6), respectively
The total rate of esterification is r = r3 + r4
(7)
Since steps (1)–(2) are presumed to be rapid, the quasiequilibrium hypothesis is applied on them cRC(OH)2 cRCOO (8) K1 = 2 cRCOOH K2 =
cA 2 cRCOOH
(9)
In the absence of water, the concentrations of the cations and anions generated in step (1) are approximately equal, i.e. cRC(OH)2 = cRCOO , and the unknown concentrations cRC(OH)2 and cA are obtained from (8) and (9) cRC(OH)2 = K1 cRCOOH
1/2
(10)
2 cA = K2 cRCOOH
(11)
The expressions (10)–(11) are inserted in rate Eqs. (5) and (6), and the overall rate Eq. (7) finally becomes 1/2
r = (k3 K1
+ k4 K2 cRCOOH )cRCOOH c R OH
(12)
It should be kept in mind that the constants K1 and K2 are strictly speaking dependent on the mixture composition. Rate Eq. (12) can qualitatively explain the experimental ob1/2 servations: in the beginning of the reaction K1 is larger than K2 cRCOOH and the reaction is of first order with respect to the carboxylic acid, in the end of the reaction K2 cRCOOH 1/2 is larger than K1 and the reaction is second order with respect to the carboxylic acid. The main disadvantage of rate Eq. (12) is that constants K1 and K2 are functions of the liquid-phase properties. A description of the changes of K1 and K2 inevitably increases the number of parameters in the system. In the characterisation of industrial polymerisation processes, the basic data consists typically of the acid numbers, i.e. the measured conversions only. Therefore, our goal is to describe the esterification kinetics with a global model being as simple as possible. We thus apply a semi-empirical approach to rate Eq. (12), and approximate the term in the parenthesis by a power-law expression kcn−1 RCOOH The exponent (n) includes the effects from the shift of the ionic equilibria and the changes of the component activity coefficients. The rate Eq. (12) becomes r = kcnRCOOH c R OH
(13)
r3 = k3 cRC(OH)2 cR OH
(5)
where the exponent n is 1 in the beginning of the reaction and it approaches the limiting value of 2 as the equilibrium is attained. The basic task is to find a suitable empirical function for the exponent to provide a smooth increase of the reaction order. We assume that the increase of the reaction order is proportional to the reaction order itself and to the change of the carboxylic acid concentration:
r4 = k4 cA cR OH
(6)
dn = −anq dcRCOOH
(14)
T. Salmi et al. / Chemical Engineering and Processing 43 (2004) 1487–1493
where a is a proportionality coefficient and q, empirical exponent. Integration of Eq. (14) with the limits cRCOOH = c0 , n = 1 and cRCOOH = c∗ , n = 2, where c0 and c∗ denote the initial and equilibrium concentrations (i.e. concentration of the carboxylic group obtained after a long experiment in a closed system, where the equilibrium is attained), respectively, gives the value of parameter a: a=
ln 2 , c0 − c ∗
q=1
(q − 1)(1 − 21−q ) , a= c0 − c ∗
(15a)
(15b)
Integration of Eq. (14) from the initial values (n = 1, cRCOOH = c0 ) to general values (n, c) and insertion of the expressions for the constant a give the following equations for n: ∗
(16a) q = 1
(16b)
As the conversion of the acid is denoted by X, X = (c0 –c)/c0 , and the equilibrium concentration c∗ is assumed to be zero (water is evaporated from the system and thus a complete conversion of the carboxylic group is obtained), Eq. (16) is simplified to n = 2X
(17a)
(17b)
The behaviour of function (17), with different values of q, is illustrated in Fig. 1. The figure shows that the rapid increase of the reaction order towards the value 2 at high conversions can be obtained by selecting a sufficiently high value of q. For stoichiometric initial amounts of the carboxylic acid and the alcohol, the concentrations in Eq. (13) are always equal, cRCOOH = cR OH = c, and rate Eq. (13) is simplified to r = kcn+1
q = 1
n = 2(c0 −c)/(c0 −c ) , q = 1 1/(1−q) 1−q c0 − c , n = 1 − (1 − 2 ) c0 − c ∗
n = [1 − (1 − 21−q )X]1/(1−q)
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(18)
For a well-stirred (semi)batch reactor, from which the reaction product, water, is continuously and instantaneously removed, the mass balance equation of the non-volatile reactant (e.g. carboxylic acid, RCOOH = A) can be written as dnA = rA V (19) dt where nA is the amount of substance and V is the liquid volume. Inserting the concentration, nA = cA V and differentiating Eq. (19) gives after rearrangement dcA dV/dt = rA − cA (20) dt V To progress further, an assumption concerning the volume change is needed. By assuming that the reaction mixture contains the carboxylic acid, the alcohol and the ester and
Fig. 1. The dependence of the reaction order (n) on the conversion (X) of the carboxylic acid according to Eq. (17) for different values of the parameter (q).
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that the contributions of partial molar volumes are negligible, it can be shown that the volume is updated from the formula V = αnA + βn0A
(21)
The coefficients α and β are explained in Appendix A. The expressions nA = cA V and n0A = c0A V0 are inserted in Eq. (21) and the differentiation dV/dt is carried out. The update of the liquid volume is V βc0A = (22) V0 1 − αcA and the balance equation becomes dcA = rA (1 − αcA ) (23) dt Parameter α is thus a measure of the importance of the volume change. A rough estimate of it is obtained from the molar masses the reactants and the esterification product (see Appendix A). For the data which will considered in the sequel, the role of the correction term αcA is not important and it will be approximated to unity. The reaction stoichiometry gives rA = –r, the subscript (A) is omitted, and equation becomes dc = −r dt
(24)
A dimensionless concentration (y) is introduced, y = c/c0 , and a combination of Eqs. (18) and (24) give dy = −kcn0 yn+1 dt
(25)
The values of y and the conversion (X) are related by X = 1 − y. Integration of Eq. (25) yields y dy F(y) = − = kt (26) n n+1 1 c0 y By choosing a suitable value of q and carrying out the integration according to Eq. (26) numerically from y = y0 to the experimental points y(t), a plot of F(y) against t should give a straight line. Alternatively, the original differential Eq. (25) is solved numerically and non-linear regression analysis is applied directly to the y − t data.
3. Application of the rate equation on the data of P. Flory We tested the proposed rate equation with the classical data of Flory [1], on the kinetic results obtained in polyesterification of adipic acid with diethylene glycol at 166 ◦ C and in esterification of adipic acid with lauryl alcohol at 202 ◦ C. The data are presented in Tables I and II of the original reference [1], and the normalised concentrations of adipic acid versus the reaction time are shown in Figs. 2 and 3. The corresponding test plots for a third order reaction, 1/(1−X)2 = 1/y2 versus t were prepared by Flory [1], and these plots were straight lines only for conversions X > 0.8, depending slightly on the specific chemical cases under consideration. On the other hand, at the initial stage of the reaction the plots of a second order reaction. 1/(1 − X) versus
Fig. 2. Polyesterification kinetics of adipic acid with diethylene glycol at 166 ◦ C according to Flory [1]. The continuous lines represent model Eqs. (17) and (25).
T. Salmi et al. / Chemical Engineering and Processing 43 (2004) 1487–1493
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Fig. 3. Polyesterification kinetics of adipic acid with lauryl alcohol at 202 ◦ C according to Flory [1]. The continuous lines represent model Eqs. (17) and (25).
Fig. 4. The plots F(y) = kt according to Eq. (26) for the polyesterification of adipic acid with diethylene glycol. The data of Flory [1].
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t gave straight lines, but failed to describe the high conversion domain. Solomon [3] prepared the test plot for the reaction order 5/2—according to the proposal of Tang and Yao [2]—and even this rate equation was unable to describe the data at high conversions [6]. The plots F(y) versus t according to our model (21) for the diethylene glycol–adipic acid and lauryl alcohol–adipic acid reactions are presented in Figs. 4 and 5, respectively. As can be seen from the figures, straight lines are obtained for q-values of approximately 709 for the diethylene glycol–adipic acid reaction and for q-values of approximately 6–7 for the lauryl alcohol–adipic acid reaction. Furthermore, we estimated the rate constant k directly from the primary data, using non-linear regression analysis with Levenberg–Marquardt method and numerical solution of the differential Eq. (25) with Rosenbrock–Wanner method [7]. The value of q was either fixed a priori or it was estimated simultaneously with the rate constant. The results are summarised in Tables 1 and 2 for the diethylene glycol–adipic acid and the lauryl alcohol–adipic acid systems, respectively. The corresponding fits of the model to the data are shown in Figs. 2 and 3. The optimal value q is about 8 for the diethylene glycol–adipic acid reaction and about 6 for the lauryl alcohol–adipic acid system; de facto these integer values could very well be used to describe the kinetics of the actual systems. These values of q provide a satisfactory description of the rapid increase of the reaction order at conversions exceeding 70–80%. As can be seen from the statistics of the regression analysis (Tables 1
Table 1 Determination of the kinetic parameters of the polyesterification of adipic acid with ethylene glycol at 166 ◦ C by using the data of Flory [1] q 8.4 1 8 9
s (%) 3.4 – – –
k 0.027 0.035 0.027 0.027
s (%) 0.8 5.9 0.7 0.7
N 23 23 23 23
M 2 1 1 1
WSRS 0.31 0.22 0.34 0.36
MRS 10−3
× × 10−1 × 10−3 × 10−3
0.15 0.10 0.15 0.16
× × × ×
10−4 10−2 10−4 10−4
q = 8.4 gives the best fit whereas the other values are shown for comparison.
Table 2 Determination of the kinetic parameters of the esterification of adipic acid with lauryl alcohol at 166 ◦ C by using the data of Flory [1] q 6.4 1 6 7
s (%) 6.4 – – –
k 0.081 0.105 0.082 0.080
s (%) 1.7 6.0 1.3 1.3
N 14 14 14 14
M 2 1 1 1
WSRS 0.39 0.73 0.41 0.46
MRS 10−3
× × 10−2 × 10−3 × 10−3
0.32 0.56 0.32 0.35
× × × ×
10−4 10−2 10−4 10−4
q = 6.4 gives the best fit whereas the other values are shown for comparison.
and 2) and from the fits (Figs. 1 and 2) of the model, the rate equation proposed by us provides an excellent description of the esterification kinetics; the differences between the experimental and the calculated conversions were typically less than 0.5%, and the increase of the reaction order was predicted correctly.
Fig. 5. The plots F(y) = kt according to Eq. (26) for the polyesterification of adipic acid with lauryl alcohol. The data of Flory [1].
T. Salmi et al. / Chemical Engineering and Processing 43 (2004) 1487–1493
4. Conclusions A new empirical rate equation (Eqs. (13) and (16)) for engineering purposes was proposed for the esterification reactions to predict the increase of the reaction order during the progress of the reaction. The model was based on the esterification mechanism (1)–(4) comprising the shifts of the ionic equilibria (1)–(2) during the reaction. The new rate equation gave an excellent description of the classical esterification data of Flory (Figs. 2 and 3). The proposed approach has perspective in the prediction of polyesterification kinetics in laboratory and industrial scale. The effect of a non-stoichiometric initial ratio of the reactants as well as the volume contraction the reaction mixture should be investigated in future, by fitting new experimental data to the model. We believe that our approach can be used as a source of inspiration in the engineering modelling of polyesterification kinetics.
Appendix A. Nomenclature a c c0 c∗ F k M n n N q r
proportionality coefficient, Eq. (14) concentration of the carboxylic acid (RCOOH) (mol/l) initial concentration of the carboxylic acid (mol/l) equilibrium concentration of the carboxylic acid (mol/l) integral function, Eq. (26) (mol/l)−n rate constant (mol/l)−n number of parameters amount of substance (mol) exponent in rate Eq. (13) number of experimental observations exponent Eq. (14) rate (mol/(l min)
s t V X y
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standard deviation of a parameter (%) time (min) volume conversion of the carboxylic acid, X = 1 − c/c0 dimensionless concentration of the carboxylic acid, y = c/c0
Greek letters α volume correction parameter (= molar mass of reactants/densities of reactants − molar mass of product/density of product) (l/mol) β volume correction parameter, (= molar mass of product/density of product) (l/mol)
References [1] P. Flory, Kinetics of polyesterification: a study of the effects of molecular weight and viscosity on reaction rate, J. Am. Chem. Soc. 61 (1939) 3334–3340. [2] A.C. Tang, K.S. Yao, Mechanism of hydrogen ion catalysis in esterification. II. Studies on the kinetics of polyesterification reactions between dibasic acids and glycols, J. Polym. Sci. 35 (1959) 219–233. [3] D.H. Solomon, Polyesterification: Kinetics and Mechanism of Polymerisation, vol. 3, New York, Dekker, 1971. [4] Y.-R. Fang, C.-G. Lai, J.-L. Lu, M.-K. Chen, The kinetics and mechanism of polyesterification of binary acid and binary alcohol, Sci. Sin. 18 (1975) 72–87. [5] T. Salmi, E. Paatero, P. Nyholm, M. Still, K. Närhi, K. Immonen, Kinetic model for the main and side reactions in the polyesterification of carboxylic acids with diols, Chem. Eng. Sci. 49 (1994) 3601–3616. [6] A. Fradet, E. Marechal, 1982. Kinetics and Mechanisms of Polyesterifications I. Diols with Diacids, in Advances in Polymer Science, vol. 43, Springer-Verlag, Berlin, Heidelberg, pp. 51–142. [7] S. Vajda, P. Valkó, Reproche—Regression Program for Chemical Engineers, Manual, European Committee for Computers in Chemical Engineering Education, Budapest, 1985.