Ind. Eng. Chem. Res. 1996, 35, 3951-3963
3951
Kinetic Model for the Homogeneously Catalyzed Polyesterification of Dicarboxylic Acids with Diols Juha Lehtonen,† Tapio Salmi,*,† Kirsi Immonen,‡ Erkki Paatero,‡ and Per Nyholm§ Laboratory of Industrial Chemistry, A° bo Akademi, FIN-20500 Turku, Finland, Laboratory of Industrial Chemistry, Lappeenranta University of Technology, FIN-53851 Lappeenranta, Finland, and Technology Centre, Neste Oy, FIN-06101 Porvoo, Finland
A generally applicable stoichiometric and kinetic model was developed for the polyesterification of unsaturated dicarboxylic acid with diols in the presence of a homogeneous acid catalyst. The model also incorporates, besides the esterification, the cis-trans isomerization and the doublebond saturation reactions. Rate equations based on plausible mechanistic reaction steps were derived, and the parameters of the rate equations were determined with nonlinear regression analysis using the polyesterification of maleic acid with propylene glycol at 150-190 °C as a demonstration system. Comparisons of the model predictions with the experimental data showed that the approach is useful in predicting the polyesterification kinetics and the product distribution. Introduction Unsaturated dicarboxylic acids are used in the industrial production of polyesters. The carboxylic acid reacts with a diol, forming the polyester:
RCOOH + R′OH f RCOOR′ + H2O Besides the main reaction, the double bond of the acid undergoes cis-trans isomerization (Vancso´-Szmercsa´nyi et al., 1966), and it can also be saturated by the alcohol, which causes branching and cross-linking in the polymer (Ordelt et al., 1968; Ordelt and Kra´tky´, 1969). Thus, these side reactions are of crucial importance for the physical properties of the polyester. The esterification is industrially carried out in the presence of an added acidic agent, which catalyzes all of the reactionssesterification, cis-trans isomerization, and doublebond saturationsbeing present in the system. Typical acid catalysts used in polyesterification are zinc acetate, p-toluenesulfonic acid, and titanium benzenesulfonate (Fradet and Marechal, 1982a). The basic chemical effects in the polyesterification of unsaturated carboxylic acids are well-known. The first quantitative polyesterification studies date to the works of Flory (1939), who investigated, e.g., the esterification of adipic acid with diethylene glycol. The cis-trans isomerization of unsaturated acids has been thoroughly studied, e.g., by Vancso´-Szmercsa´nyi et al. (1966) and Davies and Evans (1956), and the mechanism of doublebond saturation was explained by Ordelt et al. (1968) and Fradet and Marechal (1982b). In the quantitative treatment of polyesterification kinetics, the side reactions are still often discarded, and roughly simplified second- and third-order rate expressions are used, as reviewed by Fradet and Marechal (1982a). However, for the correct description of the stoichiometry and kinetics, the side reactions have to be included. They also have a large practical importance, because they consume the reagent (diol) and change the product properties. A complete kinetic †
A° bo Akademi. Lappeenranta University of Technology. § Neste Oy. ‡
S0888-5885(95)00736-6 CCC: $12.00
model can then be used to predict the performance of an industrial reactor in order to steer the product quality. The kinetics of a model system, polyesterification of maleic acid and phthalic acid with propylene glycol, was studied experimentally in the previous works of our group (Paatero et al., 1994; Salmi et al., 1994), and a kinetic model comprising the main and side reactions was developed. The kinetic model proposed previously by us is valid in the absence of an added catalyst. In the present work, we extend the theory of polyesterification kinetics to cases where an added acidic catalyst is present, affecting the main reaction and all of the side reactions. The usefulness of the kinetic model will be demonstrated with experimental data obtained for the system maleic acid-propylene glycol, a homogeneous acid catalyst. Experimental Methods The experiments for the determination of the polyesterification kinetics were carried out in a bench-scale reactor. The reactor system (Figure 1) consisted of a tank (4000 mL) equipped with an anchor stirrer and a reflux column, since water, glycol, and some byproducts were volatilized and withdrawn from the reaction vessel. Nitrogen was continuously fed through the reactor. The reactor was heated with a oil-filled jacket, and the temperature in the reactor was controlled automatically. The kinetic experiments were performed at a temperature range of 140-190 °C. The dicarboxylic acid and the catalyst were placed in the reactor vessel prior to the experiment and heated to the desired temperature. The diol was heated in a separate reservoir to the same temperature. When the diol was fed into the reaction vessel, the polyesterification was commenced. Samples were taken from the reactor regularly during the experiment. The concentration of the dicarboxylic acid (the acid value) was determined from the sample by titration with KOH, and the analytical concentrations of the isomerized units (cI) and the double bonds (cD) were determined with H+ NMR spectroscopy. The cumulative mass of the distillate (γ) was monitored during the progress of the reaction, and the distillate was analyzed with GC and HPLC. The distillate © 1996 American Chemical Society
3952 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 Table 1. Functional Groups in the Polyesterification of Unsaturated Carboxylic Acids with Diols functional group
abbrev
R′-OH H
hydroxyl group cis acid
R′OH RCOOH1D
trans acid
RCOOH2D
saturated acid
RCOOHS
cis ester
RCOOR′1D
trans ester
RCOOR′2D
saturated ester
RCOOR′S
C COOH COOH C H H
C
COOH
OR′ H C COOR′ COOR′ C H H
C
COOR′
OR′
Figure 1. Bench-scale reactor system for the polyesterification. 1, Reactor; 2, stirrer; 3, temperature measurement and control unit; 4, distillation column; 5, cooling condenser; 6, cooling reservoir for the distillate; 7, distillate reservoir; 8, inert gas inlet; 9, reservoir for the diol.
abbreviations are illustrated in Table 1. The principal reactions of the functional groups are listed below:
esterification of cis and trans isomers of the acid H
contained mostly water and glycol but also some trace components. The contents of these trace components in the distillate were typically less than 5%.
+ R′OH
C
C
H
+ H2O
C
C
COOH
(1)
COOR′
cis-trans isomerization of the original acid and its ester
Stoichiometry
H
The stoichiometry is based on the functional groups. As a functional group, the carboxylic group and the double bond are chosen in the following way:
COOH/R′
CAT
C
C
C
C
COOH/R′
(2) H
saturation of the double bond through the nucleophilic attack of the alcohol
O CH
CAT
C OH
H
+ R′OH
C
C
for the carboxylic acid and
CAT
H
H
C
C
COOH/R′
COOH/R′
(3)
COOR′ + H2O
(4)
OR′
O CH
esterification of the saturated acid
C OR′
H
for the ester. Hence, the functional group contains one carboxyl group and half of the double bond. This definition of the functional group is chosen because of the consistency of the initial state; i.e., the original carboxylic acid, O
O C HO
HC
CH
C OH
contains two functional groups, which can undergo all the reactions (esterification, isomerization, and doublebond saturation). The functional groups and their
C
H C OR′
COOH + R′OH
CAT
H
H
C
C OR′
These four kinds of reactions give the fundamentals of the stoichiometry. The basic reactions give totally nine reactions, which are represented in the closed reaction scheme in Figure 2 and in Table 2. The vector for chemical symbols (functional groups) in Tables 1 and 2 is defined as follows:
nT ) [RCOOH1D RCOOH2D RCOOHS RCOOR′1D RCOOR′2D RCOOR′S R′OH H2O] (5)
Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 3953 Table 2. Basic Reactions in the Polyesterification of Unsaturated Carboxylic Acids with Diols reaction
stoichiometry
reaction type
1 2 3 4 5 6 7 8 9
RCOOH1D + R′OH h RCOOR′1D + H2O RCOOH2D + R′OH h RCOOR′2D + H2O RCOOH1D h RCOOH2D RCOOR′1D h RCOOR′2D RCOOH1D + 0.5R′OH h RCOOHS RCOOH2D + 0.5R′OH h RCOOHS RCOOR′1D + 0.5R′OH h RCOOR′S RCOOR′2D + 0.5R′OH h RCOOR′S RCOOHS + R′OH h RCOOR′S + H2O
esterification esterification isomerization isomerization double-bond saturation double-bond saturation double-bond saturation double-bond saturation esterification
defined by vector n and A is the coefficient matrix with the elements:
[
1 A) 1 0
Figure 2. Reaction scheme for the polyesterification of an unsaturated carboxylic acid with a diol.
where 1D and 2D denote the cis and trans isomers, respectively, and S denotes the saturated compound. The stoichiometric matrix (NT) is obtained from Table 2
[
N) -1 0 0 1 0 0 -1 1
0 -1 0 0 1 0 -1 1
-1 1 0 0 0 0 0 0
0 0 0 -1 1 0 0 0
-1 0 1 0 0 0 -1/2 0
0 -1 1 0 0 0 -1/2 0
0 0 0 -1 0 1 -1/2 0
0 0 0 0 -1 1 -1/2 0
0 0 -1 0 0 1 -1 1
]
1 1 1
1 0 0
0 1 0
0 1 1
0 0 0
0 0 0
0 0 0
]
(10)
The generation rate expressions obtained from (8) and the analytical concentrations c′ are listed in the Appendix. For the generation rates of the analytical quantities (r′), the relation
r′ ) Ar
(11)
is valid. After inserting the stoichiometric relation (8) in (11), we get
r′ ) ANR
(12)
The product ANR can be developed further by using the rule of matrix algebra
(6)
(AN)T ) NTAT
(13)
r′ ) (NTAT)TR
(14)
which implies that
The reactions can now be described with the notation
Thus, the coefficient matrix can be calculated a priori from
N Tn ) 0
r′ ) MR
(15)
M ) (NTAT)T
(16)
(7)
It should be noticed that the stoichiometric coefficient -1/2 is valid for the hydroxylic group in reactions 5-8 in Table 2. This is based on the definition of the functional groups. The generation rates (r) of the functional groups are obtained directly from the stoichiometry
r ) NR
(8)
Usually all the functional group concentrations are not experimentally observable. In a typical experiment, the acid value, i.e., the sum of carboxyl groups (cCOOH) and the sum of double bonds (cD) as well as the concentration of isomerized units (cI), is measured. These analytical concentrations of the acid groups (cCOOH), double bonds (cD), and isomerized groups (cI), c′ ) [cCOOHcDcI]T, are expressed with the linear relationship
c′ ) Ac
(9)
where c is the concentration vector of chemical symbols
where
For the present case, M becomes
[
M) -1 0 0
-1 0 0
0 0 1
0 0 1
0 -1 0
0 -1 -1
0 -1 0
0 -1 -1
]
-1 0 0 (17)
The product MR is given in the Appendix. Kinetics and Mechanism Esterification. The esterification can be carried out in the presence or in the absence of an added acid catalyst. In the absence of an added acid catalyst, the carboxylic acid itself acts as a catalyst through autoprotolysis. It is generally believed that the key intermediate in the esterification is the carbenium ion RC+(OH)2, which reacts with the alcohol in a rate-
3954 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996
determining step. The nucleophilic substitution can be written as follows
diates are obtained from eqs 24-26:
cI1 ) K1cRCOOHcHA
(27)
(18)
cI2 ) xK2cRCOOH
(28)
where (v) indicates that water usually escapes from the melt phase. Reaction step (18) consists of several elementary steps as discussed in the literature (Ingold, 1953), but the steps after the nucleophilic substitution are considered to be rapid, and hence, they do not influence the experimentally observed kinetics. The slow reaction step (18) gives the rate equation
cI3 ) K3cRCOOH2
(29)
OH RC+
O
– X– + HOR′
+ H2O
RC
+ H+X–
OR′
OH
rE ) kEcRC+(OH)2cR′OH - k-EcRCOOR′cH2OcHX
(19)
RC+(OH)
The origin of the carbenium ion 2 depends on the conditions in the reaction milieu. In the presence of an added acid catalyst (HA), the carbenium ion is formed in a straightforward way
The rate equation for esterification finally becomes
rE ) (kE1K1cHA + kE2xK2 + kE3K3cRCOOH)cRCOOHcR′OH - (k-E1cHA + k-E2 + k-E3cRCOOH)cRCOOR′cH2O (30) The reaction thermodynamics implies that the rate and equilibrium constants of steps 18 and 20-22 are related to the equilibrium constant of the esterification (KE):
kE1K1 kE2xK2 kE3K3 ) ) ) KE k-E1 k-E2 k-E3
(31)
OH RCOOH + HA
A–
RC+
(20)
OH I1
In the absence of an added catalyst, the autoprotolytic formation route dominates, leading to separate ions at low and intermediate conversions of the carboxylic acid and to ion pairs at high conversions. The reaction steps can be written as follows: OH RCOOH + RCOOH
+ RCOO–
RC+
(21)
OH I2 OH RCOOH + RCOOH
RCOO–
RC+
(22)
OH I3
The esterification rate is obtained by adding the contributions of the intermediates (I): 3
rE )
∑ j)1
3
kEjcIjcR′OH -
k-EjcRCOOR′cH OcHXj ∑ j)1 2
(23)
For the backward steps of (18), H+X- is HA for the step giving I1 and RCOOH for the step giving I3; for the step giving I2, H+X- is excluded. The quasi-equilibrium approximation is applied on the rapid steps (20)-(22):
K1 )
cI1 cRCOOHcHA
K2 )
K3 )
cI2cRCOOcRCOOH2 cI3 cRCOOH2
(24)
After inserting these relationships in the rate equation (30), it gets the beautiful form
rE ) (kE1K1cHA + kE2xK2 +
(
kE3K3cRCOOH) cRCOOHcR′OH -
)
cRCOOR′cH2O KE
(32)
Rate equation (32) includes all the special cases observed in the absence of an added catalyststhe term kE2xK2 (single ions) dominates in the beginning of the reaction, and kE3K3cRCOOH (ion pairs) dominates at the end of the reactionsas well as the special case of a very strong added acid catalystskE1K1cHA dominates. For the last case, the forward reaction kinetics is de facto proportional to cHAcRCOOHcR′OH. The constants of the ionic equilibria (24)-(26) are functions of the reaction milieu, as discussed, e.g., by Fang et al. (1975). The rigorous thermodynamic treatment is, however, very cumbersome, and it inevitably implies the introduction of several new parameters. Therefore, Salmi et al. (1994) and Paatero et al. (1994) proposed a semiempirical function of the type k′EcROOHn-1, replacing the sum kE2xK2 + kE3K3cRCOOH. The empirical exponent n is a function of the carboxylic acid conversion, being n ) 1 in the beginning of the reaction and n ) 2 at equilibrium. Here a still more simplified treatment is applied, and the effect of the ion pairs (I3) at the end of the reaction is discarded. Furthermore, the catalyst concentration can be regarded as a constant during the reaction, which implies that the experimentally observable rate equation becomes
(
rE ) k′E cRCOOHcR′OH -
(25)
)
cRCOOR′cH2O KE
(33)
where
(26)
The principle of electric neutrality implies that cRCOO) cI2. Consequently, the concentrations of the interme-
k′E ) k′E0 + k′E1cRCOOH + k′′EcHA
(34)
and k′′E ) kE1K1, k′E0 ) kE2xK2, and k′E1 ) k3K3 ≈ 0. Equation 34 implies that the experimentally measured rate constants k′E for the different catalyst (HA)
Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 3955
concentrations should give a straight line as a function of the catalyst concentration. Cis-Trans Isomerization. The cis-trans isomerization of the double bond of the unsaturated carboxylic acid is known to be acid catalyzed (Vancso´-Szmercsa´nyi et al., 1966). The principal reaction can be written as follows:
contribution terms. Consequently, the experimentally observed cis-trans isomerization kinetics is given by
(
rI ) k′I ccis -
k′I ) k′I0cRCOOH + k′′IcHA
+ HA′
C COOH cis H
H
C
C+A′
(42)
where
H C
)
ctrans KI
–
COOH C
+ HA′
C
(35)
H
COOH trans
L
The effect of an added acid catalyst on the cis-trans isomerization kinetics is straightforward, the forward rate being proportional to the acid (HA) and the cisisomer concentrations. In the absence of an added catalyst, Vancso´-Szmercsa´nyi et al. (1966) propose that the isomerization rate is proportional to cRCOOH2. However, it cannot be a priori excluded that both intermediates I2 and I3 act as proton donors in the cis-trans isomerization. The proton donation step in (35) is generally believed to be rapid, whereas the second step is rate determining. Consequently, the isomerization rate can be written as
rI ) kIcL - k-IctranscHA′
and k′′I ) kI1KI′1 and k′I0 ) kI2KI′2. Double-Bond Saturation. The saturation of the double bond in the carboxylic acid also proceeds via acid catalysis. The process is initiated by a rapid protondonation step followed by the nucleophilic attack of the hydroxyl group of the diol. The subsequent steps are rapid. The reaction sequence can be written as follows: H C
+ HA′
C
1
H
H
C
C+A′
COOH
2 +R′OH
–
COOH
D
P1 H
H
C
C
COOH
3
H
H
C
C
O+R′A′ –
COOH + HA′
S
H P2
The rate of the second step is given by
Application of the quasi-equilibrium approximation on the proton donation step gives
rD ) kDcP1cR′OH - k-DcP2
cL cciscHA′
(37)
∑kIjKI′jcHA′j)ccis - ∑k-IjcHA′jctrans
(38)
For the actual case, the rate equation becomes
rI ) (kI1KI′1cHA + kI2KI′2cRCOOH)ccis (k-I1cHA + k-I2cRCOOH)ctrans (39) Again the thermodynamics gives us the fundamental relations between the forward and backward rate constants
(40)
After inserting (40) in (39), the final form of the isomerization rate equation is obtained:
(
rI ) (kI1KI′1cHA + kI2KI′2cRCOOH) ccis -
K1 ) K3 )
)
ctrans KI
(41)
A comparison of (41) with the rate equation of esterification, eq 32, reveals the similarities of the catalytic
cP1
(46)
cDcHA′ cScHA′ cP2
(47)
After inserting the concentrations of P1 and P2 from (46) and (47) in (45), the rate equation becomes
rD ) k+DK1cDcR′OHcHA′ -
k-DcScHA′ K3
(48)
In the presence of several acids, the rate expression is easily generalized to
(
k-Dj
)
∑kDjK1jcHA′j)cDcR′OH - ∑ K3j cHA′j cS
rD ) (
kI1KI′1 kI2KI′2 ) ) KI k-I1 k-I2
(45)
Application of the quasi-equilibrium hypothesis on rapid steps 1 and 3 in the above scheme implies that
The concentration cHA′ represents symbolically HA, RCOOH, and RC(OH)2+COO-, (I3). After inserting the quasi-equilibrium expression (37) in (36) and accounting for all contributions to the isomerization, eq 36 is transformed to
rI ) (
(44)
OR′
(36)
KI′ )
(43)
(49)
For the system of one carboxylic acid and one added catalyst, the rate equation obtains the following form:
rD ) (kD1K11cHA + kD2K12cRCOOH)cDcR′OH k-D1 k-D2 c + c c (50) K31 HA K32 RCOOH S
(
)
Thermodynamics gives the relation between the forward and backward rate constants:
3956 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996
kD1K11K31 kD2K12K32 ) ) KD k-D1 k-D2
(51)
After inserting (51) in (50), the final form of the rate equation becomes
(
cS KD
rD ) (kD1K11cHA + kD2K12cRCOOH) cDcR′OH -
)
(52)
The expression can be written in a compressed form
(
rD ) k′D cDcR′OH -
)
cS KD
(53)
where
k′D ) k′D0cRCOOH + k′′DcHA
(54)
and k′′D ) kD1K11 and k′D0 ) kD2K12. Rate Equations of Steps 1-9. The fundamentals introduced in the previous section are used for the derivation of the rate equations in the reaction scheme (Figure 2). The rate equations belong to the following categories: esterification (R1, R2, R9), isomerization (R3, R4), and double-bond saturation (R5, R6, R7, R8). Thus, the expressions for R1, ..., R9 are obtained from (33), (42), and (53) in a very straightforward way:
( (
R1 ) k′1 cRCOOH1DcR′OH R2 ) k′2 cRCOOH2DcR′OH -
( (
R3 ) k′3 cRCOOH1D R4 ) k′4 cRCOOR′1D -
( ( ( (
K1
cRCOOR′2DcH2O K2
K3
cRCOOR′2D K3
R6 ) k′6 cRCOOH2DcR′OH R7 ) k′7 cRCOOR′1DcR′OH R8 ) k′8 cRCOOR′2DcR′OH -
(
) )
cRCOOH2D
R5 ) k′5 cRCOOH1DcR′OH -
R9 ) k′9 cRCOOHScR′OH -
) )
cRCOOR′1DcH2O
) ) ) )
k′I ) k′3 ) k′4
(67)
k′D ) k′5 ) k′6 ) k′7 ) k′8
(68)
KE ) K1 ) K2 ) K9
(69)
KI ) K3 ) K4
(70)
(56)
KD ) K5 ) K6 ) K7 ) K8
(71)
(57)
Temperature Dependences of the Parameters. For the temperature dependences of the rate and equilibrium constants, the Arrhenius law and the general thermodynamics give the relationship
(58)
k′j ) A′j0e-E′j0/(RT) + A′j1e-E′j1/(RT)cCOOH1D +
(55)
The thermodynamic interpretation of the preexponential (A) and activation (E) parameters is given in the Nomenclature section. However, for preliminary tests of the parameters, a crude approximation
k′j ) A′je-E′j/(RT)
(73)
was used, assuming equal activation energies in (72).
(62)
)
cRCOOR′ScH2O K9
(65)
Simplifications of the Kinetic Model. The basic kinetic model, eqs 55-63, contains nine lumped rate constants and nine equilibrium constants. The most radical simplification of the model would imply that the esterification rate constants, the isomerization rate constants, and the double-bond saturation rate constants are mutually equal. An analogous principle would be applicable for the equilibrium constants. This ultimate simplification would result in three rate and three equilibrium constants only:
(61)
cRCOOR′S K8
j ) 3, ..., 8
(66)
(60)
cRCOOR′S K7
k′j ) k′j1cCOOH1D + k′′jcHA
k′E ) k′1 ) k′2 ) k′9
(59)
cRCOOHS K6
k′j ) k′j0 + k′j1cCOOH1D + k′′jcHA j ) 1, 2, 9 (64)
A′′e-E′′/(RT)cHA (72)
cRCOOHS K5
to the added catalyst and to the strongest carboxylic acid being present. In addition, the contributions of the ions I2 (eq 21) are discarded for all other cases than for the autoprotolysis of RCOOH1D. For the other esterifications, the reaction was assumed to progress through ion pairs (I3, eq 22). Thus, we obtain for the lumped parameters the following expressions:
(63)
The contents of the lumped parameters k′1, ..., k′9 are dependent on the composition of the chemical system. In principle, all carboxylic groups (COOH1D, COOH2D, and COOHS) could contribute to the acid catalysis so that, e.g., the term k′E1cRCOOH in eq 34 should be replaced by a sum consisting of the contributions of all carboxylic acids present in the system. However, for the sake of simplicity, the catalytic effect is attributed
Reactor Model In the derivation of the reactor model, some fundamental assumptions were introduced. The liquid phase in the reactor was presumed to be basicly in batch, but some volatilization of the compounds, particularly water and diol, was assumed to take place. The reaction rates were described with respect to the mass of the liquidsnot with respect to the volume, as traditionally done. However, the rates used in the present work (ri) are related to the volume-based rates (r′i) by r′i ) riF, where F is the liquid density. The analytical concentrations of carboxylic groups, double bonds, and isomerized units and the amount of distillate were measured during the reaction.
Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 3957
Based on these assumptions, the mass balance for a component (i) in the liquid phase can be written as follows:
dci ci - c′i dγ ) ri + dt 1 - γ - δ dt
(86)
where m is the mass of the liquid phase and n˘ i is the flow of the amount of substance leaving the reactor in the gas phase in the distillate. The amount of substance is expressed with the mass of liquid and the concentration,
The cumulative amounts withdrawn as samples from the liquid phase (δ) were directly included in the model from the experiments. This approach would in principle be applicable also for the cumulative mass of distillate (γ). It turned out, however, that an efficient way to describe the time evolution of the distillate is to fit an empirical model to the experimental data. The function for γ was chosen according to our previous concept (Paatero et al., 1994).
ni ) cim
γ ) A(1 - e-Bt)
dni rim ) n˘ i + dt
(74)
(75)
After differentiation of ni with time and rearrangement, the mass balance is transformed to
dci ci dm n˘ i ) ri dt m dt m
(76)
The flow leaving the reactor is expressed with the concentration in the distillate (c′i) and the distillate mass flow (m ˘ ′):
˘′ n˘ i ) c′im
m + m′ +
∑mS ) m0
(78)
where m0 is the total (initial) mass. For the distillate collector, the total mass balance is given by
m ˘ ′)
dm′ dt
(79)
Since the total mass (m0) is constant and the sampling is a discrete procedure, differentiation of eq 78 gives the simple relation
dm dm′ + )0 dt dt
dm′ dt
(81)
dci ci - c′i dm′ ) ri + dt m dt
(83)
The following dimensionless quantities are introduced:
γ ) m′/m0 δ)
∑mS/m0
(88)
The only components of global importance in the distillate were water and diol. For the weight fraction of diol, the following empirical function was introduced:
wOH ) Ce-Dt + E
(89)
where the parameters C, D, and E have following temperature dependences: C ) C1T + C2, D ) D1T + D2, and E ) E1T + E2. The water weight fraction can be calculated as follows:
wH2O ) 1 - wOH
(90)
The parameters included in eqs 87 and 89 were determined simultaneously for the experiments with equal catalyst concentrations.
The empirical parameters included in the functions for the distillate mass and the weight fractions of water and diol were obtained with nonlinear regression analysis. The data sets obtained at different temperatures were merged together, and the parameters with their temperature dependences were estimated. For example, for the cumulative mass of the distillate (γ), the objective function to be minimized was defined as
(82)
The total mass balance (82) is divided by m0, which gives for m
∑mS/m0
wi Mi
Parameter Estimation Procedure
is included in the mass balance (76), which becomes
m/m0 ) 1 - m′/m0 -
c′i )
(80)
Relation 80 is inserted in the definition of n˘ i, eq 77, after which the expression for n˘ i
n˘ i ) c′i
The empirical parameters A and B were assumed to have the following temperature dependences: A ) A1T + A2 and B ) B1eB2T. The distillate was analyzed chemically, i.e., the weight fractions (wi) of water and diol in the distillate were obtained. The concentrations c′i were calculated from the experimentally observed weight fractions (wi) and molar masses (Mi):
(77)
For the masses existing in the reactor (m) and in the distillate (m′ ) and for the samples withdrawn from the liquid phase (∑mS), the total balance is applied
(87)
(84) (85)
After inserting (83)-(85) in eq 82, the final form of the mass balance is obtained:
NT,Nk
Q)
(γi,k - γˆ i,k)2 ∑ i)1,k)1
(91)
where Nk denotes the number of experimental values at temperature Tk and NT is the number of experimental temperatures. Analogous objective functions were used for the water and diol weight fractions (wH2O, wOH). The models for the distillate mass and the weight fractions of water and diol, eqs 87 and 89 and 90, are explicit algebraic functions, which makes the objective function evaluation very straightforward. The objective function was minimized with the Levenberg-Marquardt method (Marquardt, 1963) implemented in the modeling and estimation software MODEST (Haario, 1994).
3958 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 Table 3. Empirical Parameters for the Distillate (Eqs 87 and 89) in the Polyesterification of Maleic Acid with Propylene Glycol cCAT, mol %
A1
A2
B1
B2
C1
C2
D1
D2
E1
E2
0.01 0.05 0.1
9.38 × 10-10 1.79 × 10-9 1.33 × 10-8
0.0371 0.0359 0.0310
1.03 × 10-3 9.19 × 10-4 5.76 × 10-4
-0.359 -0.315 -0.165
7.82 -64.6 6.87
8.31 71.6 6.47
-1.54 × 10-3 -1.61 × 10-3 4.54 × 10-4
1.53 1.56 0.650
1.51 × 10-3 1.71 × 10-3 -4.70 × 10-4
-0.519 -0.598 0.357
Figure 4. Experimental and predicted concentrations of the carboxyl groups (cCOOH), the isomerized units (cI), and the double bonds (cD) in the polyesterification of maleic acid with propylene glycol at 170 °C in the presence of a catalyst (0.05 mol %). The simplest model was fitted with three rate parameters (k′E, k′I, k′D).
Figure 3. Dimensionless mass of the distillate (γ) (a) and the weight fraction of the diol (wOH) (b) during the polyesterification of maleic acid with propylene glycol at 170 °C in the presence of a catalyst (0.05 mol %). The lines are calculated according to the model equations (87) (a) and (89) (b), respectively.
The estimation of the rate and equilibrium parameters was performed stagewise. First, the lumped parameters k′ were estimated separately at each temperature and catalyst concentration, and the behaviors of the parameters were checked visually with appropriate plotssArrhenius plots for the temperature dependence and k′ vs cCAT plots for the catalyst concentration dependence. The objective function was defined in the following way:
Q)
∑(cCOOH,j - cˆ COOH,j)2 + (cI,j - cˆ I,j)2 + (cD,j - cˆ D,j)2
(92)
for all cases. The concentrations cCOOH, cI, and cD were obtained from the linear transformation (9) after having solved the concentrations of the functional groups (cCOOH1D, ..., cH2O) from the mass balance (86). The mass balances were solved numerically during the objective function minimization with the backward difference method (Henrici, 1962; Gear, 1971) implemented in the software LSODE (Hindmarsh, 1983). The objective function was minimized with the Levenberg-Marquardt method (Marquardt, 1963). The methods were available in the MODEST software (Haario, 1994).
Figure 5. Experimental and predicted concentrations of the carboxyl groups (cCOOH), the isomerized units (cI), and the double bonds (cD) in the polyesterification of maleic acid with propylene glycol at 170 °C in the presence of a catalyst (0.05 mol %). The model was fitted with the parameters given in eqs 93-101.
Modeling of the Distillate The kinetic experiments were performed at five temperatures (140, 160, 170, 180, and 190 °C) varying the catalyst concentration from 0.01 to 0.1 mol %. The parameters of the empirical functions (87) and (89) were determined at the first stage. A list of the empirical parameters is given in Table 3, and an example of the fit of the model is shown in Figure 3. The general conclusion is that the exponential function (87) is suitable for the description of the amount of distillate (γ). The weight fraction of the diol (wOH) is not predicted as well, as can be seen from Figure 3b. It should, however, be remembered also that the analytical data are less accurate for the composition of the distillate. The functions (87) and (89) can be regarded to give a
Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 3959 Table 4. Rate and Equilibrium Constants in the Polyesterification of Maleic Acid with Propylene Glycol at Different Temperatures (cCAT ) 0.05 mol %) k′E, kg mol-1 min-1 1.06 × 10-3 2.0 2.13 × 10-3 3.7 3.03 × 10-3 3.8 4.83 × 10-3 1.4 6.27 × 10-3 3.9
k′I, min-1
39.7 0.0517 42.7 10 20.5 0.0450 37 7.4 18.6 0.0812 32 11 26.4 0.148 10 8.6 17.2 0.274 19.8 100
KI 6.17 12 8.71 24 10.9 25 14.1 14 19.6 45
k′D1, k′D2, k′D3, KD3, kg mol-1 min-1 KD1, kg mol-1 kg mol-1 min-1 kg mol-1 min-1 kg mol-1 1.74 × 10-5 20 6.97 × 10-6 78 1.56 × 10-5 53 5.74 × 10-5 25 1.13 × 10-4 110
9.46 × 10-5 9.46 ×
10-5
9.46 × 10-5 9.46 × 10-5 9.46 × 10-5
7.39 × 10-3 16 4.04 × 10-3 24 7.44 × 10-3 29 0.0179 16 0.0378 140
4.06 × 10-3 17 6.63 × 10-3 79 8.97 × 10-3 42 0.0290 16 0.0417 49
5.14 18 5.45 37 7.96 38 24.5 20 111 85
D2
D1
140 °C, Q ) 0.17 σ, % 160 °C, Q ) 0.43 σ, % 170 °C, Q ) 0.39 σ, % 180 °C, Q ) 0.049 σ, % 190 °C, Q ) 0.30 σ, %
KE
Figure 6. Arrhenius plots of the forward rate constants k′E (esterification), k′I (isomerization), and k′D1, k′D2, and k′D3 (double-bond saturation) in the polyesterification of maleic acid with propylene glycol in the presence of a catalyst (0.05 mol %).
satisfactory reproduction of the experimental data so that the weight fraction functions can safely be included
in the mass balance (86), which is used in the estimation of the kinetic parameters.
3960 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996
Estimation of the Kinetic Parameters The estimation of the kinetic parameters was initiated by attempting to fit the ultimately simplified model with just three rate (k′E, k′I, k′D) constants to the data (eqs 66-71). An example of the fit is presented in Figure 4, which shows the kinetic experiment carried out at 170 °C. Similar results were obtained for the other experimental temperatures, 140-180 °C. It is evident that this kind of ultimate simplification is not suitable for the description of the actual experimental datasthe model is disable to predict correctly the trends, particularly, the concentrations of the double bond (cD). At the next stage, some experimentation with the parameter fitting was carried out, avoiding the introduction of too many parameters, i.e., the overparametrization. It turned out that the rate and equilibrium parameters for the esterification (k′E, KE) on one hand and those for the isomerization on the other hand (k′I, KI) can be set equal for the cis and trans isomers, but a distinction of the parameters for the different doublebond saturation reactions is inevitable. After testing different alternatives, we arrived at the following set of parameters.
k′E ) k′1 ) k′2 ) k′9
(93)
k′I ) k′3 ) k′4
(94)
k′D1 ) k′5
(95)
k′D2 ) k′6
(96)
k′D3 ) k′7
(97)
K′E ) K′1 ) K′2 ) K′9
(98)
K′I ) K′3 ) K′4
(99)
K′D1 ) K′5
(100)
K′D3 ) K′7
(101)
Reaction 6 (Figure 2) was regarded as an irreversible reaction (K′D2 ) K′6 ≈ ∞) and reaction 8 was discarded (k′D4 ) k′8 ≈ 0). The parameters were fitted according to this concept, and the numerical values are given in Table 4 for the temperatures 140-190 °C and for the catalyst concentration 0.05 mol %. A comparison of the model prediction with the experimental data is presented in Figure 5. Table 4 reveals that the estimation statistics is very satisfactory, the standard deviations of the parameters being typically 3-50% only and the correlations between the parameters always being less than 0.900. The temperature dependences of the forward rate parameters were tested with Arrhenius plots. The plots to k′E, k′I, and the double-bond saturation rate constants (k′D1, k′D2, k′D3) are shown in Figure 6. The parts of the figure show that the rate parameters obey the Arrhenius law, except the parameter obtained for the lowest temperature, 140 °C. The reason of this observation is somewhat unclear; however, the lowest temperature was omitted from the Arrhenius plots. The equilibrium constant for esterification (KE) varies randomly with temperature (Table 4), which reflects the uncertainty in the determination of this constant. We can conclude that this equilibrium constant is virtually independent of temperature. The equilibrium constants of isomer-
Figure 7. Experimental and predicted concentrations of the carboxyl groups in the polyesterification of maleic acid with propylene glycol at 180 °C with different catalyst concentrations.
Figure 8. Dependence of the isomerization rate constant (k′I) on the catalyst concentration in the polyesterification of maleic acid with propylene glycol at 180 °C (calculated for cRCOOH1D ) 1.0 mol/ kg). Table 5. Arrhenius Parameters for the Rate and Equilibrium Constants in the Polyesterification of Maleic Acid with Propylene Glycol (T ) 140-190 °C and cCAT ) 0.05 mol %) Q ) 1.98 k′E KE k′I KI k′D1 KD1 k′D2 k′D3 KD3
A′ 23 100 kg mol-1 min-1 19.7 6210 min-1 2860 1.76 × 10-4 kg mol-1 min-1 9.46 × 10-5 kg mol-1 5.90 kg mol-1 min-1 15800 kg mol-1 min-1 2.63 × 106 kg mol-1
σA, % E′a, kJ/mol σEa, % 3.2 27 9.1 18 42 23 30 31
58.2 ≈0 41.5 21.2 10.2 ≈0 24.9 52.9 46.5
2.5 9.9 37 160 40 24 28
ization (KI) and double-bond saturation (KD3) increase with temperature, as can be seen from Table 4. The influence of the catalyst concentration on the polyesterification kinetics was studied by estimating separately the kinetic parameters for the experimental sets obtained with different catalyst concentrations varying from 0.0 to 0.1 mol %. Examples of the fits of the model (cCOOH) for experiments performed at 180 °C are presented in Figure 7. The general conclusion is that the kinetic model is suitable to describe the behavior of the system also for different catalyst concentrations. The dependences of the rate parameters on the catalyst concentrations were investigated by preparing the test plots according to eqs 64 and 65. A typical plot is shown in Figure 8. It can be concluded
Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 3961
Figure 9. Experimental and predicted concentrations of the carboxyl groups (cCOOH), the isomerized units (cI), and the double bonds (cD) in the polyesterification of maleic acid with propylene glycol at 160 °C (a, c) and 180 °C (b, d) in the presence of a catalyst (the catalyst concentrations are 0.01 mol % (a, b) and 0.05 mol % (c, d)). The model was fitted with merged data sets.
that the parameters depend on the catalyst concentration as predicted by the kinetic model. At the final step of the parameter estimation, all data sets were merged together (also the data set at 140 °C, which was omitted from the Arrhenius plots, was included), and the fundamental parameter values including their temperature dependences were estimated with regression analysis. A summary of the estimation results is presented in Table 5. It can be seen by comparing Table 4 with Table 5 that the merging of the data sets improves the estimation statistics to the same extent, making the individual parameter values statistically more reliable. Comparisons of the model predictions with experimental data are presented in Figure 9 for temperatures of 160 and 180 °C and for catalyst concentrations of 0.01 and 0.05 mol %. The activation energies for esterification and isomerization with the catalyst concentration of 0.05 mol % were estimated to 58 and 42 kJ/mol, respectively (Table 5). According to March (1985), the cis isomer usually has a lower thermochemical stability; i.e., the enthalpy for the transisomer formation is negative. As can be seen from Table 5, a positive value was obtained for the activation factor (E′a) of the cis-trans isomerization reaction in this study. This can be explained by the low value of the factor (21 kJ/mol) and by the difficulties to approach equilibrium conditions in this kind of system of slow reactions, which causes an uncertainty in the determination of the temperature dependence of the isomerization equilibrium constant. The general conclusion is that the kinetic model proposed here is completely
satisfactory to describe the tendencies in the catalyzed polyesterification of maleic acid with propylene glycol. Verification of the Kinetic Model The kinetic model was used to simulate experiments which were not included in the parameter estimation. Two verification experiments were made at isothermal conditions at 165 and 185 °C with a catalyst concentration of 0.05 mol %. The comparisons of the model predictions with experimental data are presented in parts a and b of Figure 10 for the experiments at 165 and 185 °C, respectively. The figures reveal that the kinetic model predicts the experimental trends in an excellent way; no systematic deviations are observable, and the minor differences between the model predictions and the experimental data are merely due to experimental scattering. Conclusions A kinetic scheme was proposed for homogeneously catalyzed polyesterification of unsaturated carboxylic acids with diols (Figure 2). The model comprises not only the main reaction, the esterification, but also the side reactions, the cis-trans isomerization, and the double-bond saturation. Rate laws for these reactions, based on plausible reaction mechanisms, were derived (eqs 33, 42, and 53). The kinetic model was tested with a demonstration system consisting of maleic acid with propylene glycol by determining the kinetic parameters with regression analysis (Tables 4 and 5). A comparison
3962 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 m ) mass (of the liquid phase) (kg) m′ ) distillate mass (kg) m ˘ ′ ) distillate mass flow (kg/min) M ) molar mass (kg/mol) M ) coefficient matrix, eq 16 n ) amount of substance (mol) n ) vector for the functional groups (mol) n˘ ) flow of substance (mol/min) N ) matrix for stoichiometric coefficients Q ) objective function r ) generation rate of a functional group (mol/kg min) r ) vector for generation rates of functional groups (mol/ kg min) r′ ) generation rate of an analytical quantity (mol/kg min) r′ ) vector for generation rates of analytical quantities (mol/kg min) R ) reaction rate (mol/kg min) R ) general gas constant (J/(kmol) R ) vector for reaction rates (mol/kg min) t ) time (min) T ) temperature (K) w ) weight fraction (dimensionless) Greek Letters γ ) dimensionless mass of the distillate δ ) dimensionless mass of the samples σ ) standard deviation of a parameter (%) Subscripts and Superscripts
Figure 10. Independent simulations of the polyesterification of maleic acid with propylene glycol at 165 °C (a) and 185 °C (b) in the presence of a catalyst (0.05 mol %).
of the model predictions with the actual experimental data (Figures 5 and 9) as well as an independent verification of the model (Figure 10) showed that the approach is useful and reliable in the prediction of the progress and the product distribution in the polyesterification. Acknowledgment We are grateful to Dr. Heikki Haario for his expertise in the computer implementation of the model. Nomenclature A, B, C, D, E ) empirical parameters, eqs 87-89 A ) frequency factor (the dimension depends on the case) A ) coefficient matrix, eq 9 A′, A′′ ) lumped frequency factors (the dimensions depends on the case) c ) concentration (mol/kg) c ) vector for the concentrations of the functional groups (mol/kg) c′ ) concentration (distillate), concentration of an analytical quantity (mol/kg) c′ ) vector for the concentrations of analytical quantities (mol/kg) Ea ) activation energy (J/mol) E′a, E′′a ) lumped activation energies (J/mol) k ) rate constant (the dimension depends on the case; see below) k′, k′′ ) lumped rate constants (k′j, j ) 1, ..., 2 and j ) 5, ..., 9 in kg mol-1 min-1; k′j, j ) 3, ..., 4 in min-1) K ) equilibrium constant (Kj, j ) 1, ..., 4 and j ) 9, dimensionless; Kj, j ) 5, ..., 8 in kg mol-1)
0 ) initial value cis ) cis isomer D ) double bond, double-bond saturation E ) ester, esterification i ) component index I ) isomerization j ) reaction index n ) empirical exponent S ) single bond, sample trans ) trans isomer T ) transpose of a matrix Abbreviations D ) double bond 1D ) cis isomer 2D ) trans isomer E ) ester HA ) acid catalyst HX ) acid H2O ) water I ) isomer I ) intermediate (esterification) L ) intermediate (isomerization) OH ) hydroxyl group P ) intermediate (double-bond saturation) RCOOH ) carboxylic acid RCOOR′ ) ester of carboxylic acid and diol RC+(OH)2 ) carbenium ion R′OH ) diol S ) single bond
Appendix Generation Rates (r ) NR):
rRCOOH1D ) -R1 - R3 - R5 rRCOOH2D ) -R2 + R3 - R6 rRCOOHS ) R5 + R6 - R9 rRCOOR′1D ) R1 - R4 - R7
Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 3963
rRCOOR′2D ) R2 + R4 - R8 rRCOOR′S ) R7 + R8 + R9 rR′OH ) -R1 - R2 - R9 - 1/2(R5 + R6 + R7 + R8) rH2O ) R1 + R2 + R9 Analytical Concentrations (c′ ) Ac):
c′COOH ) cRCOOH1D + cRCOOH2D + cRCOOHS c′D ) cRCOOH1D + cRCOOH2D + cRCOOR′1D + cRCOOR′2D c′I ) cRCOOH2D + cRCOOR′2D Generation Rates of the Analytical Quantities (r′ ) MR):
r′COOH ) -R1 - R2 - R9 r′D ) -R5 - R6 - R7 - R8 r′I ) R3 + R4 - R6 - R8 Literature Cited Davies, M.; Evans, F. P. The isomerization of maleic acid in aqueous solutions. Trans. Faraday Soc. 1956, 52, 74-80. Fang, Y.-R.; Lai, C.-G.; Lu, J.-L.; Chen, M.-K. The kinetics and mechanism of polyesterification of binary acid and binary alcohol. Sci. Sin. 1975, 18, 72-87. Flory, P. J. Kinetics of Polyesterification: A Study of the Effects of Molecular Weight and Viscosity on Reaction Rate. J. Am. Chem. Soc. 1939, 61, 3334-3340. Fradet, A.; Marechal, E. Kinetics and Mechanism of PolyesterificationssI. Reactions of Diols with Diacids. Adv. Polym. Sci. 1982a, 43, 51-142.
Fradet, A.; Marechal, E. Study on Models of Double Bond Saturation During the Synthesis of Unsaturated Polyesters. Macromol. Chem. 1982b, 188, 319-329. Gear, C. W. Numerical Initial Value Problems in Ordinary Differential Equations; Prentice Hall: Englewood Cliffs, NJ, 1971. Haario, H. MODEST User’s Guide, Profmath Oy, Helsinki, 1994. Henrici, P. Discrete Variable Methods in Ordinary Differential Equations; Wiley: New York, 1962. Hindmarsh, A. C. 1983, ODEPACK-A Systematized Collection of ODE-Solvers. In Scientific Computing; Stepleman, R., et al., Eds.; IMACS/North Holland: Amsterdam, pp 55-64. Ingold, C. K. Structure and Mechanism in Organic Chemistry, 1st ed.; Ithaca, NY, 1953; pp 767-770. March, J. Advanced Organic Chemistry, 3rd ed.; Wiley: New York, 1985; p 111. Marquardt, D. W. An algorithm for least squares estimation on nonlinear parameters. SIAM J. 1963, 11, 431-441. Ordelt, Z.; Kra´tky´, B. Aufgabe der Reaktionssteuerung bei der Synthese von ungesa¨ttigten Polyestern aus Maleinsa¨ureanhydrid. Farbe Lack 1969, 6, 523-531. Ordelt, Z.; Nova´k, V.; Kra´tky´, B. U ¨ ber die Umkehrbahrkeit der Diolenaddition an die Olefinische Doppelbindung der A ¨ thylen1,2-Dikarbonsa¨uren bei der Polykondensation in der Schmelze. Coll. Czech. Chem. Commun. 1968, 33, 405-415. Paatero, E.; Na¨rhi, K.; Salmi, T.; Still, M.; Nyholm, P.; Immonen, K. Kinetic Model for Main and Side Reactions in the Polyesterification of Carboxylic Acids with Diols. Chem. Eng. Sci. 1994, 49, 3601-3616. Salmi, T.; Paatero, E.; Nyholm, P.; Still, M.; Na¨rhi, K. Kintics of Melt Polymerization of Maleic acid and Phthalic Acids with Propylene Glycol. Chem. Eng. Sci. 1994, 49, 5053-5070. Vancso´-Szmercsa´nyi, I.; Maros, L. K.; Zahran, A. A. Investigations of the Kinetics of Maleate-Fumarate Isomerization during the Polyesterification of Maleic Anhydride with Different Glycols. J. Appl. Polym. Sci. 1966, 10, 513-522.
Received for review December 6, 1995 Revised manuscript received June 17, 1996 Accepted July 17, 1996X IE950736I X Abstract published in Advance ACS Abstracts, October 1, 1996.
