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Modelling of solid state polycondensation of poly(ethylene terephthalate)
Pergamon
Chemical Engineering Science, Vol. 52, No. 3, pp. 371 376, 1997
PII : S0009-2509(96)00414-9
Copyright ~c 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved
ooo9 2509/97$17.oo+ 0.OO
Modelling of solid state polycondensation of poly(ethylene terephthalate) Qiu Gao, Huang Nan-Xun] Tang Zhi-Lian *'t and L. Gerking* Chemical Engineering Department, China Textile University, 1882 West Yan-An Road. Shanghai, 200051, P.R. China; * Karl Fischer Industrieanlagen GmbH, Holzhauser Str.157, D-13509 Berlin, Germany (Received 1 June 1994) Abstract--A semi-analytical method has been proposed to analyse the mechanism of the solid state polycondensation (SSP) process. Computation results show that for the industrial SSP process for poly(ethylene terephthalate) (PET), the overall reaction rate for a single pellet can be appropriately simulated by a model controlled jointly by diffusion and reaction rates. From the core to the surface of a PET pellet, the SSP rate increases monotonically due to a gradual reduction of the concentration of such by-products as ethylene glycol and water. However, the SSP rate at any location within the pellet is observed between a maximum reaction rate when the concentration of by-products hypothetically approaches zero and a minimum reaction rate when the diffusion rate is assumed to be zero. Copyright © 1996 Elsevier Science Ltd Keywords:
Solid state polycondensation; poly(ethylene terephthalate).
reaction considerably:
INTRODUCTION
For preparing poly(ethylene terephthalate) (PET) with molecular weight greater than 20,000 and suitable for industrial textile applications, solid state polycondensation (SSP) is generally preferred (Chang et al., 1983; Chang, 1970; Chen et al., 1969; Schaaf et al., 1981). The SSP is carried out by heating the low molecular weight polymer pellets at a temperature below its melting temperature but above its glass transition temperature. The polymer is thereafter condensed at a reaction temperature between 200 and 240"(2, which is much lower than that for the melt process, thereby significantly suppressing thermal degradation, and its molecular weight increases while the by-products are removed by applying vacuum or flushing with an inert gas. Extensive work has been reported on experimental studies on the factors affecting the SSP of PET, and various authors investigated the mechanism of the process by developing diverse models, such as the diffusion model (Chang, 1970) and the purely kinetic models (Schaaf et al., 1981; Chen et al., 1969). Thereafter, Ravindranath and Mashelkar (1990) developed a comprehensive model assuming that the main polycondensation reaction is an equilibrium reaction, and that the removal of by-products, predominantly ethylene glycol, will favour the forward
* Corresponding author.
2 ~-COOCH2CH2OH
~
~e~-COOCH2CH2OOC I
+ HOCH2CH2OH I "
(1)
The diffusion- and reaction-controlling model they proposed describes a one-dimensional unsteady-state diffusion process of ethylene glycol (EG), coupled with a time rate change of the polymer end group concentration for any single particle. If the diffusion of EG is so rapid that the concentration of EG could be assumed to be nearly zero throughout the particle, the reaction is purely chemical reaction rate controlled. Incorporating the experimental data of Chen et al. (1969), Ravindranath and Mashelkar (1990) evaluated k at 160°C to be 0.1172(mol/Q)-th -~, which is surprisingly close to the value of 0.1170(mol/Q)-lh 1 derived from the data on melt polycondensation with an activation energy equal to 18.5kcalmo1-1 as reported by Yokoyama et al. (1978). When the rate of polycondensation reaclion is much faster than the diffusion of EG, the SSP could then be assumed to be governed by a diffusion controlled process. Ravindranath and Mashelkar (1990) analysed the experimental data of Chang (1970) and evaluated D, the diffusivity of EG, at various temperatures and its activation energy equal to 31.26kcalmol-L which is much higher than the 371
Qiu Gao et al.
372
activation energy of 18.5 kcalmol-1 for the forward chemical reaction, indicating that the decrease of diffusion rate of EG due to decrease of temperature would be faster than the decrease of reaction rate. It seems therefore that the lower the reaction temperature, the less credible it would be to employ the reaction rate controlling model to explain the experimental results. However, they assumed that the SSP at a temperature as low as 160°C was a reaction rate controlled process and treated the data at higher temperatures as a diffusion controlled process. This apparent discrepancy may entail some sort of suspicion upon their simplified model for description of the SSP process. The objective of the present work is to provide a set of solutions that describe the SSP process inside a particle and thereby generate suitable criteria for data analysis, which can be utilized in designing SSP reactors as well as in elucidating a number of lacune which have appeared in hitherto existing conceptions concerning the controlling mechanism in SSP. FUNDAMENTALS Chain extension (polycondensation) of PET is conventionally considered to be accomplished by two reversible reactions in SSP. Besides the ester interchange reaction already shown in eq. (1), the following esterification reaction occurs simultaneously: ~e,~-COOH
resistance in the gaseous phase is negligible. Furthermore, since polymer end groups are confined in solid state and recognized as not subject to Fickean diffusion, it is not necessary to use Ravindranath and Mashelkar's boundary conditions (1990):
Oe/Ox = O, ~z/~x = O, t > O, X = Xo and x = 0 to keep both z and e from diffusing across the boundary. It appears that an initial condition would be sufficient for a unique solution, i.e.
g=go,
e=eo,
g = gs,
t~g/~3t = D(632 g/c3x 2) - - ½(~e/t~t)
(4)
Oe/& = - 2k(e 2 - 4gz/K)
(5)
z = Zo + (Co -- e)/2 = 1 -- e/2.
(6)
where
Generally, SSP is carried out under high vacuum or high inert flow gas rate, so that the mass transfer
(9)
q=
(10) x
(11)
eqs (4)-(9) are converted to the following:
t)g/t3p = 02g/Oq 2 -- ½ (c3e/Op) ~;-
2(e 2 e=eo,
ag/Oq=O,
4g(1Ke/2! ) O<~q<~qo
p=0,
q=qo
p>0, p>0,
q=0
(12) (13, (14) (15) (16)
where q0 =
Xo.
(17)
At the very early stage of SSP, the concentration of EG in the particle is so small that the backward reaction may be ignored. Therefore, eq. (13)
Z + diester
Assuming that the diffusion process of EG is of Fickean type, isothermal, and without volume change of the particle, the one-dimensional unsteady-state diffusion process coupled with the concentration change of the polymer end groups can be described as follows (Ravindranath and Mashelkar, 1990):
x=0.
t>0,
g=gs,
~
(8)
p = kt
+ H20 ~. (2)
2E polymer end groups (carboxyl and hydroxyl)
X = Xo
By transforming the following variables:
g=go,
In as much as the order of magnitude of the esterification reaction rate constant is comparable to that of the ester interchange reaction, and the diffusivity of ethylene glycol is also of the same order of magnitude as that of water (Chen et al., 1987; Schaaf et al., 1981), it is suggested that for engineering purposes an overall equilibrium reaction in SSP could be described simply as follows:
(7)
t>0,
Og/~x=O,
+ HOCH2CH2OOC-~ '*~
,~-COOCH2CH2OOC-~,*
t = 0 , O<~x<~xo
EG T. by-products (ethylene glycol and water)
(3)
degenerates to
?e/@ = - 2e 2.
(18)
Solving this equation with the initial condition (14) and substitution into eq. (12) gives
c3g/c3p = 6~2g/~3q2 + (eo/(1 + 2e0p)) 2.
(19)
Applying Laplace transform and its inverse transform together with the relevant initial and boundary conditions, one obtains the final solution: 1 g(p, q) = g~ + 4(go -- g~) n ,=o l + 2 n
x sin
(1 + 2n)n(qo -- q) 2qo
373
Modelling of solid state polycondensation (1 + 2n)2zc2p7 4 ~ .j +
x exp
1 (1 + 2n)n(qo -- q) x y, - sin .=o 1 + 2n 2qo ×
ex
_e°
---
4q~
~2 d~.
(20)
x 1 + 2eo~ f
Being cautious that such an analytical solution would be invalid only for a short time period after the SSP process starts, we continue the simulation procedure for eqs (12) (17) by resorting to numerical calculation, which consists of a combination of the R u n g e - K u t t a method for time interval integration, and finite difference method for handling the internal divisions of the particle. We do not, however, have to make any arbitrary compromise between go and g.~ at t = t o and x = x 0, when the analytical solution is used for a short time after the commencement of SSP.
embedded in a jacket with molten salt circulating through it. The jacket is equipped with a sophisticated heating system which consists of three electrical heating units distributed at the top, middle and bottom parts along the column, together with three groups of monitoring systems made of thermocouple detectors laid separately on the core and the wall of the tube to measure both temperatures and feedback signals to the controller, so as to homogenize the temperature distribution in the fixed bed reactor. After each reaction had been completed, nitrogen purge was maintained to cool the fixed bed reactor to room temperature. Samples were collected by mixing the pellets taken from the upper, middle and lower sections of the reactor. In determining the viscosity of the polymer at the surface and the centre of a pellet, materials scraped from the skin and left in the core were used. Intrinsic viscosities were measured with the Ubblohde viscometer using a mixture of 1,1,2,2,-tetrachloroethane and phenol (1:1 by volume) as the solvent at 25 _+ 0.02°C. Thus, the average molecular weight ~ r can be calculated from the intrinsic viscosity by (Conix, 1958) [t/] = 2.1 × 10-4/~ TM.
EXPERIMENTAL
Commercial P E T precursors were supplied by a local polyester manufacturer, prepared by a melt process. The P E T pellets are of size 4 x 4 × 1.5 mm. Its initial intrinsic viscosity is 0 . 7 2 d l g -1. Before SSP, samples were vacuum dried for 12 h at 140°C to reduce the moisture content to less than 30 ppm in order to avert hydrolytic degradation during SSP. The SSP was conducted in a fixed bed reactor with a flowing stream of dry heated nitrogen gas. The tube reactor is 2.5 cm in diameter and 300 cm long,
(21)
RESULTS AND DISCUSSION
Experimental data are tabulated in Tables 1 and 2.
Influence of flow rate of nitrogen Figure 1 shows the influence of N z flow rate on the SSP reaction. When all other operating conditions are the same, average molecular weight M, of P E T pellets after SSP for 1800 min at 225°C increases with increasing nitrogen flow rate up to plateau. Similar results were reported by other investigators (Schaaf
Table 1. Influence of NE flow rate on SSP reaction Reaction temperature (°C) Reaction time (min) Flow rate (1 min- 1kg- 1 PET) Intrinsic viscosity (dl g- l) Av. molecular weight A4,
0.5 0.953 28810
1 0.999 30510
225 1800 2.0 1.057 32690
1.5 1.044 32210
3.0 1.071 33220
Table 2. Intrinsic viscosity as a function of time at various Flow rate (lmin lkg-1 PET)
Time (min)
N2
4.0 1.069 33150
5.0 1.070 33180
flOWrates Molecular weight
Intrinsic viscosity (dl g- 1) Average
Surface
Core
Average
Surface
Core
0.887 1.034 1.141
0.837 0.935 1.007
25,390 29,180 32,690
26,400 32,010 35,900
24,600 28, t 50 30,800
2.0
600 1200 1800
0.859 0.963 1.057
0.5
600 1200 1800
0.837 0.893 0.953
24,600 26,620 28,810
374
Qiu Gao et al.
........................
32500
32000~_
==
~
=
38000 !
i.....
.
~
~
i ,
.......
,~
r........
................ ~....................
'...........:...........I ..... - - , ....... ~........
iiiii iiiiiii +!i(
i
.
.5
;
1
1.5
2
2.5
3
3.5
4
4.5
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5
Na flow rate ( I/min.kg PET )
Fig. 1. Influence of nitrogen gas flow rate on SSP (eo=0.0187mol~) -1, Xo-0.75mm, reaction temp. T = 225°C, reaction time t = 1800 rain).
~ ¢D
22000
'~
20000;d , : . . i,.. i ,..i .... ~. . . . . . . . . . . 0
20 40 60
80 100 120 140 160 180
T i m e t X l O "1 ( rain. ) et al., 1981). There appears to be a critical flow rate
associated with a given value of 94 at which .~t stops increasing with increasing flow rate. Diffusion and reaction rate controllin9 model
Experimental data are processed by using the diffusion and reaction rate model, with K = 1 (Challa, 1960), to evaluate parameters D, k and Os. Referring to the data of the melt polycondensation rate constant and the diffusivity of water, a set of initial values of D, K and 9s were estimated to invoke the Simplex Program for the optimum evaluation of D, k and 9, through the best fit of experimental data with the predicted results in average M,. In Fig. 2, each curve demonstrates the increase of average molecular weight .~t vs time at various 9, for D = 7.21× 1 0 - 6 c m Z s -1 and k = 0 . 9 1 0 ( m o l / ~ ) ) - l h -1. Higher flow rate will create smaller 9s and therefore faster reaction rate. Curves 3 and 4 correspond, respectively, to a 9s at and below the critical flow rate. Curves 1 and 2 corresponding to 9s = 0 and 1 x 10-6 (mol/~)) are calculated to predict the theoretical upper limit that might be approached for SSP under very high vacuum. Figure 3 shows the pellet size and the diffusion direction of E G from the core to the surface. Figures 4 and 5 delineate, respectively, the spatial representations of E G concentration and local molecular weight at various localities in the pellet (x = O, ~ ~ Xo, ~3 4x0 and x0) from the core to the surface. It is observed that the outer part of the pellet enjoys a faster rate of increase of M , than the inner part, as a result of the decrease of concentration of E G from inner to outer part. Experimental data (Table 2) are shown in Fig. 5 as small circles. Figure 6 shows that the increase of local molecular weight vs time at various localities in a thicker pellet (Xo = 4 mm) is much slower than that in Fig. 5 (Xo = 0.75 mm), because of the greater difficulty of the diffusion of by-product E G from the inside to the surface.
Fig. 2. Average molecular weight vs time at various flow rates. [D = 7.21 x 1 0 - 6 cm2 s -1, k = 0.910 (mol~- 1)- a h -1, K = 1, 9o = 1 x 10- s mol ~ - l, Xo = 0.75 mm, eo = 0.0187 mol ~ - 1, experimental data: (O) Nz flow rate = 21 min- t kg t PET; (I--1)N2 flow rate = 0.5 1min- 1 kg- 1 PET].
X
.5
'I
anlai0n
Fig. 3. Pellet dimension (mm) and the diffusion direction of EG.
Reaction rate controllin 9 model
When the concentration of E G approaches zero, the SSP reaches maximum reaction rate. F r o m eq. (5) we obtain P , = P,o + 4kt
(22)
where P , = 2/e, and P.o stands for the initial degree of polymerization (DP). Equation (22) is plotted in Figs 5-7 as a broken line for the hypothetical maximum rate. Even the SSP occurring at the surface of a pellet [curve 6 in Fig. 5 and curve 5 in Fig. 6], where the diffusion takes place most rapidly, does not approach this m a x i m u m rate because of the insufficiently large value of D, and 9, ~ 0.
Modelling of solid state polycondensation
375
4E-005
/ ...........
O -~. 3.5E-005
_
......
...................... / :/ ~ ...... ...... : ~ . . . . . . . .
#,
3E-O05 E
iii '~ c
2.5E-005
•~
2E-005
~
1.5E4~t5
v~ 32000
1E-005
0 0
20
40
60
80
100 120 140 160 180
241100
~
Time t x 104 (min.) 22000
Fig. 4. The concentration of EG vs time at various localities in a pellet during SSP at 225°C. I-D = 7.21 × 10-6cm2s 1, k = 0 . 9 1 0 ( m o l ~ ) - l ) - ~ h -1, K = 1, 9o = l x l 0 - S m o l ~ -x, .q~ = 1.2 x 10 5 m o l ~ 1, eo = 0.0187 m o l ~ 1, Xo = 0.75 mm).
2OOOO 0
20
40
60
80 100 120 140 160 180
Time t x 10 "1 ( m i n . )
Fig. 6. Local molecular weight vs time during SSP at 225°C (Xo=4mm, 9 o = 4 × 1 0 - 5 m o 1 ~ 1 , 9 s = 3 × 1 0 5 m o l ~ ) 1, parameters D, k, K,eo same as those in Fig. 4).
38Oo0 i / / i ......................................................... ; / i i6
........... i .........
380O0 4
.......... i .............................
i-/-i
, /
:!
',
...............t .............is ~
34000
O
/
i
~
il
eTY//'
/e/ .
E ~ : 20(x)0
/
_
.
36OOO
32000 c
.
.
.
300O0 E. ._m 1: X =i0
//////, 24ooo
:
3"i2~/s
"5 260OO
........i.
"6 =E 24000 22000
20000
-'i,, ,I,., i .... i,,,, .... , .... , .... i .... 20
40
60
80 100 120 140 160 180
Time t x 101 (min.)
2O0OO 0
20
40
60
80
100 120 140 160 180
Time t x 10"1 (rain.)
Fig. 5. Molecular weight vs time at various localities in a pellet during SSP at 225°C (parameters D, k, K, go, g,, Co, xo same as those in Fig. 4).
Fig. 7. local molecular weight vs time in diffusion controlled process during SSP at 225°C (parameters D, K, go, gs, Co, Xo same as those in Fig. 4).
Non-diflusional model In the case where the diffusion term is d r o p p e d from eq. (4), c o m b i n i n g eqs (5) a n d (6) gives
e ~ leo
8go+4eo A
e x p ( - - A k t ) + - - 890 + 4Co A
(23)
where A = 4 + 4g o + 2eo.
(24)
This would be the m i n i m u m reaction rate process where diffusion is not involved. In fact, the reaction
proceeds in a closed system as if the particle were completely w r a p p e d up, a n d thereby the b y - p r o d u c t s do not diffuse o u t of the reaction system, m a k i n g the reaction reach its equilibrium in a very short time after the SSP starts [see Figs 5,6-1. W h e n plotted in Fig. 6 as a nearly h o r i z o n t a l b r o k e n line at the bottom, eq. (23) appears very close to curve 1 which represents the reaction rate of SSP at the core in a thick pellet u n d e r higher 9~. In the core region, diffusion is recognized as being slowest because of the longest distance to reach the surface. The SSP rate at
376
Qiu Gao et al.
any location within the pellet is observed between the maximum and minimum reaction rates as shown in Figs 5 and 6. Diffusion controlling model Ravindranath and Mashelkar (1990) assumed that the reaction was at equilibrium at all points when the rate of polycondensation reaction was much faster than the diffusion of EG. Thus the following relation (25) holds:
(25)
e 2 = 4gz/K.
Substituting its derivative into eq. (4) gives 1+
2
?7
=D
~-~-~
(26)
They analysed the experimental data of Chang (1970) at high reaction temperatures using their diffusion controlling model under this assumption, and evaluated the activation energy of the diffusivity of EG equal to 31.26 kcal m o l - ~ which seems to be associated with a discrepancy that we pointed out earlier. A unique diffusion controlling model would be an expression without the appearance of rate constant k in it [see eqs (25) and (26)-I. But in this case k must be extremely large, so large that it guarantees the reaction to reach a new equilibrium instantly at the instantaneous removal of certain amounts of EG. We solved eq. (26) for g, which was then substituted into eq. (25) to find e, and converting e to Mn, we finally obtained the reaction rate at any location inside the pellet expressed by a series of curves in Fig. 7. Curves 4 - 6 above the broken line should mean that the extraordinarily high rate at the outer part of the pellet is unattainable. It is not a surprise to witness such an occurrence because of the presumption of a drastic forward reaction rate constant. Usually, for polycondensation reactions, k cannot be expected to be so large as to materialize the unique diffusion controlled process in SSP. CONCLUSION A semi-analytical method to seek a solution for the SSP process has been proposed to analyse the mechanism of SSP of PET. The parameters, D, k and gs are evaluated from experimental data obtained in a fixed bed reactor. Computed results show that for industrial SSP of PET, the overall reaction rate in a single pellet is appropriately simulated by a model jointly controlled by diffusion and reaction rates. From the core to the surface of a PET pellet, SSP rate increases monotonically due to gradual reduction of the concentration of by-products. However, the SSP rate at any location within the pellet is observed between maximum and minimum reaction rates. The unique diffusion controlling model is validated only in the case where the forward reaction rate constant is extremely large, but such a requirement can hardly be fulfilled for practical polycondensation reactions, including PET.
Acknowledgement
The authors are grateful to the National United Chemical Engineering Laboratory, East China University of Science and Technology and Zhejiang University, for its financial support. NOTATION by-products overall diffusivity, cm2s-1 end group concentration, mol Q - 1 initial value of e, mol ~ concentration of by-products, mol O initial value of g, mol ~ - 1 value of g at the surface of the pellet, mol O overall polycondensation rate constant, ( m o l O - 1)-1 s-1 K overall equilibrium constant M~ molecular weight ( = 192 Pn), g m o l - ~ -M'n average molecular weight, g m o l p =kt Pn degree of polymerization ( = 2/e) q = x/(k/D) x qo = x//(k/D) Xo t reaction time, s x distance in the direction of diffusion, m x0 distance in the direction of diffusion from core to surface of pellet, m z concentration of diester groups, mol Q - 1 Zo initial concentration of diester groups, mol ~ - 1 D e eo g go gs k
REFERENCES
Challa, G. (1960) The formation of polyethylene terephthalate by ester interchange I. the polycondensation equilibrium. Makromol. Chem. 38, 105-122. Chang, T. M. (1970) Kinetics of thermally induced solid state polycondensation of poly(ethylene terephthalate). Polym. Engng Sci. 10, 364-368. Chang, S., Sheu, M. and Chen, S. (1983) Solid-state polymerization of poly(ethylene terephthalate). J. Appl. Polym. Sci. 28, 3289-3300. Chen, F. C., Griskey, R. G. and Beyer, G. H. (1969) Thermally induced solid state polycondensation of nylon 66, nylon 6-10 and polyethylene terephthalate. A.I.Ch.E.J. 15, 680-685. Conix, A. (1958) Molecular-weight determination of polyethylene terephthalate. Makromol. Chem. 26, 226-235. Ravindranath, K. and Mashelkar, R. A. (1990) Modeling of poly(ethylene terephthalate) reactor IX. Solid state polycondensation. J. Appl. Polym. Sci. 39, 1325-1345. Schaaf, E., Zimmermann, H., Dietzel, W. and Lohmann, P. (1981) Nachpolykondensation von polyethylenterephthalat in fester phase. Acta Polym. 32, 250-256. Secor, R. M. (1969) The kinetics of condensation polymerization. A.I.Ch.E.J. 15, 861-865. Yokoyama, H., Sano, T., Chijiiwa, T. and Kajiya, R. (1978) A simulation method for ethylene glycol terephthalate polycondensation process. J. Jap. Petrol. Inst. 21(4), 271-276.
Chemical Engineering Science, Vol. 52, No. 3, pp. 371 376, 1997
PII : S0009-2509(96)00414-9
Copyright ~c 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved
ooo9 2509/97$17.oo+ 0.OO
Modelling of solid state polycondensation of poly(ethylene terephthalate) Qiu Gao, Huang Nan-Xun] Tang Zhi-Lian *'t and L. Gerking* Chemical Engineering Department, China Textile University, 1882 West Yan-An Road. Shanghai, 200051, P.R. China; * Karl Fischer Industrieanlagen GmbH, Holzhauser Str.157, D-13509 Berlin, Germany (Received 1 June 1994) Abstract--A semi-analytical method has been proposed to analyse the mechanism of the solid state polycondensation (SSP) process. Computation results show that for the industrial SSP process for poly(ethylene terephthalate) (PET), the overall reaction rate for a single pellet can be appropriately simulated by a model controlled jointly by diffusion and reaction rates. From the core to the surface of a PET pellet, the SSP rate increases monotonically due to a gradual reduction of the concentration of such by-products as ethylene glycol and water. However, the SSP rate at any location within the pellet is observed between a maximum reaction rate when the concentration of by-products hypothetically approaches zero and a minimum reaction rate when the diffusion rate is assumed to be zero. Copyright © 1996 Elsevier Science Ltd Keywords:
Solid state polycondensation; poly(ethylene terephthalate).
reaction considerably:
INTRODUCTION
For preparing poly(ethylene terephthalate) (PET) with molecular weight greater than 20,000 and suitable for industrial textile applications, solid state polycondensation (SSP) is generally preferred (Chang et al., 1983; Chang, 1970; Chen et al., 1969; Schaaf et al., 1981). The SSP is carried out by heating the low molecular weight polymer pellets at a temperature below its melting temperature but above its glass transition temperature. The polymer is thereafter condensed at a reaction temperature between 200 and 240"(2, which is much lower than that for the melt process, thereby significantly suppressing thermal degradation, and its molecular weight increases while the by-products are removed by applying vacuum or flushing with an inert gas. Extensive work has been reported on experimental studies on the factors affecting the SSP of PET, and various authors investigated the mechanism of the process by developing diverse models, such as the diffusion model (Chang, 1970) and the purely kinetic models (Schaaf et al., 1981; Chen et al., 1969). Thereafter, Ravindranath and Mashelkar (1990) developed a comprehensive model assuming that the main polycondensation reaction is an equilibrium reaction, and that the removal of by-products, predominantly ethylene glycol, will favour the forward
* Corresponding author.
2 ~-COOCH2CH2OH
~
~e~-COOCH2CH2OOC I
+ HOCH2CH2OH I "
(1)
The diffusion- and reaction-controlling model they proposed describes a one-dimensional unsteady-state diffusion process of ethylene glycol (EG), coupled with a time rate change of the polymer end group concentration for any single particle. If the diffusion of EG is so rapid that the concentration of EG could be assumed to be nearly zero throughout the particle, the reaction is purely chemical reaction rate controlled. Incorporating the experimental data of Chen et al. (1969), Ravindranath and Mashelkar (1990) evaluated k at 160°C to be 0.1172(mol/Q)-th -~, which is surprisingly close to the value of 0.1170(mol/Q)-lh 1 derived from the data on melt polycondensation with an activation energy equal to 18.5kcalmo1-1 as reported by Yokoyama et al. (1978). When the rate of polycondensation reaclion is much faster than the diffusion of EG, the SSP could then be assumed to be governed by a diffusion controlled process. Ravindranath and Mashelkar (1990) analysed the experimental data of Chang (1970) and evaluated D, the diffusivity of EG, at various temperatures and its activation energy equal to 31.26kcalmol-L which is much higher than the 371
Qiu Gao et al.
372
activation energy of 18.5 kcalmol-1 for the forward chemical reaction, indicating that the decrease of diffusion rate of EG due to decrease of temperature would be faster than the decrease of reaction rate. It seems therefore that the lower the reaction temperature, the less credible it would be to employ the reaction rate controlling model to explain the experimental results. However, they assumed that the SSP at a temperature as low as 160°C was a reaction rate controlled process and treated the data at higher temperatures as a diffusion controlled process. This apparent discrepancy may entail some sort of suspicion upon their simplified model for description of the SSP process. The objective of the present work is to provide a set of solutions that describe the SSP process inside a particle and thereby generate suitable criteria for data analysis, which can be utilized in designing SSP reactors as well as in elucidating a number of lacune which have appeared in hitherto existing conceptions concerning the controlling mechanism in SSP. FUNDAMENTALS Chain extension (polycondensation) of PET is conventionally considered to be accomplished by two reversible reactions in SSP. Besides the ester interchange reaction already shown in eq. (1), the following esterification reaction occurs simultaneously: ~e,~-COOH
resistance in the gaseous phase is negligible. Furthermore, since polymer end groups are confined in solid state and recognized as not subject to Fickean diffusion, it is not necessary to use Ravindranath and Mashelkar's boundary conditions (1990):
Oe/Ox = O, ~z/~x = O, t > O, X = Xo and x = 0 to keep both z and e from diffusing across the boundary. It appears that an initial condition would be sufficient for a unique solution, i.e.
g=go,
e=eo,
g = gs,
t~g/~3t = D(632 g/c3x 2) - - ½(~e/t~t)
(4)
Oe/& = - 2k(e 2 - 4gz/K)
(5)
z = Zo + (Co -- e)/2 = 1 -- e/2.
(6)
where
Generally, SSP is carried out under high vacuum or high inert flow gas rate, so that the mass transfer
(9)
q=
(10) x
(11)
eqs (4)-(9) are converted to the following:
t)g/t3p = 02g/Oq 2 -- ½ (c3e/Op) ~;-
2(e 2 e=eo,
ag/Oq=O,
4g(1Ke/2! ) O<~q<~qo
p=0,
q=qo
p>0, p>0,
q=0
(12) (13, (14) (15) (16)
where q0 =
Xo.
(17)
At the very early stage of SSP, the concentration of EG in the particle is so small that the backward reaction may be ignored. Therefore, eq. (13)
Z + diester
Assuming that the diffusion process of EG is of Fickean type, isothermal, and without volume change of the particle, the one-dimensional unsteady-state diffusion process coupled with the concentration change of the polymer end groups can be described as follows (Ravindranath and Mashelkar, 1990):
x=0.
t>0,
g=gs,
~
(8)
p = kt
+ H20 ~. (2)
2E polymer end groups (carboxyl and hydroxyl)
X = Xo
By transforming the following variables:
g=go,
In as much as the order of magnitude of the esterification reaction rate constant is comparable to that of the ester interchange reaction, and the diffusivity of ethylene glycol is also of the same order of magnitude as that of water (Chen et al., 1987; Schaaf et al., 1981), it is suggested that for engineering purposes an overall equilibrium reaction in SSP could be described simply as follows:
(7)
t>0,
Og/~x=O,
+ HOCH2CH2OOC-~ '*~
,~-COOCH2CH2OOC-~,*
t = 0 , O<~x<~xo
EG T. by-products (ethylene glycol and water)
(3)
degenerates to
?e/@ = - 2e 2.
(18)
Solving this equation with the initial condition (14) and substitution into eq. (12) gives
c3g/c3p = 6~2g/~3q2 + (eo/(1 + 2e0p)) 2.
(19)
Applying Laplace transform and its inverse transform together with the relevant initial and boundary conditions, one obtains the final solution: 1 g(p, q) = g~ + 4(go -- g~) n ,=o l + 2 n
x sin
(1 + 2n)n(qo -- q) 2qo
373
Modelling of solid state polycondensation (1 + 2n)2zc2p7 4 ~ .j +
x exp
1 (1 + 2n)n(qo -- q) x y, - sin .=o 1 + 2n 2qo ×
ex
_e°
---
4q~
~2 d~.
(20)
x 1 + 2eo~ f
Being cautious that such an analytical solution would be invalid only for a short time period after the SSP process starts, we continue the simulation procedure for eqs (12) (17) by resorting to numerical calculation, which consists of a combination of the R u n g e - K u t t a method for time interval integration, and finite difference method for handling the internal divisions of the particle. We do not, however, have to make any arbitrary compromise between go and g.~ at t = t o and x = x 0, when the analytical solution is used for a short time after the commencement of SSP.
embedded in a jacket with molten salt circulating through it. The jacket is equipped with a sophisticated heating system which consists of three electrical heating units distributed at the top, middle and bottom parts along the column, together with three groups of monitoring systems made of thermocouple detectors laid separately on the core and the wall of the tube to measure both temperatures and feedback signals to the controller, so as to homogenize the temperature distribution in the fixed bed reactor. After each reaction had been completed, nitrogen purge was maintained to cool the fixed bed reactor to room temperature. Samples were collected by mixing the pellets taken from the upper, middle and lower sections of the reactor. In determining the viscosity of the polymer at the surface and the centre of a pellet, materials scraped from the skin and left in the core were used. Intrinsic viscosities were measured with the Ubblohde viscometer using a mixture of 1,1,2,2,-tetrachloroethane and phenol (1:1 by volume) as the solvent at 25 _+ 0.02°C. Thus, the average molecular weight ~ r can be calculated from the intrinsic viscosity by (Conix, 1958) [t/] = 2.1 × 10-4/~ TM.
EXPERIMENTAL
Commercial P E T precursors were supplied by a local polyester manufacturer, prepared by a melt process. The P E T pellets are of size 4 x 4 × 1.5 mm. Its initial intrinsic viscosity is 0 . 7 2 d l g -1. Before SSP, samples were vacuum dried for 12 h at 140°C to reduce the moisture content to less than 30 ppm in order to avert hydrolytic degradation during SSP. The SSP was conducted in a fixed bed reactor with a flowing stream of dry heated nitrogen gas. The tube reactor is 2.5 cm in diameter and 300 cm long,
(21)
RESULTS AND DISCUSSION
Experimental data are tabulated in Tables 1 and 2.
Influence of flow rate of nitrogen Figure 1 shows the influence of N z flow rate on the SSP reaction. When all other operating conditions are the same, average molecular weight M, of P E T pellets after SSP for 1800 min at 225°C increases with increasing nitrogen flow rate up to plateau. Similar results were reported by other investigators (Schaaf
Table 1. Influence of NE flow rate on SSP reaction Reaction temperature (°C) Reaction time (min) Flow rate (1 min- 1kg- 1 PET) Intrinsic viscosity (dl g- l) Av. molecular weight A4,
0.5 0.953 28810
1 0.999 30510
225 1800 2.0 1.057 32690
1.5 1.044 32210
3.0 1.071 33220
Table 2. Intrinsic viscosity as a function of time at various Flow rate (lmin lkg-1 PET)
Time (min)
N2
4.0 1.069 33150
5.0 1.070 33180
flOWrates Molecular weight
Intrinsic viscosity (dl g- 1) Average
Surface
Core
Average
Surface
Core
0.887 1.034 1.141
0.837 0.935 1.007
25,390 29,180 32,690
26,400 32,010 35,900
24,600 28, t 50 30,800
2.0
600 1200 1800
0.859 0.963 1.057
0.5
600 1200 1800
0.837 0.893 0.953
24,600 26,620 28,810
374
Qiu Gao et al.
........................
32500
32000~_
==
~
=
38000 !
i.....
.
~
~
i ,
.......
,~
r........
................ ~....................
'...........:...........I ..... - - , ....... ~........
iiiii iiiiiii +!i(
i
.
.5
;
1
1.5
2
2.5
3
3.5
4
4.5
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5
Na flow rate ( I/min.kg PET )
Fig. 1. Influence of nitrogen gas flow rate on SSP (eo=0.0187mol~) -1, Xo-0.75mm, reaction temp. T = 225°C, reaction time t = 1800 rain).
~ ¢D
22000
'~
20000;d , : . . i,.. i ,..i .... ~. . . . . . . . . . . 0
20 40 60
80 100 120 140 160 180
T i m e t X l O "1 ( rain. ) et al., 1981). There appears to be a critical flow rate
associated with a given value of 94 at which .~t stops increasing with increasing flow rate. Diffusion and reaction rate controllin9 model
Experimental data are processed by using the diffusion and reaction rate model, with K = 1 (Challa, 1960), to evaluate parameters D, k and Os. Referring to the data of the melt polycondensation rate constant and the diffusivity of water, a set of initial values of D, K and 9s were estimated to invoke the Simplex Program for the optimum evaluation of D, k and 9, through the best fit of experimental data with the predicted results in average M,. In Fig. 2, each curve demonstrates the increase of average molecular weight .~t vs time at various 9, for D = 7.21× 1 0 - 6 c m Z s -1 and k = 0 . 9 1 0 ( m o l / ~ ) ) - l h -1. Higher flow rate will create smaller 9s and therefore faster reaction rate. Curves 3 and 4 correspond, respectively, to a 9s at and below the critical flow rate. Curves 1 and 2 corresponding to 9s = 0 and 1 x 10-6 (mol/~)) are calculated to predict the theoretical upper limit that might be approached for SSP under very high vacuum. Figure 3 shows the pellet size and the diffusion direction of E G from the core to the surface. Figures 4 and 5 delineate, respectively, the spatial representations of E G concentration and local molecular weight at various localities in the pellet (x = O, ~ ~ Xo, ~3 4x0 and x0) from the core to the surface. It is observed that the outer part of the pellet enjoys a faster rate of increase of M , than the inner part, as a result of the decrease of concentration of E G from inner to outer part. Experimental data (Table 2) are shown in Fig. 5 as small circles. Figure 6 shows that the increase of local molecular weight vs time at various localities in a thicker pellet (Xo = 4 mm) is much slower than that in Fig. 5 (Xo = 0.75 mm), because of the greater difficulty of the diffusion of by-product E G from the inside to the surface.
Fig. 2. Average molecular weight vs time at various flow rates. [D = 7.21 x 1 0 - 6 cm2 s -1, k = 0.910 (mol~- 1)- a h -1, K = 1, 9o = 1 x 10- s mol ~ - l, Xo = 0.75 mm, eo = 0.0187 mol ~ - 1, experimental data: (O) Nz flow rate = 21 min- t kg t PET; (I--1)N2 flow rate = 0.5 1min- 1 kg- 1 PET].
X
.5
'I
anlai0n
Fig. 3. Pellet dimension (mm) and the diffusion direction of EG.
Reaction rate controllin 9 model
When the concentration of E G approaches zero, the SSP reaches maximum reaction rate. F r o m eq. (5) we obtain P , = P,o + 4kt
(22)
where P , = 2/e, and P.o stands for the initial degree of polymerization (DP). Equation (22) is plotted in Figs 5-7 as a broken line for the hypothetical maximum rate. Even the SSP occurring at the surface of a pellet [curve 6 in Fig. 5 and curve 5 in Fig. 6], where the diffusion takes place most rapidly, does not approach this m a x i m u m rate because of the insufficiently large value of D, and 9, ~ 0.
Modelling of solid state polycondensation
375
4E-005
/ ...........
O -~. 3.5E-005
_
......
...................... / :/ ~ ...... ...... : ~ . . . . . . . .
#,
3E-O05 E
iii '~ c
2.5E-005
•~
2E-005
~
1.5E4~t5
v~ 32000
1E-005
0 0
20
40
60
80
100 120 140 160 180
241100
~
Time t x 104 (min.) 22000
Fig. 4. The concentration of EG vs time at various localities in a pellet during SSP at 225°C. I-D = 7.21 × 10-6cm2s 1, k = 0 . 9 1 0 ( m o l ~ ) - l ) - ~ h -1, K = 1, 9o = l x l 0 - S m o l ~ -x, .q~ = 1.2 x 10 5 m o l ~ 1, eo = 0.0187 m o l ~ 1, Xo = 0.75 mm).
2OOOO 0
20
40
60
80 100 120 140 160 180
Time t x 10 "1 ( m i n . )
Fig. 6. Local molecular weight vs time during SSP at 225°C (Xo=4mm, 9 o = 4 × 1 0 - 5 m o 1 ~ 1 , 9 s = 3 × 1 0 5 m o l ~ ) 1, parameters D, k, K,eo same as those in Fig. 4).
38Oo0 i / / i ......................................................... ; / i i6
........... i .........
380O0 4
.......... i .............................
i-/-i
, /
:!
',
...............t .............is ~
34000
O
/
i
~
il
eTY//'
/e/ .
E ~ : 20(x)0
/
_
.
36OOO
32000 c
.
.
.
300O0 E. ._m 1: X =i0
//////, 24ooo
:
3"i2~/s
"5 260OO
........i.
"6 =E 24000 22000
20000
-'i,, ,I,., i .... i,,,, .... , .... , .... i .... 20
40
60
80 100 120 140 160 180
Time t x 101 (min.)
2O0OO 0
20
40
60
80
100 120 140 160 180
Time t x 10"1 (rain.)
Fig. 5. Molecular weight vs time at various localities in a pellet during SSP at 225°C (parameters D, k, K, go, g,, Co, xo same as those in Fig. 4).
Fig. 7. local molecular weight vs time in diffusion controlled process during SSP at 225°C (parameters D, K, go, gs, Co, Xo same as those in Fig. 4).
Non-diflusional model In the case where the diffusion term is d r o p p e d from eq. (4), c o m b i n i n g eqs (5) a n d (6) gives
e ~ leo
8go+4eo A
e x p ( - - A k t ) + - - 890 + 4Co A
(23)
where A = 4 + 4g o + 2eo.
(24)
This would be the m i n i m u m reaction rate process where diffusion is not involved. In fact, the reaction
proceeds in a closed system as if the particle were completely w r a p p e d up, a n d thereby the b y - p r o d u c t s do not diffuse o u t of the reaction system, m a k i n g the reaction reach its equilibrium in a very short time after the SSP starts [see Figs 5,6-1. W h e n plotted in Fig. 6 as a nearly h o r i z o n t a l b r o k e n line at the bottom, eq. (23) appears very close to curve 1 which represents the reaction rate of SSP at the core in a thick pellet u n d e r higher 9~. In the core region, diffusion is recognized as being slowest because of the longest distance to reach the surface. The SSP rate at
376
Qiu Gao et al.
any location within the pellet is observed between the maximum and minimum reaction rates as shown in Figs 5 and 6. Diffusion controlling model Ravindranath and Mashelkar (1990) assumed that the reaction was at equilibrium at all points when the rate of polycondensation reaction was much faster than the diffusion of EG. Thus the following relation (25) holds:
(25)
e 2 = 4gz/K.
Substituting its derivative into eq. (4) gives 1+
2
?7
=D
~-~-~
(26)
They analysed the experimental data of Chang (1970) at high reaction temperatures using their diffusion controlling model under this assumption, and evaluated the activation energy of the diffusivity of EG equal to 31.26 kcal m o l - ~ which seems to be associated with a discrepancy that we pointed out earlier. A unique diffusion controlling model would be an expression without the appearance of rate constant k in it [see eqs (25) and (26)-I. But in this case k must be extremely large, so large that it guarantees the reaction to reach a new equilibrium instantly at the instantaneous removal of certain amounts of EG. We solved eq. (26) for g, which was then substituted into eq. (25) to find e, and converting e to Mn, we finally obtained the reaction rate at any location inside the pellet expressed by a series of curves in Fig. 7. Curves 4 - 6 above the broken line should mean that the extraordinarily high rate at the outer part of the pellet is unattainable. It is not a surprise to witness such an occurrence because of the presumption of a drastic forward reaction rate constant. Usually, for polycondensation reactions, k cannot be expected to be so large as to materialize the unique diffusion controlled process in SSP. CONCLUSION A semi-analytical method to seek a solution for the SSP process has been proposed to analyse the mechanism of SSP of PET. The parameters, D, k and gs are evaluated from experimental data obtained in a fixed bed reactor. Computed results show that for industrial SSP of PET, the overall reaction rate in a single pellet is appropriately simulated by a model jointly controlled by diffusion and reaction rates. From the core to the surface of a PET pellet, SSP rate increases monotonically due to gradual reduction of the concentration of by-products. However, the SSP rate at any location within the pellet is observed between maximum and minimum reaction rates. The unique diffusion controlling model is validated only in the case where the forward reaction rate constant is extremely large, but such a requirement can hardly be fulfilled for practical polycondensation reactions, including PET.
Acknowledgement
The authors are grateful to the National United Chemical Engineering Laboratory, East China University of Science and Technology and Zhejiang University, for its financial support. NOTATION by-products overall diffusivity, cm2s-1 end group concentration, mol Q - 1 initial value of e, mol ~ concentration of by-products, mol O initial value of g, mol ~ - 1 value of g at the surface of the pellet, mol O overall polycondensation rate constant, ( m o l O - 1)-1 s-1 K overall equilibrium constant M~ molecular weight ( = 192 Pn), g m o l - ~ -M'n average molecular weight, g m o l p =kt Pn degree of polymerization ( = 2/e) q = x/(k/D) x qo = x//(k/D) Xo t reaction time, s x distance in the direction of diffusion, m x0 distance in the direction of diffusion from core to surface of pellet, m z concentration of diester groups, mol Q - 1 Zo initial concentration of diester groups, mol ~ - 1 D e eo g go gs k
REFERENCES
Challa, G. (1960) The formation of polyethylene terephthalate by ester interchange I. the polycondensation equilibrium. Makromol. Chem. 38, 105-122. Chang, T. M. (1970) Kinetics of thermally induced solid state polycondensation of poly(ethylene terephthalate). Polym. Engng Sci. 10, 364-368. Chang, S., Sheu, M. and Chen, S. (1983) Solid-state polymerization of poly(ethylene terephthalate). J. Appl. Polym. Sci. 28, 3289-3300. Chen, F. C., Griskey, R. G. and Beyer, G. H. (1969) Thermally induced solid state polycondensation of nylon 66, nylon 6-10 and polyethylene terephthalate. A.I.Ch.E.J. 15, 680-685. Conix, A. (1958) Molecular-weight determination of polyethylene terephthalate. Makromol. Chem. 26, 226-235. Ravindranath, K. and Mashelkar, R. A. (1990) Modeling of poly(ethylene terephthalate) reactor IX. Solid state polycondensation. J. Appl. Polym. Sci. 39, 1325-1345. Schaaf, E., Zimmermann, H., Dietzel, W. and Lohmann, P. (1981) Nachpolykondensation von polyethylenterephthalat in fester phase. Acta Polym. 32, 250-256. Secor, R. M. (1969) The kinetics of condensation polymerization. A.I.Ch.E.J. 15, 861-865. Yokoyama, H., Sano, T., Chijiiwa, T. and Kajiya, R. (1978) A simulation method for ethylene glycol terephthalate polycondensation process. J. Jap. Petrol. Inst. 21(4), 271-276.