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The dynamic behavior of continuous solution polymerization reactors—IX. Effects of inhibition
Pergamon
Chemical Engineerin 0 Science, Vol. 51, No. 1, pp. 63-79, 1996 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0009-2509/96 $9.50 + 0.00
0009-2509(95)00209-X
THE DYNAMIC BEHAVIOR OF CONTINUOUS SOLUTION POLYMERIZATION REACTORS--IX. EFFECTS OF INHIBITION J. C. P I N T O 1 and W. H. RAY* Department of Chemical Engineering, University of Wisconsin, 1415 Johnson Drive, Madison, WI 53706, U.S.A. (First received 5 January 1995; revised manuscript received and accepted 20 June 1995)
Abstract--In a previous paper Pinto and Ray (1995, Chem. Engng Sci. 50, 715-736) developed a mathematical model to describe the dynamic behavior of continuous free-radical solution copolymerization reactors and validated the model with experimental results obtained for the copolymerization of vinyl acetate (VA) and methyl methacrylate (MMA) in tert-butanol (TB). The model is now extended to allow the analysis of the dynamic effects induced by the presence ofinhibitors in the reaction environment. It is shown both theoretically and experimentally that the existence of small amounts of inhibitors in the feed stream may lead to the development of unstable operation; in particular, oscillatory behavior and multiple steady-state conditions.
l. I N T R O D U C T I O N
feedstocks, the only results published in the literature about the effects caused by inhibitors on the operation of continuous polymerization reactors are the ones written by Kirchner and coworkers [see Stolzenberg and Kirchner (1981), Rintelen et al. (1993) and Kirchner and Rintelen (1986)]. In these papers it is shown how monomer conversion and molecular weight distribution change when different inhibitors are added to an AIBN initiated styrene solution polymerization reactor. Particularly it is shown that monomer conversion and average molecular weight decrease continuously as the inhibitor feed concentration increases. Only steady-state results are analyzed. Very few results have been presented on the dynamic behavior of continuous inhibited polymerization reactors. A preliminary study was presented by Clinch (1983) using a simple mathematical model (similar to the one presented in the next section) to describe experimental results obtained for the VA homopolymerization in TB, initiated by AIBN. In this work, the observed effect of inhibitor on steady-state behavior was adequately predicted by the model. However, for dynamic oscillations, the mathematical model did not adequately match the experimental results in some cases. It seems that there are two important reasons which explain the relative absence of works in this field. The first one is the large use of batch reactors in industry, as inhibition effects are somewhat simple in these processes, leading most of the times to the appearance of an induction period of reaction only (Flory, 1953). The second one is the usual complexity of the inhibition kinetic mechanism, which generally involves a large number of steps for which kinetic parameters are almost never available, even when the kinetic mechanism is properly established. As it was
In previous work Pinto and Ray (1990, 1995) developed a mathematical model to describe the dynamic behavior of continuous free-radical solution copolymerization reactors. Using vinyl acetatemethyl methacrylate (VA-MMA) copolymerization in tert-butanol (TB) initiated by 2,2'-azobis-2-methylpropionitrile (AIBN) as an example, it was shown both theoretically and experimentally that these reactors may present oscillatory behavior at operation conditions where both homopolymerization reactors are stable. It was also shown that model predictions and experimental temperature and conversion dynamic profiles agreed reasonably well. It was found that an important reason that V A - M M A copolymerization reactors oscillate at small concentrations of M M A is that the M M A acts as an inhibitor in the polymerization. In addition, in some cases the model predictions and the experimental results obtained would agree only if additional assumptions were made regarding the presence of inhibitors in the reaction medium. As the development of unexpected oscillatory behavior and/or reaction extinction was observed in these particular experiments, it was concluded that inhibition reactions may lead to unstable operation of continuous polymerization reactors (Pinto and Ray, 1991). In spite of the widespread usage of inhibitors in the polymer industry for the stabilization of monomer
t Corresponding author. 1Present address: Programa de Engenharia Quimica/ COPPE, Universidade Federal do Rio de Janeiro, Cidade Universithria-CP: 68502, Rio de Janeiro, RJ 21945-970, Brazil. 63 CES 51-I-E
64
J. C. PINTO and W. H. RAY
written by Eastmond (1976), inhibition is the "least well-understood aspect of free-radical polymerization". For the reasons noted above, the large majority of the papers on inhibition reactions published in the literature attempt to establish a certain kinetic mechanism or evaluate certain kinetic parameters for a specific polymerization system. Some recent works are listed below. Bhanu and Kishore (1991) studied oxygen inhibition in various polymerization systems. Rieumont and Vega (1991) studied the inhibition of VA homopolymerization caused by furan and its derivatives. Levy (1992) studied the catalytic inhibition of acrylic acid polymerization by phenothiazine. Suddaby et al. (1992) studied the catalytic inhibition of MMA and methacrylamide polymerizations by a boron fluoride derivative of cobaloxime. Mohanty et al. (1994) studied the retarded polymerization of acrylonitrile by phenol. Except for the works presented by Chen and Tsai (1986) on the inhibition of MMA/styrene copolymerization by hydroquinone (HQ) and p-tert-butylpyrocatechol, and that of works of Tiid6s and coworkers [see Tfinczos et al. (1983a, b)] on the inhibition of MMA-styrene-acrylonitrilemethyl acrylate copolymerization by organic nitrocompounds, kinetic studies on the inhibition of copolymerization reactions are not available. It is important to say that in the three papers cited above, the authors observed that the rates of inhibition depended strongly on the composition of the reaction medium. Given the important industrial use of HQ as an inhibitor of free-radical polymerization reactions, a reasonably large collection of experimental data of inhibition caused by HQ and its derivatives in freeradical polymerization systems can be found in the literature (Kice, 1954; Bevington et al., 1995; Bagdasar'ian and Sinitsina 1961; Goldfinger et al., 1967; Brandrup and Immergut, 1975; Deb and Kapoor, 1980; Simfindi and Tfidos, 1985; Chen and Tsai, 1986; Bartofi and Jurani6ovfi, 1989). It is important to say that Breitenbach et al. (1938, 1941) showed in their classical papers that HQ is not an inhibitor in the absence of oxygen and that its inhibiting properties are due to the oxidation of HQ to benzoquinone. They also showed that HQ is completely converted to benzoquinone in the presence of air. Regarding the inhibition of VA free-radical polymerization, the few results available in the literature include some recent studies carried out by Rieumont et al. (1988) and Rieumont and Vega (1991) on the inhibition caused by furan and its derivatives, and those presented by Bartlett and Kwart (1950, 1952) and Bagdasar'ian and Sinitsina (1961) on the inhibition caused by organic nitro and sulfur compounds. No results about the inhibition caused by HQ in VA free-radical polymerization were found. As this paper was being prepared for publication, we became aware of work by Kim (1991). The main objective of this paper is to study the dynamic behavior of continuous inhibited solution
polymerization reactors. Given its industrial importance, VA-MMA solution copolymerization in TB, initiated by AIBN and inhibited by HQ, was chosen as an example. Due to the lack of data, this study was divided into two parts. In the first part, the mathematical model was developed and analyzed, based on previous results presented by Pinto and Ray (1995): In the second part, experiments were carried out and experimental results were compared against the theoretical results obtained with the model. It is shown that the presence of small amounts of inhibitors in the feed stream may lead to unstable operation--more specifically, to multiple steady states and to the development of oscillatory behavior. 2. MATHEMATICAL MODELING
As the mathematical model has been described before, we limit ourselves to presenting the equations and emphasizing the main points. The process is described by a set of five ordinary differential equations, as follows: Solvent mass balance: . . dv~
.
dr
. p,yv,:
p~
dTd(1/p,) v ~ q o q i + p , v ~ dz dT
(1)
Monomer 1 mass balance: dvt
Plf Vlf
dz
Pl
v I (qoqi + ® Rate0 + Pl vx
dT d(1/p,) d~ dT
(2) Monomer 2 mass balance: dv2 _
dr
dT d(1/V2) v2(qoqi + ORate2) + p2v2-dz dT
p2fv2f if2
(3) Initiator mass balance:
dci
-~z = ci: - ci (qoqi + kd O)
(4)
Energy balance: (1 + e)
dT
p : ( T : -- T)
dr
p
o
+--(AHtlVlPlkpllP pcp
+ AH12v2p2kpI2P + AH21Vlplkj,21Q + A n 2 2 v 2 P 2 k p 2 2 Q ) - ~ [UA(T - T~) pcp
+ 0tUA(T - Ta) + Qlat]
(5)
where it was assumed that the reactor volume is constant (experimental constraint), that volume additivity holds and that the kinetic mechanism is the
Dynamic behavior of continuous solution polymerization reactors--IX usual free-radical copolymerization reaction mechanism, described below:
faster than the other ones, the following can be written:
Initiation:
k~
I
65
~2R
Rate1 = kpl 1 P + kp21 Q -t- R~k,
(8)
Rate2 = kpl 2 P + kp22 Q + Rckr
(9)
where k,l = k,2 is assumed in eqs (8) and (9) and R + M1
krl
' PlO
R~k, = 2kdCi/(Ml/fl + M2/f2)
(10)
is the normalized rate of initiation. The denominator accounts for changes in the initiation efficiency due to changes in the comonomer composition. The concentration of species P and Q can be written as
R + M 2 kr2 ,Q01 Propagation: Pij + M1 k~ll Pi+I,j
Q= Qij + M~ kp21 Pi+l,r P~i + M2 kp12 ~ Qi,r+ 1
/R~k,(M1 + M2) -- [ H Q ] I fItQ/O . --~ ~/ ktll k~ + 2ktl2ks + kt22
(11)
P = ksQ
(12)
ks = (kp21 M1)/(kp12 Mz).
(13)
where
Qij + M2 kr'22 * Qi,j+ 1
The termination and propagation constants in eqs (1)-(13) include the gel effect and so have the general form ktij = kaj o gtij
ktl 1
o kpij = kpijgpl j
Termination: Pij + P.,~
'
mi+n.r+m
where k°r and k°ir are the kinetic constants at zero polymer concentration and #,i and gpo are the gel effect correlations that take into account the effects of increasing polymer concentration. The gel effect correlations are shown in Appendix A. In the energy balance, qoqi is a term that takes into account changes in the density of the reactive mixture and is given by
Pit + Q~m ktl2 ~.Ai+n,r+m
Qij + Qn~
kt22
, Ai+n,r+m
Inhibition: fnQPo + H Q fHQQij + H Q
kH kH
(14)
' Air qoqi = Psf v~f + P l f Vlf -I- P2f •2f Ps P1 P2
' Air
+ O [ R a t e 1 v 1(P 1/Pp -- 1 ) + Rate2 v2(p 2/Pp - 1)]
where the reactivity ratios are defined as rij = kpii/kplr
d T [(6)
and the cross termination constants are defined as IpO
=
k,o/~
d(1/ps)
Lp, v s - w -
+
d(1/pl) +
+
d(1/pp)l + p, V p ~ j .
d(1/p2) d---T (15)
(7)
It was assumed that the kinetic constants of inhibition reactions of different types of free-radicals were the same and that one molecule of H Q might terminate more than one free-radical species, due to its bifunctionality. It was also assumed that transfer reactions could be neglected because our main interest was the dynamic behavior of global variables, such as reactor temperature and conversion. It is known (Ray, 1972) that these additional reactions are either uncoupled or very weakly coupled with the dynamics of these global variables. Additionally, if one assumes that the quasi-steadystate assumption is valid for the free-radical species (R, P and Q), and that the inhibition step is much
Some global variables in the energy balance may be computed as p = Vsps + vlp 1 d'- v2p 2 "~ ~ppp
(16)
pep = vsp, cp~ + vlplcpl + Vzp2Cp2 + VpppCpp. (17) Regarding the term Q,,t in eq. (5), it was assumed that it could be conveniently written as Qlat = ( 0 ,
2(T - Tb) 2,
T < Tb
T >I Tb
(18)
where Tb is the boiling point of the reactive solution and 2 is a numerical parameter that controls the
66
J. C. PINTOand W. H. RAY
strength of the boiling constraint. In order to calculate the boiling temperature, the classical Flory-Huggins equation for solvent activity in polymer solutions was used as ln(l-I#H7 ~t) = In(v/) + vp + Zi v2
(19)
with the additional assumption that the mixture of solvent and monomers behaved ideally. Most of the computations presented in this paper were carried out with routines provided by AUTO (Doedel, 1986) for continuation of steady states and bifurcation points. All derivatives were calculated analytically. Dynamic simulations were carried out with routines provided by DDASSL (Petzold, 1982),
which uses BDF methods to solve systems of algebraic-differential equations. The model parameters are the same as used by Pinto and Ray 0995) which were validated by experiment. This reference should be consulted for additional details.
3. BIFURCATIONANALYSIS Before initiating the theoretical bifurcation analysis, it is important to show an example of how inclusion of inhibitors in the model allows good agreement with experiment. Consider the results presented in Fig. 1, where the jacket temperature was
80.0
~[~Simula[ed I" Reactor Temperature ] (..)
% 70.0
/ ;
<
aJ
I%.
60.0
[ Experimental Reactor Temperature I [ Air Coolant Temperature I
50.0
40.0 900
i000
ii00
1200
1300
Time (min) 80.0-
I
LJ
1[ Simulated Reactor Temperature I
~ 70.0-
60.0-
.
Experimental Reactor Temperature
%,h
50.0Air Coolant Temperature
40.0900
i000
Ii00
1200
1300
Time (min) Fig. 1. Experimental and theoretical temperature profiles (Vsf = 0.6000, Cif = 0.04 gmol/l, UA = 0.94(UA)o, O = 120 rain): (a) no inhibition assumed; (b) [HQ] s [nQ = 1.7 x 10-3 gmol/l.
Dynamic behavior of continuous solution polymerization reactors---IX changed at the steady state obtained with a residence time of 120 min. If no inhibitor is assumed to be present in the feed stream, the model is not able to reproduce the huge oscillation and extinction observed experimentally [Fig. l(a)]. However, by postulating a small amount of inhibitor, the model matches the experiment quite closely [Fig. l(b)]. Bifurcation analysis of the model yields a series of bifurcation diagrams which show how the system dynamics evolves as the inhibitor feed concentration increases. The most important points are: (1) The increase of inhibitor feed concentration in the VA solution homopolymerization may lead to the
67
development of additional steady states, to self-sustained oscillations and to total extinction operation, as the limit cycles may also lose stability at higher inhibitor feed concentrations (Fig. 2). (2) The appearance of an additional pair of steady states, leads to a maximum multiplicity of five different steady states (Fig. 3). This phenomenon had already been detected by Clinch (1983). (3) The presence of inhibitors in the feed stream may induce the development of multiple stable periodic solutions at certain ranges of residence time (Figs 4 and 5). Figure 6 shows dynamic simulation results, confirming the existence of multiple periodic solutions.
Different Inhibitor Concentratlons(gmol/I} 36O o
o
0t.0e-3
350-
r.5.-4
u~lE
,
""340-
~
o°
•
•
~
•
o
"
o
o
0000
oO
0
.2
~
o
0001.25e-3 _0
.
t.50e-o
ou
2o,-3 o
o
o
,,c, o
330-
320-
5.0e-4
l.Oe-4
0.000
310 0
7.5e-4
2.5e-4 i 5 000
1.0e-3
t.25e-3
I to 00o
4.50°-3
I t5 ooo
20 00o
Residence Time (s) Fig. 2. Bifurcation diagrams for inhibited VA polymerization for various inhibitor feed concentrations ([HQ]/ fno) (v,: = 0.6000, vly = 0.4000, v2y = 0.0000, c~.t = 0.04 gmol/l, Tc = 48°C) [-(--) stable steady states; (---) unstable steady states; (e) stable oscillations; (©) unstable oscillations].
Different Inhibitor Concentrations ( g m d / I )
3513-
-~~..-.~ ~.~__~ 1.o°-3 -...~ : ~ . ~ ~-~ i.z~°-3 . . ^ ~.~.~--..~-~/-~._~-=~ ~
_ ~
~.ue-o
340-
~. 3 3 0 -
\\
~5.0e-4
~-.
1.0e /
0.000
320J
3t0
0
I
500
I
t000
I
1 500
2000
Residence Time (s) Fig. 3. Bifurcation diagrams for inhibited VA polymerization showing five steady state multiplicity (vs: = 0.6000, vt: = 0.4000, v 2 / = 0.0000, c~y = 0.04 gmol/i, Tc = 48°C) (symbols described in Fig. 2).
68
J. C. PINTO a n d W. H. RAY
Different Inhibitor Concentratk)ns ( g r n d / I ) 3~0
. 000qD
. . . . . gO O0 OOOOOOOO00~
350-
000000000000oU II QOo. '
~340~
~OQOOoOq D q D ~ : 1.0e-3
• 30 . . . . . . ct.ee e ~ t e e ' 9 . . . . . . -: :Z~_ - - LT_--- _--,LL~_ o-o_oo_xrtzo-o--o-0_-- - # -
M'e" *~" ~" "J il;i'i'~?i~E--7
32o-
3t0 8000
i . . . . . 8 250
I
t 850O
...... 8750
9000
Residence Time (s}
Fig. 4. Bifurcation diagrams for inhibited VA polymerization showing multiplicity of periodic orbits at higher residence times (v,y = 0.6000, vl: = 0.4000, v2: = 0.0000, el: = 0.04 gmolA, Tc = 48° C) (symbols described in Fig. 2) Different
Inhibitor Concentrations ( g r a d / I )
36o
~ OOOOQO
•OO0••OOO0•O0
350-
j
.
.
.
.
.
.
.
.
.
.
.
•
330-
o
o
o
e
~
~
~
3203'10
45oo
1
4eoo
I 4~'00
'
'
'~
4eoo
'
1 4900
5ooo
Residence Time (s)
Fig. 5. Bifurcation diagrams for inhibited VA polymerization showing multiplicity of periodic orbits at lower residence times (v~,: = 0.6000, vl: = 0.4000, v2: = 0.0000, ct/-- 0.04 gmol/l, Tc = 48°C) (symbols described in Fig. 2)
(4) As already expected, the presence of small amounts of M M A in the feed stream may make the reaction system even more sensitive to the presence of inhibitors (Fig. 7), so that multiple steady states and periodic solutions occur at lower inhibitor concentrations. As in the case of copolymerization without inhibitors (Pinto and Ray, 1995), the system dynamics may be extremely sensitive to changes in the jacket temperature, heat transfer coefficient or initiator feed concentration, as it is shown in Figs 8-12. These bifurcation diagrams should be compared with Fig. 2, the nominal case. These results indicate that the ex-
perimental detection of stable limit cycles may be extremely difficult, if they occur at narrow ranges of experimental conditions, which are very sensitive to small changes in To, UA and ci:. An interesting result is shown in Fig. 12, where an ISOLA of periodic solutions is present. This preliminary analysis shows how complex the reactor dynamics may be in the presence of inhibitors and how important the dynamic effects may be in practical situations. The presence of inhibitors in an industrial environment is almost inevitable. (Inhibitors are generally added to monomers to allow safe stocking of raw materials, avoiding spontaneous thermal polymerization in the storage tanks. Besides,
Dynamic behavior of continuous solution polymerization reactors--IX
low inhibitor feed concentrations, below 100 ppm, the proper operation may be completely disturbed by the presence of inhibitors or be even impossible. Assuming that H Q is a strong inhibitor, the theoretical analysis suggests that oscillatory behavior and reaction extinction may be observed during real plant operation if the inhibitor feed concentration varies in the range of 0-100ppm. Observing these dynamic patterns experimentally is the main objective of the next sections.
monomers are generally used in this inhibited form during the polymerization, with no additional purification.) Because the appearance of unstable steady states and oscillatory behavior occurs at very
9O 0
69
1
£ 70
J
4. EXPERIMENTAL
5O 0 ....... ~W~ ...... ~A~ ...... 6~ Time (rain)
...... a~
.....
t.00
o,80-
8
0.60-
90 Temperoture (*C)
Fig. 6. Dynamic simulation showing multiple periodic solutions Iv,/= 0.6000, lYtf = 0 . 4 0 0 0 , ~12f = 0 . 0 0 0 0 , C i f : 0.04 gmol/l, T, = 48° C, [I-IQ]/ fnQ = 1.0 x 10- 3 gmol/l, (0) O = 8350 s; (l)O = 8000 s; (2) O = 8350 s].
SET-UP
The basic experimental set-up and process operation have been described by Pinto and Ray (1995) and we will avoid replicating the information. Nevertheless, we emphasize some important points and constraints related to the experimental procedure. Figure 13 shows a diagram of the copolymerization process and general equipment. It is important to notice that air was used as coolant medium, that the whole experimental set-up was kept at inert (nitrogen) atmosphere to avoid oxygen inhibition, that the feed conditions were properly controlled and that reactor temperature and conversion were measured in line. In the feed section, two different containers were used to store a mixture of solvent (TB) and initiator (2,2'-azobis-2-methyl-propionitrile--AIBN) and the comonomer mixture (VA/MMA and HQ). The initiator was dissolved in the solvent to avoid polymerization in the storage tanks. The monomers were premixed because of the low M M A feed concentrations used in the experiments, which made it difficult to control the feed composition otherwise. It is important to say that the fractional reactor volume occupied by vapor was negligible at all experiments, which means that residence time was measured with good precision at all experimental conditions.
Different Inhibitor Concentrotions (g mol/I ) 36O
1.0e-4
5e-4~..
~'
~'--~'~
500-4 •
r.oe-q - -
8
o
~,~.',,% ~ = ~_.:; ~ ~
350-
~b?'eeo
~ll~OOO ~ ~ o ,~. ~ O 0 •
,e
•
•
•
•
•
,~¢..
°~
oo
o,0._, oo oo ~ o,.25._3 ° o° o
• • •
~ o • -
:.~
o o
o
d ~ _
o
~
0 t50o-3
•
o
oo
o'
0.~ 320-
E.5e-4
5.00-4
7.5e-4
t.0o-3
t.25o-3
t.50e-3
t.Oe-4
310
0
i 5000
i 10OO0
t 15O00
2O000
Residence Time (s)
Fig. 7. Bifurcation diagrams for inhibited VA/MMA copolymerization for various inhibitor feed concentrations ([HQ] I fue) (v,s = 0.6000, v~i = 0.3985, v2/= 0.0015, c~y = 0.04 gmol/l, Tc = 48° C) (symbols described in Fig. 2).
70
J. C. PINTO and W. H. RAY
Different Inhibitor Concentrations (gmol/I) 5.0o-4 7.5o-4
360
~ - - - - , ~',al~, O o o • ~=" " • | ~ " S • ,
350-
o
o 0
o
%,
:
~340-
o° o0 o,.2~°-3 00
/
-
,
o
-.
-.
\\x,
9
""'~-~
0o o0
ooo _o°t,'5e-3
o°
o o~ o
- - - _ % ! -~"- " ~--_S
o0 o
z
oo°
"''"'='~--~-'~-'~i ',, "x ",.~.,~-''~e,. " ' ~ a
320-
oo
o
o°
.%~.<%.~_~ , ~
i330-
o
%.oo°-s
o
o
-'~-tcu:£
~'o4x~V4ooo
n -
I 15000
20000
~
-u--
t 75e-3
O0
75e 4
Pure VA 310
i 5000
0
i 10000
I
Residence Time (s) Fig. 8. Bifurcation diagrams for inhibited VA polymerization for various inhibitor feed concentrations ([HQ]I fnQ) [vsI = 0.6000, vii = 0.4000, v2i = 0.0000, cii = 0.04 gmol/l, Tc = 48°C, UA = 1.10(UA)o] (symbols described in Fig. 2).
Different Inhibitor Concentrations (gmol/I) 36O
~00 1.50e-5
350-
0 00 O00 _0002.00e_3.
t.75e-5
0O0 t25e-3
340a
~
00~
~,},<<.~.~.
, ~
,(~.~?~.<.\\
~ ~
#.
~o °
o°
,
oO o
~o
o
~o
,.o
y ~.-3
o o~
o o~
o
~
~~-fa.....---~ ,~.. £.<-.~~
330-
o°
o°
~oo
Oo7 o
320--
5.0e-4
7.5e-4
t.00e-3
1.25e-3
.50e-3
Pure VA 3t0 0
[ 5000
I 10000
I 15000
20000
Residence Time (s) Fig. 9. Bifurcation diagrams for inhibited VA polymerization for various inhibitor feed concentrations ([HQ]I fnQ) r,v,i = 0.6000, vii = 0.4000, v2i = 0.0000, q / = 0.04 gmol/I, Tc = 48°C, UA = 0.90(UA)0] (symbols described in Fig. 2).
In order to evaluate conversion, an on-line process refractometer was used (model SSR-72, from Electron Machine Corporation). It must be emphasized, however, that dynamic conversion profiles were never used for parameter evaluation and quantitative analysis because:
internal walls of the output line and measurement compartment. (3) The presence of nitrogen bubbles in the output stream, due to the mechanism used to control the reactor volume, caused a considerable amount of noise in the measured refractive index.
(1) Refractive indices are extremely sensitive to temperature changes and rigorous temperature control inside the measurement compartment was difficult. (2) The high viscosities of concentrated polymer solutions caused a damping effect on the conversion dynamics, due to the formation of a gel layer on the
As shown by Teymour (1989) other analytical techniques for on-line evaluation of conversion were not well suited (for similar and other reasons) for application in the system we have been working with. Attempts have been made to install an NIR spectrophotometer for on-line evaluation of the composition of the reaction medium, hut the equipment and
Dynamic behavior of continuous solution polymerization reactors--IX
71
Different InhibitorConcentrations(gmolll) 36o
5.00-4
,~_~x~o,oo,o |
~
~o- ~'B,
o
~***, ~• S~i° e
r~
~o
~ ~
•
~ ejE= ~ wj~O~
~\\[\<,~
" \ - ~ "-..
\,\ \
"-
I.ZSe-3
o
o
o o 0
•
o
ou 0
-o t -~.~oe- o
o-
o
o
_"~.%
o°
o
ooOo
-~--7--~- . . . .
-
~
2.50-4
oo
oOO°
~
e ° ,,=
! '~.2~'e."~ee-"-\~ /
o
•
• ~ • ° w
°o
°o °
°o •
"
O',"~'Zl~i%d:~.--~i.
~33o
t o0%3
7~.-4
Z.Se-4 e
0
5.00-4
1.00o-3
7.50-4
150e-3 1.25e-3
Pure VA 310
I
i 10o00
5OOO
0
= 15000
20000
Residence Time ( s )
Fig. 10. Bifurcation diagrams for inhibited VA polymerization for various inhibitor feed concentrations ([HQ]: fno) (vsy = 0.6000, h : = 0.4000, v2: = 0.0000, ely = 0.04 gmol/l, Tc = 46°C, UA = 1.00(UA)o) (symbols described in Fig. 2).
Different InhibitorConcentrations(gmolll) 36O
C~O 350 -
000 00~)
0
\
OOo
k
~
°o
oo
o
o
o
oo
340-
0
0
0
o°
t.5oe-3o o,~o3
oo
"
0
0
~6 ° z o o o - 3
/ ooooOO ^o
o~
.9.=
l
~
3~--
/ ~ ' , " . . ' - : . ' ~ . ' . - ' . _ ~ _ _ ~-O.o~Oo o 320--
~
-
4
7.50-4 t.00e-3 ].25e-3 t.50e-3
t.75e-3
Pure VA 310 0
I
i
5000
10000
i
15000
20000
Residence Time ( s )
Fig. 1l. Bifurcation diagrams for inhibited VA polymerization for various inhibitor feed concentrations ([HQ]/ fnQ) (vs: = 0.6000, vl: = 0.4000, v2: = 0.0000, el: = 0.04 gmol/l, Tc = 50°C, UA = 1.00(UA)0) (symbols described in Fig. 2). technique necessary were not available by the time this work was done. F o r all the reasons discussed above, dynamic conversion measurements were used for qualitative model analysis only. Finally, it must be pointed out that the experimental set up was designed for operation at atmospheric pressure so that the reactor temperature was not allowed to rise above the boiling temperature of the reaction mixture. This important constraint has already been included into the mathematical model. Regarding the sources of the chemicals used, VA was bought from Pfaltz and Bauer, Inc., M M A , TB (99.5% + pure) and H Q (99.0% + pure) from Eastman K o d a k Co. and nitrogen (99.996% + pure) from Liquid Carbonic. VA and M M A were distilled under
vacuum and inert atmosphere one day before the experiment was carried out and were stored in a refrigerator at 10-15°C. The other chemicals were used as received.
5. EXPERIMENTAL RESULTS 5.1. Effects cause by the presence of small amounts of water and HQ in the feed stream As observed experimentally through N I R spectrephotometry (Pinto, 1991), VA, M M A , TB, PVA and P M M A absorb water from the atmosphere extremely fast. So, it is quite possible that a certain amount of water be present in the reaction medium at both experimental conditions and industrial operation.
72
J.C. PINTOand W. H. RAY 36O
350340\\\\\
2
,.
~_~ 330-
320k 310/ 0
I
I
1
IO0OO
5000
20000
15000
ResidenceTime(s)
Fig. 12. Bifurcation diagram for inhibited VA polymerization showing an isola of periodic solutions fHQ) = 1.5 x 10 - 3 gmol/1 [•sf = 0.6000, Vlf = 0.4000, v2f = 0.0000, cif = 0.04055 gmol/l, Tc = 50°C, UA = 1.00(UA)o](symbols described in Fig. 2).
([HQ]f
.....
Nitrogen Feed To DA system
Solution Polymerization
I Nitrogen Monomerand solvent (+AIBN)containers ~" -
I
i !
Monomer
Rotameter
i i
Waste
line
o
Nitrogen
0
I Electrical resistance
Feed i
Products
'
I ~ ~
i r'-/~ '
Waste
PID controller and VARIAC
Burets
Volumetric PID controller VARIAC
and
Fig. 13. Experimental set-up.
Very few studies have been done on the effects caused by water in free-radical polymerization reactions. Among them we can refer to the papers written by De Schryver and coworkers [see De Schryver e t al. (1986) and Jacob e t al. (1972)-1, where the authors observed that the presence of small amounts of water in the reaction medium was beneficial to the polymerization of MMA, methacrylic acid and acrylamide, provoking an increase of the reaction rates.
As already said, monomers used for large-scale polymerization generally contain small amounts of inhibitors. In the particular case of MMA and VA polymerization, monomers generally contain 10-15 ppm of HQ. In order to analyze the importance of these small amounts of water and HQ during real operation, a series of experiments were carried out. Figure 14 shows the results obtained when the feed, containing
73
Dynamic behavior of continuous solution polymerization reactors--IX
ChangingFeed Composition Reslden(~Time= 90mln
800 f ~ eo.o-
~
_
.
~
Reactortemperature
I I
1
I
Nr Temperature 4QO
0
too
2o0
30o
I 100
I 200
I I 300 400 Tlme(mln)
I
6°.0- I
ao I o
I 500
I 600
700
Fig. 14. Experimental temperature and conversion profiles (vs: = 0.6000, vl: = 0.3990, v2: = 0.0010, c~: = 0.04 gmol/l, ® = 90 min. Arrow shows when the uninhibited and dry feed is changed for a water saturated, 15 ppm HQ inhibited feed stream.)
dry monomers and solvent, was changed for a new feed, containing water-saturated monomers and solvent and 6 ppm of HQ (corresponding to a 15 ppm HQ inhibited comonomer feed stream), after steady state had been reached. It is clear that, at the conditions analyzed, the operation is stable.
80.
5.2. Multiplicity and domains of attraction A series of experiments were carried out in order to show that inhibitors may change the number of steady states at certain operation conditions, also changing their domains of attraction. The reference case is shown in Fig. 15, where it may be seen that reaction ignition occurs spontaneously at residence times of 60 min, with no need to increase the jacket temperature in order to reach the upper steady state, when the reactor does not contain any initiator at the beginning of the run. This result agrees with the model predictions presented for VA homopolymerization in Section 3. Figure 16 shows, however, that reaction does not ignite spontaneously when the feed stream contains 40 ppm of HQ, even when the reactor contains 0.04 gmol/1 of AIBN at the beginning of the run and the residence time is as high as 180 min. According to the results shown in Section 3, this occurs because the presence of inhibitors in the feed stream extends the existence of the lower branch of steady states to higher residence times, reducing the domain of attraction of the upper steady-state solution. Figure 17 clearly shows the existence of the two stable steady states at residence times of 90 min. After igniting the reaction with uninhibited VA, the feed was changed to a 40 ppm HQ inhibited feed stream
4oi, o
i
~
'''500
Time(rain)
1oo
~w
C t~
o
100
2oo
300
400
5oo
Time(rain)
Fig. 15. Experimental temperature and conversion profiles (vs.r= 0.6000, v1: = 0.4000, v2: = 0.0000, c~f = 0.04 gmolfl,
@ = 60 min. Reaction ignites spontaneously, even when the initial initiator concentration is equal to zero.)
(the arrow shows the exact point where the feed composition was changed). It may be seen that the oscilllations introduced by changing the feed composition were negligible and that the new steady-state conditions was not significantly different from the previous one. After showing that steady-state conditions would not change, the jacket temperature was reduced to
J. C. PINTO and W. H. RAY
74
Changing Feed Composition
Changing Residence Time 8(1o
~
70.0
'='\1
~.0
i~ 50.0
..... [ 0
~00
~.......... I Air l'emperature ]
40O
600
800
t000
I
I 6o0 Time (rain)
I 8oo
lOOO
I t
1200
toQo| eo.o
1
40.0
20.0 0.0
I
0
200
400
I
12oo
Fig. 16. Experimental temperature and conversion profiles [v~I = 0.6000, vii = 0.4000, v21 = 0.0000, c~r = 0.04 gmoi/l, 40 ppm HQ, (0) ® = 60 rain; (1) ® = 90 min; (2) (9 = 120 min; (3) (9 = 150 min; (4) (9 = 180 min; (5) (9 = 90 min. Reaction does not ignite spontaneously.]
............
,~ I\
,---JJ-'i r / - ....... I
20
........ 40O
y,° ............. 8OO
I t~00
Time (rain)
Table 1. Dynamic experimental profiles obtained for HQ inhibited VA polymerization HQ feed concentration (ppm)
Experimental dynamic profile (Replicates)
40
Damped oscillations of small amplitude (2) Damped oscillations of small amplitude (2) Damped oscillation of small amplitude (2) Oscillation followed by extinction (2) Extinction (2) Possibly sustained oscillations of large amplitude (2) Possibly sustained oscillations of large amplitude (2) Extinction (3)
50 60
tO0
70
i
80
50
100
O'
400
8O0
1200
Tlmelmln) Fig. 17. Experimental temperature conversion profiles. Steady state multiplicity. (v,i = 0.6000, vii = 0.4000, v2I = 0.0000, ci¢ = 0.04 gmol/1, ® = 90 min. Arrow shows when uninhibited feed is changed for 40 ppm HQ inhibited feed stream.)
25°C, which was followed by reaction extinction. Increasing the jacket temperature to the original value, however, did not lead the system to the former steady state again. As it may be seen, operation was
stabilized at much lower temperature and conversion. The original steady-state conditions could only be attained after introducing a peak in the jacket temperature. The second arrow shows the exact point where reaction ignites again. 5.3. Oscillator behaviour and instabilities According to Fig. 2, an increase of the inhibitor feed concentration may cause destabilization of the reactor operation, leading to the development of self-sustained oscillations or reaction extinction. F o r this reason, many experiments were carried out in order to define the ranges of H Q feed concentrations that
75
Dynamic behavior of continuous solution polymerization reactors--IX might lead to unstable operation and oscillatory behavior. In all these experiments reaction was started with an inhibitor-free feed stream, which was changed for a new feed feed stream, with known inhibitor concentration, after approximate steady-state conditions had been attained. Experimental results are
summarized in Table 1. Figures 18-20 show some experimental profiles obtained. The results presented in Table 1 show that H Q feed concentrations below 60 ppm do not change the stability of the reactor operation significantly, at the experimental conditions analyzed. F o r H Q at
Changing Feed Cemposltlon Residence Time = 90 rain
8GO
t L"~
~ 5o.o40.0
Air Temperature I
200
0
I
i
4O0
600
I
800
I
tooo
t200
100.0 ,,...,.
ae 80.060.0-
"~> 40.0C_~ 20.0-0.0
I
2O0
0
I
I
400
6O0
I
800
I
t 000
t~00
Time(rain) Fig. 18. Experimental temperature and conversion profiles (vsl = 0.6000, v l f = 0.4000, v2I = 0.0000, c~I = 0.04 gmol/l, O = 90 min. Arrow shows when uninhibited feed is changed for 60 ppm HQ inhibited feed stream.)
Changing Feed Composition Residence Time = 90 mln 80.0
I f"~
7o.o-I~ I~
I'= 40,0 t ~ 0
Air Temperature I
I
200
400
I 2OO
I 40O
1
I
I
600
800
1000
t200
600
8O0
t000
1200
100.0
80.o60.0QrJ
~ 40.0¢~ 20.00.0 0
Time(rain) Fig. 19. Experimental temperature and conversion profiles (vsy = 0.6000, V l f = 0.4000, V2f = 0 . 0 0 0 0 , c~y = 0.04 gmol/l, O = 90 min. Arrow shows when uninhibited feed is changed for 80 ppm HQ inhibited feed stream.)
76
J.C. PINTOand W. H. RAY Changing Feed C~moosltlon Residence Time = 90 rain 80.0~
•--.
Ji'
~,f"N
• r°.°--ll
J/i
•~ 60.0
j ool/ '
"°.°-I'-
T Reactor Temperature
V
0
_ r
r
Air Temperature
.
r
T~T
\ r
t00
200
300
400
500
600
700
I t00
I 200
1 300
I 400
I 500
I rN;)O
700
too.o 8o.o-
60.0--, ~ 40.0-0.0 0
Time(rain) Fig. 20. Experimental temperature and conversion profiles (vs/= 0.6000, vii = 0.4000, v2f = 0 , 0 0 0 0 , cis = 0.04 gmol/l, ® = 90 min. Arrow shows when uninhibited feed is changed for 100 ppm HQ inhibited feed stream.)
70-80 ppm levels, oscillations are observed. However, when the HQ feed concentration is above 100 ppm, reaction extinction was detected. Because of the long period of the oscillations (more than 3 h), it was extremely difficult to keep conditions completely constant during this period. As shown in Section 3, the experimental observation of limit cycles in inhibited reactions may be difficult, due to the narrow ranges of operation conditions where they occur and to the high sensitivity of these regions with respect to small variations in the operation conditions. Thus we are not able to show unequivocally that the observed oscillations are sustained. However we believe this to be true. Comparing the results obtained with those presented in Fig. 2, we conclude that fnQ is approximately equal to 1.5, showing that HQ molecules may terminate more than one growing free radical molecule. This result agrees with stoichiometric results presented by Goldfinger et al. (1967) and with the bifunctional nature of HQ molecules.
that our kinetic representation of inhibition is oversimplified. In order to understand the difference between predicted and observed results, the experimental temperature profiles were used to estimate experimental inhibition rates, using a second-order filter (Derenzo, 1990). If it is assumed that the differences between model and experiment are due to errors in the kinetic mechanism of inhibition, then one can assume --Rinh = 2fl kdcl -- kttl i-p]2
This hypothesis is supported by previous results that show that model and experiments agree very well in the absence of HQ (Pinto and Ray, 1995). Figures 21 and 22 show experimental inhibition rates obtained in the same experiments presented by Figs 18 and 19. It is clear that the rates of inhibition are not constant throughout the experimental run. Actually, if Ri,h we assumed (as is usually done) Rinh = kn [P'I [HQ] kn --- exp( - A E u / R T
6. ADDITIONALQUANTITATIVEANALYSIS The experimental results obtained were used to test the model adequacy to describe the phenomena quantitatively. Only the experiments where oscillations of more than 3°C developed were used, because the small oscillation case may be easily fitted with small adjustments of the heat transfer coefficient (Pinto, 1991). For the larger oscillations the a priori model was not successful to reproduce either the period or the amplitude of oscillations. This may indicate
(20)
+ ASu)
(21) (22)
(Brandrup and Immergut, 1975), it is possible to fit all experimental data with a proper adjustment of AEn and ASh (Pinto, 1991). However, this is just an empirical fit. It is clear that more fundamental kinetic studies have yet to be carried out. 7. CONCLUSIONS The theoretical and experimental analysis carried out in this work show that the stability of continuous solution polymerization reactors may be extremely
Dynamic behavior of continuous solution polymerization reactors--IX
reactor qualitatively, but failed to reproduce the experimental results quantitatively. A more detailed analysis show that the inhibition rates may be temperature dependent and that the inclusion of the temperature dependence through an ordinary Arrhenius expression allows the reproduction of the experimental results. More fundamental kinetic studies have yet to be done, however, if a consistent inhibition kinetic mechanism is to be obtained.
1.00E-006-
Ex0orim~t 8.00E-007.
!!
77
,"
&OOE-007
4.00E-007 t-i
== --
2.00E-007
Simulation O00E+OOC
. . . . . . . . .
i
. . . . . . . . .
60O
i . . . . . . . . .
800
i , , ,
I000
. . . . .
t200
Tlmelmln)
Fig. 21. Experimental and theoretical inhibition rate profiles. [Experimental conditions described in Fig. 18. UA = 0.99 (UA)o, AEn/R = -6789.6 K, ASn/R = 20.7649 for simulation.]
2.00E-OO6Simolation i it)
o~ .,a
vv L
O.OOE+O00 . . . . . . . . . , . . . . . . . . ~, . . . . . . . . . , . . . . . . . . . , . . . . . . . . . 2O0 4OO 6O0 8OO I000 1200 T i m o ( rain )
Fig. 22. Experimental and theoretical inhibition rate profiles. [Experimental conditions described in Fig. 19. UA = 0.90 (UA)o, AEn/R = -11710.0 K, ASn/R = 35.8035 for simulations.]
sensitive to the presence of small amounts of inhibitors in the feed stream. Particularly, it was shown that small amounts of HQ, in the range 60-100 ppm, may introduce vigorous oscillatory behavior and lead to reaction extinction in the operation of VA homopolymerization reactors. It was also shown that reactor operation may become even more sensitive to the HQ feed concentration if small amounts of MMA are present in the feed stream. It was shown that the presence of inhibitors may change the number ol steady states of the system and modify the basins of attraction of a desired steady-state condition, which may eventually require changes of the reactor operation. A simple mathematical model was developed to describe the dynamic behavior of inhibited polymerization reactors. The model was reasonably successful to describe the dynamic behavior of an experimental
Acknowledoements--We thank James Schneider for his valuable help and support. We also thank Conselho National de Desenvolvimento Cientifico e Tecnol6gico (CNPq--Brazil) for supporting J.C. Pinto's staying in Madison, WI. We are grateful to the industrial sponsors of the University of Wisconsin Polymer Reaction Laboratory (UWPREL) and to the National Science Foundation for support of this research. NOTATION molar concentration of initiator heat capacity initiator efficiency for initiation of monomer i inhibitory functionality of HQ gel effect correlation for km gel effect correlation for k, u initiator kinetic constant for initiator decomposition kinetic constant for inhibition reaction kinetic constant for propagation of free radical i with monomer j k,~ kinetic constant for initiation of free radical i ks kinetic "equilibrium constant", as defined in eq. (13) ktu kinetic constant for termination of free radical i with free radical j Mi monomer i, molar concentration of monomer i P total molar concentration of free radicals Pu Pu molar concentration of free radical I (having monomer 1 at the active site), containing i units of monomer 1 and j units of monomer 2 qoqi volume shrinkage, as in eq. (15) Q total molar concentration of free radicals Q# Qu molar concentration of free radical 2 (having monomer 2 at the active site), containing i units of monomer 1 and j units of monomer 2 Qlat rate of heat removal due to condensation ro reactivity ratio, as defined in eq. (1) R initiator fragment Rate~ normalized rate of reaction of monomer i R,k, normalized rate of initiation T temperature To ambient temperature Tb boiling temperature T, coolant air temperature Tgi glass transition temperature of species i UA global heat transfer coefficient to coolant jacket v/ free volume vyF critical free volume for propagation v;t, critical free volume for termination V reactor volume
ci cp f/ fnQ gm g.j I kd kH km
J. C. PINTO and W. H. RAY
78
Greek letters •i expansion coefficient of species i, for free volume calculation • U A global heat transfer coefficient to the ambient AH o heat of reaction of propagation kp~j e external capacitance ® residence time 2 numerical constant in eq. (18) v~ volume fraction of species i I-ll partial pressure of species i H7at saturation pressure of species i p density dimensionless time (t/O) q~ conversion Xi interaction parameter for species i ~bo cross-termination constant, as defined in eq. (7)
Special subscripts 1 monomer 1 2 monomer 2 f feed p polymer s solvent
REFERENCES
Bagdasar'ian, K. H. S and Sinitsina, Z. A., 1961, J. Polym. Sci. 52, 31-38. Bartlett, P. D. and Kwart, H., 1950, J. Am. Chem. Soc. 72, 1051-1059. Bartlett, P. D. and Kwart, H., 1952, J. Am. Chem. Soc. 74, 3969-3973. Bartofi, J and Jurani~ov/t, V., 1989, Makromol. Chem. 190, 769-775. Bevington, J. C., Ghanem, N. A. and Melville, H. W., 1955, J. Chem. Soc. London 2822-2830. Bhanu, V. A. and Kishore, K., 1991, Chem. Rev. 91, 99-117. Brandrup, J. and Immergut, E. H., 1975, Polymer Handbook, 2nd Edition. Wiley New York. Breitenbach, J. W., Springer, A. and Horeischy, K., 1941, Ber. Dtsch. Chem. Ges. 71, 1438. Breitenbach, J. W., Springer, A. and Horeischy, K., 1941, Bet. Dtsch. Chem. Ges. 74, 1386. Chen, S.-A. and Tsai, L.-C., 1986, Makromol. Chem. 187, 653-666. Clinch, A. B., 1983, Phenomena of the nonisothermal solution homopolymerization of methyl methacrylate or vinyl acetate in a CSTR. M.Sc. thesis, University of Wisconsin, Madison. Deb, P. C. and Kapoor, S. K., 1980, Eur. Polym. J. 16, 763-767. Derenzo, S. E., 1990, INTERFACING: A Laboratory Approach Usino the Microcomputer for Instrumentation, Data Analysis, and Control. Prentice-Hall International, London. De Schryver, F. C., Smets, G. and Van Thielen, J., 1968, Polym. Lett. 6, 547-550. Doedel, E. J., 1986, AUTO: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations. California Institute of Technology, Pasadena. Eastmond, G. C., 1976, Comprehensive Chemical Kinetics, Vol. 14a: Free-Radical Polymerization. Elsevier, New York. Flory, P. J., 1953, Principles of Polymer Chemistry. Cornell University Press, Ithaca. Goldfinger, G., Yee, W. and Gilbert, R. D. 1967, in Encyclopeida of Polymer Science and Technology (Edited by H. F. Mark and N. G. Gaylord), Vol. 7, pp. 6444564. Wiley Interscience, New York.
Jacob, M., Smets, G. and De Schryver, F. C., 1972, Polym. Lett. 10, 669-673. Kice, J. L., 1954, J. Am. Chem. Soc. 76, 6274-6280. Kim, K. J., 1991, Modelling and control of continuous free radical polymerization reactors. Ph.D. thesis, University of Maryland. Kirchner, K. and Rintelen, T., 1986, Angew. Makromol. Chem. 141, 85-93 Levy, L. B., 1992, J. Polym. Sci. A: Polym. Chem. 30, 569-576. Mohanty, A. K., Misra, M. and Singh B. C., 1994, Polym. Plast. Technol. Eng. 33, 207-219. Petzold, L. R., 1982, A description of DDASSL: a differential algebraic system solver. Sandia National Laboratories, Report ~ SAND82-8637. Pinto, J. C., 1991, Anfilise de Din~rnica de Sistemas de Polimerizagfio Pela Teoria de Bifurca~6es. Ph.D. thesis, PEQ/COPPE/Universidade Federal do Rio de Janeiro, Rio de Janeiro (in Portuguese). Pinto, J. C. and Ray, W. H., 1990, presented at the 1990 Annual A.1.Ch.E. Meeting, Chicago. Pinto, J. C. and Ray, W. H., 1991, presented at the 1991 Annual A.I.Ch.E. Meeting, Seattle. Pinto, J. C. and Ray, W. H., 1995, Chem. Engng Sci. 50, 715 736. Ray, W. H., 1972, J, Macromol. Sci. Rev. Macromol. Chem. C8, 1 56. Reid, R. C., Prausnitz, J. M. and Polina, B. E., 1986, The Properties of Gases and Liquids, 4th Edition. McGrawHill, New York. Rieumont, J. and Vega, R., 1991, Makromol. Chem. 192, 1387 1397. Rieumont, J., Vega, R., Davidenko, N. and Paz, J. A., 1988, Eur. Polym. J. 24, 909-911. Rintelen, T., Riederle, K. and Kirchner, K., 1983, in Polymer Reaction Engineering (Edited by K.H. Reichert and W. Geiseler), pp. 269-286. Hanser, New York. Ross, R. T. and Laurence, R. L., 1976, A.I.Ch.E. Syrup. Ser. 72, 74. Simfindi, T. L. and Tiid6s, F., 1985, Eur. Polym. J. 21, 865-869. Stolzenberg, K. and Kirchner, K., 1981, Angew. Makromol. Chem. 95, 185-197. Suddaby K. G., O'Driscoll, K. F. and Rudin, A., 1992, J. Polym. Sci. A: Polym. Chem. 30, 643~a48. Tfinczos, I., F61des-Berezsnich, T. and Tiid6s, F., 1983a, Eur. Polym. J. 19, 153-157. Tfinczos, 1., F61des-Berezsnich, T. and Tiid6s, F., 1983b, Eur. Polym. J. 19, 593-595. Teymour, F., 1989, The dynamic behavior of free radical solution polymerization reactions in a continuous stirred tank reactor. Ph.D. thesis, University of Wisconsin, Madison. APPENDIX A
In order to account for the gel effect (the fact that the kinetic constants depend on composition when the polymer concentration is high due to diffusion limitations), the following equations were used: A.1. VA gel1 = 1 Yt~ = exp( -0.4407 ~b -6.7530~b 2 -0.3495~b 3) (A1) where ~b is the total conversion defined by q~ = pp Vp/(plvl + p2v2 + p~vs + ppVp).
(A2)
Equations (A 1) are empirical but were used successfully by Teymour (1989) and Pinto and Ray (1992a) to describe the VA homopolymerization in TB. A.2. MMA The Ross and Laurence equations, based on the freevolume theory (Ross and Laurence, 1976), were used to
Dynamic behavior of continuous solution polymerization reactors--IX describe the gel effect of MMA homopolymerization in TB, as presented by Pinto and Ray (1992a). The equations are: t 0.10575exp(17.15v~ - 0.01715(T - 273.15)), [JY > ~ftc gl22 =
2.3 x 10-%xp(75vl),
Vf~Vfp c
0.1856 - 2.965 × 10-+(T -273.15) vf~ = 0.05
CE$ 51-1-F
vf = v l t v t + vfi =
(A3)
vs2v2 + vy, v, + VypVp
(A7)
0.025 + ~ i ( T - Tgi)
(A8)
and ~ and T v are presented by Pinto and Ray (1995), for some chemical species. (A4)
where vftc =
and the total free volume vy is given by
where
vl ~< vft+
~1, vf > Vfpc gp22 = ~7.1× 10_Sexp(171.53vl) '
79
(A5) (A6)
A.3. V A / M M A
copolymerization
The gel effect correlation used for copolymerization reactions were g,12 = ~
+,22
(A9)
Chemical Engineerin 0 Science, Vol. 51, No. 1, pp. 63-79, 1996 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0009-2509/96 $9.50 + 0.00
0009-2509(95)00209-X
THE DYNAMIC BEHAVIOR OF CONTINUOUS SOLUTION POLYMERIZATION REACTORS--IX. EFFECTS OF INHIBITION J. C. P I N T O 1 and W. H. RAY* Department of Chemical Engineering, University of Wisconsin, 1415 Johnson Drive, Madison, WI 53706, U.S.A. (First received 5 January 1995; revised manuscript received and accepted 20 June 1995)
Abstract--In a previous paper Pinto and Ray (1995, Chem. Engng Sci. 50, 715-736) developed a mathematical model to describe the dynamic behavior of continuous free-radical solution copolymerization reactors and validated the model with experimental results obtained for the copolymerization of vinyl acetate (VA) and methyl methacrylate (MMA) in tert-butanol (TB). The model is now extended to allow the analysis of the dynamic effects induced by the presence ofinhibitors in the reaction environment. It is shown both theoretically and experimentally that the existence of small amounts of inhibitors in the feed stream may lead to the development of unstable operation; in particular, oscillatory behavior and multiple steady-state conditions.
l. I N T R O D U C T I O N
feedstocks, the only results published in the literature about the effects caused by inhibitors on the operation of continuous polymerization reactors are the ones written by Kirchner and coworkers [see Stolzenberg and Kirchner (1981), Rintelen et al. (1993) and Kirchner and Rintelen (1986)]. In these papers it is shown how monomer conversion and molecular weight distribution change when different inhibitors are added to an AIBN initiated styrene solution polymerization reactor. Particularly it is shown that monomer conversion and average molecular weight decrease continuously as the inhibitor feed concentration increases. Only steady-state results are analyzed. Very few results have been presented on the dynamic behavior of continuous inhibited polymerization reactors. A preliminary study was presented by Clinch (1983) using a simple mathematical model (similar to the one presented in the next section) to describe experimental results obtained for the VA homopolymerization in TB, initiated by AIBN. In this work, the observed effect of inhibitor on steady-state behavior was adequately predicted by the model. However, for dynamic oscillations, the mathematical model did not adequately match the experimental results in some cases. It seems that there are two important reasons which explain the relative absence of works in this field. The first one is the large use of batch reactors in industry, as inhibition effects are somewhat simple in these processes, leading most of the times to the appearance of an induction period of reaction only (Flory, 1953). The second one is the usual complexity of the inhibition kinetic mechanism, which generally involves a large number of steps for which kinetic parameters are almost never available, even when the kinetic mechanism is properly established. As it was
In previous work Pinto and Ray (1990, 1995) developed a mathematical model to describe the dynamic behavior of continuous free-radical solution copolymerization reactors. Using vinyl acetatemethyl methacrylate (VA-MMA) copolymerization in tert-butanol (TB) initiated by 2,2'-azobis-2-methylpropionitrile (AIBN) as an example, it was shown both theoretically and experimentally that these reactors may present oscillatory behavior at operation conditions where both homopolymerization reactors are stable. It was also shown that model predictions and experimental temperature and conversion dynamic profiles agreed reasonably well. It was found that an important reason that V A - M M A copolymerization reactors oscillate at small concentrations of M M A is that the M M A acts as an inhibitor in the polymerization. In addition, in some cases the model predictions and the experimental results obtained would agree only if additional assumptions were made regarding the presence of inhibitors in the reaction medium. As the development of unexpected oscillatory behavior and/or reaction extinction was observed in these particular experiments, it was concluded that inhibition reactions may lead to unstable operation of continuous polymerization reactors (Pinto and Ray, 1991). In spite of the widespread usage of inhibitors in the polymer industry for the stabilization of monomer
t Corresponding author. 1Present address: Programa de Engenharia Quimica/ COPPE, Universidade Federal do Rio de Janeiro, Cidade Universithria-CP: 68502, Rio de Janeiro, RJ 21945-970, Brazil. 63 CES 51-I-E
64
J. C. PINTO and W. H. RAY
written by Eastmond (1976), inhibition is the "least well-understood aspect of free-radical polymerization". For the reasons noted above, the large majority of the papers on inhibition reactions published in the literature attempt to establish a certain kinetic mechanism or evaluate certain kinetic parameters for a specific polymerization system. Some recent works are listed below. Bhanu and Kishore (1991) studied oxygen inhibition in various polymerization systems. Rieumont and Vega (1991) studied the inhibition of VA homopolymerization caused by furan and its derivatives. Levy (1992) studied the catalytic inhibition of acrylic acid polymerization by phenothiazine. Suddaby et al. (1992) studied the catalytic inhibition of MMA and methacrylamide polymerizations by a boron fluoride derivative of cobaloxime. Mohanty et al. (1994) studied the retarded polymerization of acrylonitrile by phenol. Except for the works presented by Chen and Tsai (1986) on the inhibition of MMA/styrene copolymerization by hydroquinone (HQ) and p-tert-butylpyrocatechol, and that of works of Tiid6s and coworkers [see Tfinczos et al. (1983a, b)] on the inhibition of MMA-styrene-acrylonitrilemethyl acrylate copolymerization by organic nitrocompounds, kinetic studies on the inhibition of copolymerization reactions are not available. It is important to say that in the three papers cited above, the authors observed that the rates of inhibition depended strongly on the composition of the reaction medium. Given the important industrial use of HQ as an inhibitor of free-radical polymerization reactions, a reasonably large collection of experimental data of inhibition caused by HQ and its derivatives in freeradical polymerization systems can be found in the literature (Kice, 1954; Bevington et al., 1995; Bagdasar'ian and Sinitsina 1961; Goldfinger et al., 1967; Brandrup and Immergut, 1975; Deb and Kapoor, 1980; Simfindi and Tfidos, 1985; Chen and Tsai, 1986; Bartofi and Jurani6ovfi, 1989). It is important to say that Breitenbach et al. (1938, 1941) showed in their classical papers that HQ is not an inhibitor in the absence of oxygen and that its inhibiting properties are due to the oxidation of HQ to benzoquinone. They also showed that HQ is completely converted to benzoquinone in the presence of air. Regarding the inhibition of VA free-radical polymerization, the few results available in the literature include some recent studies carried out by Rieumont et al. (1988) and Rieumont and Vega (1991) on the inhibition caused by furan and its derivatives, and those presented by Bartlett and Kwart (1950, 1952) and Bagdasar'ian and Sinitsina (1961) on the inhibition caused by organic nitro and sulfur compounds. No results about the inhibition caused by HQ in VA free-radical polymerization were found. As this paper was being prepared for publication, we became aware of work by Kim (1991). The main objective of this paper is to study the dynamic behavior of continuous inhibited solution
polymerization reactors. Given its industrial importance, VA-MMA solution copolymerization in TB, initiated by AIBN and inhibited by HQ, was chosen as an example. Due to the lack of data, this study was divided into two parts. In the first part, the mathematical model was developed and analyzed, based on previous results presented by Pinto and Ray (1995): In the second part, experiments were carried out and experimental results were compared against the theoretical results obtained with the model. It is shown that the presence of small amounts of inhibitors in the feed stream may lead to unstable operation--more specifically, to multiple steady states and to the development of oscillatory behavior. 2. MATHEMATICAL MODELING
As the mathematical model has been described before, we limit ourselves to presenting the equations and emphasizing the main points. The process is described by a set of five ordinary differential equations, as follows: Solvent mass balance: . . dv~
.
dr
. p,yv,:
p~
dTd(1/p,) v ~ q o q i + p , v ~ dz dT
(1)
Monomer 1 mass balance: dvt
Plf Vlf
dz
Pl
v I (qoqi + ® Rate0 + Pl vx
dT d(1/p,) d~ dT
(2) Monomer 2 mass balance: dv2 _
dr
dT d(1/V2) v2(qoqi + ORate2) + p2v2-dz dT
p2fv2f if2
(3) Initiator mass balance:
dci
-~z = ci: - ci (qoqi + kd O)
(4)
Energy balance: (1 + e)
dT
p : ( T : -- T)
dr
p
o
+--(AHtlVlPlkpllP pcp
+ AH12v2p2kpI2P + AH21Vlplkj,21Q + A n 2 2 v 2 P 2 k p 2 2 Q ) - ~ [UA(T - T~) pcp
+ 0tUA(T - Ta) + Qlat]
(5)
where it was assumed that the reactor volume is constant (experimental constraint), that volume additivity holds and that the kinetic mechanism is the
Dynamic behavior of continuous solution polymerization reactors--IX usual free-radical copolymerization reaction mechanism, described below:
faster than the other ones, the following can be written:
Initiation:
k~
I
65
~2R
Rate1 = kpl 1 P + kp21 Q -t- R~k,
(8)
Rate2 = kpl 2 P + kp22 Q + Rckr
(9)
where k,l = k,2 is assumed in eqs (8) and (9) and R + M1
krl
' PlO
R~k, = 2kdCi/(Ml/fl + M2/f2)
(10)
is the normalized rate of initiation. The denominator accounts for changes in the initiation efficiency due to changes in the comonomer composition. The concentration of species P and Q can be written as
R + M 2 kr2 ,Q01 Propagation: Pij + M1 k~ll Pi+I,j
Q= Qij + M~ kp21 Pi+l,r P~i + M2 kp12 ~ Qi,r+ 1
/R~k,(M1 + M2) -- [ H Q ] I fItQ/O . --~ ~/ ktll k~ + 2ktl2ks + kt22
(11)
P = ksQ
(12)
ks = (kp21 M1)/(kp12 Mz).
(13)
where
Qij + M2 kr'22 * Qi,j+ 1
The termination and propagation constants in eqs (1)-(13) include the gel effect and so have the general form ktij = kaj o gtij
ktl 1
o kpij = kpijgpl j
Termination: Pij + P.,~
'
mi+n.r+m
where k°r and k°ir are the kinetic constants at zero polymer concentration and #,i and gpo are the gel effect correlations that take into account the effects of increasing polymer concentration. The gel effect correlations are shown in Appendix A. In the energy balance, qoqi is a term that takes into account changes in the density of the reactive mixture and is given by
Pit + Q~m ktl2 ~.Ai+n,r+m
Qij + Qn~
kt22
, Ai+n,r+m
Inhibition: fnQPo + H Q fHQQij + H Q
kH kH
(14)
' Air qoqi = Psf v~f + P l f Vlf -I- P2f •2f Ps P1 P2
' Air
+ O [ R a t e 1 v 1(P 1/Pp -- 1 ) + Rate2 v2(p 2/Pp - 1)]
where the reactivity ratios are defined as rij = kpii/kplr
d T [(6)
and the cross termination constants are defined as IpO
=
k,o/~
d(1/ps)
Lp, v s - w -
+
d(1/pl) +
+
d(1/pp)l + p, V p ~ j .
d(1/p2) d---T (15)
(7)
It was assumed that the kinetic constants of inhibition reactions of different types of free-radicals were the same and that one molecule of H Q might terminate more than one free-radical species, due to its bifunctionality. It was also assumed that transfer reactions could be neglected because our main interest was the dynamic behavior of global variables, such as reactor temperature and conversion. It is known (Ray, 1972) that these additional reactions are either uncoupled or very weakly coupled with the dynamics of these global variables. Additionally, if one assumes that the quasi-steadystate assumption is valid for the free-radical species (R, P and Q), and that the inhibition step is much
Some global variables in the energy balance may be computed as p = Vsps + vlp 1 d'- v2p 2 "~ ~ppp
(16)
pep = vsp, cp~ + vlplcpl + Vzp2Cp2 + VpppCpp. (17) Regarding the term Q,,t in eq. (5), it was assumed that it could be conveniently written as Qlat = ( 0 ,
2(T - Tb) 2,
T < Tb
T >I Tb
(18)
where Tb is the boiling point of the reactive solution and 2 is a numerical parameter that controls the
66
J. C. PINTOand W. H. RAY
strength of the boiling constraint. In order to calculate the boiling temperature, the classical Flory-Huggins equation for solvent activity in polymer solutions was used as ln(l-I#H7 ~t) = In(v/) + vp + Zi v2
(19)
with the additional assumption that the mixture of solvent and monomers behaved ideally. Most of the computations presented in this paper were carried out with routines provided by AUTO (Doedel, 1986) for continuation of steady states and bifurcation points. All derivatives were calculated analytically. Dynamic simulations were carried out with routines provided by DDASSL (Petzold, 1982),
which uses BDF methods to solve systems of algebraic-differential equations. The model parameters are the same as used by Pinto and Ray 0995) which were validated by experiment. This reference should be consulted for additional details.
3. BIFURCATIONANALYSIS Before initiating the theoretical bifurcation analysis, it is important to show an example of how inclusion of inhibitors in the model allows good agreement with experiment. Consider the results presented in Fig. 1, where the jacket temperature was
80.0
~[~Simula[ed I" Reactor Temperature ] (..)
% 70.0
/ ;
<
aJ
I%.
60.0
[ Experimental Reactor Temperature I [ Air Coolant Temperature I
50.0
40.0 900
i000
ii00
1200
1300
Time (min) 80.0-
I
LJ
1[ Simulated Reactor Temperature I
~ 70.0-
60.0-
.
Experimental Reactor Temperature
%,h
50.0Air Coolant Temperature
40.0900
i000
Ii00
1200
1300
Time (min) Fig. 1. Experimental and theoretical temperature profiles (Vsf = 0.6000, Cif = 0.04 gmol/l, UA = 0.94(UA)o, O = 120 rain): (a) no inhibition assumed; (b) [HQ] s [nQ = 1.7 x 10-3 gmol/l.
Dynamic behavior of continuous solution polymerization reactors---IX changed at the steady state obtained with a residence time of 120 min. If no inhibitor is assumed to be present in the feed stream, the model is not able to reproduce the huge oscillation and extinction observed experimentally [Fig. l(a)]. However, by postulating a small amount of inhibitor, the model matches the experiment quite closely [Fig. l(b)]. Bifurcation analysis of the model yields a series of bifurcation diagrams which show how the system dynamics evolves as the inhibitor feed concentration increases. The most important points are: (1) The increase of inhibitor feed concentration in the VA solution homopolymerization may lead to the
67
development of additional steady states, to self-sustained oscillations and to total extinction operation, as the limit cycles may also lose stability at higher inhibitor feed concentrations (Fig. 2). (2) The appearance of an additional pair of steady states, leads to a maximum multiplicity of five different steady states (Fig. 3). This phenomenon had already been detected by Clinch (1983). (3) The presence of inhibitors in the feed stream may induce the development of multiple stable periodic solutions at certain ranges of residence time (Figs 4 and 5). Figure 6 shows dynamic simulation results, confirming the existence of multiple periodic solutions.
Different Inhibitor Concentratlons(gmol/I} 36O o
o
0t.0e-3
350-
r.5.-4
u~lE
,
""340-
~
o°
•
•
~
•
o
"
o
o
0000
oO
0
.2
~
o
0001.25e-3 _0
.
t.50e-o
ou
2o,-3 o
o
o
,,c, o
330-
320-
5.0e-4
l.Oe-4
0.000
310 0
7.5e-4
2.5e-4 i 5 000
1.0e-3
t.25e-3
I to 00o
4.50°-3
I t5 ooo
20 00o
Residence Time (s) Fig. 2. Bifurcation diagrams for inhibited VA polymerization for various inhibitor feed concentrations ([HQ]/ fno) (v,: = 0.6000, vly = 0.4000, v2y = 0.0000, c~.t = 0.04 gmol/l, Tc = 48°C) [-(--) stable steady states; (---) unstable steady states; (e) stable oscillations; (©) unstable oscillations].
Different Inhibitor Concentrations ( g m d / I )
3513-
-~~..-.~ ~.~__~ 1.o°-3 -...~ : ~ . ~ ~-~ i.z~°-3 . . ^ ~.~.~--..~-~/-~._~-=~ ~
_ ~
~.ue-o
340-
~. 3 3 0 -
\\
~5.0e-4
~-.
1.0e /
0.000
320J
3t0
0
I
500
I
t000
I
1 500
2000
Residence Time (s) Fig. 3. Bifurcation diagrams for inhibited VA polymerization showing five steady state multiplicity (vs: = 0.6000, vt: = 0.4000, v 2 / = 0.0000, c~y = 0.04 gmol/i, Tc = 48°C) (symbols described in Fig. 2).
68
J. C. PINTO a n d W. H. RAY
Different Inhibitor Concentratk)ns ( g r n d / I ) 3~0
. 000qD
. . . . . gO O0 OOOOOOOO00~
350-
000000000000oU II QOo. '
~340~
~OQOOoOq D q D ~ : 1.0e-3
• 30 . . . . . . ct.ee e ~ t e e ' 9 . . . . . . -: :Z~_ - - LT_--- _--,LL~_ o-o_oo_xrtzo-o--o-0_-- - # -
M'e" *~" ~" "J il;i'i'~?i~E--7
32o-
3t0 8000
i . . . . . 8 250
I
t 850O
...... 8750
9000
Residence Time (s}
Fig. 4. Bifurcation diagrams for inhibited VA polymerization showing multiplicity of periodic orbits at higher residence times (v,y = 0.6000, vl: = 0.4000, v2: = 0.0000, el: = 0.04 gmolA, Tc = 48° C) (symbols described in Fig. 2) Different
Inhibitor Concentrations ( g r a d / I )
36o
~ OOOOQO
•OO0••OOO0•O0
350-
j
.
.
.
.
.
.
.
.
.
.
.
•
330-
o
o
o
e
~
~
~
3203'10
45oo
1
4eoo
I 4~'00
'
'
'~
4eoo
'
1 4900
5ooo
Residence Time (s)
Fig. 5. Bifurcation diagrams for inhibited VA polymerization showing multiplicity of periodic orbits at lower residence times (v~,: = 0.6000, vl: = 0.4000, v2: = 0.0000, ct/-- 0.04 gmol/l, Tc = 48°C) (symbols described in Fig. 2)
(4) As already expected, the presence of small amounts of M M A in the feed stream may make the reaction system even more sensitive to the presence of inhibitors (Fig. 7), so that multiple steady states and periodic solutions occur at lower inhibitor concentrations. As in the case of copolymerization without inhibitors (Pinto and Ray, 1995), the system dynamics may be extremely sensitive to changes in the jacket temperature, heat transfer coefficient or initiator feed concentration, as it is shown in Figs 8-12. These bifurcation diagrams should be compared with Fig. 2, the nominal case. These results indicate that the ex-
perimental detection of stable limit cycles may be extremely difficult, if they occur at narrow ranges of experimental conditions, which are very sensitive to small changes in To, UA and ci:. An interesting result is shown in Fig. 12, where an ISOLA of periodic solutions is present. This preliminary analysis shows how complex the reactor dynamics may be in the presence of inhibitors and how important the dynamic effects may be in practical situations. The presence of inhibitors in an industrial environment is almost inevitable. (Inhibitors are generally added to monomers to allow safe stocking of raw materials, avoiding spontaneous thermal polymerization in the storage tanks. Besides,
Dynamic behavior of continuous solution polymerization reactors--IX
low inhibitor feed concentrations, below 100 ppm, the proper operation may be completely disturbed by the presence of inhibitors or be even impossible. Assuming that H Q is a strong inhibitor, the theoretical analysis suggests that oscillatory behavior and reaction extinction may be observed during real plant operation if the inhibitor feed concentration varies in the range of 0-100ppm. Observing these dynamic patterns experimentally is the main objective of the next sections.
monomers are generally used in this inhibited form during the polymerization, with no additional purification.) Because the appearance of unstable steady states and oscillatory behavior occurs at very
9O 0
69
1
£ 70
J
4. EXPERIMENTAL
5O 0 ....... ~W~ ...... ~A~ ...... 6~ Time (rain)
...... a~
.....
t.00
o,80-
8
0.60-
90 Temperoture (*C)
Fig. 6. Dynamic simulation showing multiple periodic solutions Iv,/= 0.6000, lYtf = 0 . 4 0 0 0 , ~12f = 0 . 0 0 0 0 , C i f : 0.04 gmol/l, T, = 48° C, [I-IQ]/ fnQ = 1.0 x 10- 3 gmol/l, (0) O = 8350 s; (l)O = 8000 s; (2) O = 8350 s].
SET-UP
The basic experimental set-up and process operation have been described by Pinto and Ray (1995) and we will avoid replicating the information. Nevertheless, we emphasize some important points and constraints related to the experimental procedure. Figure 13 shows a diagram of the copolymerization process and general equipment. It is important to notice that air was used as coolant medium, that the whole experimental set-up was kept at inert (nitrogen) atmosphere to avoid oxygen inhibition, that the feed conditions were properly controlled and that reactor temperature and conversion were measured in line. In the feed section, two different containers were used to store a mixture of solvent (TB) and initiator (2,2'-azobis-2-methyl-propionitrile--AIBN) and the comonomer mixture (VA/MMA and HQ). The initiator was dissolved in the solvent to avoid polymerization in the storage tanks. The monomers were premixed because of the low M M A feed concentrations used in the experiments, which made it difficult to control the feed composition otherwise. It is important to say that the fractional reactor volume occupied by vapor was negligible at all experiments, which means that residence time was measured with good precision at all experimental conditions.
Different Inhibitor Concentrotions (g mol/I ) 36O
1.0e-4
5e-4~..
~'
~'--~'~
500-4 •
r.oe-q - -
8
o
~,~.',,% ~ = ~_.:; ~ ~
350-
~b?'eeo
~ll~OOO ~ ~ o ,~. ~ O 0 •
,e
•
•
•
•
•
,~¢..
°~
oo
o,0._, oo oo ~ o,.25._3 ° o° o
• • •
~ o • -
:.~
o o
o
d ~ _
o
~
0 t50o-3
•
o
oo
o'
0.~ 320-
E.5e-4
5.00-4
7.5e-4
t.0o-3
t.25o-3
t.50e-3
t.Oe-4
310
0
i 5000
i 10OO0
t 15O00
2O000
Residence Time (s)
Fig. 7. Bifurcation diagrams for inhibited VA/MMA copolymerization for various inhibitor feed concentrations ([HQ] I fue) (v,s = 0.6000, v~i = 0.3985, v2/= 0.0015, c~y = 0.04 gmol/l, Tc = 48° C) (symbols described in Fig. 2).
70
J. C. PINTO and W. H. RAY
Different Inhibitor Concentrations (gmol/I) 5.0o-4 7.5o-4
360
~ - - - - , ~',al~, O o o • ~=" " • | ~ " S • ,
350-
o
o 0
o
%,
:
~340-
o° o0 o,.2~°-3 00
/
-
,
o
-.
-.
\\x,
9
""'~-~
0o o0
ooo _o°t,'5e-3
o°
o o~ o
- - - _ % ! -~"- " ~--_S
o0 o
z
oo°
"''"'='~--~-'~-'~i ',, "x ",.~.,~-''~e,. " ' ~ a
320-
oo
o
o°
.%~.<%.~_~ , ~
i330-
o
%.oo°-s
o
o
-'~-tcu:£
~'o4x~V4ooo
n -
I 15000
20000
~
-u--
t 75e-3
O0
75e 4
Pure VA 310
i 5000
0
i 10000
I
Residence Time (s) Fig. 8. Bifurcation diagrams for inhibited VA polymerization for various inhibitor feed concentrations ([HQ]I fnQ) [vsI = 0.6000, vii = 0.4000, v2i = 0.0000, cii = 0.04 gmol/l, Tc = 48°C, UA = 1.10(UA)o] (symbols described in Fig. 2).
Different Inhibitor Concentrations (gmol/I) 36O
~00 1.50e-5
350-
0 00 O00 _0002.00e_3.
t.75e-5
0O0 t25e-3
340a
~
00~
~,},<<.~.~.
, ~
,(~.~?~.<.\\
~ ~
#.
~o °
o°
,
oO o
~o
o
~o
,.o
y ~.-3
o o~
o o~
o
~
~~-fa.....---~ ,~.. £.<-.~~
330-
o°
o°
~oo
Oo7 o
320--
5.0e-4
7.5e-4
t.00e-3
1.25e-3
.50e-3
Pure VA 3t0 0
[ 5000
I 10000
I 15000
20000
Residence Time (s) Fig. 9. Bifurcation diagrams for inhibited VA polymerization for various inhibitor feed concentrations ([HQ]I fnQ) r,v,i = 0.6000, vii = 0.4000, v2i = 0.0000, q / = 0.04 gmol/I, Tc = 48°C, UA = 0.90(UA)0] (symbols described in Fig. 2).
In order to evaluate conversion, an on-line process refractometer was used (model SSR-72, from Electron Machine Corporation). It must be emphasized, however, that dynamic conversion profiles were never used for parameter evaluation and quantitative analysis because:
internal walls of the output line and measurement compartment. (3) The presence of nitrogen bubbles in the output stream, due to the mechanism used to control the reactor volume, caused a considerable amount of noise in the measured refractive index.
(1) Refractive indices are extremely sensitive to temperature changes and rigorous temperature control inside the measurement compartment was difficult. (2) The high viscosities of concentrated polymer solutions caused a damping effect on the conversion dynamics, due to the formation of a gel layer on the
As shown by Teymour (1989) other analytical techniques for on-line evaluation of conversion were not well suited (for similar and other reasons) for application in the system we have been working with. Attempts have been made to install an NIR spectrophotometer for on-line evaluation of the composition of the reaction medium, hut the equipment and
Dynamic behavior of continuous solution polymerization reactors--IX
71
Different InhibitorConcentrations(gmolll) 36o
5.00-4
,~_~x~o,oo,o |
~
~o- ~'B,
o
~***, ~• S~i° e
r~
~o
~ ~
•
~ ejE= ~ wj~O~
~\\[\<,~
" \ - ~ "-..
\,\ \
"-
I.ZSe-3
o
o
o o 0
•
o
ou 0
-o t -~.~oe- o
o-
o
o
_"~.%
o°
o
ooOo
-~--7--~- . . . .
-
~
2.50-4
oo
oOO°
~
e ° ,,=
! '~.2~'e."~ee-"-\~ /
o
•
• ~ • ° w
°o
°o °
°o •
"
O',"~'Zl~i%d:~.--~i.
~33o
t o0%3
7~.-4
Z.Se-4 e
0
5.00-4
1.00o-3
7.50-4
150e-3 1.25e-3
Pure VA 310
I
i 10o00
5OOO
0
= 15000
20000
Residence Time ( s )
Fig. 10. Bifurcation diagrams for inhibited VA polymerization for various inhibitor feed concentrations ([HQ]: fno) (vsy = 0.6000, h : = 0.4000, v2: = 0.0000, ely = 0.04 gmol/l, Tc = 46°C, UA = 1.00(UA)o) (symbols described in Fig. 2).
Different InhibitorConcentrations(gmolll) 36O
C~O 350 -
000 00~)
0
\
OOo
k
~
°o
oo
o
o
o
oo
340-
0
0
0
o°
t.5oe-3o o,~o3
oo
"
0
0
~6 ° z o o o - 3
/ ooooOO ^o
o~
.9.=
l
~
3~--
/ ~ ' , " . . ' - : . ' ~ . ' . - ' . _ ~ _ _ ~-O.o~Oo o 320--
~
-
4
7.50-4 t.00e-3 ].25e-3 t.50e-3
t.75e-3
Pure VA 310 0
I
i
5000
10000
i
15000
20000
Residence Time ( s )
Fig. 1l. Bifurcation diagrams for inhibited VA polymerization for various inhibitor feed concentrations ([HQ]/ fnQ) (vs: = 0.6000, vl: = 0.4000, v2: = 0.0000, el: = 0.04 gmol/l, Tc = 50°C, UA = 1.00(UA)0) (symbols described in Fig. 2). technique necessary were not available by the time this work was done. F o r all the reasons discussed above, dynamic conversion measurements were used for qualitative model analysis only. Finally, it must be pointed out that the experimental set up was designed for operation at atmospheric pressure so that the reactor temperature was not allowed to rise above the boiling temperature of the reaction mixture. This important constraint has already been included into the mathematical model. Regarding the sources of the chemicals used, VA was bought from Pfaltz and Bauer, Inc., M M A , TB (99.5% + pure) and H Q (99.0% + pure) from Eastman K o d a k Co. and nitrogen (99.996% + pure) from Liquid Carbonic. VA and M M A were distilled under
vacuum and inert atmosphere one day before the experiment was carried out and were stored in a refrigerator at 10-15°C. The other chemicals were used as received.
5. EXPERIMENTAL RESULTS 5.1. Effects cause by the presence of small amounts of water and HQ in the feed stream As observed experimentally through N I R spectrephotometry (Pinto, 1991), VA, M M A , TB, PVA and P M M A absorb water from the atmosphere extremely fast. So, it is quite possible that a certain amount of water be present in the reaction medium at both experimental conditions and industrial operation.
72
J.C. PINTOand W. H. RAY 36O
350340\\\\\
2
,.
~_~ 330-
320k 310/ 0
I
I
1
IO0OO
5000
20000
15000
ResidenceTime(s)
Fig. 12. Bifurcation diagram for inhibited VA polymerization showing an isola of periodic solutions fHQ) = 1.5 x 10 - 3 gmol/1 [•sf = 0.6000, Vlf = 0.4000, v2f = 0.0000, cif = 0.04055 gmol/l, Tc = 50°C, UA = 1.00(UA)o](symbols described in Fig. 2).
([HQ]f
.....
Nitrogen Feed To DA system
Solution Polymerization
I Nitrogen Monomerand solvent (+AIBN)containers ~" -
I
i !
Monomer
Rotameter
i i
Waste
line
o
Nitrogen
0
I Electrical resistance
Feed i
Products
'
I ~ ~
i r'-/~ '
Waste
PID controller and VARIAC
Burets
Volumetric PID controller VARIAC
and
Fig. 13. Experimental set-up.
Very few studies have been done on the effects caused by water in free-radical polymerization reactions. Among them we can refer to the papers written by De Schryver and coworkers [see De Schryver e t al. (1986) and Jacob e t al. (1972)-1, where the authors observed that the presence of small amounts of water in the reaction medium was beneficial to the polymerization of MMA, methacrylic acid and acrylamide, provoking an increase of the reaction rates.
As already said, monomers used for large-scale polymerization generally contain small amounts of inhibitors. In the particular case of MMA and VA polymerization, monomers generally contain 10-15 ppm of HQ. In order to analyze the importance of these small amounts of water and HQ during real operation, a series of experiments were carried out. Figure 14 shows the results obtained when the feed, containing
73
Dynamic behavior of continuous solution polymerization reactors--IX
ChangingFeed Composition Reslden(~Time= 90mln
800 f ~ eo.o-
~
_
.
~
Reactortemperature
I I
1
I
Nr Temperature 4QO
0
too
2o0
30o
I 100
I 200
I I 300 400 Tlme(mln)
I
6°.0- I
ao I o
I 500
I 600
700
Fig. 14. Experimental temperature and conversion profiles (vs: = 0.6000, vl: = 0.3990, v2: = 0.0010, c~: = 0.04 gmol/l, ® = 90 min. Arrow shows when the uninhibited and dry feed is changed for a water saturated, 15 ppm HQ inhibited feed stream.)
dry monomers and solvent, was changed for a new feed, containing water-saturated monomers and solvent and 6 ppm of HQ (corresponding to a 15 ppm HQ inhibited comonomer feed stream), after steady state had been reached. It is clear that, at the conditions analyzed, the operation is stable.
80.
5.2. Multiplicity and domains of attraction A series of experiments were carried out in order to show that inhibitors may change the number of steady states at certain operation conditions, also changing their domains of attraction. The reference case is shown in Fig. 15, where it may be seen that reaction ignition occurs spontaneously at residence times of 60 min, with no need to increase the jacket temperature in order to reach the upper steady state, when the reactor does not contain any initiator at the beginning of the run. This result agrees with the model predictions presented for VA homopolymerization in Section 3. Figure 16 shows, however, that reaction does not ignite spontaneously when the feed stream contains 40 ppm of HQ, even when the reactor contains 0.04 gmol/1 of AIBN at the beginning of the run and the residence time is as high as 180 min. According to the results shown in Section 3, this occurs because the presence of inhibitors in the feed stream extends the existence of the lower branch of steady states to higher residence times, reducing the domain of attraction of the upper steady-state solution. Figure 17 clearly shows the existence of the two stable steady states at residence times of 90 min. After igniting the reaction with uninhibited VA, the feed was changed to a 40 ppm HQ inhibited feed stream
4oi, o
i
~
'''500
Time(rain)
1oo
~w
C t~
o
100
2oo
300
400
5oo
Time(rain)
Fig. 15. Experimental temperature and conversion profiles (vs.r= 0.6000, v1: = 0.4000, v2: = 0.0000, c~f = 0.04 gmolfl,
@ = 60 min. Reaction ignites spontaneously, even when the initial initiator concentration is equal to zero.)
(the arrow shows the exact point where the feed composition was changed). It may be seen that the oscilllations introduced by changing the feed composition were negligible and that the new steady-state conditions was not significantly different from the previous one. After showing that steady-state conditions would not change, the jacket temperature was reduced to
J. C. PINTO and W. H. RAY
74
Changing Feed Composition
Changing Residence Time 8(1o
~
70.0
'='\1
~.0
i~ 50.0
..... [ 0
~00
~.......... I Air l'emperature ]
40O
600
800
t000
I
I 6o0 Time (rain)
I 8oo
lOOO
I t
1200
toQo| eo.o
1
40.0
20.0 0.0
I
0
200
400
I
12oo
Fig. 16. Experimental temperature and conversion profiles [v~I = 0.6000, vii = 0.4000, v21 = 0.0000, c~r = 0.04 gmoi/l, 40 ppm HQ, (0) ® = 60 rain; (1) ® = 90 min; (2) (9 = 120 min; (3) (9 = 150 min; (4) (9 = 180 min; (5) (9 = 90 min. Reaction does not ignite spontaneously.]
............
,~ I\
,---JJ-'i r / - ....... I
20
........ 40O
y,° ............. 8OO
I t~00
Time (rain)
Table 1. Dynamic experimental profiles obtained for HQ inhibited VA polymerization HQ feed concentration (ppm)
Experimental dynamic profile (Replicates)
40
Damped oscillations of small amplitude (2) Damped oscillations of small amplitude (2) Damped oscillation of small amplitude (2) Oscillation followed by extinction (2) Extinction (2) Possibly sustained oscillations of large amplitude (2) Possibly sustained oscillations of large amplitude (2) Extinction (3)
50 60
tO0
70
i
80
50
100
O'
400
8O0
1200
Tlmelmln) Fig. 17. Experimental temperature conversion profiles. Steady state multiplicity. (v,i = 0.6000, vii = 0.4000, v2I = 0.0000, ci¢ = 0.04 gmol/1, ® = 90 min. Arrow shows when uninhibited feed is changed for 40 ppm HQ inhibited feed stream.)
25°C, which was followed by reaction extinction. Increasing the jacket temperature to the original value, however, did not lead the system to the former steady state again. As it may be seen, operation was
stabilized at much lower temperature and conversion. The original steady-state conditions could only be attained after introducing a peak in the jacket temperature. The second arrow shows the exact point where reaction ignites again. 5.3. Oscillator behaviour and instabilities According to Fig. 2, an increase of the inhibitor feed concentration may cause destabilization of the reactor operation, leading to the development of self-sustained oscillations or reaction extinction. F o r this reason, many experiments were carried out in order to define the ranges of H Q feed concentrations that
75
Dynamic behavior of continuous solution polymerization reactors--IX might lead to unstable operation and oscillatory behavior. In all these experiments reaction was started with an inhibitor-free feed stream, which was changed for a new feed feed stream, with known inhibitor concentration, after approximate steady-state conditions had been attained. Experimental results are
summarized in Table 1. Figures 18-20 show some experimental profiles obtained. The results presented in Table 1 show that H Q feed concentrations below 60 ppm do not change the stability of the reactor operation significantly, at the experimental conditions analyzed. F o r H Q at
Changing Feed Cemposltlon Residence Time = 90 rain
8GO
t L"~
~ 5o.o40.0
Air Temperature I
200
0
I
i
4O0
600
I
800
I
tooo
t200
100.0 ,,...,.
ae 80.060.0-
"~> 40.0C_~ 20.0-0.0
I
2O0
0
I
I
400
6O0
I
800
I
t 000
t~00
Time(rain) Fig. 18. Experimental temperature and conversion profiles (vsl = 0.6000, v l f = 0.4000, v2I = 0.0000, c~I = 0.04 gmol/l, O = 90 min. Arrow shows when uninhibited feed is changed for 60 ppm HQ inhibited feed stream.)
Changing Feed Composition Residence Time = 90 mln 80.0
I f"~
7o.o-I~ I~
I'= 40,0 t ~ 0
Air Temperature I
I
200
400
I 2OO
I 40O
1
I
I
600
800
1000
t200
600
8O0
t000
1200
100.0
80.o60.0QrJ
~ 40.0¢~ 20.00.0 0
Time(rain) Fig. 19. Experimental temperature and conversion profiles (vsy = 0.6000, V l f = 0.4000, V2f = 0 . 0 0 0 0 , c~y = 0.04 gmol/l, O = 90 min. Arrow shows when uninhibited feed is changed for 80 ppm HQ inhibited feed stream.)
76
J.C. PINTOand W. H. RAY Changing Feed C~moosltlon Residence Time = 90 rain 80.0~
•--.
Ji'
~,f"N
• r°.°--ll
J/i
•~ 60.0
j ool/ '
"°.°-I'-
T Reactor Temperature
V
0
_ r
r
Air Temperature
.
r
T~T
\ r
t00
200
300
400
500
600
700
I t00
I 200
1 300
I 400
I 500
I rN;)O
700
too.o 8o.o-
60.0--, ~ 40.0-0.0 0
Time(rain) Fig. 20. Experimental temperature and conversion profiles (vs/= 0.6000, vii = 0.4000, v2f = 0 , 0 0 0 0 , cis = 0.04 gmol/l, ® = 90 min. Arrow shows when uninhibited feed is changed for 100 ppm HQ inhibited feed stream.)
70-80 ppm levels, oscillations are observed. However, when the HQ feed concentration is above 100 ppm, reaction extinction was detected. Because of the long period of the oscillations (more than 3 h), it was extremely difficult to keep conditions completely constant during this period. As shown in Section 3, the experimental observation of limit cycles in inhibited reactions may be difficult, due to the narrow ranges of operation conditions where they occur and to the high sensitivity of these regions with respect to small variations in the operation conditions. Thus we are not able to show unequivocally that the observed oscillations are sustained. However we believe this to be true. Comparing the results obtained with those presented in Fig. 2, we conclude that fnQ is approximately equal to 1.5, showing that HQ molecules may terminate more than one growing free radical molecule. This result agrees with stoichiometric results presented by Goldfinger et al. (1967) and with the bifunctional nature of HQ molecules.
that our kinetic representation of inhibition is oversimplified. In order to understand the difference between predicted and observed results, the experimental temperature profiles were used to estimate experimental inhibition rates, using a second-order filter (Derenzo, 1990). If it is assumed that the differences between model and experiment are due to errors in the kinetic mechanism of inhibition, then one can assume --Rinh = 2fl kdcl -- kttl i-p]2
This hypothesis is supported by previous results that show that model and experiments agree very well in the absence of HQ (Pinto and Ray, 1995). Figures 21 and 22 show experimental inhibition rates obtained in the same experiments presented by Figs 18 and 19. It is clear that the rates of inhibition are not constant throughout the experimental run. Actually, if Ri,h we assumed (as is usually done) Rinh = kn [P'I [HQ] kn --- exp( - A E u / R T
6. ADDITIONALQUANTITATIVEANALYSIS The experimental results obtained were used to test the model adequacy to describe the phenomena quantitatively. Only the experiments where oscillations of more than 3°C developed were used, because the small oscillation case may be easily fitted with small adjustments of the heat transfer coefficient (Pinto, 1991). For the larger oscillations the a priori model was not successful to reproduce either the period or the amplitude of oscillations. This may indicate
(20)
+ ASu)
(21) (22)
(Brandrup and Immergut, 1975), it is possible to fit all experimental data with a proper adjustment of AEn and ASh (Pinto, 1991). However, this is just an empirical fit. It is clear that more fundamental kinetic studies have yet to be carried out. 7. CONCLUSIONS The theoretical and experimental analysis carried out in this work show that the stability of continuous solution polymerization reactors may be extremely
Dynamic behavior of continuous solution polymerization reactors--IX
reactor qualitatively, but failed to reproduce the experimental results quantitatively. A more detailed analysis show that the inhibition rates may be temperature dependent and that the inclusion of the temperature dependence through an ordinary Arrhenius expression allows the reproduction of the experimental results. More fundamental kinetic studies have yet to be done, however, if a consistent inhibition kinetic mechanism is to be obtained.
1.00E-006-
Ex0orim~t 8.00E-007.
!!
77
,"
&OOE-007
4.00E-007 t-i
== --
2.00E-007
Simulation O00E+OOC
. . . . . . . . .
i
. . . . . . . . .
60O
i . . . . . . . . .
800
i , , ,
I000
. . . . .
t200
Tlmelmln)
Fig. 21. Experimental and theoretical inhibition rate profiles. [Experimental conditions described in Fig. 18. UA = 0.99 (UA)o, AEn/R = -6789.6 K, ASn/R = 20.7649 for simulation.]
2.00E-OO6Simolation i it)
o~ .,a
vv L
O.OOE+O00 . . . . . . . . . , . . . . . . . . ~, . . . . . . . . . , . . . . . . . . . , . . . . . . . . . 2O0 4OO 6O0 8OO I000 1200 T i m o ( rain )
Fig. 22. Experimental and theoretical inhibition rate profiles. [Experimental conditions described in Fig. 19. UA = 0.90 (UA)o, AEn/R = -11710.0 K, ASn/R = 35.8035 for simulations.]
sensitive to the presence of small amounts of inhibitors in the feed stream. Particularly, it was shown that small amounts of HQ, in the range 60-100 ppm, may introduce vigorous oscillatory behavior and lead to reaction extinction in the operation of VA homopolymerization reactors. It was also shown that reactor operation may become even more sensitive to the HQ feed concentration if small amounts of MMA are present in the feed stream. It was shown that the presence of inhibitors may change the number ol steady states of the system and modify the basins of attraction of a desired steady-state condition, which may eventually require changes of the reactor operation. A simple mathematical model was developed to describe the dynamic behavior of inhibited polymerization reactors. The model was reasonably successful to describe the dynamic behavior of an experimental
Acknowledoements--We thank James Schneider for his valuable help and support. We also thank Conselho National de Desenvolvimento Cientifico e Tecnol6gico (CNPq--Brazil) for supporting J.C. Pinto's staying in Madison, WI. We are grateful to the industrial sponsors of the University of Wisconsin Polymer Reaction Laboratory (UWPREL) and to the National Science Foundation for support of this research. NOTATION molar concentration of initiator heat capacity initiator efficiency for initiation of monomer i inhibitory functionality of HQ gel effect correlation for km gel effect correlation for k, u initiator kinetic constant for initiator decomposition kinetic constant for inhibition reaction kinetic constant for propagation of free radical i with monomer j k,~ kinetic constant for initiation of free radical i ks kinetic "equilibrium constant", as defined in eq. (13) ktu kinetic constant for termination of free radical i with free radical j Mi monomer i, molar concentration of monomer i P total molar concentration of free radicals Pu Pu molar concentration of free radical I (having monomer 1 at the active site), containing i units of monomer 1 and j units of monomer 2 qoqi volume shrinkage, as in eq. (15) Q total molar concentration of free radicals Q# Qu molar concentration of free radical 2 (having monomer 2 at the active site), containing i units of monomer 1 and j units of monomer 2 Qlat rate of heat removal due to condensation ro reactivity ratio, as defined in eq. (1) R initiator fragment Rate~ normalized rate of reaction of monomer i R,k, normalized rate of initiation T temperature To ambient temperature Tb boiling temperature T, coolant air temperature Tgi glass transition temperature of species i UA global heat transfer coefficient to coolant jacket v/ free volume vyF critical free volume for propagation v;t, critical free volume for termination V reactor volume
ci cp f/ fnQ gm g.j I kd kH km
J. C. PINTO and W. H. RAY
78
Greek letters •i expansion coefficient of species i, for free volume calculation • U A global heat transfer coefficient to the ambient AH o heat of reaction of propagation kp~j e external capacitance ® residence time 2 numerical constant in eq. (18) v~ volume fraction of species i I-ll partial pressure of species i H7at saturation pressure of species i p density dimensionless time (t/O) q~ conversion Xi interaction parameter for species i ~bo cross-termination constant, as defined in eq. (7)
Special subscripts 1 monomer 1 2 monomer 2 f feed p polymer s solvent
REFERENCES
Bagdasar'ian, K. H. S and Sinitsina, Z. A., 1961, J. Polym. Sci. 52, 31-38. Bartlett, P. D. and Kwart, H., 1950, J. Am. Chem. Soc. 72, 1051-1059. Bartlett, P. D. and Kwart, H., 1952, J. Am. Chem. Soc. 74, 3969-3973. Bartofi, J and Jurani~ov/t, V., 1989, Makromol. Chem. 190, 769-775. Bevington, J. C., Ghanem, N. A. and Melville, H. W., 1955, J. Chem. Soc. London 2822-2830. Bhanu, V. A. and Kishore, K., 1991, Chem. Rev. 91, 99-117. Brandrup, J. and Immergut, E. H., 1975, Polymer Handbook, 2nd Edition. Wiley New York. Breitenbach, J. W., Springer, A. and Horeischy, K., 1941, Ber. Dtsch. Chem. Ges. 71, 1438. Breitenbach, J. W., Springer, A. and Horeischy, K., 1941, Bet. Dtsch. Chem. Ges. 74, 1386. Chen, S.-A. and Tsai, L.-C., 1986, Makromol. Chem. 187, 653-666. Clinch, A. B., 1983, Phenomena of the nonisothermal solution homopolymerization of methyl methacrylate or vinyl acetate in a CSTR. M.Sc. thesis, University of Wisconsin, Madison. Deb, P. C. and Kapoor, S. K., 1980, Eur. Polym. J. 16, 763-767. Derenzo, S. E., 1990, INTERFACING: A Laboratory Approach Usino the Microcomputer for Instrumentation, Data Analysis, and Control. Prentice-Hall International, London. De Schryver, F. C., Smets, G. and Van Thielen, J., 1968, Polym. Lett. 6, 547-550. Doedel, E. J., 1986, AUTO: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations. California Institute of Technology, Pasadena. Eastmond, G. C., 1976, Comprehensive Chemical Kinetics, Vol. 14a: Free-Radical Polymerization. Elsevier, New York. Flory, P. J., 1953, Principles of Polymer Chemistry. Cornell University Press, Ithaca. Goldfinger, G., Yee, W. and Gilbert, R. D. 1967, in Encyclopeida of Polymer Science and Technology (Edited by H. F. Mark and N. G. Gaylord), Vol. 7, pp. 6444564. Wiley Interscience, New York.
Jacob, M., Smets, G. and De Schryver, F. C., 1972, Polym. Lett. 10, 669-673. Kice, J. L., 1954, J. Am. Chem. Soc. 76, 6274-6280. Kim, K. J., 1991, Modelling and control of continuous free radical polymerization reactors. Ph.D. thesis, University of Maryland. Kirchner, K. and Rintelen, T., 1986, Angew. Makromol. Chem. 141, 85-93 Levy, L. B., 1992, J. Polym. Sci. A: Polym. Chem. 30, 569-576. Mohanty, A. K., Misra, M. and Singh B. C., 1994, Polym. Plast. Technol. Eng. 33, 207-219. Petzold, L. R., 1982, A description of DDASSL: a differential algebraic system solver. Sandia National Laboratories, Report ~ SAND82-8637. Pinto, J. C., 1991, Anfilise de Din~rnica de Sistemas de Polimerizagfio Pela Teoria de Bifurca~6es. Ph.D. thesis, PEQ/COPPE/Universidade Federal do Rio de Janeiro, Rio de Janeiro (in Portuguese). Pinto, J. C. and Ray, W. H., 1990, presented at the 1990 Annual A.1.Ch.E. Meeting, Chicago. Pinto, J. C. and Ray, W. H., 1991, presented at the 1991 Annual A.I.Ch.E. Meeting, Seattle. Pinto, J. C. and Ray, W. H., 1995, Chem. Engng Sci. 50, 715 736. Ray, W. H., 1972, J, Macromol. Sci. Rev. Macromol. Chem. C8, 1 56. Reid, R. C., Prausnitz, J. M. and Polina, B. E., 1986, The Properties of Gases and Liquids, 4th Edition. McGrawHill, New York. Rieumont, J. and Vega, R., 1991, Makromol. Chem. 192, 1387 1397. Rieumont, J., Vega, R., Davidenko, N. and Paz, J. A., 1988, Eur. Polym. J. 24, 909-911. Rintelen, T., Riederle, K. and Kirchner, K., 1983, in Polymer Reaction Engineering (Edited by K.H. Reichert and W. Geiseler), pp. 269-286. Hanser, New York. Ross, R. T. and Laurence, R. L., 1976, A.I.Ch.E. Syrup. Ser. 72, 74. Simfindi, T. L. and Tiid6s, F., 1985, Eur. Polym. J. 21, 865-869. Stolzenberg, K. and Kirchner, K., 1981, Angew. Makromol. Chem. 95, 185-197. Suddaby K. G., O'Driscoll, K. F. and Rudin, A., 1992, J. Polym. Sci. A: Polym. Chem. 30, 643~a48. Tfinczos, I., F61des-Berezsnich, T. and Tiid6s, F., 1983a, Eur. Polym. J. 19, 153-157. Tfinczos, 1., F61des-Berezsnich, T. and Tiid6s, F., 1983b, Eur. Polym. J. 19, 593-595. Teymour, F., 1989, The dynamic behavior of free radical solution polymerization reactions in a continuous stirred tank reactor. Ph.D. thesis, University of Wisconsin, Madison. APPENDIX A
In order to account for the gel effect (the fact that the kinetic constants depend on composition when the polymer concentration is high due to diffusion limitations), the following equations were used: A.1. VA gel1 = 1 Yt~ = exp( -0.4407 ~b -6.7530~b 2 -0.3495~b 3) (A1) where ~b is the total conversion defined by q~ = pp Vp/(plvl + p2v2 + p~vs + ppVp).
(A2)
Equations (A 1) are empirical but were used successfully by Teymour (1989) and Pinto and Ray (1992a) to describe the VA homopolymerization in TB. A.2. MMA The Ross and Laurence equations, based on the freevolume theory (Ross and Laurence, 1976), were used to
Dynamic behavior of continuous solution polymerization reactors--IX describe the gel effect of MMA homopolymerization in TB, as presented by Pinto and Ray (1992a). The equations are: t 0.10575exp(17.15v~ - 0.01715(T - 273.15)), [JY > ~ftc gl22 =
2.3 x 10-%xp(75vl),
Vf~Vfp c
0.1856 - 2.965 × 10-+(T -273.15) vf~ = 0.05
CE$ 51-1-F
vf = v l t v t + vfi =
(A3)
vs2v2 + vy, v, + VypVp
(A7)
0.025 + ~ i ( T - Tgi)
(A8)
and ~ and T v are presented by Pinto and Ray (1995), for some chemical species. (A4)
where vftc =
and the total free volume vy is given by
where
vl ~< vft+
~1, vf > Vfpc gp22 = ~7.1× 10_Sexp(171.53vl) '
79
(A5) (A6)
A.3. V A / M M A
copolymerization
The gel effect correlation used for copolymerization reactions were g,12 = ~
+,22
(A9)