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Steady-state multiplicity during solution polymerization of methyl methacrylate in a CSTR
Chemical Enymecrtng Saence, Vol. 44, No. IO, pp. 2269 2281, 1989. Prmfed m Great Britain.
CQOS-2509/89 $3 uo + 0.M) 0 1989 Pergamon Press pk
STEAbY-STATE MULTIPLICITY DURING SOLUTION POLYMERIZATION OF METHYL METHACRYLATE IN A CSTR A. K. ADEBEKUN, K. M. KWALIK and F. J. SCHORK’ School of Chemical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0100, (Receioed 14.April 1988; accepted 7 December
U.S.A.
1988)
Abstract- This paper examines the steady-state behavior of a CSTR during solution polymerization of methyl methacrylate. Simulation studies reveal the hitherto unreported existence of muhiple isolas and unusual multiplicity features, arising from isola evolution for some regions of reactor parameter space. The stability ofsome of these unusual steady states is analyzed. The influence of the reactor operating conditions on the molecular weight distribution (MWD) is also investigated and the various multiplicity features associated with the MWD are reported. The behavior of the reactor at very high conversion (> 95%) and the associated MWD profiles are presented. Finally, the practical consequences of these phenomena on reactor control and MWD manipulation are further discussed.
INTRODUCTION
The dynamic and steady-state behavior of CSTRs has received a great deal of attention in the literature (Van Heerden, 1953; Aris and Amundson, 1958a, b; Bilous and Amundson, 1955; Poore, 1973). These reactors, because of their inherent nonlinearities arising from complex kinetics, temperature effects and heat dissipation constraints, exhibit steady-state multiplicity and pose difficult control problems. Reactor behavior arising from nonisothermal operation for certain restricted kinetics has been well studied by Poore (1973), Uppal et al. (1974, 1976) and Balakotaiah and Luss (1981). The multiplicity features of some other reactions and the unusual steady-state conversion profiles arising from such have also been investigated by several workers including Gray and Scott (1983) and Hlavacek et al. (1970). The review paper by Raz6n and Schmitz (1987) is an excellent reference for other related work. Polymerization in a CSTR presents a formidable challenge to the reaction engineer, and, in contrast to the classical chemical reactors, the existing literature on polymerization reactors is not quite as extensive. Hoftyzer and Zwietering (1961) studied the multiplicity phenomena in a bulk polymerization reactor and concluded that as many as five steady states could exist for certain reactor parameters. Perhaps, the first systematic investigation of polymerization reactor dynamics was the work of Jaisinghani and Ray (1977), in which the authors studied the continuous bulk polymerization of methyl methacrylate (MMA). They found as many as three steady states were possible for a wide range of operating conditions. Limit cycle behavior was found for some combinations of reactor parameters. Balaraman et al. (1982) also reported the existence of three steady states in the continuous freeradical bulk copolymerization of styrene and acrylonitrile. Further work involving design considerations ‘Author
to whom correspondence
should be directed. 2269
for copolymerization reactors with recycle can be found in Balaraman et al. (1983). The experimental results of Schmidt and Ray (198 1) for the isothermal CSTR solution polymerization of MMA show the existence of three steady states during isothermal operation. The multiplicity observed in the isothermal reactor results from the autoacceleration induced
by
the
gel
effect
(Odian,
1981;
Tulig
and
The Schmidt-Ray model was further studied by Schmidt et al. (1984) and Hamer et al. (1981). In these investigations, the authors predicted and confirmed the existence of isola-type multiplicities. Some simulation results corresponding to five steady states were reported, and a parametric analysis of the system was also carried out. It is of interest to note that in all these investigations, the existence of more than one isola has never been reported. Moreover, in none of these studies, has the molecular weight distribution (MWD) been discussed. However, the influence of the MWD on polymer end-use is of great significance and should not be overlooked. Hence, in this investigation, we carry out a steadystate simulation of MMA solution polymerization in a CSTR and present results showing some unusual behavior. We demonstrate the existence of multiple isolas for certain reactor operating conditions, and show the unusual feature of the isola-tuck, which arises from a shift in isola origin and leads to a region of five steady states. We examine the effects on the MWD and show the importance of the chain propagation probability at very high conversions. From this viewpoint, we reestablish the fact that, for design purposes, when the MWD is of interest, operating a single CSTR might not be desirable when operating close to complete conversion. Finally, we present the multiplicity patterns in the MWD and justify the existence of these by clarifying the link between the MWD moment equations and the NACL-polydispersity representations of Tanner et al. (1987). Tirrel,
1981).
A.
2270 REACTOR
I (VJ > 0.05)
MODEL
The polymerization of MMA via a free-radical scheme with ethyl acetate solvent and benzoyl peroxide initiator is assumed. The kinetic scheme includes reactions for chain transfer to solvent and monomer. By invoking the quasi-steady-state approximation, long-chain hypothesis and assuming constant density, the following dynamical equations can be derived (Schmidt and Ray, 1981; Tanner, 1985): Vk,MP
V$=y(“,-h+
V,C~$=ypC,(T,-
et al.
K. ADEBEKUN
“=
V;=q(I,-I)-
V
0
T
(21
Vk,I
(3)
-42
M, M
-l, -
MfO
MD
A,
12
P
MS0
M,o
M,a
X5=-.-,X6=-,X,=-,W=-
If
S __
-,_$Z-,XqZ
X2=
T,)
M,o 1
X 2e =
MfO
E,
ED
Ed
Y~=RT,Y1=E,Yd=E.Y,d=E. = -qqA,
+ V[K,M
+ k,P+
V%
= -+I,
I
kfsSJaP
+ + Vk,, P2
P
+ k,,p+
4 exp-
h&r A= ~
k,S)
x(2cr-cr’)+K,,P]P/(l-a)
PC, V “=k6,exp
(6)
4 =k:, ev V[(k,M+k,,P+k,,S) x (a3 -
Da, =&
(7)
CL= k, M/(k, M + k, M + k,S co 1 n’M,
A-k=
Da,-,=QexP
+ k, P)
Dardo
(k=O,
p= c
ev (-_yJ
exP (-Y,JM~~L = kb ew
ydyl
Dad,,= tlDa,,
(-yIyp)M,Otr
Da,, = k,, Ml, t., Darco = k:,, exp (-_y,~,,)M,,t, t-YJY,)MJoL
=4,,
exp
(-~Yp~td)MJ,~,.
1, 2)
n=2
2fkJ __ k,
P
(-y,)M,,
(-Y,Y,)/~~~
Dar0 =t&Da,
3a2 + 4a) + k,,P
x(a+2)]P/(1-a)2
where
Et*
P
(5)
+ V[(k,M
V%=-L$~+
(9)
M
t/t,
e=-,T=-,f--,x,=-, 4 tr
(1)
X
V%
< 0.05).
k,, is the termination rate constant corresponding to zero conversion with k,, similarly defined. V, is the free volume, calculated from the volume fractions of monomer, polymer and solvent in the reactor. Similar to Jaisinghani and Ray (1977) and Wppal et al. (1976) the following dimensionless variables are introduced:
T)+(-AH)Vk,MP
- hA,( T-
(V,
7.1 x 10e5 exp(171.53Vf)
A list of kinetic constants is shown in Table 1. The symbols t, and M,, represent arbitrary reference values of residence time and monomer feed concentration, respectively, and, hence, the dimensionless model equations are
‘.’ >
P is the total concentration
of live radicals and k, (= k,, termination rate constant. The ith moment of the dead-polymer MWD is denoted by ;ii (i=O, 1, 2), and M is the monomer concentration. T, I and S are the reactor temperature, initiator concentration and solvent concentration, respectively. The rate constants with the exception of k,= are assumed to follow an Arhenius dependence on temperature. The correlations for the gel effect (gC= k,/k,,, gp = k,/k,) as proposed by Ross and Laurence (1976) and extended by Schmidt and Ray (1981) have been employed. The relevant equations are + kid) is the overall
0.10575exp[17.15Vf-O.O1715(T-273.2)] I’, > CO.1856 - 2.965 x lo-+( T-
dx, p=x dt’ dxz dt’=
dx, -= dt'
@a) 273.2)]
273.2)]
- x1 - gpDap Wx, E,
(10)
--x,+Bg,Da,Wx,y,E,--B,z(x,-x,,) (11)
2.3 x lo-“exp(75V,) V/ Q CO.1856 - 2.965 x 10-4( T-
”
(8b)
xjf
-x3
-Dadx3&
(12)
in a CSTR
Solution polymerization of methyl methacrylate
2271
Table 1.
Kinetic rate constants Literature
kj
value (mol, 1, s)
Reference1
1.69 x lO’*exp (-30,OOO/RT) 4.92 x 105exp (-4353/RT) 9.80 x lO’exp(-7Ol/RT)
kd k,
k 10
4.92 exp (-4353JRT) 0.091 a.23
k, k/s k,lk,,
Brandrup
and Immergut (1975)
Reactor and reactor medium constants j” =0.5 h = 135 cal/m2-sK A, = 2.8 m2 p = 1038 g/l
Rodriguez (1982) Jaisinghani and Ray (1977) Jaisinghani and Ray (1977)
C, = 0.4 Cal/g-K
(-AH)=
13.8 kcal/gmol
‘Except where noted, values for the kinetic rate constants are available in Schmidt and Ray (1981).
dx, -= dt’ dx, p= dt’
dxs dt’=
X4f
-x5
-
(13)
x4
-+ [(Da/x, Ex, + Da,+
+ gt Da,d @‘&,d )
x (a W)] + 0.5 W’g, DateE,,
(14)
define
-x6
+ [(Da/x,
+ g,Da,
z2 = St D% L,
WExtd
(19) W)
then
dw - = Z, [2-a-p(1 (15)
dx,
Z, = Da/ x, E,, + Da,x,
E,, + Da,x,
+ grDa,,WE,,,)(2~-aaZ)
dr’=
refer to the number average chain length (NACL or /I) and the polydispersity (D) of the MWD. It can be shown that these variables have the following dynamical equations (Tanner et al., 1987):
-x,
+ [(Da,xl
dt’
dD -= dt’
- 3a2 + 4~)
+(91Da,=E,,,W)(a+2)]------
-a)]
+Z+;(:-l)]}&
E,, + Da,x,
+ g,Da,, WE,,)(cr3
(
W (1-a)2
(16)
(21)
-3a+4)+pD(l-a)*
-
and a = Qa,xl
~,l(g,Da,~,x~
+ Da/%x,
+ &Da, WE,, + Da,x,)
(17)
=exPCx2/(1+~d~Jl Exj=exPCYjXz/(1+ xzhp)I-
(184
where E,
Daj = Dajoz [j 4 = (f; d, s, tc, td etc.)] as appropriate.
(18b) Vc)
Note that the system is parameterized as a function of the dimensionless residence time (2). This is evident from eq. (18~). We shall in subsequent sections have occasion to
Equations (8)-(22) represent the dynamic model of the MMA reactor. As is often done in the estimation literature (Schuler and Suzhen, 1985; Gilles, 1986), it is convenient to view the process model as two subsystems in series. Subsystem 1 corresponds to the monomer, temperature, initiator and solvent levels while subsystem 2 corresponds to the MWD components represented by rl,, il, and 1, or, equivalently, h,, p and D. The need for this partition will become apparent later.
2272
A. K.
The
steady-state
ADEBEKUN
AND DISCUSS[ON
RESULTS
results
for
the system can be computed by solving the model eqs (8H22) with time derivatives of all dynamical equations set to zero. Equation (13) for the solvent balance is included in the interest of generality and is simply a washout equation. In general, one solves these equations at various values of dimensionless residence time (t) in order to generate the conversion-residence time profiles for a unique set of operating conditions (Kwalik, 1988). In the interest of brevity, the details of the solution procedure are omitted. The results are organized as follows; Section 1 addresses conversion-residence time profiles for both the isothermal and nonisothermal behavior of the system. Section 2 discusses the stability of selected steady states from Section 1. Section 3 presents the behavior of the MWD, and the high-conversion behavior of the system is examined in Section 4. 1. Steady-state results and discussion Isothermal reactor. An interesting feature of the CSTR isothermal polymerization of MMA is the existence of multiple steady states, As pointed out by Knorr and O’Driscoll (1970), this behavior is analogous to that observed in isothermal autocatalytic and heterogeneously catalysed reactions, and is induced by the somewhat autocatalytic “gel-effect” in polymerization reactions. Indeed, as polymerization proceeds, the reaction medium becomes increasingly viscous, thus impeding the diffusion of growing polymer molecules. Consequently, termination of the growing polymer chains does not occur as frequently as propagation, and this results in a net increase in the observed rate of polymerization. This autoacceleration can, in turn, lead to more complicated reactor behavior. Multiplicity in the isothermal polymerization of MMA can be demonstrated over ranges of important operating conditions. Figure 1 illustrates the steadystate monomer conversion as a function of dimensionless residence time for solvent volume fractions (0,) varying from 0.20 to 0.70. For each of these cases, the reactor is maintained at 340 K and the molar ratio Stationery
Isothermal
Solutions:
er
al.
of monomer to initiator in the feed stream is constant. Clearly, in this example multiple steady states are possible for solvent volume fractions reduced below 4, = 0.5. Increasing the solvent concentration makes the gel-effect less pronounced, as expected. These results are in good agreement with the simulated and experimental results for the identical system, reported by Schmidt and Ray (1981). Note also that at the lowest solvent concentration ($, = 0.2), the maximum attainable conversion is lower than that of the other curves in Fig. 1. The highly viscous reaction medium present at high conversions reduces the propagation rate and effectively limits the maximum conversion. As demonstrated, it is the autoacceleration induced by the gel-effect that gives rise to steady-state multiplicities. The gel-effect phenomenon is directly and nonlinearly dependent on the operating temperature and solvent concentration. These combined nonlinear dependences produce steady-state branches which are not everywhere differentiable or, in other words, they give rise to the “bended-knee” present on each of the steady-state branches shown in Fig. 1. However, reducing the operating temperature from 340 K produces smooth branches as illustrated in Fig. 2. Conversely, at higher temperatures, the “bended-knee” behavior is more evident. In Fig. 2, the solvent volume fraction is maintained relatively high at 4, = 0.5 in order to avoid multiplicity induced at low solvent concentrations and to illustrate the nonlinear dependence of this process on operating temperature. Thus, we attempt to isolate the influence of these two variables on reactor behavior. The resuhs presented above demonstrate the range of steady-state behavior observed for the isothermal polymerization models. An analytical treatment shows that multiplicity greater than three and isolated branches will never be expected for the isothermal process (Kwalik, 1988). Previous work by Schmidt and Ray (1981) are in agreement with this result. It is also of interest to note that these results can be qualitatively compared with those of Gray and Scott (1983) in their study of isothermal autocatalytic reactions with restricted kinetics in a CSTR.
Model
Stationary
Solutlons:
isothermal
Model
_ ___-------
T=320-360~
___---.
;p,=O.Z-0.7 T,=340~ 0.5
1.0
Residence
1.5
Time =
2.5
2.0
f3 /
3.0
t,
(4,). 4, varies from 0.2 to 0.7 in steps of 0.1 for curves A, B, C, and
F,
respectively.
M ,/I
s=
0.5
1.0
Residence
Fig. 1. Isothermal polymerization: variation of conversion with residence time as a function of solvent volume fraction D, E
0.0
95.
1.5
Time =
2.0
0 /
2.5
3.0
t,
Fig. 2. Isothermal polymerization: variation of conversion with residence time as function of reactor operating temperature, T. Tvaries from 320 to 360 K in steps of 10 K for curves A. 3. C. D and E. resoectivelv. M,lZ, = 95.
Solution polymerization
of methyl methacrylate
Nonisothermal reactors: isolas and isola-tucks. The nonisothermal polymerization reactor can be expected to demonstrate behavior similar to that of the classic nonisothermal CSTR, with the understanding that the steady-state behavior of the polymerization reaction may be further complicated by the gel-effect. Various types of behavior are illustrated as specific operating conditions are varied, namely solvent volume fraction (b,), feed and cooling water temperature, and the molar ratio of monomer feed concentration to initiator feed concentration. Figure 3 shows steady-state branches traced for a progression
of solvent
volume
fractions
ranging
from
0.45 to 0.48, while feed and cooling water temperatures
are maintained constant at 315 K. At the lowest solvent volume fraction, 4, = 0.45, a single distorted “S” curve is formed. As 4, is increased to 0.46, a small isolated branch is cleaved from the upper branch of the “S” curve. By further increasing d,T, two isolas are pinched from the “s” curve. The small isola evapol’ates and the larger isola deflates as the solvent volume fraction approaches 0.48. Increasing the solvent concentration further leaves only ‘Y-type multiplicities. These
too
are
eliminated
at
much
higher
concen-
of solvent. The steady-state reactor temperature also forms solution branches corresponding to each conversion branch. These are also illustrated in Fig. 3. In some instances, the steady-state temperature is extremely high and it is questionable whether or not the system can exist at such elevated temperatures. One could perhaps in practice make a case for operating the reactors at high pressure, if necessary, in order to trations
Stationary Soluttonw Nonlaothermrl Modal
in a CSTR
observe these steady states. Care must be exercised, however, since some rate constants and the reaction medium viscosity are highly dependent on the operating pressure (Odian, 1981). Moreover, in considerations involving the MWD, the effect of pressure on transfer agents is dependent on the agent and general conclusions are hard to draw. We, however, observe that in bulk polymerizations, where the ceiling temperature is approximately 520 K, this type of steadystate profile may possibly exist. The simultaneous occurrence of two isolated branches has not been reported previously for any reaction carried out in a CSTR. However, multiple isolas have been observed by Adomaitis and Cinar (1987) for a tubular reactor under feedback control. Additional case studies demonstrating the occurrence of multiple isolas are shown in Fig. 4. Isolated branches are formed as the temperature of the feed and cooling water are varied between 325 and 315 K, while 4, is maintained at 0.47. Tt can also be shown that this process has regions where five steady-state solutions exist for a given residence time. This behavior is illustrated in Fig. 5 where the feed stream and cooling water temperatures vary between 330 and 340 K, and the solvent volume fraction is constant at 4, = 0.43. Maintaining the feed temperature at 330 K yields multiple steady states only in the familiar “s” curve. An increase of 5 or 8 K in the feed temperature produces steady-state solution branches that form distorted “S” curves exhibiting the “bended-knee” behavior. A single isolated branch is also formed. Operating the reactor with cooling water and feed temperatures at 340 K yields an “S” curve having a severe bend and an isola tucked behind the “knee.” In the narrow region of the “isola-tuck,” five steady-state monomer conversions are possible for a single residence time. Corresponding to each of the five steady-state conversions, there is a distinct steadystate temperature, as shown in Fig. 5. The isola is absorbed into the “S” curve as the feed temperature is further increased, or, conversely, as the temperature is reduced the isola is cleaved from both the upper and
Stationary
d.5
LO
2.0
15
Residence Time =
8 /
2.5
2273
Solutions: Nonisothermal Model
9
f,
0.5
1.0
6.5
io
Residence lime
is
=
i.0
0 /
2.5
t,
Fig. 3. Nonisothermal polyme&ation: formation of multiple isolas. 4, varies from 0.45 to 0.48 in steps of 0.01 for isolas and/or “S” curves A, B, C and D, respectively. M,/I, = 95.
i.0
1.5
Residence Time =
B /
2.5
t,
Fig. 4. Nonisothermal polymerization: parametric sensitivity of reactor to changes in cooling jacket temperature (r, = T,) at fixed solvent volume fraction (I$, = 0.47). For isolas and “s” curves A, B and C, 7” equals 313, 315 and 3 18 K, respectively. Curves D and E correspond to r, having values of 320 and 325 K, respectively. M,/I, = 95.
A. K. ADEBEKUN er al. StrttlonarySolutlons:Nonisothermal
+,=0.43
0.5
1.0
1.5
Residence Time =
Solutions:
Nonlsothermal
T,=330-340K
2.0 0 /
Stationary
Model
2.5
p,=o.43
3.0
t,
c&=0.43
Model
0.0
0.5
1.0
1.5
Residence Time =
2.0
0 /
T,=340
2.5
K
3.0
t,
Fig. 6. Nonisothermal polymerization: evolution of the isola-tuck at fixed 4, as molar ratio of monomer feed concentration to initiator feed concentration (M//I,) is varied. For curve A, M,/I, = 500. Isola-tuck and “S” curve B correspond to M,/I, of 540. “S curyes and/or isolas C, D and E occur at MT/I, values of 600, 700 and 800, respect-
i,=330-34OK
ively.
b.0
d.5
i.0
1.5
Residence Time =
i.0
0 /
3.5
3.0
t,
Fig. 5. Nonisothermal polymerization: formation of the isola-tuck at fixed 4, as feed temperature (T, = 7’,) is varied, Curve A corresponds to r, = 330 K. “S” curves and isolas B and C occur at T, values of 335 and 338 K, respectively. “S’ curve and isola-tuck D occur at T, = 340 K. M,/I, = 95.
lower branches of the “s” curve. This behavior significantly contrasts with the behavior illustrated in Figs 3 and 4 showing detachment of isolas from the upper branch of the “S” curve. Once again, the high-temperature steady states may not represent realistic operating conditions. The appearance of the isola-tuck behavior, which results in a region of five steady states, is a new result. Schmidt et al. (1984) also illustrate an example of a distorted “s” curve which produces a region of five steady-state conversions. This behavior could perhaps have been a precursor to the formation of an isola in the form of an isola-tuck. It should be notkd that the unusual steady-state behavior of this system is revealed over a very narrow range of operating conditions. In the previous example, the range of operating temperatures over which the isola-tuck behavior appears is only 10 K. This case is particularly interesting because it might be of practical importance in implementing feedback control. A closed-loop control scheme could potentially move
the system
into a regime where the steady-
state behavior is much different than that of the open-
system. Hence, as shown by our results and others (Schmidt et al., 1984), care must be taken when designing control schemes for these reactors. As an illustrative example, one might be interested in controlling conversion by manipulating initiator feed concentration. The steady-state reactor behavior of the open-loop system is shown in Fig. 6. These case studies trace the solution branches for various molar loop
ratios of monomer to initiator feed concentrations, and they indicate that the isola-tuck behavior mani-
fests itself as this ratio varies. Studies are currently in progress to further assess the consequences of closedloop control on this steady-state behavior. The results from the nonisothermal polymerization of MMA correlate well with similar modeling and experimental work performed by Schmidt et al. (1984). Their work demonstrates ‘Y-shaped multiplicities in conversion and the existence of a single isola detached from the upper branch of the “s” curve. This behavior is analogous to that illustrated in Figs 3 and 4. However, the modeling results in that study do not reveal the existence of a second isola. Their selection of operating conditions may have precluded the formation of two isolated branches. 2. Reactor stabikty The question of the stability of the various steadystate profiles will now be addressed. Only representative examples depicting the stability features of the stationary solutions will be discussed. It is well known from linear stability analysis that the eigenvalues of the characteristic polynomial dictate the local stability of the stationary solution. A stationary solution is unstable if any of the eigenvalues have a positive real part. Time-periodic solutions bifurcate from the stationary solution branches at Hopf bifurcation points. Hopf points occur when a single pair of eigenvalues is purely imaginary and the remaining eigenvalues are left half plane stable. An exchange of stability along the stationary solution branch will occur at a Hopfpoint or at a turning(limit) point. Only the stability of the stationary solutions is considered here. The stable and unstable periodic behavior exhibited by this system is presented by Kwalik (1988). Changes in the stability of the stationary solutions are illustrated for two of the more interesting cases previously discussed. Figure 7 shows the stable and unstable behavior of the nonisothermal polymeriz-
Solution Stationary
b.o
Solutions:
0.5
polymerization
Nanlsothsrmal
1.0
1.5
Residence Time =
2.5
t,
Fig. 7. Nonisothermal polymerization: stability regions corresponding to curves in Fig. 4. Solid portions of curves indicate stable regions while dotted parts represent unstable steady
Stationary
0.0
0.5
Solutions:
1.0
states.
Nonisothermal
1.5
Residence Time =
2.0
0 /
Model
25
methacrylate
in a CSTR
2275
trast, Schmidt et al. (1984) indicate when five steady states exist in monomer conversion, possibly three of the states are stable.
Model
2.0
f3 /
of methyl
3.0
tr
Fig. 8. Nonisothermal polymerization: stability plot corresponding to curves in Fig. 6. Solid lines indicate stable steady states while dotted portions represent unstable ones.
ation model as two isolas are formed. The plot corresponds to the monomer conversion solution branches presented in Fig. 4. The lower and upper branches of the “s” curve are stable, while the middle branch is unstable. Other researchers (Uppal et al., 1974) observed this same stability behavior whenever “s”shaped multiplicities were encountered. An unstable middle branch is similarly observed in this study of the isothermal model. Notice also that the stable portion of an isola, if it exists, is formed from high-conversion solutions. This should not be surprising since the isolas evolve from a stable high-conversion branch. As the isolas evaporate, their region of stability shrinks and disappears. Schmidt et al. (1984) demonstrated experimentally that stable operation can be achieved for selected points on an isolated branch. The stability of the process exhibiting the “isolatuck” behavior is shown in Fig. 8. This case is equivalent to the results illustrated in Fig. 6. Once again the region of stability on the isolated branch shrinks as the size of the isola diminishes. Note that within the fivesteady-state region only the upper and lower stationary branch points on the “s” curve are stable, so, despite the appearance of multiple solutions beyond three, there still remain only two stable stationary solutions. No cases were uncovered for this process where more than two stable states coexisted. In con-
3. MWD In this section, we examine the effects of reactor conditions on the MWD. The results presented here are for nonisothermal operation of the reactor only, and we assume complete characterization of the MWD is available from information contained in the three leading moments. As will be discussed later, our classification of the various MWD multiplicity profiles obviates the need for computing the MWD during isothermal operation. From these classifications, the MWD behavior during isothermal polymerization can be inferred. Figure 9 shows the computed solutions for fixed operating conditions at various solvent fractions (4, = 0.424.5). These results are again similar to the characteristic “S” curves of Schmidt ef nl. (1981, 1984). The curves for the NACL and polydispersity corresponding to the conditions in Fig. 9 are shown in Fig. 10, where the influence of the multiplicity in the reactor subsystem 1 on the MWD subsystem 2 is apparent. The MWD behavior is not entirely unexpected since, for a given residence time, one would expect to make different polymers at the three corresponding conversion levels. Note that, over a range of low residence times, the system with the lowest volume fraction of solvent has the highest value of NACL, ’ possibly because of the reduced tendency for chain transfer to solvent. However, at higher residence times, the reverse is true. This may be attributed to viscous effects associated with increase in conversion. It is also interesting to observe the relative insensitivity of the polydispersity to residence time variations; this has been noted by various authors (Hicks et nl., 1969). Figure 11 shows the effect of variations in the molar ratio of monomer feed concentration to initiator feed concentration at a constant solvent volume fraction. Figure 1 I, associated with Fig. 6, shows that, at a given residence time, the system with the lowest aforementioned ratio produces the shortest chains. This
Stdonary
SoWIons: Nonbothermal
Residence Time = -0 /
Model
t,
Fig. 9. Nonisothermal polymerization: conversion-residence time profile at fixed r,( = r,) as function of solvent volume fraction, 4,. For curves A, B, C, D and E, 4, varies from 0.42 to 0.50 in steps of 0.02, respectkvely. M,/I, = 500.
A. K. Stationary
Solullon~:
Nonlsotharmal
ADEBEKUN
et al.
given in eqs (21) and (22), as this is of some practical importance. These equations indicate that the NACL and polydispersity are nonlinear functions of the subsystem 1 state variables: therein lies their significance. The problem of MWD manipulation is of importance in the manufacture of polymers with a desired distribution. In order to provide a partial solution to this problem, we introduce the following notation and definitions: Let S, (eR4) define reactor subsystem 1 comprising monomer, temperature, initiator and solvent levels, and S, (E@) be the vector containing information on the MWD subsystem 2 which is assumed to be characterized by x5, p and D, or x5, x6 and x,. Specifically, the elements of S, are the zeroth moment, NACL (p) and polydispersity (D). Then we classify the MWD multiplicity as follows:
Model
Dejinition 1
8
z
, 0.0
1.0
0.5
Residence
1.5
Time
=
2.0
8 /
2.5
t,
Fig. 10. Nonisothermal polymerization: NACL and polydispersity behawor corresponding to Fig. 9.
.o
0.5
1.0
Residence
1.5
Time
2.0
=
6 /
A type 1 MWD multiplicity is said to occur if corresponding to given values of one or more elements of S,, two or more values of monomer conversion (and consequently S, ) exist.
2.5
t,
Fig. 11. Nonisothermal polymerization: NACL and polydispersity behavior corresponding to Fig. 6. observation is reasonable in that, at higher initiator levels, more chains are initiated. Remarks are in order here about the information contained in the alternative MWD representation
Dejinition 2 A type 2 MWD multiplicity is said to occur if corresponding to given values of monomer conversion, one or more elements of S,, has at least two values. These definitions are further clarified below and for the most part, it is assumed the reactor designer is more interested in values of the NACL and D. Hence, no explicit reference to the zeroth moment is made in subsequent discussions, A type 1 MWD multiplicity indicates that a polymer with specified NACL and/or D values could .be produced at two or more values of conversion. Figure 12 shows this behavior clearly, and it corresponds to the MWD-conversion plots associated with Figs 9 and 10. Hence, in practice, one might have the flexibility of operating the reactor at conversions suitable for economic operation in order to make polymer of a specified distribution. In many cases though, as is seen in Fig. 12, the variations in polydispersity may be hard to detect in practice. It appears from our simulations that type 1 MWD multiplicity is to be expected for a wide range of operating parameters and it will, in most cases, occur when the conversion-residence time ‘5” curve exists. It should be pointed out that type 1 MWD multiplicity could exist in batch polymerkations. An illustrative example can be found in the work of Achilias and Kiparissides (1987). The following question now arises: given a vaIue of conversion and a set of operating conditions, will the MWD always be uniquely determined? The answer is affirmative if type 2 multiplicities do not exist or, in other words, if, for a given set of reactor parameters, every value of conversion corresponds to one, and only one value of residence time. In effect, the existence
Solution
b.0
0.2
b.0
6.2
0.4
polymerization
of methyl
0.6
0.8
1.0
0.6
0.8
1
0
d.4 Monomer
Conversion
Fig. 12. Nonisothermal polymerization: type 1 MWD plicity corresponding to Figs 9 and 10.
8 ni
44/10-M
227-l
From the above discussion, it, however, should not be concluded that the nonexistence of an isola is sufficient to guarantee that conversion uniquely determines the MWD. Figure 14, included to further clarify this point, contains the MWD plots associated with the simulation conditions of curves D and E in Fig. 4; they indicate that, even though an isola has not formed, type 2 MWD multiplicity exists at given conversions. Furthermore, these can be seen to arise at those values of conversion which can be attained by carrying out the reaction at two or more different values or residence time. Notice again that type 1 multiplicity also exists in this plot over certain conversion ranges. With the above discussion in mind, we can as stated earlier now make a conclusion about the isothermal system. It should by now be clear that, if MWD multiplicity will occur in an isothermal reactor, it will be of necessity of type 1 origin. The relative insensitivity of the poiydispersity over a wide range of operating conditions is made more apparent from our results. It would appear that the chain propagation probability (CL)is more intimately connected to the polydispersity than earlier realized, in that both of these variables are relatively insensitive with respect to conversion, and also had virtually the same profiles (as “functions” of conversion). As a case in point, the plots of CLvs conversion associated with Fig. 14 are shown in Fig. 15, which when compared with the polydispersity curve in Fig. 14 confirms this observation. It would then appear from these plots that perhaps in order to narrow the distribution, we should create conditions in the reactor which would give rise to relatively low values of c(. With the benefit of hindsight, we, however, conclude such a policy might
Conversion
polymerization: type 2 MWD multito “s” curve and isolas “B” in Fig. 4.
of an isola will in general lead to several values of the NACL (and/or D) at a given value of conversion. The plots in Fig. 13 illustrate the type 2 MWD multiplicity for the multiple-isola case in Fig. 4, and they show that, over some range of conversion, a given value of conversion corresponds to three or five values of the MWD. One should also observe the simultaneous occurrence of both type 1 and type 2 multiplicities in these plots. CES
in a CSTR
I
Monomer
Fig. 13. Nonisothermal plicity corresponding
multi-
methacrylate
s5
Monomer
Conversion
Fig. 14. Nonisothermal polymerization: type MWD multiplicity corresponding to curves Fig. 4, respectively.
1 and type 2 D and E of
A. K. ADEBEKUN
2278
et al.
V,j =0.025
u
Tgj)(.j 4 m, p, s)
and values for a, and Tgi (j A m, p, s) are available in Schmidt and Ray (1981). We expand the exponential . term contaming x3, in eq. (25) and truncate after the first-order term to obtain
Monomer Conversion
Fig. 15. Nonisothermal
probability
+ cc,(T-
polymerization: chain propagation as function of conversion. This figure corresponds to curves D and E in Fig. 4.
(26)
St = g;(l f&x,,) where s; = cl exp (k, + klXls + k,+,
be undesirable sections.
and
this
is discussed
in subsequent
4. M WD behavior at high conversions In this section, we consider the high-conversion behavior of the system. In doing this, we obtain analytical solutions of the conversion-residence time profiles for an approximate restricted model of the system, establish the validity of the approximate model via simulations and predict the MWD behavior at arbitrarily high conversions (close to 100%). We then show from the model equations that, at these conversion levels, both. the real and approximate restricted systems have the same MWD. In using this approach, we simultaneously solve the problem of investigating the MWD behavior for solution polymerizations using the parameterization of Jaisinghani and Ray (1977). Setting the gel-effect factor gp to unity is known in most cases to be a good assumption, and this is done in the approximate model. We also assume that the heat transfer coefficient bOz is held fixed, independent of residence time (7). With this restriction, we effectively reduce the system parameterization to that of Jaisinghani and Ray (1977). From this parameterization, and using eqs (lo)-( 12), we obtain Da, = C(X,l -
~~,~l~~,lC~~s,~l~~~fX3,)3”2
x CevC+dy,- yn- W2 (1 +x~JYJI). Now a generalized
form for g, is given by
gr = c, exp [cZ V, - cg( T-
273.2)], (24)
that
exp (~1 v,) = exp (kO + klxls + Lx,,)
+ kZXZs
exp (k,x,,)
where k. = ~~(0.025 - T,-apxzsp/YI1) k, =M,,c,M,,(V,,,k, =cz T,u,lv,
exp [(c3(T-
273.2)].
(27)
Now similar to Jaisinghani’s work, substitute for x3, in eq. (23) from eq. (12), replace gr in eq. (23) by the expression in eq. (26) and squaring the final equation, we obtain Da;,?
-gg:R,(~Da,,E,,)r-g:R,(l
+kax,,)
= 0 (28)
with R, = [(Xl, - xl,)/n,,1’(~/21Sx,~-){exP -2)/U
+%JYJll.
[Xz,(Y, -
l;d
(29)
Equation (28) is a quadratic equation in 5. We now make the additional assumption that the constants c,, c2 and cj are largely determined by x1, x2 and xq in which case eq. (28) has coefficients independent of T, and the solution on physical grounds will be 7 = {s;R,qE,,+C(g;R,~~,,)'
+4gjR,(l+k,x3r)]“2~/2Da,,.
(30)
Furthermore we also show that for practical purposes, the term Ik3x,,I is in fact always much smaller than 1. We have
Assuming a maximum temperature corresponding to the ceiling temperature of bulk MMA, and even at initiator levels of 0.01 mol/l, which is reasonable in practice, we obtain for MMA: Ik3x3,1max = 0.016 < 1.
ci > 0 (i = 1, 2, 3) and it can be shown
(23)
+ k,x,,)j
y,p)/~n
(25)
Incidentally, this also justifies the expansion made in eq. (26) since Ik3x,SI.< Ik,x,,l. Some comments on eq. (30) are now in order. Equation (30) has a right-hand side essentially independent of 7 (notice that the dependence on z comes in through V/during the determination of the constants, Thus, Cl, Cl and cJ in the gel-effect correlation). effectively, eq. (30) indicates that, given the subsystem 1 vector S,, we can uniquely determine a value of r. Consequently, no isolas are expected for this system. This result is hence consistent with that of the bulk
Solution polymerization of methyl methacrylate in a CSTR polymerization model of Jaisinghani and Ray (1977) and the classical chemical reactor theory (Uppal et al., 1974). We emphasize that eq. (30) was derived based on fixing f105, setting g,, equal to unity and using the linearized gel-effect correlation for g, [eq. (26)]. Hence, we refer to the model under these conditions as the “approximate restricted” model-approximate in the sense that both gel-effect correlations used are inexact, and restricted in that B,T is fixed. In order to assess the quality of the solutions of the apprnximate restricted model, we need to compare the results obtained from solving eq. (30) with these of a model in which &7 is fixed and the rigorous expressions for gP and gt [eqs (8a) and (8b)j are employed. We refer to such a model as the rigorous restricted model. Consequently, eq. (30) was solved for a variety of conditions in which the solvent volume fraction varied from 0.2 to 0.6 and compared with the solution obtained for the rigorous restricted model. The results were virtually identical at solvent volume fraction above 0.2, and it was impossible to draw separate curves. However, at 4, = 0.2, the effect of including gP in the rigorous restricted model was apparent in that the maximum attainable conversion was limited. (This is seen in Fig. 16.) It should also be pointed out that eq. (30), together with the boundary condition T = 0 and x 1.5= X11’ enables us to solve directly for 7 at any
Stetlonory
Solutlone: Approx and RLgoroum
2279
given value .of xls_ This is in contrast to the solution procedure for the rigorous restricted model where we solve for xl8 at given values of r. The complete results (approximate
restricted
for this example are further shown in Fig. 17. Notice that only type 1 MWD multiplicities exist since, as noted earlier, we have one value ofconversion corresponding to exactly one value of z. It is also significant to point out that even though, the conversion-residence time profile for 4. = 0.5 (curve D) indicates a multiplicity of 3, according to our MWD classification scheme no MWD multiplicity exists. Finally, we observe from these plots that, at high conversion, both the polydispersity and the NACL (p) begin to drop sharply and ultimately go to values close to 1 and 2, respectively. Evidently, we come to the interesting conclusion that a narrow distribution can only be achieved at the expense of making a low molecular weight product. The reason for this not too obvious behavior at higher conversions can be attributed to the drop in chain propagation probability (a) which can be seen to approach zero in the limit as xls approaches zero. In this situation, eqs (15) and (16) and the NACL-polydispersity eqs (21) and (22) indeed indicate this result. In effect, we have demonstrated that, irrespective of other reactor parameters, at high conversions, because a -+ 0, the NACL (p) and polydispersity D approach the above limiting values. This result is then seen to be valid for the more general model (with fl.7 varying) considered in Sections 1-3. model)
UWD
Multlpfklty
: ?P*
1
1
0.5
1.0
Residence
2.0
1.5
Time =
0 /
2.5
t,
nonisothermal model: conversionFig. 16. Restricted residence time profiles as function of solvent volume fraction, c$,, for approximate restricted model (top) and rigorous restricted model (bottom). In both plots, curves A, B, C, D and E correspond to 4, varying from 0.2 to 0.6 in steps ofO.1, respectively. M,/l, = 500.
0.0
Monomer
Conversion
Fig. 17. Restricted nonisothermal model:
corresponding
to approximate restricted M,/I, = 500.
MWD behavior model in Fig. 16.
A. K.
2280
ADEBEKCTN
et al.
jacket temperature feed temperature glass transition temperature for species j [j P monomer (m), solvent (s), polymer (P)] volume of reactor total free volume specific free volume contributions of species j [j a monomer (m), solvent (s), polymer
CONCLUSIONS
We have presented in this paper predictions of new steady-state behavior in continuous solution polymerization reactors. The issues relating to the existence and stability of multiple isolas and isola-tucks have been discussed. In this study a maximum of two stable steady states were found. Finally, a broad classification of the various kinds of multiplicity patterns possible in the molecular weight distribution has also been introduced as a means towards resolving the manipulation problem during polymer manufacture.
V Vf 6)
(P)l dimensionless live-polymer concentration dimensionless reactor state variable (i= 1,2, 7) steady-state value of xi dimensionless monomer feed concentration dimensionless jacket temperature dimensionless initiator feed concentration dimensionless solvent feed concentration
W Xi
NOTATION
4
GP D
Daj Ej f gr h
AH I If 4 kf kfs 4 4
k tc k td k2 k; k;, k; k:,, k:,, km k;, M Mf Mro Mwi M wm M, P 4 R t t,
T
heat transfer area of reactor specific heat capacity of reactor contents polydispersity Damkohler n.umber, ‘j defined in text activation energy, j defined in text initiator efficiency gel-effect factor heat transfer coefficient to cooling jacket heat of reaction initiator concentration in reactor, gmol/l initiator feed concentration gmol/l dissociation rate constant of initiator rate constant for chain transfer to monomer rate constant for chain transfer to solvent propagation rate constant overall termination fate constant rate constant for termination by combination rate constant for termination by disproportionation frequency factor for dissociation reactor of initiator frequency factor for chain transfer reaction to monomer frequency factor for propagation reaction frequency factor for overall termination frequency factor for termination by combination reaction at zero conversion frequency factor for termination by disproportionation reaction at zero conversion overall termination rate constant at zero conversion frequency factor for overall termination rate constant at zero conversion monomer concentration in reactor monomer feed concentration reference monomer feed concentration at beginning of simulation molecular weight of initiator molecular weight of monomer molecular weight of solvent concentration of live polymer volumetric flow rate universal gas constant time reference time reactor temperature
Greek u
letters
B
chain propagation probability coefficients of volumetric expansion for species j (j 9 m, P. s) dimensionless heat transfer coefficient
P 0 20, 4, 4 P Pi, Pmr P. Z E R”
(= I&T) number average chain length residence time zeroth, first and second moments of dead polymer density of reacting medium density of initiator, monomer and solvent, respectively dimensionless residence time belongs to real space of dimension n
aj
Subscripts
dissociation transfer to monomer propagation transfer to solvent termination combination termination disproportionation monomer temperature initiator solvent zeroth moment first moment second rnument
: P
s
tc td
1 2 3 4
5 6 7
REFERENCES
Achilias, D. and Kiparissides, C., 1987, Mathematical modeling
of
thermal copolymerization of styrene/pPolym. Proc. Engng 5, 23-36. R. and Char, A., 1987. Proc. Am. Control Cant
methylstyrene. Adomaitis,
1419-1424. Ark, R. and Amundson, N. R., 1958a, An analysis ofchemical reactor stability and control-I. Chem. Engng Sci. 7, 121-131. Aris, R. and Amundson, N. R., 1958b, An analysis of chemical reactor stability and control-II. Chem. Engng Sci. 7, 132-147.
Solution
polymerization
of methyl
Balakotaiah, V. and Luss, D., 1981, Analysis of the multiplicity patterns of a CSTR. Chem. Engng Commun. 13, 111-132. Balaraman, K. S., Kulkarni, B. D. and Mashelkar, R. A., i982, Multiplicity of states in continuous stirred copolymerization reactors: its existence and cunsequznces. Chem. Engng Commun. 16, 34%360. Balaraman, K. S., Kulkarni, B. D. and Mashelkar, R. A., 1983, Bulk copolymerization of styrene-acryhc esters: some analysis and design considerations. Polym. Engng Sci. 23, 719-725. Bilous, 0. and Amundson, N. R., 1955, Chemical reactor stability and sensitivity. A.2.CL.E. J. 1, 513-521. Brandrup, J. and Immergut, E. H., 1975, Polymer Handbook, 2nd Edition. John Wiley, New York. Gilles, E. D., 1986, Some new approaches for controlling complex processes in chemical engineering, in Proc. Third Jnt. Conf Chem. Process ControI(Edited by M. Morari and T. J. MacAvoy), pp. 689-747. Elsevier, Amsterdam. Gray, P. and Scott, S. K., 1983, Autocatalytic reactions in the isothermal continuous stirred tank reactor. Chem. Engng Sci. 38, 2943. Hamer, J. W., Akramov, T. A. and Ray, W. H., 1981, The dynamic behavior of continuous polymerization rcactors-II. Chem. Engng Sci. 36, 1897-1913. Hicks, J., Mohan, A. and Ray, W. H., 1969, The optimal control of polymerization reactors. Can. J. them. Engng 47. 590-597. Hlavacek, V., Kubicek, M. and Jelinek, J., 1970, Stability and oscillatory behavior of the CSTR. Chem. Engng Sci. 25, 1441-1461. Hoftyzer, P. J. and Zwletering, T. N., 1961, The characteristics of a homogenized reactor for the polymerization of ethylene. Chem. Engng Sci. 14,241-251. Jaisinghani, R. and Ray, W. H., 1977, On the dynamic behavior of a class of homogeneous continuous stirred tank polymerization reactors. Chem. Engng Sci. 32, 811-825. Knorr, R. S. and O’Driscoll, K. F. J., 1970, Multiple steady states, viscosity and high conversion in continuous freeradical polymerization. J. appi. Polym. Sci. 14,2683-2696.
methacrylate
in a CSTR
2281
Kwalik, K. M., 1988, Bifurcation characteristics in closedloop polymerization reactors. Ph.D. thesis, Georgia Institute of Technology. Odian, G., 1981, Principles of Polymerization. John Wiley, New York. Poore, A. B., 1973, A model equation arising from chemical reactor theory. Archs ration. Mech. Analysis 52, 358-388. Raz6n, L. F. and Schmitz, R. A., 1987, Multiplicities and instabilities m chermcally reacting systems-a review. Chem. Engng Sci. 42, 1005-1047. Rodriguez, F., 1982, Principles of Polymer Systems, 2nd Edition. McGraw-Hill, New York. Ross, R. T. and Laurence, R. L., 1976, Gel effect and free volume in the bulk polymerization of methyl methacrylate. A.1.Ch.E. Symp. Ser. 72, 74-79. Schmidt, A. D., Clinch, A. B. and Ray, W. H., 1984, The dynamic behavior of continuous polymerization reactors-111. Chem. Engng Sci. 39, 419-432. Schmidt, A. D. and Ray, W. H., 1981, The dynamic behavior of continuous polymerization reactors--I. Chem. Erqng Sci. 36, 1401-1410. Schuler, S. and Suzhen, Z., 1985, Real-time estimation of the chain length distribution of a polymer reactor. Chem. Engng sci. 40, 1891-1904. Tanner, B. M., 1985, Optimal control of the molecular weight distribution in a continuous stirred tank reactor. MS. thesis, Georgia Institute of Technology. Tanner, B. M., Adebekun, A. K. and Schork, F. I., 1987, Feedback control of molecular weight distribution during continuous polymerization. Polym Proc. Engng 5,75-l 18. Tulig, T. J. and Tirrel, M., 1981, Toward a molecular theory of the Trommsdorff effect. Macromolecules 14, 1501-1511. Uppal, A., Ray, W. H. and Poore, A. B., 1974, On the dynamic behavior of continuous stirred tank reactors. Chem. Engng Sci. 29,967-985. Uppal, A., Ray, W. H. and Poore, A. B., 1976, The classification of the dynamic behavior of continuous stirred tank reactors-influence of reactor residence time. Chem. Engng Sci. 31, 205-214. Van Heerden, C., 1953, Autothermic processes. lnd. Engng Chem. 45, 1242-1247.
CQOS-2509/89 $3 uo + 0.M) 0 1989 Pergamon Press pk
STEAbY-STATE MULTIPLICITY DURING SOLUTION POLYMERIZATION OF METHYL METHACRYLATE IN A CSTR A. K. ADEBEKUN, K. M. KWALIK and F. J. SCHORK’ School of Chemical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0100, (Receioed 14.April 1988; accepted 7 December
U.S.A.
1988)
Abstract- This paper examines the steady-state behavior of a CSTR during solution polymerization of methyl methacrylate. Simulation studies reveal the hitherto unreported existence of muhiple isolas and unusual multiplicity features, arising from isola evolution for some regions of reactor parameter space. The stability ofsome of these unusual steady states is analyzed. The influence of the reactor operating conditions on the molecular weight distribution (MWD) is also investigated and the various multiplicity features associated with the MWD are reported. The behavior of the reactor at very high conversion (> 95%) and the associated MWD profiles are presented. Finally, the practical consequences of these phenomena on reactor control and MWD manipulation are further discussed.
INTRODUCTION
The dynamic and steady-state behavior of CSTRs has received a great deal of attention in the literature (Van Heerden, 1953; Aris and Amundson, 1958a, b; Bilous and Amundson, 1955; Poore, 1973). These reactors, because of their inherent nonlinearities arising from complex kinetics, temperature effects and heat dissipation constraints, exhibit steady-state multiplicity and pose difficult control problems. Reactor behavior arising from nonisothermal operation for certain restricted kinetics has been well studied by Poore (1973), Uppal et al. (1974, 1976) and Balakotaiah and Luss (1981). The multiplicity features of some other reactions and the unusual steady-state conversion profiles arising from such have also been investigated by several workers including Gray and Scott (1983) and Hlavacek et al. (1970). The review paper by Raz6n and Schmitz (1987) is an excellent reference for other related work. Polymerization in a CSTR presents a formidable challenge to the reaction engineer, and, in contrast to the classical chemical reactors, the existing literature on polymerization reactors is not quite as extensive. Hoftyzer and Zwietering (1961) studied the multiplicity phenomena in a bulk polymerization reactor and concluded that as many as five steady states could exist for certain reactor parameters. Perhaps, the first systematic investigation of polymerization reactor dynamics was the work of Jaisinghani and Ray (1977), in which the authors studied the continuous bulk polymerization of methyl methacrylate (MMA). They found as many as three steady states were possible for a wide range of operating conditions. Limit cycle behavior was found for some combinations of reactor parameters. Balaraman et al. (1982) also reported the existence of three steady states in the continuous freeradical bulk copolymerization of styrene and acrylonitrile. Further work involving design considerations ‘Author
to whom correspondence
should be directed. 2269
for copolymerization reactors with recycle can be found in Balaraman et al. (1983). The experimental results of Schmidt and Ray (198 1) for the isothermal CSTR solution polymerization of MMA show the existence of three steady states during isothermal operation. The multiplicity observed in the isothermal reactor results from the autoacceleration induced
by
the
gel
effect
(Odian,
1981;
Tulig
and
The Schmidt-Ray model was further studied by Schmidt et al. (1984) and Hamer et al. (1981). In these investigations, the authors predicted and confirmed the existence of isola-type multiplicities. Some simulation results corresponding to five steady states were reported, and a parametric analysis of the system was also carried out. It is of interest to note that in all these investigations, the existence of more than one isola has never been reported. Moreover, in none of these studies, has the molecular weight distribution (MWD) been discussed. However, the influence of the MWD on polymer end-use is of great significance and should not be overlooked. Hence, in this investigation, we carry out a steadystate simulation of MMA solution polymerization in a CSTR and present results showing some unusual behavior. We demonstrate the existence of multiple isolas for certain reactor operating conditions, and show the unusual feature of the isola-tuck, which arises from a shift in isola origin and leads to a region of five steady states. We examine the effects on the MWD and show the importance of the chain propagation probability at very high conversions. From this viewpoint, we reestablish the fact that, for design purposes, when the MWD is of interest, operating a single CSTR might not be desirable when operating close to complete conversion. Finally, we present the multiplicity patterns in the MWD and justify the existence of these by clarifying the link between the MWD moment equations and the NACL-polydispersity representations of Tanner et al. (1987). Tirrel,
1981).
A.
2270 REACTOR
I (VJ > 0.05)
MODEL
The polymerization of MMA via a free-radical scheme with ethyl acetate solvent and benzoyl peroxide initiator is assumed. The kinetic scheme includes reactions for chain transfer to solvent and monomer. By invoking the quasi-steady-state approximation, long-chain hypothesis and assuming constant density, the following dynamical equations can be derived (Schmidt and Ray, 1981; Tanner, 1985): Vk,MP
V$=y(“,-h+
V,C~$=ypC,(T,-
et al.
K. ADEBEKUN
“=
V;=q(I,-I)-
V
0
T
(21
Vk,I
(3)
-42
M, M
-l, -
MfO
MD
A,
12
P
MS0
M,o
M,a
X5=-.-,X6=-,X,=-,W=-
If
S __
-,_$Z-,XqZ
X2=
T,)
M,o 1
X 2e =
MfO
E,
ED
Ed
Y~=RT,Y1=E,Yd=E.Y,d=E. = -qqA,
+ V[K,M
+ k,P+
V%
= -+I,
I
kfsSJaP
+ + Vk,, P2
P
+ k,,p+
4 exp-
h&r A= ~
k,S)
x(2cr-cr’)+K,,P]P/(l-a)
PC, V “=k6,exp
(6)
4 =k:, ev V[(k,M+k,,P+k,,S) x (a3 -
Da, =&
(7)
CL= k, M/(k, M + k, M + k,S co 1 n’M,
A-k=
Da,-,=QexP
+ k, P)
Dardo
(k=O,
p= c
ev (-_yJ
exP (-Y,JM~~L = kb ew
ydyl
Dad,,= tlDa,,
(-yIyp)M,Otr
Da,, = k,, Ml, t., Darco = k:,, exp (-_y,~,,)M,,t, t-YJY,)MJoL
=4,,
exp
(-~Yp~td)MJ,~,.
1, 2)
n=2
2fkJ __ k,
P
(-y,)M,,
(-Y,Y,)/~~~
Dar0 =t&Da,
3a2 + 4a) + k,,P
x(a+2)]P/(1-a)2
where
Et*
P
(5)
+ V[(k,M
V%=-L$~+
(9)
M
t/t,
e=-,T=-,f--,x,=-, 4 tr
(1)
X
V%
< 0.05).
k,, is the termination rate constant corresponding to zero conversion with k,, similarly defined. V, is the free volume, calculated from the volume fractions of monomer, polymer and solvent in the reactor. Similar to Jaisinghani and Ray (1977) and Wppal et al. (1976) the following dimensionless variables are introduced:
T)+(-AH)Vk,MP
- hA,( T-
(V,
7.1 x 10e5 exp(171.53Vf)
A list of kinetic constants is shown in Table 1. The symbols t, and M,, represent arbitrary reference values of residence time and monomer feed concentration, respectively, and, hence, the dimensionless model equations are
‘.’ >
P is the total concentration
of live radicals and k, (= k,, termination rate constant. The ith moment of the dead-polymer MWD is denoted by ;ii (i=O, 1, 2), and M is the monomer concentration. T, I and S are the reactor temperature, initiator concentration and solvent concentration, respectively. The rate constants with the exception of k,= are assumed to follow an Arhenius dependence on temperature. The correlations for the gel effect (gC= k,/k,,, gp = k,/k,) as proposed by Ross and Laurence (1976) and extended by Schmidt and Ray (1981) have been employed. The relevant equations are + kid) is the overall
0.10575exp[17.15Vf-O.O1715(T-273.2)] I’, > CO.1856 - 2.965 x lo-+( T-
dx, p=x dt’ dxz dt’=
dx, -= dt'
@a) 273.2)]
273.2)]
- x1 - gpDap Wx, E,
(10)
--x,+Bg,Da,Wx,y,E,--B,z(x,-x,,) (11)
2.3 x lo-“exp(75V,) V/ Q CO.1856 - 2.965 x 10-4( T-
”
(8b)
xjf
-x3
-Dadx3&
(12)
in a CSTR
Solution polymerization of methyl methacrylate
2271
Table 1.
Kinetic rate constants Literature
kj
value (mol, 1, s)
Reference1
1.69 x lO’*exp (-30,OOO/RT) 4.92 x 105exp (-4353/RT) 9.80 x lO’exp(-7Ol/RT)
kd k,
k 10
4.92 exp (-4353JRT) 0.091 a.23
k, k/s k,lk,,
Brandrup
and Immergut (1975)
Reactor and reactor medium constants j” =0.5 h = 135 cal/m2-sK A, = 2.8 m2 p = 1038 g/l
Rodriguez (1982) Jaisinghani and Ray (1977) Jaisinghani and Ray (1977)
C, = 0.4 Cal/g-K
(-AH)=
13.8 kcal/gmol
‘Except where noted, values for the kinetic rate constants are available in Schmidt and Ray (1981).
dx, -= dt’ dx, p= dt’
dxs dt’=
X4f
-x5
-
(13)
x4
-+ [(Da/x, Ex, + Da,+
+ gt Da,d @‘&,d )
x (a W)] + 0.5 W’g, DateE,,
(14)
define
-x6
+ [(Da/x,
+ g,Da,
z2 = St D% L,
WExtd
(19) W)
then
dw - = Z, [2-a-p(1 (15)
dx,
Z, = Da/ x, E,, + Da,x,
E,, + Da,x,
+ grDa,,WE,,,)(2~-aaZ)
dr’=
refer to the number average chain length (NACL or /I) and the polydispersity (D) of the MWD. It can be shown that these variables have the following dynamical equations (Tanner et al., 1987):
-x,
+ [(Da,xl
dt’
dD -= dt’
- 3a2 + 4~)
+(91Da,=E,,,W)(a+2)]------
-a)]
+Z+;(:-l)]}&
E,, + Da,x,
+ g,Da,, WE,,)(cr3
(
W (1-a)2
(16)
(21)
-3a+4)+pD(l-a)*
-
and a = Qa,xl
~,l(g,Da,~,x~
+ Da/%x,
+ &Da, WE,, + Da,x,)
(17)
=exPCx2/(1+~d~Jl Exj=exPCYjXz/(1+ xzhp)I-
(184
where E,
Daj = Dajoz [j 4 = (f; d, s, tc, td etc.)] as appropriate.
(18b) Vc)
Note that the system is parameterized as a function of the dimensionless residence time (2). This is evident from eq. (18~). We shall in subsequent sections have occasion to
Equations (8)-(22) represent the dynamic model of the MMA reactor. As is often done in the estimation literature (Schuler and Suzhen, 1985; Gilles, 1986), it is convenient to view the process model as two subsystems in series. Subsystem 1 corresponds to the monomer, temperature, initiator and solvent levels while subsystem 2 corresponds to the MWD components represented by rl,, il, and 1, or, equivalently, h,, p and D. The need for this partition will become apparent later.
2272
A. K.
The
steady-state
ADEBEKUN
AND DISCUSS[ON
RESULTS
results
for
the system can be computed by solving the model eqs (8H22) with time derivatives of all dynamical equations set to zero. Equation (13) for the solvent balance is included in the interest of generality and is simply a washout equation. In general, one solves these equations at various values of dimensionless residence time (t) in order to generate the conversion-residence time profiles for a unique set of operating conditions (Kwalik, 1988). In the interest of brevity, the details of the solution procedure are omitted. The results are organized as follows; Section 1 addresses conversion-residence time profiles for both the isothermal and nonisothermal behavior of the system. Section 2 discusses the stability of selected steady states from Section 1. Section 3 presents the behavior of the MWD, and the high-conversion behavior of the system is examined in Section 4. 1. Steady-state results and discussion Isothermal reactor. An interesting feature of the CSTR isothermal polymerization of MMA is the existence of multiple steady states, As pointed out by Knorr and O’Driscoll (1970), this behavior is analogous to that observed in isothermal autocatalytic and heterogeneously catalysed reactions, and is induced by the somewhat autocatalytic “gel-effect” in polymerization reactions. Indeed, as polymerization proceeds, the reaction medium becomes increasingly viscous, thus impeding the diffusion of growing polymer molecules. Consequently, termination of the growing polymer chains does not occur as frequently as propagation, and this results in a net increase in the observed rate of polymerization. This autoacceleration can, in turn, lead to more complicated reactor behavior. Multiplicity in the isothermal polymerization of MMA can be demonstrated over ranges of important operating conditions. Figure 1 illustrates the steadystate monomer conversion as a function of dimensionless residence time for solvent volume fractions (0,) varying from 0.20 to 0.70. For each of these cases, the reactor is maintained at 340 K and the molar ratio Stationery
Isothermal
Solutions:
er
al.
of monomer to initiator in the feed stream is constant. Clearly, in this example multiple steady states are possible for solvent volume fractions reduced below 4, = 0.5. Increasing the solvent concentration makes the gel-effect less pronounced, as expected. These results are in good agreement with the simulated and experimental results for the identical system, reported by Schmidt and Ray (1981). Note also that at the lowest solvent concentration ($, = 0.2), the maximum attainable conversion is lower than that of the other curves in Fig. 1. The highly viscous reaction medium present at high conversions reduces the propagation rate and effectively limits the maximum conversion. As demonstrated, it is the autoacceleration induced by the gel-effect that gives rise to steady-state multiplicities. The gel-effect phenomenon is directly and nonlinearly dependent on the operating temperature and solvent concentration. These combined nonlinear dependences produce steady-state branches which are not everywhere differentiable or, in other words, they give rise to the “bended-knee” present on each of the steady-state branches shown in Fig. 1. However, reducing the operating temperature from 340 K produces smooth branches as illustrated in Fig. 2. Conversely, at higher temperatures, the “bended-knee” behavior is more evident. In Fig. 2, the solvent volume fraction is maintained relatively high at 4, = 0.5 in order to avoid multiplicity induced at low solvent concentrations and to illustrate the nonlinear dependence of this process on operating temperature. Thus, we attempt to isolate the influence of these two variables on reactor behavior. The resuhs presented above demonstrate the range of steady-state behavior observed for the isothermal polymerization models. An analytical treatment shows that multiplicity greater than three and isolated branches will never be expected for the isothermal process (Kwalik, 1988). Previous work by Schmidt and Ray (1981) are in agreement with this result. It is also of interest to note that these results can be qualitatively compared with those of Gray and Scott (1983) in their study of isothermal autocatalytic reactions with restricted kinetics in a CSTR.
Model
Stationary
Solutlons:
isothermal
Model
_ ___-------
T=320-360~
___---.
;p,=O.Z-0.7 T,=340~ 0.5
1.0
Residence
1.5
Time =
2.5
2.0
f3 /
3.0
t,
(4,). 4, varies from 0.2 to 0.7 in steps of 0.1 for curves A, B, C, and
F,
respectively.
M ,/I
s=
0.5
1.0
Residence
Fig. 1. Isothermal polymerization: variation of conversion with residence time as a function of solvent volume fraction D, E
0.0
95.
1.5
Time =
2.0
0 /
2.5
3.0
t,
Fig. 2. Isothermal polymerization: variation of conversion with residence time as function of reactor operating temperature, T. Tvaries from 320 to 360 K in steps of 10 K for curves A. 3. C. D and E. resoectivelv. M,lZ, = 95.
Solution polymerization
of methyl methacrylate
Nonisothermal reactors: isolas and isola-tucks. The nonisothermal polymerization reactor can be expected to demonstrate behavior similar to that of the classic nonisothermal CSTR, with the understanding that the steady-state behavior of the polymerization reaction may be further complicated by the gel-effect. Various types of behavior are illustrated as specific operating conditions are varied, namely solvent volume fraction (b,), feed and cooling water temperature, and the molar ratio of monomer feed concentration to initiator feed concentration. Figure 3 shows steady-state branches traced for a progression
of solvent
volume
fractions
ranging
from
0.45 to 0.48, while feed and cooling water temperatures
are maintained constant at 315 K. At the lowest solvent volume fraction, 4, = 0.45, a single distorted “S” curve is formed. As 4, is increased to 0.46, a small isolated branch is cleaved from the upper branch of the “S” curve. By further increasing d,T, two isolas are pinched from the “s” curve. The small isola evapol’ates and the larger isola deflates as the solvent volume fraction approaches 0.48. Increasing the solvent concentration further leaves only ‘Y-type multiplicities. These
too
are
eliminated
at
much
higher
concen-
of solvent. The steady-state reactor temperature also forms solution branches corresponding to each conversion branch. These are also illustrated in Fig. 3. In some instances, the steady-state temperature is extremely high and it is questionable whether or not the system can exist at such elevated temperatures. One could perhaps in practice make a case for operating the reactors at high pressure, if necessary, in order to trations
Stationary Soluttonw Nonlaothermrl Modal
in a CSTR
observe these steady states. Care must be exercised, however, since some rate constants and the reaction medium viscosity are highly dependent on the operating pressure (Odian, 1981). Moreover, in considerations involving the MWD, the effect of pressure on transfer agents is dependent on the agent and general conclusions are hard to draw. We, however, observe that in bulk polymerizations, where the ceiling temperature is approximately 520 K, this type of steadystate profile may possibly exist. The simultaneous occurrence of two isolated branches has not been reported previously for any reaction carried out in a CSTR. However, multiple isolas have been observed by Adomaitis and Cinar (1987) for a tubular reactor under feedback control. Additional case studies demonstrating the occurrence of multiple isolas are shown in Fig. 4. Isolated branches are formed as the temperature of the feed and cooling water are varied between 325 and 315 K, while 4, is maintained at 0.47. Tt can also be shown that this process has regions where five steady-state solutions exist for a given residence time. This behavior is illustrated in Fig. 5 where the feed stream and cooling water temperatures vary between 330 and 340 K, and the solvent volume fraction is constant at 4, = 0.43. Maintaining the feed temperature at 330 K yields multiple steady states only in the familiar “s” curve. An increase of 5 or 8 K in the feed temperature produces steady-state solution branches that form distorted “S” curves exhibiting the “bended-knee” behavior. A single isolated branch is also formed. Operating the reactor with cooling water and feed temperatures at 340 K yields an “S” curve having a severe bend and an isola tucked behind the “knee.” In the narrow region of the “isola-tuck,” five steady-state monomer conversions are possible for a single residence time. Corresponding to each of the five steady-state conversions, there is a distinct steadystate temperature, as shown in Fig. 5. The isola is absorbed into the “S” curve as the feed temperature is further increased, or, conversely, as the temperature is reduced the isola is cleaved from both the upper and
Stationary
d.5
LO
2.0
15
Residence Time =
8 /
2.5
2273
Solutions: Nonisothermal Model
9
f,
0.5
1.0
6.5
io
Residence lime
is
=
i.0
0 /
2.5
t,
Fig. 3. Nonisothermal polyme&ation: formation of multiple isolas. 4, varies from 0.45 to 0.48 in steps of 0.01 for isolas and/or “S” curves A, B, C and D, respectively. M,/I, = 95.
i.0
1.5
Residence Time =
B /
2.5
t,
Fig. 4. Nonisothermal polymerization: parametric sensitivity of reactor to changes in cooling jacket temperature (r, = T,) at fixed solvent volume fraction (I$, = 0.47). For isolas and “s” curves A, B and C, 7” equals 313, 315 and 3 18 K, respectively. Curves D and E correspond to r, having values of 320 and 325 K, respectively. M,/I, = 95.
A. K. ADEBEKUN er al. StrttlonarySolutlons:Nonisothermal
+,=0.43
0.5
1.0
1.5
Residence Time =
Solutions:
Nonlsothermal
T,=330-340K
2.0 0 /
Stationary
Model
2.5
p,=o.43
3.0
t,
c&=0.43
Model
0.0
0.5
1.0
1.5
Residence Time =
2.0
0 /
T,=340
2.5
K
3.0
t,
Fig. 6. Nonisothermal polymerization: evolution of the isola-tuck at fixed 4, as molar ratio of monomer feed concentration to initiator feed concentration (M//I,) is varied. For curve A, M,/I, = 500. Isola-tuck and “S” curve B correspond to M,/I, of 540. “S curyes and/or isolas C, D and E occur at MT/I, values of 600, 700 and 800, respect-
i,=330-34OK
ively.
b.0
d.5
i.0
1.5
Residence Time =
i.0
0 /
3.5
3.0
t,
Fig. 5. Nonisothermal polymerization: formation of the isola-tuck at fixed 4, as feed temperature (T, = 7’,) is varied, Curve A corresponds to r, = 330 K. “S” curves and isolas B and C occur at T, values of 335 and 338 K, respectively. “S’ curve and isola-tuck D occur at T, = 340 K. M,/I, = 95.
lower branches of the “s” curve. This behavior significantly contrasts with the behavior illustrated in Figs 3 and 4 showing detachment of isolas from the upper branch of the “S” curve. Once again, the high-temperature steady states may not represent realistic operating conditions. The appearance of the isola-tuck behavior, which results in a region of five steady states, is a new result. Schmidt et al. (1984) also illustrate an example of a distorted “s” curve which produces a region of five steady-state conversions. This behavior could perhaps have been a precursor to the formation of an isola in the form of an isola-tuck. It should be notkd that the unusual steady-state behavior of this system is revealed over a very narrow range of operating conditions. In the previous example, the range of operating temperatures over which the isola-tuck behavior appears is only 10 K. This case is particularly interesting because it might be of practical importance in implementing feedback control. A closed-loop control scheme could potentially move
the system
into a regime where the steady-
state behavior is much different than that of the open-
system. Hence, as shown by our results and others (Schmidt et al., 1984), care must be taken when designing control schemes for these reactors. As an illustrative example, one might be interested in controlling conversion by manipulating initiator feed concentration. The steady-state reactor behavior of the open-loop system is shown in Fig. 6. These case studies trace the solution branches for various molar loop
ratios of monomer to initiator feed concentrations, and they indicate that the isola-tuck behavior mani-
fests itself as this ratio varies. Studies are currently in progress to further assess the consequences of closedloop control on this steady-state behavior. The results from the nonisothermal polymerization of MMA correlate well with similar modeling and experimental work performed by Schmidt et al. (1984). Their work demonstrates ‘Y-shaped multiplicities in conversion and the existence of a single isola detached from the upper branch of the “s” curve. This behavior is analogous to that illustrated in Figs 3 and 4. However, the modeling results in that study do not reveal the existence of a second isola. Their selection of operating conditions may have precluded the formation of two isolated branches. 2. Reactor stabikty The question of the stability of the various steadystate profiles will now be addressed. Only representative examples depicting the stability features of the stationary solutions will be discussed. It is well known from linear stability analysis that the eigenvalues of the characteristic polynomial dictate the local stability of the stationary solution. A stationary solution is unstable if any of the eigenvalues have a positive real part. Time-periodic solutions bifurcate from the stationary solution branches at Hopf bifurcation points. Hopf points occur when a single pair of eigenvalues is purely imaginary and the remaining eigenvalues are left half plane stable. An exchange of stability along the stationary solution branch will occur at a Hopfpoint or at a turning(limit) point. Only the stability of the stationary solutions is considered here. The stable and unstable periodic behavior exhibited by this system is presented by Kwalik (1988). Changes in the stability of the stationary solutions are illustrated for two of the more interesting cases previously discussed. Figure 7 shows the stable and unstable behavior of the nonisothermal polymeriz-
Solution Stationary
b.o
Solutions:
0.5
polymerization
Nanlsothsrmal
1.0
1.5
Residence Time =
2.5
t,
Fig. 7. Nonisothermal polymerization: stability regions corresponding to curves in Fig. 4. Solid portions of curves indicate stable regions while dotted parts represent unstable steady
Stationary
0.0
0.5
Solutions:
1.0
states.
Nonisothermal
1.5
Residence Time =
2.0
0 /
Model
25
methacrylate
in a CSTR
2275
trast, Schmidt et al. (1984) indicate when five steady states exist in monomer conversion, possibly three of the states are stable.
Model
2.0
f3 /
of methyl
3.0
tr
Fig. 8. Nonisothermal polymerization: stability plot corresponding to curves in Fig. 6. Solid lines indicate stable steady states while dotted portions represent unstable ones.
ation model as two isolas are formed. The plot corresponds to the monomer conversion solution branches presented in Fig. 4. The lower and upper branches of the “s” curve are stable, while the middle branch is unstable. Other researchers (Uppal et al., 1974) observed this same stability behavior whenever “s”shaped multiplicities were encountered. An unstable middle branch is similarly observed in this study of the isothermal model. Notice also that the stable portion of an isola, if it exists, is formed from high-conversion solutions. This should not be surprising since the isolas evolve from a stable high-conversion branch. As the isolas evaporate, their region of stability shrinks and disappears. Schmidt et al. (1984) demonstrated experimentally that stable operation can be achieved for selected points on an isolated branch. The stability of the process exhibiting the “isolatuck” behavior is shown in Fig. 8. This case is equivalent to the results illustrated in Fig. 6. Once again the region of stability on the isolated branch shrinks as the size of the isola diminishes. Note that within the fivesteady-state region only the upper and lower stationary branch points on the “s” curve are stable, so, despite the appearance of multiple solutions beyond three, there still remain only two stable stationary solutions. No cases were uncovered for this process where more than two stable states coexisted. In con-
3. MWD In this section, we examine the effects of reactor conditions on the MWD. The results presented here are for nonisothermal operation of the reactor only, and we assume complete characterization of the MWD is available from information contained in the three leading moments. As will be discussed later, our classification of the various MWD multiplicity profiles obviates the need for computing the MWD during isothermal operation. From these classifications, the MWD behavior during isothermal polymerization can be inferred. Figure 9 shows the computed solutions for fixed operating conditions at various solvent fractions (4, = 0.424.5). These results are again similar to the characteristic “S” curves of Schmidt ef nl. (1981, 1984). The curves for the NACL and polydispersity corresponding to the conditions in Fig. 9 are shown in Fig. 10, where the influence of the multiplicity in the reactor subsystem 1 on the MWD subsystem 2 is apparent. The MWD behavior is not entirely unexpected since, for a given residence time, one would expect to make different polymers at the three corresponding conversion levels. Note that, over a range of low residence times, the system with the lowest volume fraction of solvent has the highest value of NACL, ’ possibly because of the reduced tendency for chain transfer to solvent. However, at higher residence times, the reverse is true. This may be attributed to viscous effects associated with increase in conversion. It is also interesting to observe the relative insensitivity of the polydispersity to residence time variations; this has been noted by various authors (Hicks et nl., 1969). Figure 11 shows the effect of variations in the molar ratio of monomer feed concentration to initiator feed concentration at a constant solvent volume fraction. Figure 1 I, associated with Fig. 6, shows that, at a given residence time, the system with the lowest aforementioned ratio produces the shortest chains. This
Stdonary
SoWIons: Nonbothermal
Residence Time = -0 /
Model
t,
Fig. 9. Nonisothermal polymerization: conversion-residence time profile at fixed r,( = r,) as function of solvent volume fraction, 4,. For curves A, B, C, D and E, 4, varies from 0.42 to 0.50 in steps of 0.02, respectkvely. M,/I, = 500.
A. K. Stationary
Solullon~:
Nonlsotharmal
ADEBEKUN
et al.
given in eqs (21) and (22), as this is of some practical importance. These equations indicate that the NACL and polydispersity are nonlinear functions of the subsystem 1 state variables: therein lies their significance. The problem of MWD manipulation is of importance in the manufacture of polymers with a desired distribution. In order to provide a partial solution to this problem, we introduce the following notation and definitions: Let S, (eR4) define reactor subsystem 1 comprising monomer, temperature, initiator and solvent levels, and S, (E@) be the vector containing information on the MWD subsystem 2 which is assumed to be characterized by x5, p and D, or x5, x6 and x,. Specifically, the elements of S, are the zeroth moment, NACL (p) and polydispersity (D). Then we classify the MWD multiplicity as follows:
Model
Dejinition 1
8
z
, 0.0
1.0
0.5
Residence
1.5
Time
=
2.0
8 /
2.5
t,
Fig. 10. Nonisothermal polymerization: NACL and polydispersity behawor corresponding to Fig. 9.
.o
0.5
1.0
Residence
1.5
Time
2.0
=
6 /
A type 1 MWD multiplicity is said to occur if corresponding to given values of one or more elements of S,, two or more values of monomer conversion (and consequently S, ) exist.
2.5
t,
Fig. 11. Nonisothermal polymerization: NACL and polydispersity behavior corresponding to Fig. 6. observation is reasonable in that, at higher initiator levels, more chains are initiated. Remarks are in order here about the information contained in the alternative MWD representation
Dejinition 2 A type 2 MWD multiplicity is said to occur if corresponding to given values of monomer conversion, one or more elements of S,, has at least two values. These definitions are further clarified below and for the most part, it is assumed the reactor designer is more interested in values of the NACL and D. Hence, no explicit reference to the zeroth moment is made in subsequent discussions, A type 1 MWD multiplicity indicates that a polymer with specified NACL and/or D values could .be produced at two or more values of conversion. Figure 12 shows this behavior clearly, and it corresponds to the MWD-conversion plots associated with Figs 9 and 10. Hence, in practice, one might have the flexibility of operating the reactor at conversions suitable for economic operation in order to make polymer of a specified distribution. In many cases though, as is seen in Fig. 12, the variations in polydispersity may be hard to detect in practice. It appears from our simulations that type 1 MWD multiplicity is to be expected for a wide range of operating parameters and it will, in most cases, occur when the conversion-residence time ‘5” curve exists. It should be pointed out that type 1 MWD multiplicity could exist in batch polymerkations. An illustrative example can be found in the work of Achilias and Kiparissides (1987). The following question now arises: given a vaIue of conversion and a set of operating conditions, will the MWD always be uniquely determined? The answer is affirmative if type 2 multiplicities do not exist or, in other words, if, for a given set of reactor parameters, every value of conversion corresponds to one, and only one value of residence time. In effect, the existence
Solution
b.0
0.2
b.0
6.2
0.4
polymerization
of methyl
0.6
0.8
1.0
0.6
0.8
1
0
d.4 Monomer
Conversion
Fig. 12. Nonisothermal polymerization: type 1 MWD plicity corresponding to Figs 9 and 10.
8 ni
44/10-M
227-l
From the above discussion, it, however, should not be concluded that the nonexistence of an isola is sufficient to guarantee that conversion uniquely determines the MWD. Figure 14, included to further clarify this point, contains the MWD plots associated with the simulation conditions of curves D and E in Fig. 4; they indicate that, even though an isola has not formed, type 2 MWD multiplicity exists at given conversions. Furthermore, these can be seen to arise at those values of conversion which can be attained by carrying out the reaction at two or more different values or residence time. Notice again that type 1 multiplicity also exists in this plot over certain conversion ranges. With the above discussion in mind, we can as stated earlier now make a conclusion about the isothermal system. It should by now be clear that, if MWD multiplicity will occur in an isothermal reactor, it will be of necessity of type 1 origin. The relative insensitivity of the poiydispersity over a wide range of operating conditions is made more apparent from our results. It would appear that the chain propagation probability (CL)is more intimately connected to the polydispersity than earlier realized, in that both of these variables are relatively insensitive with respect to conversion, and also had virtually the same profiles (as “functions” of conversion). As a case in point, the plots of CLvs conversion associated with Fig. 14 are shown in Fig. 15, which when compared with the polydispersity curve in Fig. 14 confirms this observation. It would then appear from these plots that perhaps in order to narrow the distribution, we should create conditions in the reactor which would give rise to relatively low values of c(. With the benefit of hindsight, we, however, conclude such a policy might
Conversion
polymerization: type 2 MWD multito “s” curve and isolas “B” in Fig. 4.
of an isola will in general lead to several values of the NACL (and/or D) at a given value of conversion. The plots in Fig. 13 illustrate the type 2 MWD multiplicity for the multiple-isola case in Fig. 4, and they show that, over some range of conversion, a given value of conversion corresponds to three or five values of the MWD. One should also observe the simultaneous occurrence of both type 1 and type 2 multiplicities in these plots. CES
in a CSTR
I
Monomer
Fig. 13. Nonisothermal plicity corresponding
multi-
methacrylate
s5
Monomer
Conversion
Fig. 14. Nonisothermal polymerization: type MWD multiplicity corresponding to curves Fig. 4, respectively.
1 and type 2 D and E of
A. K. ADEBEKUN
2278
et al.
V,j =0.025
u
Tgj)(.j 4 m, p, s)
and values for a, and Tgi (j A m, p, s) are available in Schmidt and Ray (1981). We expand the exponential . term contaming x3, in eq. (25) and truncate after the first-order term to obtain
Monomer Conversion
Fig. 15. Nonisothermal
probability
+ cc,(T-
polymerization: chain propagation as function of conversion. This figure corresponds to curves D and E in Fig. 4.
(26)
St = g;(l f&x,,) where s; = cl exp (k, + klXls + k,+,
be undesirable sections.
and
this
is discussed
in subsequent
4. M WD behavior at high conversions In this section, we consider the high-conversion behavior of the system. In doing this, we obtain analytical solutions of the conversion-residence time profiles for an approximate restricted model of the system, establish the validity of the approximate model via simulations and predict the MWD behavior at arbitrarily high conversions (close to 100%). We then show from the model equations that, at these conversion levels, both. the real and approximate restricted systems have the same MWD. In using this approach, we simultaneously solve the problem of investigating the MWD behavior for solution polymerizations using the parameterization of Jaisinghani and Ray (1977). Setting the gel-effect factor gp to unity is known in most cases to be a good assumption, and this is done in the approximate model. We also assume that the heat transfer coefficient bOz is held fixed, independent of residence time (7). With this restriction, we effectively reduce the system parameterization to that of Jaisinghani and Ray (1977). From this parameterization, and using eqs (lo)-( 12), we obtain Da, = C(X,l -
~~,~l~~,lC~~s,~l~~~fX3,)3”2
x CevC+dy,- yn- W2 (1 +x~JYJI). Now a generalized
form for g, is given by
gr = c, exp [cZ V, - cg( T-
273.2)], (24)
that
exp (~1 v,) = exp (kO + klxls + Lx,,)
+ kZXZs
exp (k,x,,)
where k. = ~~(0.025 - T,-apxzsp/YI1) k, =M,,c,M,,(V,,,k, =cz T,u,lv,
exp [(c3(T-
273.2)].
(27)
Now similar to Jaisinghani’s work, substitute for x3, in eq. (23) from eq. (12), replace gr in eq. (23) by the expression in eq. (26) and squaring the final equation, we obtain Da;,?
-gg:R,(~Da,,E,,)r-g:R,(l
+kax,,)
= 0 (28)
with R, = [(Xl, - xl,)/n,,1’(~/21Sx,~-){exP -2)/U
+%JYJll.
[Xz,(Y, -
l;d
(29)
Equation (28) is a quadratic equation in 5. We now make the additional assumption that the constants c,, c2 and cj are largely determined by x1, x2 and xq in which case eq. (28) has coefficients independent of T, and the solution on physical grounds will be 7 = {s;R,qE,,+C(g;R,~~,,)'
+4gjR,(l+k,x3r)]“2~/2Da,,.
(30)
Furthermore we also show that for practical purposes, the term Ik3x,,I is in fact always much smaller than 1. We have
Assuming a maximum temperature corresponding to the ceiling temperature of bulk MMA, and even at initiator levels of 0.01 mol/l, which is reasonable in practice, we obtain for MMA: Ik3x3,1max = 0.016 < 1.
ci > 0 (i = 1, 2, 3) and it can be shown
(23)
+ k,x,,)j
y,p)/~n
(25)
Incidentally, this also justifies the expansion made in eq. (26) since Ik3x,SI.< Ik,x,,l. Some comments on eq. (30) are now in order. Equation (30) has a right-hand side essentially independent of 7 (notice that the dependence on z comes in through V/during the determination of the constants, Thus, Cl, Cl and cJ in the gel-effect correlation). effectively, eq. (30) indicates that, given the subsystem 1 vector S,, we can uniquely determine a value of r. Consequently, no isolas are expected for this system. This result is hence consistent with that of the bulk
Solution polymerization of methyl methacrylate in a CSTR polymerization model of Jaisinghani and Ray (1977) and the classical chemical reactor theory (Uppal et al., 1974). We emphasize that eq. (30) was derived based on fixing f105, setting g,, equal to unity and using the linearized gel-effect correlation for g, [eq. (26)]. Hence, we refer to the model under these conditions as the “approximate restricted” model-approximate in the sense that both gel-effect correlations used are inexact, and restricted in that B,T is fixed. In order to assess the quality of the solutions of the apprnximate restricted model, we need to compare the results obtained from solving eq. (30) with these of a model in which &7 is fixed and the rigorous expressions for gP and gt [eqs (8a) and (8b)j are employed. We refer to such a model as the rigorous restricted model. Consequently, eq. (30) was solved for a variety of conditions in which the solvent volume fraction varied from 0.2 to 0.6 and compared with the solution obtained for the rigorous restricted model. The results were virtually identical at solvent volume fraction above 0.2, and it was impossible to draw separate curves. However, at 4, = 0.2, the effect of including gP in the rigorous restricted model was apparent in that the maximum attainable conversion was limited. (This is seen in Fig. 16.) It should also be pointed out that eq. (30), together with the boundary condition T = 0 and x 1.5= X11’ enables us to solve directly for 7 at any
Stetlonory
Solutlone: Approx and RLgoroum
2279
given value .of xls_ This is in contrast to the solution procedure for the rigorous restricted model where we solve for xl8 at given values of r. The complete results (approximate
restricted
for this example are further shown in Fig. 17. Notice that only type 1 MWD multiplicities exist since, as noted earlier, we have one value ofconversion corresponding to exactly one value of z. It is also significant to point out that even though, the conversion-residence time profile for 4. = 0.5 (curve D) indicates a multiplicity of 3, according to our MWD classification scheme no MWD multiplicity exists. Finally, we observe from these plots that, at high conversion, both the polydispersity and the NACL (p) begin to drop sharply and ultimately go to values close to 1 and 2, respectively. Evidently, we come to the interesting conclusion that a narrow distribution can only be achieved at the expense of making a low molecular weight product. The reason for this not too obvious behavior at higher conversions can be attributed to the drop in chain propagation probability (a) which can be seen to approach zero in the limit as xls approaches zero. In this situation, eqs (15) and (16) and the NACL-polydispersity eqs (21) and (22) indeed indicate this result. In effect, we have demonstrated that, irrespective of other reactor parameters, at high conversions, because a -+ 0, the NACL (p) and polydispersity D approach the above limiting values. This result is then seen to be valid for the more general model (with fl.7 varying) considered in Sections 1-3. model)
UWD
Multlpfklty
: ?P*
1
1
0.5
1.0
Residence
2.0
1.5
Time =
0 /
2.5
t,
nonisothermal model: conversionFig. 16. Restricted residence time profiles as function of solvent volume fraction, c$,, for approximate restricted model (top) and rigorous restricted model (bottom). In both plots, curves A, B, C, D and E correspond to 4, varying from 0.2 to 0.6 in steps ofO.1, respectively. M,/l, = 500.
0.0
Monomer
Conversion
Fig. 17. Restricted nonisothermal model:
corresponding
to approximate restricted M,/I, = 500.
MWD behavior model in Fig. 16.
A. K.
2280
ADEBEKCTN
et al.
jacket temperature feed temperature glass transition temperature for species j [j P monomer (m), solvent (s), polymer (P)] volume of reactor total free volume specific free volume contributions of species j [j a monomer (m), solvent (s), polymer
CONCLUSIONS
We have presented in this paper predictions of new steady-state behavior in continuous solution polymerization reactors. The issues relating to the existence and stability of multiple isolas and isola-tucks have been discussed. In this study a maximum of two stable steady states were found. Finally, a broad classification of the various kinds of multiplicity patterns possible in the molecular weight distribution has also been introduced as a means towards resolving the manipulation problem during polymer manufacture.
V Vf 6)
(P)l dimensionless live-polymer concentration dimensionless reactor state variable (i= 1,2, 7) steady-state value of xi dimensionless monomer feed concentration dimensionless jacket temperature dimensionless initiator feed concentration dimensionless solvent feed concentration
W Xi
NOTATION
4
GP D
Daj Ej f gr h
AH I If 4 kf kfs 4 4
k tc k td k2 k; k;, k; k:,, k:,, km k;, M Mf Mro Mwi M wm M, P 4 R t t,
T
heat transfer area of reactor specific heat capacity of reactor contents polydispersity Damkohler n.umber, ‘j defined in text activation energy, j defined in text initiator efficiency gel-effect factor heat transfer coefficient to cooling jacket heat of reaction initiator concentration in reactor, gmol/l initiator feed concentration gmol/l dissociation rate constant of initiator rate constant for chain transfer to monomer rate constant for chain transfer to solvent propagation rate constant overall termination fate constant rate constant for termination by combination rate constant for termination by disproportionation frequency factor for dissociation reactor of initiator frequency factor for chain transfer reaction to monomer frequency factor for propagation reaction frequency factor for overall termination frequency factor for termination by combination reaction at zero conversion frequency factor for termination by disproportionation reaction at zero conversion overall termination rate constant at zero conversion frequency factor for overall termination rate constant at zero conversion monomer concentration in reactor monomer feed concentration reference monomer feed concentration at beginning of simulation molecular weight of initiator molecular weight of monomer molecular weight of solvent concentration of live polymer volumetric flow rate universal gas constant time reference time reactor temperature
Greek u
letters
B
chain propagation probability coefficients of volumetric expansion for species j (j 9 m, P. s) dimensionless heat transfer coefficient
P 0 20, 4, 4 P Pi, Pmr P. Z E R”
(= I&T) number average chain length residence time zeroth, first and second moments of dead polymer density of reacting medium density of initiator, monomer and solvent, respectively dimensionless residence time belongs to real space of dimension n
aj
Subscripts
dissociation transfer to monomer propagation transfer to solvent termination combination termination disproportionation monomer temperature initiator solvent zeroth moment first moment second rnument
: P
s
tc td
1 2 3 4
5 6 7
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