2003_pinto

Chemical Engineering Science 58 (2003) 2805–2821www.elsevier.com/locate/ces

The bifurcation behavior of continuous free-radical solution looppolymerization reactors

Pr()amo A. Melo∗, Evaristo C. Biscaia Jr., Jos(e Carlos PintoPrograma de Engenharia Qu��mica/COPPE, Universidade Federal do Rio de Janeiro, Cidade Universit�aria, CP 68502, Rio de Janeiro,

RJ 21945-970, Brazil

Received 5 November 2002; received in revised form 10 February 2003; accepted 18 February 2003

Abstract

The bifurcation behavior of continuous free-radical solution loop polymerization reactors is analyzed in this work. A mathematical modelis developed in order to describe the impact of the recycling pump and other external reactor parts upon the process dynamics and stability.Stability analysis is performed using bifurcation theory and continuation methods. It is shown that under certain operational conditions asmany as seven steady states are predicted for the loop polymerization reactor. Oscillatory behavior is observed for a wide range of processparameters and onset of oscillations is observed during the transition from operation without material recycling to operation with partialrecirculation of the polymer solution. Besides, at certain constrained range of operation conditions, complex dynamics can be observed,including the onset of chaotic behavior. It is also shown that the thermal parameters of the reactor and recycling pump exert a profounde7ect upon the process stability. For this reason it is shown that oscillatory behavior is very unlikely to occur in actual industrial reactors.? 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Loop reactor; Free-radical polymerization; Bifurcation; Steady-state multiplicity; Periodic oscillations

1. Introduction

Polymerization reaction engineering is a very intriguing9eld of research and investigation for many reasons. Firstand foremost, polymerization processes di7er from otherchemical processes because they are concerned with the syn-thesis of macromolecules. Macromolecules of very di7erentchain lengths are produced in polymerization reactions andthe resulting molecular mass distribution plays a fundamen-tal role for the determination of the end-use properties ofthe polymer produced. Second, several kinetic mechanismsare available for the synthesis of the macromolecules andthe same monomer may produce di7erent polymers depend-ing on the selected reaction mechanism. Third, in some pro-cesses very small amounts of impurities, of the order of partsper million, may compromise catalysts performance even atplant site. Fourth, the reactor non linear behavior may bean important issue in many processes and should not be ne-glected during reactor design. And 9nally, polymerization

∗ Corresponding author. Tel.: +55-21-2562-8339;fax: +55-21-2562-8300.

E-mail address: [email protected] (P. A. Melo).

process engineers have to cope routinely with operationaldiAculties related to dramatic changes in the viscosity ofliquid-phase reactions, need to remove high amounts of heatreleased by reaction and the challenge to keep mixing pat-terns close to the desired ones.

As a consequence of the reasons pointed above, poly-merization reaction engineering has always been a verydynamic 9eld, where old processes are continuously re-placed by new technologies which aim at optimizing bothproduction and product quality while minimizing opera-tional costs. The free-radical polymerization technologyhas followed a similar trend. The 9rst commercial, indus-trial scale plants for poly(styrene), for example, were builtback in the 1920s and used bulk batch stirred tank reactorsto perform the reaction until completion, as removal ofmonomer residuals was not practiced at that time (Gerrens,1982). As early as 1936, BASF Aktiengesellschaft devel-oped the 9rst continuous process for poly(styrene), usinga combination of stirred tanks and tower reactors, butremoval of residual monomer was still an open matter.Process improvements included the development of a so-lution process with downstream removal of monomer andsolvent by the Dow Chemical Co. (Gerrens, 1982), and the

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2806 P. A. Melo et al. / Chemical Engineering Science 58 (2003) 2805–2821

combination of stirred tanks and tubular reactors to pro-duce crystal poly(styrene) by Crown Products, Inc. (Chen,1994).

The continuous search for new polymerization processesattracted the attention of both industry and academia toloop reactors in the 1960s, when Phillips Petroleum Co.presented the 9rst loop reactor technology for the produc-tion of polyole9ns (Ferrero & Chiovetta, 1990). Today,loop reactors are used for large-scale liquid-phase produc-tion of poly(propylene) (cf. Zacca & Ray, 1993) and is ex-pected to carry out heterogeneous gas-phase polymerizationsin the near future (De Vries & Izzo-Iammarrone, 2001).Other well-established applications of loop reactor technol-ogy, both in the scienti9c and patent literature, concern thefree-radical emulsion and suspension polymerization pro-cesses (e.g. Abad, De La Cal, & Asua, 1994; Paquet &Ray, 1994; Ara(ujo, Abad, De La Cal, Pinto, & Asua, 1999;Lanthier, 1970; Geddes, 1983; Adams, 1984).

Regarding free-radical solution polymerization processes,Lynn and Hu7 (1971) seem to be the 9rst to propose theuse of polymer recycling in tubular reactors. However,much of the work published in the literature is due toRenken and coworkers, who investigated in detail issuesregarding process performance and viability for both homo-and co-polymerization reactions in loop reactors (cf. Tien,Flaschel, & Renken, 1983; Meyer & Renken, 1990; Fleury,Meyer, & Renken, 1992; Belkhiria, Meyer, & Renken,1994). As far as the patent literature is concerned, appli-cations date back to 1980, when Montedison SPA (1980)patented a chemical reactor in the form of a closed loopwith a section comprising a bundle of tubes leading backto a circulation pump. Later on, BASF Aktiengesellschaft(Reiner, 1992) patented a chemical process for bulk poly-merization or solution polymerization of vinyl monomers inan essentially back-mixing reaction apparatus designed as aforced-circulation reactor with the forced-circulation reactordesigned as a shell-and-tube reactor. More recently, BASFAktiengesellschaft patented a process for the free-radicalcopolymerization reaction of styrene and acrylonitrile in aloop reactor (Fischer & Baumgartel, 1998).

The stability of free-radical polymerizations in stirredtank reactors has been studied extensively (see, for in-stance Melo, Sampaio, Biscaia Jr., & Pinto, 2001c). It iswell known that these reactors can present multiple steadystates, self-sustained oscillations and chaotic behavior.However, very little is known about the dynamic behav-ior of free-radical loop polymerization reactors. The mostcomprehensive study available about the dynamic behaviorof loop polymerization reactors was presented by Zaccaand Ray (1993) for Ziegler–Natta slurry polymerizations.However, in this case both the reaction mechanism and thereactor operation conditions are quite di7erent from theones normally used to perform solution free-radical poly-merizations. Besides, Zacca and Ray (1993) did not takeinto consideration the possible inKuence of external ther-mal capacitance upon the reactor dynamics, which may be

very signi9cant even at full-scale industrial facilities (Melo,Biscaia Jr., & Pinto, 2001a).

The stability of free-radical loop polymerization pro-cesses was addressed by Melo, Biscaia Jr., Pinto, & Ray(2000), who used a simpli9ed mathematical model todescribe free-radical solution polymerizations in loop reac-tors. Multiple steady states and the onset of self-sustainedperiodic oscillations were analyzed by Melo et al. (2000),but the macromixing behavior and thermal dynamics in-troduced by the recycling pump and external reactor partswere not taken into account.

In this paper, the bifurcation behavior of continuousfree-radical solution loop polymerization reactors is ana-lyzed in detail. A novel mathematical model is developedin order to describe the impact of the recycling pump andother external reactor parts upon the process dynamicsand stability. It is shown that multiple steady states arepredicted for the loop polymerization reactor and that os-cillatory behavior is observed for a wide range of processparameters. It is also shown that the thermal parameters ofthe reactor and recycling pump exert a profound e7ect uponthe process stability.

2. Reactor mathematical model

The loop polymerization reactor investigated here con-sists of a pair of continuous tubular reactors whose ends areconnected, at one side, to the inlet feed and, at the other side,to the exit stream and to a recycling pump, as presented inFig. 1. Perfect mixing is assumed at the feed point whereasthe exit stream point is assumed to be a simple splitter.

The most comprehensive study presented in the literatureabout the mathematical modeling of loop polymerizationreactors is that of Zacca and Ray (1993). Zacca and Rayinvestigated the liquid-phase polymerization of ole9ns inloop reactors and developed a detailed model to describe thedynamic behavior of this class of reactors. The mathematicalmodel of Zacca and Ray was the starting point for the modelpresented in this section. However, some modi9cations werenecessary in order to allow for a proper description of themixing patterns inside the reactor and to take into accountthermal issues appropriately, as reported by Melo, Pinto, andBiscaia Jr. (2001b) and Melo et al. (2001a).

The main di7erence between the loop reactor model of Za-cca and Ray and the loop reactor model presented here is that

Tubular Section 2

Tubular Section 1

Recycle Pump

Reactor Inlet Reactor Outlet

Fig. 1. Loop reactor diagram (after Melo et al., 2001b).

P. A. Melo et al. / Chemical Engineering Science 58 (2003) 2805–2821 2807

the recycling pump is not considered as a zero-capacitancevessel but, instead, as a piece of equipment whose behav-ior plays a fundamental part on the overall process dynam-ics. In addition to that, the external thermal capacitance ofboth the reactor tubular sections and the recycling pumpwere also taken into account. These modi9cations were mo-tivated by previous experimental and modeling results ob-tained in a lab-scale loop reactor (Melo et al., 2001a, b),that emphasized the need of adding the recycling pump intothe model in order to be able to describe appropriately themacromixing behavior of the reactor as well as its thermaldynamics.

As depicted in Fig. 1, the loop reactor considered hereis composed of three distinct reaction sections, namely, thetubular section 1, that connects the inlet feed point to theexit stream and the recycling pump; the recycling pumpitself, that recirculates part of the polymer solution and isconnected to the tubular section 2; and the tubular section2, that connects the exit of the recycling pump to the inletfeed point. The tubular sections were described accordingto the axial dispersion model (Levenspiel, 1962; Froment& Bischo7, 1990) using closed–closed boundary conditions(Danckwerts, 1953).

2.1. Mass balances

For the tubular sections, mass balances for the componenti in the reaction medium is given by

@Cji@t′

+ vzj@Cji@z′j

= Dm@2Cji@z2j′

− Ri; (1)

valid for 0¡z′j ¡Lj, whereCji is the molar concentration ofcomponent i (M for monomer and I for initiator) in sectionj; vzj is the axial Kow velocity in the tubular section j; Dmis the overall mass dispersion coeAcient; Ri is the reactionrate of component i due to polymerization; z′j is the axialcoordinate; Lj is the total length of tubular section j; and t′

is the time.The recycling pump is described as an ideal continuous

stirred tank reactor. The mass balance for component i isgiven below:

VpdCpidt′

= Q2(C1i |z′1=L1 − Cpi ) − VpRi; (2)

where Cpi is the molar concentration of component i inthe recycling pump; Vp is the internal volume of the recy-cling pump; and Q2 is the volumetric Kow rate in tubularsection 2.

For the tubular section 1, the boundary conditions aregiven by

−Dm @C1i

@z′1

∣∣∣∣z′1=0

+ vz1C1i |z′1=0 = vz2C

2i |z′2=L2 + vfCif (3)

for the inlet feed point, and

@C1i

@z′1

∣∣∣∣z′1=L1

= 0 (4)

for the exit point.For the tubular section 2, the boundary conditions are

−Dm @C2i

@z′2

∣∣∣∣z′2=0

+ vz2C2i |z′2=0 = vz2C

pi (5)

for the inlet point, and

@C2i

@z′2

∣∣∣∣z′2=L2

= 0 (6)

for the outlet point.The equations presented above may be rendered dimen-

sionless, so that the important dimensionless groups can beidenti9ed and state variables can be re-scaled appropriately.In dimensionless form, the model becomes

�@y1

i

@t+ (1 + Rec)

@y1i

@z1=

1�Pem

@2y1i

@z21− �DaRi ; (7)

valid for 0¡z1¡ 1,

(1 − �)@y2i

@t+ Rec

@y2i

@z2

=1Pem

11 − �

@2y2i

@z22− (1 − �)DaRi ; (8)

valid for 0¡z2¡ 1, and

�pdypidt

= Rec(y1i |z1=1 − ypi ) − �p DaRi ; (9)

where yji = Cji =CMf; zj = z′j=Lj; t = t′=Lvf, Rec = vz2 =vf,Pem = vfL=Dm, �=L1=L, Da=RMfL=vfCMf, Ri =Ri=RMf,and L is the total length of the tubular sections. The dimen-sionless boundary conditions are given by

− 1� Pem

@y1i

@z1

∣∣∣∣z1=0

+ (1 + Rec)y1i |z1=0

=Recy2i |z2=1 + �i; (10)

@y1i

@z1

∣∣∣∣z1=1

= 0; (11)

− 1(1 − �)Pem

@y2i

@z2

∣∣∣∣z2=0

+ Recy2i |z2=0 = Recypi (12)

and

@y2i

@z2

∣∣∣∣z2=1

= 0; (13)

where �M = 1, if i =M , and �I = CIf=CMf, if i = I .

2.2. Energy balance

The energy balance is obtained by neglecting energychanges due to expansion work, viscous Kow, external


9elds, radiation and heat of mixing. The energy balance foreach tubular section can be given by Froment and Bischo7(1990):

[�Cp + (�Cp)r]@T j

@t′+ �Cpvzj

@T j

@z′j

=Dt�Cp@2T j

@z′2j+ (OHr)RM +

4Urd

[Tc − T j]; (14)

valid for 0¡z′j ¡Lj, where T j is the reactor temperaturein tubular section j. The term �Cp in Eq. (14) stands for thethermal capacitance of the polymer solution Kowing throughtubular section j, whereas the term (�Cp)r represents theoverall thermal capacitance of the reactor externals such astube walls, 9ttings, valves, recycling pump, etc. The bound-ary conditions for the tubular section 1 are given by

−Dt�Cp @T1

@z′1

∣∣∣∣z′1=0

+ vz1�CpT1|z′1=0

=vz2�CpT2|z′2=L2 + vf�CpTf (15)

and

@T 1

@z′1

∣∣∣∣z′1=L1

= 0: (16)

For the tubular section 2 the boundary conditions are

−Dt�Cp @T2

@z′2

∣∣∣∣z′2=0

+ vz2�CpT2|z′2=0 = vz2�CpT

p (17)

and

@T 2

@z′2

∣∣∣∣z′2=1

= 0: (18)

The energy balance for the recycling pump is given by

[Vp�CP + (V�CP)p]dTp

dt′

=�CpQ2(T 1|z′1=L1 − Tp) + (OHr)VpRi

+ (UA)p(Tc − Tp); (19)

where Vp is the internal volume of the recycling pump.Analogously to the mass balance equations, the energy

balance equations may be rendered dimensionless. For thetubular sections 1 and 2 and the recycling pump one maywrite

�(1 + %r)@& 1

@t+ (1 + Rec)

@& 1

@z1

=1

� Pet

@2& 1

@z21− �BDaRM + (r(&c − & 1); (20)

valid for 0¡z1¡ 1,

(1 − �)(1 + %r)@& 2

@t+ Rec

@& 2

@z2

=1

(1 − �)Pet@2& 2

@z22− (1 − �)BDaRM

+ (r(&c − & 2); (21)

valid for 0¡z2¡ 1, and

�p(1 + %p)d& p

dt

=Rec(& 1|z1=1 − & p) − �pBDaRi + (p(&c − & p);(22)

where & j =T j=Tf, B=(−OHr)CMf=�CpTf, Pet = vfL=Dt ,(r =4UrL=�Cpvfd, (p =(UA)p=�CpQf, %r =(�CP)r=�CP ,%p = (V�CP)p=V�CP , and �p represents the ratio of the in-ternal volume of the recycling pump to the total internalvolume of the tubular sections.

The corresponding boundary conditions for the energybalance equations are

− 1� Pet

@& 1

@z1

∣∣∣∣z1=0

+ (1 + Rec)& 1|z1=0

=Rec & 2|z2=1 + 1; (23)

@& 1

@z1

∣∣∣∣z1=1

= 0; (24)

− 1(1 − �)Pet

@& 2

@z2

∣∣∣∣z2=0

+ Rec & 2|z2=0 = Rec & p (25)

and

@& 2

@z2

∣∣∣∣z2=1

= 0: (26)

It may be noticed in Eqs. (20)–(22) that external ther-mal capacitance factors for the reactor and for the recyclingpump have been considered in the reactor modeling. Thesefactors represent the ratio of the thermal capacitance of ex-ternals to that of the reaction mixture Kowing through thetubes and pump. The formulation for the external thermalcapacitance presented above assumes that the externals arein thermal equilibrium with the reactor contents. As typi-cally most of real reactor parts are built with a very low heatcapacity and very high thermal conductivity material (stain-less steel), the assumption of thermal equilibrium betweenreactor externals and the reaction mixture seems to be quitereasonable.

3. Kinetic model

The polymerization reaction investigated in this work isdescribed according to the free-radical mechanism. Follow-ing Hamer and Ray (1982), the reaction mechanism wasdevised as general as possible. The kinetic steps of the mech-anism are provided below.


Radical initiation

Ikd→ 2R;

R+MkpR→ P′

1:

Chain propagation

Pn +Mkp→ Pn+1:

Chain transfer to monomer

Pn +Mktr;M→ Mn +M∗;

M∗ +MkpM→ P1 (re-initiation):

Chain transfer to solvent

Pn + Sktr;S→ Mn + S∗;

S∗ +MkpS→ P′

1 (re-initiation)

Chain termination by combination

Pn + Pmktc→Mn+m:

Chain termination by disproportionation

Pn + Pmktd→Mn +Mm:

In the equations above, I represents an initiator molecule,R is a dissociated initiator radical,M is a monomer molecule,S is a solvent or chain transfer agent molecule, Pi is a grow-ing or live polymer chain of size i, andMi is a dead polymerchain of size i.

As provided by Hamer and Ray (1982), the reaction ratesfor all the species in the reaction medium are given by

RM =−[kp,T0M + ktr;M ,T0M + kpMMM∗

+kpSMS∗ + kpRMR]; (27)

RS = −ktr; S,T0S; (28)

RI = −kdI; (29)

RR = 2fkdI − kpRMR; (30)

RM∗ = ktr;M ,T0M − kpMMM∗; (31)

RS∗ = ktr; S,T0S − kpSMS∗; (32)

where

,T0 =(

2fkdIktc + ktd

)1=2

: (33)

Adopting the quasi-steady-state approximation for all rad-icals (Hamielec, Hodgins, & Tebbens, 1967; Ray, 1969),Eqs. (27)–(29) can simpli9ed to

RM =−[2fkdI + kpM,T0 + 2ktr;M ,T0M

+ ktr; S,T0S]; (34)

Table 1Kinetic parameters for the homopolymerization of vinyl acetate inmethanol, using AIBN as free-radical initiator component

Parameter Reference

kd = 1:58 × 1015e−30800=RgT (1/s) Stevens (1988)kp = 3:2 × 1010e−6300=RgT (cm3=gmol=s) Kroschwits (1985)ktc0 = 3:7 × 1012e−3200=RgT (cm3=gmol=s) Kroschwits (1985)ktd0 = 0:0 cm3=gmol=s Hamer (1983)ktr;M = 2:08 × 10−2e−2950=RgT (cm3=gmol=s) Stevens (1988)ktr;S = 0:22e−4340=RgT kp (cm3=gmol=s) Stevens (1988)f = 0:8 Stevens (1988)

RS = −ktr; S,T0S; (35)

RI = −kdI: (36)

In this work, the free-radical solution polymerization ofvinyl acetate is investigated. The reaction is carried out inmethanol using AIBN as initiator. At high conversions, thewell-known gel-e7ect may a7ect polymerization rates, evenfor the case of solution processes. This e7ect was consideredhere according to the equation given by Teymour and Ray(1989) for vinyl acetate free-radical solution polymerization

gT =ktc + ktdktc0 + ktd0

= exp(a1xT + a2x2T + a3x3T ); (37)

where T is the reactor temperature in Kelvin, xT is the poly-mer weight fraction and a1 = −0:047; a2 = −6:735, anda3 = −0:3495.

All kinetic parameters for the free-radical polymerizationof vinyl acetate used in the present investigation were col-lected from the literature and are listed in Table 1. Physicalproperties are given in Table 2. Temperature is given inKelvin.

4. Numerical methods and continuation parameters

Eqs. (7)–(13) and (20)–(26) constitute a set of coupled,nonlinear parabolic partial and ordinary di7erential equa-tions. The axial coordinates of the tubular sections, z1 andz2, were discretized according to a polynomial approxima-tion on 9nite elements, as described by Melo et al. (2001b).The method was capable of transforming the original sys-tem of equations into a much simpler though with higherdimension set of ordinary di7erential equations in the form

dy(t; /)dt

=G · y(t; /) + u(t) + f(y; /); (38)

where y stands for the vector of the reactor state variables,G is the characteristic matrix of the system, u represents thereactor inputs vector, / is the vector of model parametersand f is the vector function enclosing the terms related tochemical reaction. Numerical method convergence issueshave been discussed in Melo et al. (2001b).


Table 2Physical properties

Component Property Reference

Densities (g=cm3)Monomer �M (T ) = (0:6210 + 1:54 × 10−3T )−1 Hamer (1983)Solvent �S(T ) = (0:9318 + 1:60 × 10−3T )−1 Hamer (1983)Polymer �P(T ) = (0:6514 + 6:30 × 10−4T )−1 Hamer(1983)

Speci9c heats (cal/g/K)Monomer cpM (T ) = 0:4479 + 5:635 × 10−4(T − 273:15) Kroschwits (1985)Solvent cpS (T ) = 0:6086 Teymour (1989)Polymer cpP (T ) = 0:3543 + 9:55 × 10−4(T − 273:15) Teymour (1989)

Regarding the model stability analysis, the well-knowncontinuation and bifurcation package AUTO of Doedelet al. (1997) was used. AUTO uses a pseudoarc-lengthcontinuation procedure in order to trace out branches ofsteady states and of periodic solutions. AUTO is, though,designed for carrying out low-dimensional continuationproblems. Besides, depending on the number of 9nite el-ements necessary to guarantee the numerical convergenceof the discretization method, the resulting system of or-dinary di7erential equations may become quite large (infact, the total number of equations to be solved is precisely6N + 3, where N is the number of 9nite elements per tubu-lar section). For these reasons, only steady-state branchesof solutions are presented here, once the problem of 9ndingbranches of periodic solutions relies on the resolution of amuch higher-dimensional system of equations (Kub()Rcek &Marek, 1983).

Since Uppal, Ray, and Poore (1976), the reactor aver-age residence time has been used as the main continuationparameter in many bifurcation problems of chemical engi-neering. This is due to the fact that this is an easily manip-ulated operation parameter, both at plant site and speciallyin lab-scale apparatuses. Therefore, the average reactor res-idence time was chosen as the main continuation parameterin the present investigation. In spite of that, other continua-tion parameters have also been considered here. The recycleratio was considered as a continuation parameter because itbonds smoothly the transition between the Kow patterns ofa dispersed plug-Kow reactor (Rec → 0) and an ideal con-tinuous stirred tank reactor (Rec → ∞). Other continuationparameters considered were the overall heat transfer coeA-cients in the tubes and the cooling jacket temperature, be-cause both play a fundamental part on the reactor thermaldynamics.

For all calculations presented in the next section, a num-ber of seven 9nite elements per tubular section was used. Formost of the range of the operational parameters analyzed,the convergence of the steady-state branches of solutionswas always achieved. For some bifurcation diagrams, con-tinuation of steady-state branches at high temperatures pre-sented diAcult convergence, indicating that a larger numberof 9nite elements per tubular section would be necessary.Because a compromise between the completeness of the

bifurcation diagrams and the continuation performance wassought, steady-state branches of solutions where conver-gence problems were found are ended with the symbol ×. Inwhat follows, stable and unstable branches of steady-statesolutions are represented by the line types — and - - -,respectively. Hopf bifurcation points are indicated by thesymbol �.

5. Results

Simulation data are presented in Table 3. These valuesshould be regarded simply as reference numbers for the sim-ulations, once they are allowed to vary within certain rangesduring the calculations. In all simulations presented below,the bifurcation diagrams are represented in terms of the exitreactor temperature because of space limitation. Besides, un-less stated otherwise, simulations were performed using thedata given by Table 3.

5.1. E7ect of thermal and mass dispersion

Fig. 2 shows the e7ect of thermal dispersion coeA-cients on the loop reactor stability. Curve 2a presents atypical S-shaped steady-state response for variations ofthe average reactor residence time. It can be observed thatself-sustained oscillatory behavior is possible for a reason-ably wide range of residence times, given the existence ofthe pair of Hopf bifurcation points. Multiple steady statesare predicted for low residence times. Small values of Petimply in improved homogenization of the temperature pro-9les along the tubular sections. Fig. 2 shows that increasein the thermal dispersion coeAcient results in Katteningof the bifurcation diagram, thus decreasing the upper limitfor steady-state temperatures. The explanation for this be-havior may be given by taking into consideration the limitcase when Pet → 0. In this case, because axial thermalmixing is perfect, dissipation of the heat of reactor isfacilitated.

Fig. 3 shows dynamic simulations of the reactor modeloperating with Rec = 10, under di7erent conditions of ther-mal internal mixing for a residence time where oscilla-tory behavior is expected. Independent dynamic simulations


Table 3Simulation data of the loop reactor

Parameter Symbol Value Unit

Total reactor length L 317.2 cmReactor diameter D 1.7272 cmTotal volume of tubular sections V 855.80 cm3

Feed temperature Tf 42.0 ◦CCooling jacket temperature Tc 55.0 ◦CFeed monomer concentration CMf 7:3 × 10−03 gmol=cm3=sFeed solvent concentration CSf 7:3 × 10−03 gmol=cm3=sFeed initiator concentration CIf 3:0 × 10−05 gmol=cm3=sOverall heat transfer coeAcient in the tubes Ur 2:0 × 10−04 cal=cm2=s=KOverall heat transfer coeAcient in the pump (UA)p 0.2 cal/s/KMass Peclet number Pem 10.0 dimensionlessThermal Peclet number Pet 0.5 dimensionlessVolumetric capacitance factor of the pump �p 0.151 dimensionlessReactor exit dimensionless point � 0.50 dimensionlessExternal thermal capacitance factors %r ; %p 0.0 dimensionless

0.0 0.5 1.0 1.5 2.0 2.5 3.040

60

80

100

120

140

160

180

d

c

b

a

Residence Time (h)

Tem

pera

ture

(°C

)

Fig. 2. E7ect of the thermal dispersion on the loop reactor stability(Rec=10:0, (a) Pet=1:0, (b) Pet=0:5, (c) Pet=0:1, and (d) Pet=0:01).

0 2 4 6 8

40

60

80

100

120

140

160

180

200 a b c

Time (h)

Tem

pera

ture

(°C

)

Fig. 3. Dynamic simulations of the loop reactor—e7ect of Pet (Rec=10:0,� = 5000 s, (a) Pet = 1:0, (b) Pet = 0:1, and (c) Pet = 0:01).

0.0 0.5 1.0 1.5 2.0 2.5 3.040

60

80

100

120

140

160

180

c

b

a

Residence Time (h)

Tem

pera

ture

(°C

)

Fig. 4. E7ect of the mass dispersion on the loop reactor stability(Rec = 10:0, (a) Pem = 100:0, (b) Pem = 10:0, and (c) Pem = 4:0).

con9rm the existence of self-sustained oscillatory responses.Although the increase of thermal mixing decreases station-ary temperatures, the maximum amplitude of the oscillationsis approximately the same for all cases analyzed. However,the e7ect of Pet upon the oscillations is remarkable. Bothresults can be explained in terms of the simple bang–bangoscillatory pattern, where oscillatory responses can be di-vided into sequences of reactor charging and adiabatic re-action when Pet is smaller. Temperature information Kowsfaster from reactor entrance to reactor output in this case.

Contrary to the behavior described above, increase of ax-ial mass mixing in the loop reactor leads to increase ofsteady-state temperatures, as presented in Fig. 4. In thiscase, both the monomer and initiator homogenous concen-tration pro9les contribute to increasing average polymer-ization rates which, in turn, results in higher steady-statetemperatures.


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Fig. 5. E7ect of the initiator feed stream concentration on the loop reactor stability (Rec = 50, (a) CIf = 3:0 × 10−05 gmol=cm3,(b) CIf = 0:22 × 10−05 gmol=cm3, (c) CIf = 0:212 × 10−05 gmol=cm3, and (d) CIf = 0:2 × 10−05 gmol=cm3).

5.2. E7ect of the feed initiator concentration

The initiator concentration in the feed stream is also ma-nipulated easily in real continuous free-radical polymeriza-tion systems. Fig. 5 presents the e7ect of the initiator feedstream concentration on the reactor stability. According toFig. 5, the range of residence times where self-sustained os-cillatory behavior is possible is enlarged while decreasingthe concentration of initiator in the feed stream. Besides,at suAciently low initiator concentrations, Fig. 5 showsthe possible occurrence of isolated steady-state branches ofsteady-state solutions—isolas. The presence of stationaryisolas has practical consequences upon the reactor perfor-mance. When predicted appropriately, the presence of isolasmeans that high monomer steady-state conversion may beattained at relatively low residence times. Experimentally,though, detection of steady-state isolas depends on the sup-ply of additional energy to the system, in order to promoteits ignition (Schmidt, Clinch, & Ray, 1984).

As previously pointed out, the importance of the recycleratio lays in the fact that it is a parameter responsible for themacroscopic mixing in the loop reactor. Fig. 6 presents con-tinuation diagrams for the loop reactor as both the recycle ra-tio and initiator feed stream concentration are varied. Within

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Fig. 6. E7ect of the initiator feed stream concentration on the loop reactorstability using the recycle ratio as the continuation parameter (�=5000 s,Ur = 6:0 × 10−04 cal=cm2=s=K, (a) CIf = 4:0 × 10−05 gmol=cm3,(b) CIf = 3:5× 10−05 gmol=cm3, (c) CIf = 3:0× 10−05 gmol=cm3, and(d) CIf = 2:0 × 10−05 gmol=cm3).

the range of initiator feed stream concentrations analyzed,one may observe the existence of unstable steady-state solu-tions during the transition of operation without recycling of


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Fig. 7. Dynamic simulations of the loop reactor in the neighborhood of the upper Hopf bifurcation point (� = 5000 s, CIf = 3:0 × 10−05 gmol=cm3,Ur = 6:0 × 10−04 cal=cm2=s=K, (a) Rec = 0:265, (b) Rec = 0:27, (c) Rec = 0:272).

material to operation with partial recirculation of the poly-mer solution. In the cases analyzed, operation without recy-cle is always stable as well as the operation for suAcientlyhigh recycle ratios. The transition from one mode of reac-tor operation to the other is marked by the appearance ofself-sustained oscillatory behavior. This behavior should bealready expected, since the addition of a recycling streamline to tubular reactors adds oscillatory modes on the reac-tor dynamic responses, as observed early by Schmeal andAmundson (1966). This may be a very important practicalissue, as the process engineer is not allowed to modify therecycle ratio at will.

Fig. 6 also shows that the decrease in the initiator feedstream concentration shrinks the range of recycle ratioswhere unstable steady-state solutions are found. This is dueto a decrease of the overall polymerization rate, resultingin lower reactor temperatures. Therefore, lower dissipationof reaction heat narrows the region of oscillatory behavior.

Interesting dynamic behavior was observed in the neigh-borhood of the Hopf bifurcation point found in the up-per branch of steady-state solutions. Consider, for example,curve c of Fig. 6. Dynamic simulations in the neighbor-hood of the upper Hopf bifurcation point are presented inFig. 7. For low recycle ratios, the attainment of a single, sta-ble steady state is expected. However, for Rec=0.265, twostationary solutions are predicted (cf. Fig. 7a). The conver-

gence to a given solution depends on the initial conditions.Small perturbation on the recycle ratio leads to the develop-ment of oscillatory behavior around the upper steady-statesolution, as presented in Fig. 7b. The increase of the recy-cle ratio even further leads to the development of oscillatorypatterns as presented in Fig. 7c.

Fig. 7c presents a qualitative change in the oscillatorybehavior. Dynamic simulations showed that the Hopf bifur-cation gave rise to a supercritical branch of periodic orbits.Lower oscillations in the vicinity of the upper Hopf bifur-cation point were replaced by very larger oscillations aftera small perturbation of the recycle ratio. Oscillations areso large that the minimum temperature of the oscillatoryresponse is lower than the temperature of the lower stablesteady state. This indicates that the oscillatory trajectory ap-proaches the stable steady state very closely (and eventually“touches” it, giving birth to a homoclinic orbit). Besides,the period of oscillation increases very signi9cantly aftera small perturbation of the recycle ratio, which may be re-garded as an additional indication of the probable existenceof a homoclinic orbit. Therefore, based on the previous re-marks, it is possible to conjecture that the sudden increaseof the oscillatory responses observed may be linked tothe existence of homoclinic trajectories. These issues dis-cussed above are explored in more details in the followingparagraphs.


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Fig. 8. Bifurcation diagram for the loop reactor (V =500 cm3, �=5000 s,L = 213:4 cm, Ur = 6:0 × 10−04 cal=cm2=s=K).

Fig. 8 presents a bifurcation diagram similar to that shownin Fig. 6c after a change in the reactor volumetric capac-ity. Analogously, the analysis is performed around the upperHopf bifurcation point. Comparing Fig. 8 with Fig. 6c, andbased on the previous paragraph, one may conjecture aboutthe existence of a homoclinic orbit in the neighborhood ofthe unstable equilibrium point, as discussed previously. (Ac-tually, homoclinic orbits are also likely to exist in Figs. 2and 5 when the number of Hopf bifurcation points is odd,as in these cases at least one branch of periodic orbits willnot connect a pair of Hopf bifurcation points. However, dif-ferently from the simulation cases presented in Fig. 8, theexistence of homoclinic orbits is not important to explainthe dynamic behavior of the loop reactor at conditions pre-sented in Figs. 2 and 5.)

Analysis of the characteristic values of the unstable branchof steady-state solutions reveals that for a range of recy-cle ratios we have one real positive characteristic value (1),while many of the remaining characteristic values are com-plex conjugate pairs with negative real part. Let the leadingcomplex conjugate characteristic values pair be denoted by−�±(i, with �¿ 0 and (¿ 0. The analysis also reveals that1¿�¿ 0. Then, the RSilnikov theorem guarantees that thereis a neighborhood of the homoclinic orbit which containsan in9nitely countable number of unstable periodic orbits(cf. Feroe, 1993; Arn(eodo, Argoul, Elezgaray, & Richetti,1993; Pinto, 1995; Hale & KoWcak, 1991). RSilnikov cyclesare related to the appearance of homoclinic saddle-focus or-bits and are, invariably, unstable. Therefore, its computa-tion cannot be made through direct integration of the modelequations (the authors are aware that a number of numericaltechniques can be used to compute unstable dynamic trajec-tories based on iterative techniques, such as those discussedby Ourique, Biscaia Jr., & Pinto, 2002). It may be empha-sized that the presence of RSilnikov cycles imply in chaoticdynamic behavior. If the vector 9eld of a saddle-focus or-bit undergoes small perturbations, then the RSilnikov may bebroken-up and originate chaotic attractors in the phase space

(Hale & KoWcak, 1991). Fig. 9 presents complex dynamicbehavior veri9ed in the neighborhood of the upper Hopf bi-furcation point. Figs. 9a and b present the onset of periodicoscillations around the upper Hopf bifurcation point. Fig. 9cpresents the destruction of the RSilnikov cycle, giving birthto a likely chaotic attractor in the phase space (cf. Fig. 10).Figs. 9d and e present the regularization of the reactor dy-namic behavior after the critical recycle ratios are passed.

5.3. E7ect of the overall heat transfer coe:cient andcooling jacket temperature

One of the principal advantages of using loop reactors inpolymerization processes is the large heat transfer capabil-ities of these vessels, an appreciated feature for the highlyexothermic polymerization reactions. Fig. 11 presents thedynamic simulation of the failure of the cooling system inthe loop polymerization reactor investigated here. Decreas-ing the overall heat transfer coeAcient imply in the devel-opment of self-sustained periodic oscillations. This behav-ior may be catastrophic at plant site. Assuming operation atatmospheric pressure, the bubble point of the solvent usedin the simulations is ≈ 65◦C. Therefore, sharp elevation ofthe temperature may lead to the undesired bubbles in thereactor and general equipment failure.

Due to its importance on the bifurcation behavior of loopreactors, the overall heat transfer coeAcient was used asthe main bifurcation behavior. Fig. 12 shows that, for verylow values of Ur , self-sustained periodic oscillations are notexpected. This result is consistent, for in the limit whenUr → 0 (i.e., adiabatic operation) self-sustained oscillationsare not possible (cf. Reichert & Moritz, 1989). SuAcientlylarge overall heat transfer coeAcients also decrease the pos-sibility of observing reactor oscillatory dynamics. In thiscase, the thermal dynamic behavior of the loop reactor isdominated by the elevated heat transfer rates of the reac-tor walls with the cooling jacket. However, for intermedi-ate values of Ur , periodic oscillations should be expected.Besides, steady-state multiplicity becomes a relevant phe-nomenon while the recycle ratio is decreased (cf. Figs. 12fand g).

The inKuence of the cooling jacket temperature on theloop bifurcation diagram is analyzed in Fig. 13. At low recy-cle ratios it may be observed that as many as 9ve steady-statesolutions are predicted for the loop reactor. For the casesinvestigated, the reactor operation is always stable atsuAciently low and suAciently high cooling jacket tem-peratures. At these extreme situations, the reactor thermaldynamic behavior is completely controlled by the heat ex-change through the reactor walls, thus implying in trivialreactor behavior. However, at intermediate cooling temper-atures, unstable steady states solutions are observed.

If one slightly perturbs the overall heat transfer coeAcientused in the simulations of Fig. 13, and use a recycle ratioof 0.75, Fig. 14 shows that the loop polymerization reactor


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Fig. 9. Dynamic simulations of the loop reactor in the neighborhood of the upper Hopf bifurcation point (V = 500 cm3, � = 5000 s,CIf = 3:0 × 10−05 gmol=cm3, Ur = 6:0 × 10−04 cal=cm2=s=K, (a) Rec = 0:01, (b) Rec = 0:265, (c) Rec = 0:27, (d) Rec = 0:272, and(e) Rec = 0:5).

may present as many as seven steady-state solutions. Allintermediate stationary solutions are though unstable. Thisobservation only stresses out the fact that transition from lowto large recycle ratios may be accompanied by extremelyundesired nonlinear phenomena.

5.4. E7ect of the solvent concentration and purity

It is well known that continuous solution polymerizationreactions are attractive because of the good control of the

reactor temperature (Odian, 1991). The higher the solventfeed fraction, the lower the maximum observed steady-statereactor temperatures, as shown in Fig. 15. It may also be ob-served that the range of residence times where self-sustainedperiodic oscillations are found is moved eastwards. Besides,oscillatory behavior was not observed for feed solvent frac-tions lower that 0.4 and larger than 0.76.

The presence of impurities in the solvent and or mono-mer can also modify the stability of loop polymerizationreactors. This is particularly important for free-radical


Fig. 10. Strange attractor of the loop reactor (Rec = 0:27, V = 500 cm3, � = 5000 s, CIf = 3:0 × 10−05 gmol=cm3, Ur = 6:0 × 10−04 cal=cm2=s=K).

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Fig. 11. Dynamic simulation of the loop reactor: failure of the cooling sys-tem (�=5000 s, Rec=10, Ur=6:0×10−04 → 2:0×10−05 cal=cm2=s=K).

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gfedcba

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Fig. 12. Bifurcation diagram for the loop reactor (�=5000 s, (a) Rec=5,(b) Rec = 3, (c) Rec = 2, (d) Rec = 1, (e) Rec = 0:75, (f) Rec = 0:5, and(g) Rec = 0:2).

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Fig. 13. Bifurcation diagram for the loop reactor (� = 5000 s,Ur=3:0×10−04 cal=cm2=s=K, (a) Rec=1:0, (b) Rec=0:8, (c) Rec=0:7,and (d) Rec = 0:6).

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Fig. 14. Bifurcation diagram for the loop reactor (� = 5000 s,Ur = 2:4 × 10−04 cal=cm2=s=K, and Rec = 0:75).


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Fig. 15. Bifurcation diagram for the loop reactor (Rec = 50, Ur = 4:0 × 10−04 cal=cm2=s=K, (a) xS = 0:3, (b) xS = 0:4, (c) xS = 0:5, (d) xS = 0:7, (e)xS = 0:74, and (f) xS = 0:76).

polymerization because inhibition agents are typicallyadded to storage containers of monomers in order to preventundesired and dangerous thermal polymerization reactions.Small amounts of inhibitor molecules in the feed streamelongate the lower branch of steady-state solutions of theloop polymerization reactor towards larger residence times,as shown in Fig. 16. This phenomenon imparts relevance tosteady state multiplicity in the loop polymerization reactor.

These results are very similar to those previously reportedby Pinto and Ray (1996), who analyzed dynamical responses

of a continuous stirred tank polymerization reactor in thepresence of hydroquinone. As illustrated in Fig. 16, the pres-ence of a inhibition agent increases the range of residencetimes where multiple steady states coexist. It is well knownthat inhibitor molecules retard polymerization reaction rates.In a batch reactor, the larger the inhibition agent concentra-tion, the larger the time required to achieve a given monomerconversion, when compared to the uninhibited case. For acontinuous system, larger residence times are necessary tocompensate for the presence of inhibition agents, as far as


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Fig. 16. Bifurcation diagram for the loop reactor(Ur = 4:0 × 10−04 cal=cm2=s=K, CIf = 1:0 × 10−05 gmol=cm3,(a) CH = 0:0 gmol=cm3 (b) CH = 5:0 × 10−08 gmol=cm3,(c) CH = 7:3 × 10−08 gmol=cm3, (d) CH = 3:0 × 10−07 gmol=cm3,(e) CH = 5:0× 10−07 gmol=cm3, and (f) CH = 1:0× 10−06 gmol=cm3).

monomer conversion is concerned. Therefore, it is observedthe elongation of the lower steady-state solutions branch.

The increase in the concentration of hydroquinone alsoinduce the appearance of stationary isola, thus shrinkingthe region of oscillatory responses. It may be noticed (cf.Fig. 16) that there is a range of residence times where asmany as 9ve steady-state solutions are predicted for the looppolymerization reactor.

5.5. E7ect of the recycling pump thermal parameters

The thermal dynamics of loop reactors was recently ana-lyzed by Melo et al. (2001b). In short, Melo et al. showedboth experimentally and theoretically that the recyclingpump mass capacitance and the external parts of the loopreactor (e.g. tube walls, 9ttings, valves, recycling pumpbody, etc.) must be considered when describing this classof reactors. Three parameters were shown to be of funda-mental importance to 9t theoretical predictions to lab-scaleexperimental responses: the overall heat transfer coeAcientof the recycling pump, and the external thermal capacitancefactors for both the recycling pump and the loop reactoritself. Besides, industrial scale reactors were also shownto be a7ected by these parameters. Therefore, the impactof these parameters upon the bifurcation diagrams of looppolymerization reactors is analyzed in this section.

Fig. 17 presents bifurcation diagrams for the loop poly-merization reactor under di7erent levels of heat dissipationin the recycling pump. It is shown that non linear responsesare annihilated while the overall heat transfer coeAcient ofthe recycling pump is increased. This may be regarded asa positive feature for industrial scale reactors, because os-cillations are undesired in this case once attention to safetyoperational issues must always be paid. On the other hand,

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Fig. 17. Bifurcation diagram for the loop reactor (Rec = 50,Pet = 0:1, (a) (UA)p = 0:2 cal=s=K, (b) (UA)p = 2:0 cal=s=K, and (c)(UA)p = 10:0 cal=s=K).

this result implies that exploratory investigation of nonlin-ear behavior at lab-scale reactors is only possible providedthat thermal issues are appropriately taken into account.

When external capacitance factors are neglected duringthe development of the reactor model, Fig. 18a shows atypical bifurcation diagram where a range of self-sustainedoscillatory behavior is expected. Fig. 18b, in turn, shows thatassuming some considerably thermal capacitance for thereactor externals only, the range of residence times whereoscillations are possible vanish. Accordingly, oscillatoryresponses are not possible when the external thermalcapacitance of the recycling pump exceeds a critical value(cf. Fig. 18c).

6. Conclusions

The bifurcation behavior of loop polymerization reactorswas investigated in this work. A novel mathematical modelfor the reactor was developed by taking into account partic-ipation of the recycling pump and reactor external parts onthe overall process dynamics. Branches of steady-state reac-tor solutions were traced out in order to classify the possibleresponses of these reactors. For various combinations of thereactor operational parameters, typical nonlinear responses,such as steady-state multiplicity and self-sustained periodicoscillations, were observed.

Transition from operation without recycling of materialto partial recirculation of polymer solution was found to becharacterized by the development of oscillatory behavior. Atlow recycle ratios, complex dynamics including chaos wasobserved. Besides, it was found that under certain operationconditions, as many as seven steady states are predicted bythe reactor model.

The nonlinear analysis presented here also permitted tounderstand the importance played by the recycling pumpupon the reactor stability. Although steady-state reactor


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Fig. 18. Bifurcation diagram for the loop reactor (Rec = 50, (a) %r = 0; %p = 0, (b) %r = 0:25, %p = 0, and (c) %r = 0; %p = 6).

responses are not a7ected by this piece of equipment, thethermal dynamical behavior of loop polymerization re-actors may be strongly perturbed by the recycling pumpmass capacitance. Furthermore, it was also found that theexternal parts of both the reactor tubes and the recyclingpump present major implications on the reactor stability.Even relatively low values of the external thermal capaci-tance factors are capable of annihilating oscillatory reactorresponses. For industrial scale reactors, this may be re-garded as a positive feature, both economically and as faras process safety is concerned.

Notation

A reactor transversal section area, cm2

B adiabatic increase of reactor temperatureC molar concentration of species i in the

tubular section j, gmol=cm3

Cp mixture thermal capacityd diameter of tubular sections, cm2

Da DamkXohler numberDm mass dispersion coeAcient, cm2=sDt thermal dispersion coeAcient, cm2=sL total length of tubular sections, cmN number of 9nite element per tubular sectionPem mass Peclet number

Pet thermal Peclet numberQ Kow rate, cm3=sRec recycle ratioRi reaction rate of species i, gmol=cm3=sRi dimensionless reaction rate of species it dimensionless timet′ time, sT temperature, KUr overall heat transfer coeAcients in the

tubular sections, cal=cm2=s=K(UA)p overall heat transfer coeAcients in the

recycling pump, cal=s=KV volume, cm3

v Kow velocity, cm=syj dimensionless concentration for species jz dimensionless axial coordinate of the

tubular sectionz′ axial coordinate of the tubular sections, cm

Greek letters

� reactor exit point(p dimensionless overall heat transfer

coeAcient in the recycling pump(r dimensionless overall heat transfer

coeAcient in the tubular sections


(−OHr) reaction enthalpy, cal/gmol% external thermal capacitance factor& dimensionless temperature&c reactor walls dimensionless temperature� mixture density, g=cm3

� average residence time, h�p volumetric capacitance factor for the

recycling pump

Superscripts

1 related to tubular section 12 related to tubular section 2p related to the recycling pump

Subscripts

1 related to tubular section 12 related to tubular section 2c related to the cooling jacketf related to the feed streamH hydroquinoneI related to initiatorM related to monomerp related to the recycling pumpr related to the loop reactorz related to the axial Kow velocity

Acknowledgements

The authors would like to thank CNPQ—Conselho Na-cional de Desenvolvimento Cient��@co e Tecnol�ogico, forproviding scholarships and supporting this research.

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