Chemical Engineering Science 58 (2003) 2973–2984www.elsevier.com/locate/ces

State multiplicity in PFR–separator–recycle polymerization systems

Anton A. Kissa ;∗, Costin S. Bildeab, Alexandre C. Dimiana, Piet D. Iedemaa

aChemical Engineering Department, University of Amsterdam, Nieuwe Achtergracht 166, 1018 WV Amsterdam, NetherlandsbDelft University of Technology, Julianalaan 136, 2628 BL Delft, Netherlands

Received 8 April 2002; received in revised form 20 February 2003; accepted 13 March 2003

Abstract

This article explores the non-linear behaviour of isothermal and non-isothermal plug-6ow reactor (PFR)–separator–recycle systems,with reference to radical polymerization. The steady-state behaviour of six reaction systems of increasing complexity, from one-reactant7rst-order reaction to chain-growth polymerization, is investigated. In PFR–separator–recycle systems feasible steady states exist onlyif the reactor volume exceeds a critical value. For one-reaction systems, one stable steady state is born at a transcritical bifurcation. Incase of consecutive-reaction systems, including polymerization, a fold bifurcation can lead to two feasible steady states. The transcriticalbifurcation is destroyed when two reactants are involved. In addition, the thermal e8ects also introduce state multiplicity. When multiplesteady states exist, the instability of the low-conversion branch sets a lower limit on the conversion achievable at a stable operating point.A low-density polyethylene process is presented as a real plant example.

The results obtained in this study are similar to CSTR–separator–recycle systems. This suggests that the behaviour is dictated by thechemical reaction and 6owsheet structure, rather than by the reactor type.? 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Reaction engineering; Polymerization; Kinetics; Recycle systems; Non-linear dynamics

1. Introduction

Reaction systems involving material recycles are veryusual in chemical industry. The 6exibility of such systemsmust be ensured by the reactor design and its operationpolicy. The study presented in this article originated froman unexpected bifurcation diagram (Fig. 11), obtained fora low-density polyethylene plant (Fig. 10). Figs. 10 and 11will be discussed in detail in a following section. To un-derstand the origins of this surprising behaviour, we startedthe investigation with the CSTR reactor model (Kiss,Bildea, Dimian, & Iedema, 2002) that is computationallyless-demanding. The present work concludes the endeav-our, considering the non-linear behaviour of plug-6owreactor (PFR)–separator–recycle systems, with reference toradical polymerization. In the following, we explain howanalysing systems involving PFRs, and comparing the re-sults with those obtained for CSTRs, contributes to betterunderstanding of the e8ect of mass recycle.

∗ Corresponding author. Tel.: +31-20-525-5026;fax: +31-20-525-5604.E-mail addresses: ktony@science.uva.nl (A. A. Kiss),

c.s.bildea@tnw.tudelft.nl (C. S. Bildea), alexd@science.uva.nl(A. C. Dimian), piet@science.uva.nl (P. D. Iedema).

The dependence of reaction rate vs. temperature and con-centration is a non-linearity that is present in chemical reac-tors. Coupled with a feedback mechanism, this dependencecan lead to state multiplicity, isolated solution branches, in-stability, sustained oscillations, strange attractors or chaoticbehaviour. Energy feedback is often present due to mixing(Balakotaiah & Luss, 1984), axial dispersion (Jensen &Ray,1982), recycle of reactor’s eDuent (Reilly & Schmitz, 1966,1967), autothermal operation (Lovo & Balakotaiah, 1992)and heat integration (Bildea & Dimian, 1998). In practice,the above-mentioned phenomena are undesired. For this rea-son, energy feedback is often excluded by means of controlloops. These manipulate, for example, the coolant 6ow rate,the by-pass around a feed-eDuent heat exchanger, or theduty of a heater/cooler upstream the reactor.In many chemical plants, feedback is also present due to

material recycle (Fig. 1). In such systems, the reactor eDu-ent is 7rst processed by a separation section, and afterwardsrecycled. Product (4) and recycle (3) streams have 7xedcomposition. This is achieved by local control of the sepa-ration units. Hence, the composition at reactor inlet is notdirectly dependent on the reactor eDuent. Moreover, thetemperature at the reactor inlet is either the ambient temper-ature or has a constant value due to a heat exchanger placed

0009-2509/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0009-2509(03)00162-3

2974 A. A. Kiss et al. / Chemical Engineering Science 58 (2003) 2973–2984

T2

f0, zM,0 CC

CC

Separation

Recycle

CC

fI,0, zI,0 Ifeed

f1, z1

TC1

θ 1= 0

θc

PFR1

Coolant

0

f2, z2

2

f3, z33

z3=zM,0

zM,4=0

f4, z44

Product

Feed

Fig. 1. General structure of reactor–separator–recycle systems. The reactor eDuent is processed by the separation section, and afterwards recycled. Theproduct and recycle streams have 7xed composition. Dashed drawing refers to systems with two reactants.

upstream the reactor. Luss and Amundson (1967) recog-nized this structure as representative for chemical plants.Pareja and Reilly (1969) acknowledged that recycle reactorscould become unstable solely as a result of material recycle,but they did not present any result. The interest in reactor–separator–recycle systems is further motivated by plantwidecontrol studies (Luyben & Luyben, 1997) that reported con-trol diIculties due to the “snowball e8ect”. Bildea, Dimian,and Iedema (2000) analysed the relationship between designand control of several systems, showing that the non-linearbehaviour leads to control problems. However, the studywas restricted to the steady-state behaviour of an isothermalreactor.The non-linear behaviour of CSTR–separator–recycle

systems was the subject of few recent articles. Pushpavanamand Kienle (2001) considered a 7rst-order reaction anddemonstrated that state multiplicity, isolas, instability andlimit cycles can easily arise. Although the 6owsheet in-cluded inter-units heat exchangers, energy feedback wasstill present inside the CSTR due to the mixing. Kiss et al.(2002) studied the case of complex reactions, includingpolymerization, in isothermal CSTR–separator–recycle sys-tems. It was shown that the non-linearity of a consecutivemechanism is suIcient for multiplicity and instability tooccur. The same phenomena are exempli7ed by Sagale andPushpavanam (2002), who considered di8erent modes ofoperation for the autocatalytic reaction A + 2B → 3B, inan isothermal CSTR–6ash–recycle system.Such phenomena, induced by heat e8ects (for example,

Balakotaiah & Luss, 1984) or complex kinetics (Gray &Scott, 1983, 1984), can also occur in a stand-alone CSTR.For this reason, in this article we consider the PFR. Althoughthe generic structure of PFR–separator–recycle is present in

many industrial plants, no results concerning its non-linearbehaviour have been reported. In contrast to the stand-aloneCSTR, the model of the stand-alone PFR is an initial-valueODE problem that always has a unique solution. Axial dis-persion and radial temperature gradients are important fordetailed mathematical modelling. However, these phenom-ena were deliberately neglected because they are sourcesof state multiplicity and instability in stand-alone reactors(Jensen & Ray, 1982). Therefore, our analysis is expectedto reveal non-linear phenomena that can be attributed exclu-sively to the recycle of mass. In addition, when heat e8ectsare included, we are certain that state multiplicity and in-stability are not the result of heat feedback. It should alsobe remarked that the plug-6ow model has been applied withsuccess for kinetic studies, design and simulation of manyindustrial reactors, including polymerization (for exampleChen, Vermeyduk, Howel, & Ehrlich, 1976; Gupta, Kumar,& Krishnamurthy, 1985; Zabisky, Chan, Gloor, & Hamielec,1992; Kiparissides, Verros, Pertsinidis, & Goossens, 1996).The non-linearity introduced by the reaction stoichiome-

try clearly a8ects the type of behaviour. In order to assessthis e8ect, we choose six reaction systems, of increasingcomplexity, denoted by S1–S6 and shown in Table 1. S1 andS2 are the simplest cases, representing the global stoichiom-etry of self-initiated and chemically initiated polymeriza-tion. These systems show regions of unfeasible operation,a feasible solution of the balance equations being born at atranscritical bifurcation. A fold bifurcation can occur whenheat e8ects are introduced, leading to state multiplicityand instability. An additional but important result is thatstate multiplicity is not restricted to exothermal reactions incooled reactors, but can also occur in heated reactors carry-ing on endothermal reactions. S3 and S4 take into account

A. A. Kiss et al. / Chemical Engineering Science 58 (2003) 2973–2984 2975

Table 1Investigated reactions mechanisms

Self-initiationa Chemical initiationa;b

S1. One reactant S2. Two reactants

Mk1→P M + I

k1→P

S3. Consecutive reactions S4. Consecutive reactions

Mk1→R I

k1→R

M + Rk2→P M + R

k2→P

S5. Chain-growth polymerization S6. Chain-growth polymerization

Mkd→ I · I2

kd→ 2I ·M + I · kI→R1· M + I · kI→R1·Rn · +M

kp→Rn+1· Rn · +Mkp→Rn+1·

Rn · +Rm · kt→Pn + Pm Rn · +Rm · kt→Pn + Pm

aFrom the reactor eDuent only the reactant M is recycled.bConcentration of I at reactor inlet is 7xed.

the initiation step. The autocatalytic, consecutive-reactionmechanism is enough to induce state multiplicity and insta-bility, even in the isothermal case. Moreover, the transcriti-cal bifurcation is destroyed when two reactants are involved.Heat e8ects do not change the qualitative picture, althoughthe unstable operating points cover a larger conversionrange. Finally, detailing the propagation–termination steps(S5 and S6) does not add anything new to the qualitativebehaviour.Interestingly, the bifurcation diagrams are qualitatively

similar to recycle systems involving a CSTR (Kiss et al.,2002). Moreover, there is also a quantitative agreement withrespect to the parameter values at which di8erent bifurcationphenomena occur. This suggests that the type of behaviour isdictated mainly by the reaction stoichiometry and 6owsheetstructure, rather than by the reactor type.

2. One-reactant recycle systems (S1)

We start our study with one-reactant 7rst-order reactionM → P, carried out in a PFR–separator–recycle sys-tem. This is the simplest case, but having a stoichiometryequivalent to the global stoichiometry of a self-initiatedpolymerization. A steady-state model, containing the massand energy balance, can be written in the following dimen-sionless form:

M balance:dMd�

=−Daf2

M exp(

��1 + �

): (1a)

Heat balance:

d�d�

=Daf2

(BM exp

(��

1 + �

)− H (� − �c)

); (1b)

M (0) = 1; �(0) = 0: (1c)

Reactor 6ow rate: f2 = 1=(1−M (1)): (1d)

The dimensionless variables and parameters, de7nedwith the feed 6ow rate F0, feed concentration cM;0 andreactor inlet temperature T1 as reference values, are: axialcoordinate 06 �6 1, concentration M (�) = cM (�)=cM;0,temperature �(�) = (T (�) − T1)=T1, reactor 6ow rate,f2 =F2=F0; plant DamkQohler number Da= k(T1)VcM;0=F0,Arrhenius temperature � = TA=T1, adiabatic tempera-ture rise B = (−RH)cM;0=(�cpT1), heat-transfer ca-pacity H = UA=(V�cpk(T1)), and coolant temperature�c = (Tc − T1)=T1. In this study we consider a sharp sep-aration; therefore, the recycle consists of pure reactant M ,while no M is found in the product stream.The model equations can be solved by a shooting tech-

nique: start with an initial guess for M (1), calculate thereactor outlet 6ow rate f2 using Eq. (1d), integrate the PFRequations (1a)–(1c), check and update the guess M (1). Bysetting equal reactor-inlet and coolant temperatures (�c =�1=0) and zero adiabatic temperature rise (B=0), the modelof the reactor operated isothermally at � = 0 is obtained.Then, integration of Eq. (1a), followed by the substitutionof f2 from Eq. (1d) leads to

lnM (1) =−Da(1−M (1)): (2)

It follows from Eq. (2) that M (1)=1 is a trivial solution,satisfying the balance equation irrespective of the Da value.Although unfeasible because of the corresponding in7nite6ow rates, accounting for this solution helps understandingthe mechanism by which feasible states occur.The non-trivial solution, given by

Da=− 11−M (1)

ln(M (1)) (3)

is feasible (positive 6ow rates) if, and only if Da¿DaT ,where

DaT = limM (1)→1

Da= 1: (4)

The relationship expressing the limiting value is identi-cal to the one given for CSTR–separator–recycle system byBildea et al. (2000). Such a feasibility constraint does notexist in the case of stand-alone reactors. As consequence,this is characteristic to recycle systems. The explanation isthat the separation section does not allow the reactant toleave the process. Therefore, for a given reactant feed 6owrate (F0), a large reactor volume (V ) or fast kinetics (k(T1))are necessary to consume completely the reactant fed in theprocess, and consequently to avoid reactant accumulation.The dimensionless plant DamkQohler number convenientlyincludes these three variables (6ow rate, reactor volume andkinetics).

2976 A. A. Kiss et al. / Chemical Engineering Science 58 (2003) 2973–2984

-0.2

0

0.2

0.4

0.6

0.8

1

0.1 1 10Da

X

α = 25H = 5θ c = 0

B = 0.30.15

0.1

0.20.25

0.05stableunstable

0

unfeasible

Fig. 2. One-reactant, 7rst-order reaction (S1). One solution is unfeasible, corresponding to zero conversion and in7nite recycle 6ow rate. For isothermaloperation (B = 0) a feasible solution exists if, and only if Da¿ 1. For the non-isothermal operation and �B¿ 1, multiple steady states exist.

Fig. 2 shows the dependence of reaction conversion,X = 1 − M (1), vs. Da number (bifurcation diagram) fordi8erent values of the adiabatic temperature rise (B¿ 0).At Da = DaT , the trivial and non-trivial steady-statemanifolds cross each other, in the combined space ofstate variable (X ) and parameter (Da). Simultaneously,one eigenvalue of the linearized dynamic model is zero(this can be easily shown by using a CSTR model)and the two solution branches exchange their stability.Thus, (X;Da) = (0; 1) is a transcritical bifurcation point(Guckenheimer & Holmes, 1983). This type of bifurca-tion occurs only in special cases, and is expected thatit will disappear under model perturbation, for examplemore complex kinetics (see Kiss et al. for a more detaileddiscussion).The entire bifurcation diagram (including the unfeasible

domain) contains one fold point. This enters the feasi-ble region at a boundary limit singularity (Golubitsky &Schae8er, 1985). For the CSTR–separator–recycle sys-tem, Pushpavanam and Kienle (2001) assumed in7-nite activation energy and equal coolant and reactorinlet temperatures, and derived an analytical expres-sion of this singularity. If these assumptions are re-moved, it can be shown that the fold point is feasiblewhenever

(H��c)¿ 1− B�: (5)

Remarkably, this relationship also applies to PFR–separator–recycle systems. Eq. (5) demonstrates that statemultiplicity is not restricted to exothermal reactions incooled reactors, but it can also occur in heated reactorscarrying on endothermal reactions.In the non-isothermal case, feasible non-trivial solutions

are possible for Da¡DaT =1. The trivial solution (X =0)is stable when Da¡DaT , but loses stability at Da = DaT .

Simultaneously, for increasing Da values,

• if the fold point is unfeasible (B�¡ 1), the non-trivialsolution gets physically meaningful values and gains sta-bility;

• if the fold point is located in the feasible region (B�¿ 1),the low-conversion branch of the non-trivial solutiongains stability, but becomes unfeasible.

Analysing more complex systems, we will disregard thetrivial, unfeasible solution (X = 0, in7nite 6ow rates).In the multiplicity region, the instability of the low-

conversion state can be proven using the steady-state model.For a stand-alone reactor, the amount of reactant consumed(F1X ) depends on the reactor’s feed 6ow rate. This depen-dence is shown in Fig. 3, using the dimensionless variablesF1=(k1(T1)cM;0V )=1=Da1 and F1X=(k1(T1)cM;0V )=X=Da1.In a reactor–separator–recycle system, the intersectionsof this curve with the dimensionless amount of reactantfed in the process F0=(k1(T1)cM;0V ) = 1=Da, providesthe steady-state values of the reactor-inlet 6ow rate. Twosteady states exist for H = 30; B = 0:25 � = 25; �c = 0 andDa = 0:6. To analyse the stability of the low-conversionstate B, let us consider an increase of the reactor inlet6ow rate. At the right of point B, the amount of reactantfed in the process is larger than the amount of reactantconsumed. Reactant accumulation occurs, leading to a fur-ther increase of the recycle and reactor-inlet 6ow rates.These arguments do not guarantee the stability of thehigh-conversion state, because stability can be lost at Hopfbifurcation points, simultaneously with occurrence of a limitcycle. Identifying such phenomena is beyond the scopeof this article, because it requires a full dynamic model,including the composition-controlled separation, the delayintroduced by recycle and the temperature-controlled heatexchanger.

A. A. Kiss et al. / Chemical Engineering Science 58 (2003) 2973–2984 2977

1

1.5

2

2.5

1 10 100

1 / Da 1

X / D

a1

1 /

Da

consumption rate

reactant feed rate

A B

α = 25

H = 5

θ c = 0

B = 0.25

Fig. 3. Instability of the low-conversion steady state. Steady-state analysis gives a necessary, but not suIcient stability condition.

3. Two-reactants recycle systems (S2)

The second-order reactionM+I → P has a stoichiometrythat is identical to the global stoichiometry of a chemicallyinitiated polymerization. From the reactor eDuent, only one

reactant (M) is recycled. The feed 6ow rate of the secondreactant (I) is adjusted in order to keep its reactor-inletconcentration at a prescribed value (zI;1). The dimensionlessmodel is given by

M balance:dMd�

=−Daf2

MI exp(

��1 + �

): (6a)

I balance:dId�

=−Daf2

MI exp(

��1 + �

): (6b)

Heat balance:

d�d�

=Daf2

(BMI exp

(��

1 + �

)− H (� − �c)

); (6c)

Fixed zI;1: zI;1 − fI;0zI;0f2

= 0: (6d)

Flow rate: f2 − (1 + f2M (1) + fI;0) = 0; (6e)

M (0) = 1− zI;1=zI;0; I(0) = zI;1; �(0) = 0: (6f)

where, in addition to the variables already de7ned

Da= k1(T1)V

F0=cM;0cM;0; I = cI (�)=cM;0;

zI;0 = cI;0=cM;0; zI;1 = cI;1=cM;0:

For isothermal operation (B = 0; � = 0) a closed-formparametric solution can be obtained:

M (1) = m; (7a)

Da=− z2I;0 ln((zI;0 − zI;1)(zI;0zI;1 + zI;1 − zI;0 + zI;0m)=z2I;0zI;1m)

(zI;0zI;1 − zI;0 + zI;1)(zI;0m− zI;0 + zI;1); (7b)

I(1) = zI;1 −(1− zI;1

zI;0− m

); (7c)

f2 =1

1− m− zI;1=zI;0; (7d)

fI;0 =zI;1

zI;0 − zI;0m− zI;1; (7e)

where m∈ [max(0; 1− zI;1=zI;0 − zI;1); 1− zI;1=zI;0].Fig. 4, diagram A, presents the conversion X vs. the plant

DamkQohler number, for zI;0 = 1 and di8erent values of zI;1.Similarly to S1, a feasible solution occurs at a transcriti-cal bifurcation. This results by setting m = 1 − zI;1=zI;0 inEq. (7b):

DaT =zI;0

zI;1(zI;0 − zI;1): (8)

Moreover, the minimum value of DaT is given by

minDaT = (DaT ) zI; 1=1=2zI; 0=1

= 4: (9)

Relationships (8) and (9) are identical to the ones foundfor isothermal CSTR–separator–recycle system (Kiss et al.,2002).DiagramB of Fig. 4 presents results for the non-isothermal

case. Similarly to the one-reactant system S1, multiple

2978 A. A. Kiss et al. / Chemical Engineering Science 58 (2003) 2973–2984

0

0.2

0.4

0.6

0.8

1

1 10 100

Da

X

z I,1 = 0.1

0.2

0.3

0.4

0.5

0.60.7

0.8

0.9

z I,0 = 1

( )I,0

T

I,1 I,0 I,1

zDa

z z z=

−

0

0.2

0.4

0.6

0.8

1

X

z I0=1z I1=0.5α =25

θH =5

c=0B = 0

0.6

0.4

0.2

Stable

Unstable

0.8

1

(A)

1 10 100

Da(B)

Fig. 4. Two-reactants, second-order reaction (S2). (A) Isothermal: afeasible solution exists only if Da exceeds the transcritical bifurcation,Da¿DaT . (B) Non-isothermal: heat e8ects induce state multiplicity.

steady states occur as the result of thermal e8ects, whilethe transcritical bifurcation remains intact. A high value ofB has a severe e8ect on state multiplicity, shifting the foldpoint to high conversions. When multiple steady states exist,the low-conversion branch is unstable.

4. Consecutive-reactions recycle systems

In this section, more complexity is introduced by consid-ering two consecutive reactions and an intermediate compo-nent (R). In a chain-growth polymerization, these roughlycorrespond to the initiation and propagation–terminationsteps, and the radical species, respectively.The dimensionless parameters are de7ned following the

procedure presented by Kiss et al. (2002), where the reactantrate constant used in the DamkQohler number is related tothe consumption rate of reactant M under quasi-steady-stateapproximation (QSSA). A second parameter, !, is relatedto the QSSA concentration of the intermediate component.We assume small activation energies, with the exception ofthe initiation step (dimensionless parameter �). Moreover,we consider that heat is released only by the propagationstep (dimensionless parameter B).

4.1. Self-initiation (S3)

The stoichiometry and kinetic model of self-initiated con-secutive reactions are given in Table 1. The dimensionlessmodel of the PFR–separator–recycle system can be writtenas:

M balance:

dMd�

=− Da2f2

(M exp

(��

1 + �

)+ !MR

); (10a)

R balance:

dRd�

=Da2f2

(M exp

(��

1 + �

)− !MR

); (10b)

Heat balance:d�d�

=Da2f2

(B!MR− H (� − �c)); (10c)

M (0) = 1; R(0) = 0; �(0) = 0; (10d)

Flow rate: f2 = 1=(1−M (1)); (10e)

where Da = 2k1(T1)V=F0=cM;0; ! = k2cM;0=k1(T1); � =TA;1=T1; B = [(−RH2)cM;0=�cp]1=T1; R(�) = cR=cM;0.

Diagram A of Fig. 5 presents the dependence of reactionconversion vs. plant DamkQohler number, for di8erent valuesof ! and isothermal conditions. All curves pass through thetranscritical bifurcation point (Da; X )=(2; 0). When the sec-ond reaction is slow (!¡ 1), the system behaves similarlyto the 7rst-order isothermal reaction in a PFR–separator–recycle system: a unique, feasible steady state (X ¿ 0) ex-ists if, and only if, Da¿ 2. When the second reaction is fast(!¿ 1) the system exhibits multiple steady states. For verylarge !, the conversion on the unstable branch has very smallvalues. As in the case of CSTR–separator–recycle systems(Kiss et al., 2002),

lim!→∞

DaF = 1: (11)

In diagram B of Fig. 5, the bifurcation diagram for thenon-isothermal case is presented. State multiplicity appearsas a result of thermal e8ects. A higher value of B shifts thefold point to higher conversions. The Da–X space is dividedin two regions with di8erent stability.In order to illustrate the change of stability the dynamic

model for system S3 is being used:

M balance:

dMd"

=−dMd�

− Da2f2

(M exp

(��

1 + �

)+ !MR

);

(12a)

R balance:

dRd"

=−dRd�

+Da2f2

(M exp

(��

1 + �

)− !MR

); (12b)

A. A. Kiss et al. / Chemical Engineering Science 58 (2003) 2973–2984 2979

0

0.2

0.4

0.6

0.8

1

1 10Da

X

β =1α Η

=25 =5

θ c=0

B = 0

0.6

2

0.2Stable

Unstable

1.51

0

0.2

0.4

0.6

0.8

1

1 10Da

X

102

β =10-1110

(A)

(B)

Fig. 5. Consecutive reactions with self-initiation (S3). (A) Isothermal: allthe curves pass through the point (Da; X ) = (2; 0) that is a transcriticalbifurcation point. For !¿ 1, state multiplicity is possible. No steady stateexists for Da¡ 1. (B) Non-isothermal: state multiplicity appears as aresult of thermal e8ects.

Heat balance:

d�d"

=−d�d�

+Da2f2

(B!MR − H (� − �c)); (12c)

M (0) = 1; R(0) = 0; �(0) = 0; (12d)

Flow rate: f2 = 1=(1−M (1)): (12e)

Diagram A of Fig. 6 shows the dynamic simulation re-sults. The system is initially in the unstable low-conversionstate. At " = 10, the DamkQohler number Da is changed by±1% for a limited period of time (R" = 1). Depending onthe disturbance direction, the high- or zero-conversion sta-ble states are reached. In diagram B of the same 7gure thedimensionless concentration pro7le of reactant M is plottedalong the reactor length at di8erent times.

4.2. Chemical initiation (S4)

This system extends S2, by including an intermediatecomponent (Table 1). The dimensionless model is given by

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50τ τ

X

Da = 1.75 + 1%

Da = 1.75 - 1%

Da =1.75β θ

= 1

c= 0α = 25H = 5B = 0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1ξ ξ

M

τ =0..10

40

30

25

3231

2928

33

(Α)

(Β)

Fig. 6. Instability of the low-conversion steady state (S3). (A) Small dis-turbances lead to zero conversion (reactant accumulation) or higher con-version state (reactant depletion). (B) Concentration pro7les of reactantM along the reactor, during the transition from low- to high-conversionbranch.

the following equations:

M balance :dMd�

=−Daf2

!MR; (13a)

I2 balance:dId�

=−Daf2

I exp(

��1 + �

); (13b)

R balance:dRd�

=Daf2

(I exp

(��

1 + �

)− !MR

); (13c)

Heat balance:d�d�

=Daf2

(B!MR− H (� − �c)); (13d)

Fixed zI;1: zI;1 − fI;0zI;0f2

= 0; (13e)

Flow rate: f2 − (1 + f2M (1) + fI;0) = 0; (13f)

M (0) = 1− zI;1=zI;0; I(0) = zI;1;

R(0) = 0; �(0) = 0; (13g)

where Da= k1(T1)V=F0=cM;0, ! = k2cM;0=k1(T1).

2980 A. A. Kiss et al. / Chemical Engineering Science 58 (2003) 2973–2984

0

0.2

0.4

0.6

0.8

1

1 10 100Da

X

β = 104

102

10-1

10

1

zI,0 = 1zI,1 = 0.5

103

FI,1

1lim Da

zβ ∞ =

0

0.2

0.4

0.6

0.8

1

0.1 1 10

Da

X

β =100α =25Η =5θ c=0

B = 0

0.5

2 0.21.5

1

zI,0 = 1zI,1 = 0.5

30.7

0.3

(A)

(B)

→

Fig. 7. Consecutive reactions with chemical initiation (S4). (A) Isother-mal: all the curves exhibit a fold point, presented by the dotted lines fordi8erent values of !. For large !, the conversion on the lower branchhas small values. (B) Non-isothermal: heats e8ect shift the fold point tohigher conversion.

Diagram A of Fig. 7 presents the dependence of reactionconversion vs. plant DamkQohler number, for di8erent valuesof !, zI;0=1, and zI;1=0:5. All curves exhibit a turning (fold)point, represented by the dotted line. When the QSSA for theintermediate component R is valid (large !), the conversionat the turning point has very small values. Moreover, similarto CSTR–separator–recycle systems,

lim!→∞

DaF =1zI;1

: (14)

Comparing Fig. 7 with Figs. 2, 4A and 5A, we see adramatic e8ect due to the introduction of the second reactantI , in the reaction mechanism. For 7nite ! (!¡∞), thetranscritical bifurcation has vanished, and state multiplicityturns out to be a generic feature.When thermal e8ects are considered the unstable branch

extends to high conversions, as shown in diagram B of Fig.7. The pattern of the curves is similar to the one observedfor previous systems.We conclude that state multiplicity occurs generically

in isothermal reactor–separator–recycle system where con-secutive reactions involving two reactants take place.

Nevertheless, when the QSSA of the intermediate compo-nent is valid, the conversion on the unstable branch is verylow.

5. Polymerization recycle systems

In this section we consider radical polymerization sys-tems (Table 1, S5 and S6). Only the main steps of radi-cal polymerization are included: initiation, propagation andtermination. Pn (length n) and Rn· denote dead and livingpolymer, respectively. The concentration of initiator radi-cals (I) is subject to the quasi-steady-state approximation(QSSA). This is usually valid for polymerization systems.Furthermore, the results do not change by removing this as-sumption. One manifestation of chain reactions is that thetotal concentration of the active intermediates is very small.A typical value for radical polymerization is 10−8 mol=l.The quasi-steady-state approximation considers that initia-tion and termination rate of the intermediates are virtuallyequal. The QSSA is remarkable accurate under most con-ditions. Most kinetic analysis of chain reactions rely on theQSSA for computation of intermediate concentrations. Fora detailed explanation about the QSSA validity the reader isreferred to the book of Biesenberger and Sebastian (1993).In the case of polymerization systems, the dimensionless

parameter � represents the activation energy of the disso-ciation step. B is heat of reaction for the propagation step.The model parameters Da and ! are related to the QSSAmonomer consumption rate and radical concentration, re-spectively. The parameter # is the ratio between terminationand propagation rate constants.

5.1. Chain-growth polymerization, with self-initiation(S5)

In case of self-initiated chain-growth polymerization themonomer is also acting as an initiator. The kinetic modelis given in Table 1 (S5). The dimensionless equations aregiven by

M balance:

dMd�

=−Daf2

(2#!

M exp(

��1 + �

)− !MR

); (15a)

R balance:

dRd�

=Daf2

(#!

M exp(

��1 + �

)− #!R2

); (15b)

Heat balance:d�d�

=Daf2

(B!MR− H (� − �c)); (15c)

M (0) = 1; R(0) = 0; �(0) = 0; (15d)

Flow rate: f2 − (1 + f2M (1)) = 0; (15e)

A. A. Kiss et al. / Chemical Engineering Science 58 (2003) 2973–2984 2981

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5Da

X

β = 106

γ = 106

105

104, 103

10210

1

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5Da

X

β =106

γ =10 =25

Η α

=5θ c =0

B = 0

0.6

0.40.2

0.81

(A)

(B)

Fig. 8. Chain-growth polymerization systems with self-initiation (S5).(A) Isothermal: two feasible steady states appear at a fold bifurcationpoint. For large # values (fast termination), the fold point corresponds tolow conversion. (B) Non-isothermal: heat e8ect shifts the fold point tohigher conversion.

where Da = kp√(kdcM;0=2kt)V=F0=cM;0; ! =

√2ktcM;0=kd;

#= 2kt=kp; R(�) =∑

n cRn(�)=cM;0.Diagram A of Fig. 8 shows the monomer conver-

sion (X ) vs. the plant DamkQohler number, for ! = 106

and di8erent termination/propagation ratios, #. Feasi-ble steady states do not exist for low values of Da.Two feasible steady states appear at a fold bifurcationpoint:

• for large values of # (fast termination, for example#¿ 103), the corresponding conversion has extremelysmall values. In this case, the low-conversion unstablebranch has no practical implication;

• for low values of # (slow termination), the foldpoint corresponds to a large value of the conver-sion. This means that there is a practical range ofconversion values that results in an unstable reactordesign.

The thermal e8ects have the expected e8ect of shift-ing the fold point to higher values of conversion (Fig. 8,diagram B).

5.2. Chain-growth polymerization, with chemicalinitiation (S6)

For this system the kinetic model is presented in Table 1.The dimensionless model, Eqs. (16), assumes small initiatorconcentration at reactor inlet, zI;1. Therefore, the initiatorfeed 6ow rate fI;0 can be neglected.

I2 balance:dI2d�

=−Daf2

#2!

I2 exp(

��1 + �

); (16a)

M balance:

dMd�

=−Daf2

(!MR+

#!

I2exp(

��1 + �

)); (16b)

R balance:

dRd�

=Daf2

(#!

I2 exp(

��1 + �

)− #!R2

); (16c)

Heat balance:d�d�

=Daf2

(B!MR− H (� − �c)); (16d)

M (0) = 1− zI;1=zI;0; I(0) = zI;1;

R(0) = 0; �(0) = 0; (16e)

Flow rate: f2 − (1 + f2M (1)) = 0; (16f)

where Da=kp√(kd(T1)cM;0=kt)V=F0=cM;0; !=

√ktcM;0=kd;

#= 2kt=kp.The diagram A of Fig. 9 shows the solution of Eqs.

(16), for di8erent termination/propagation ratios, dimen-sionless termination/propagation ratios and zI2 ;1 =10−4, !=103. Each bifurcation diagram exhibits one fold (turning)point. Thus, no steady states exist for Da¡DaF and twosteady states exist for Da¿DaF . For small values of #(slow termination) the conversion at the fold point is highand the multiplicity of states is important. For large val-ues of # (for example #¿ 102) the conversion of the lowerbranch has a very small value. Hence, only the upper state,existing for Da¿DaF , has practical signi7cance. How-ever, when the termination/propagation ratio is further in-creased, the conversion becomes limited by the amount ofinitiator fed in the process. As in CSTR–separator–recyclesystems,

lim#→∞DaF =

1√zI;1

: (17)

Diagram B of Fig. 9 presents the results for the casewhen the thermal e8ects are considered. As predictedby the behaviour of previous non-isothermal systems,higher reaction heat B shifts the fold point to higherconversions.

2982 A. A. Kiss et al. / Chemical Engineering Science 58 (2003) 2973–2984

0

0.2

0.4

0.6

0.8

1

100 1000 10000

Da

X

β = 103

z I ,1 = 10-4

1

102

103

5010

γ = 10-1

0

0.2

0.4

0.6

0.8

1

30 60 90 120 150

Da

X

zI ,1 = 10-4

β = 106

γ = 102

α = 25H = 1θ c = 0

2

2

B = 0

5

14 3 2

(A)

(B)

Fig. 9. Chain-growth polymerization systems with chemical initiation(S6). (A) Isothermal: two feasible steady states appear at a fold bifurcationpoint. For slow termination (small #), the unstable branch extends to highconversion. (B) Non-isothermal: heat e8ect shifts the fold point to higherconversion.

5.3. Example—low-density polyethylene (LDPE)process

In this section, we use the high-pressure LDPE processas a case study, in order to show that state multiplicityand instability can occur in realistic PFR–separator–recyclesystems. In addition, we also check the validity of severalassumptions that were employed throughout the previoussections.The process and most important speci7cations are pre-

sented in Fig. 10. Fresh ethylene is fed at 100◦C and 2025bar to the mixing section, where the recycle is added. Themixed stream is heated in a pre-heater where the tempera-ture increases to 170◦C. The heated stream is passed thenthrough the reactor. This has a diameter of 0:05 m anda total length of 2000 m. In view of the large value ofthe reactor length, resulting in a small value of the Pecletnumber, the PFR model is realistic. The initiator (benzoylperoxide) is fed at four di8erent locations, at a concentra-tion of 0:18 mol=m3. The reaction mixture leaves the reac-tor and goes to the high-pressure separation 6ash operatedat 250 bar. Here, about 95% of the monomer is separatedand sent to the recycle mixer. The second 6ash operates at alower pressure (2 bar) and makes the 7nal separation. The

rest of the monomer is sent to the recycle mixer, while theLDPE is recovered at the bottom, at purity higher than 99%.The high- and low-pressure recycles of monomer are mixedand then compressed at the same pressure as the fresh ethy-lene stream. The compressed recycle stream is mixed withthe fresh ethylene and fed again to the reactor.The (simpli7ed) model is very similar to Eqs. (16),

adding the reactions of chain transfer to monomer andsolvent, initiator feed at four points along the reactor, andtemperature dependence for all reaction constants. Thisis compared with a rigorous Aspen Polymers PlusTM

(AspenTech, 2001) simulation that accounts for variablephysical properties, imperfect separation, and heat e8ects forall reaction steps. The kinetic parameters are chosen fromMavridis and Kiparissides (1985). The physical propertieswere calculated using the Sanchez-Lacombe model, withthe parameter values available in Aspen Polymer PlusTM.Similar results were obtained using the SAFT model withregressed parameter data (Orbey, Bokis, & Chen, 1998).Vadapalli and Seader (2001) described the framework

for tracing arbitrary bifurcation diagrams using commercialsimulators. In our study we used a simpler, although lessgeneral approach: the user speci7es the reaction conversion,and the simulator calculates the corresponding feed 6owrate. This procedure can be easily implemented by means ofa design speci7cation block.The dimensionless parameters indicate large termina-

tion/propagation ratio (# = 9:35 × 104), but also highheat of reaction (B = 3:595) and near adiabatic operation(H =5:87×10−3). Therefore, a fold point is expected, witha signi7cant value of the corresponding conversion. Unex-pectedly, the bifurcation diagram (Fig. 11A) looks as thesuperposition of four (the number of initiator feeds) bifur-cation diagrams similar to the ones presented in Fig. 9, andshows seven fold points. A rather large range of conver-sion (0...0.07, 0.09...0.11, and 0.15–0.20) correspond to anunstable operating point. High sensitivity of the conversionto changes of plant DamkQohler number, around Da = 80,should be also remarked.The agreement between the simpli7ed and the rigorous

models is good, except the region of very low conversions.As X → 0, the solution of the simpli7ed model goes toDa=∞, while the solution of the rigorous model turns backtowards Da = 0. This di8erence has no practical implica-tion because of the extremely low value of the conversion.It can be explained by the di8erence in modelling the sep-aration unit. In the simpli7ed model, separation is perfectand no monomer is allowed to leave the plant. In practice,perfect separation is not possible because some monomeris dissolved in the polymer product. This amount is signi7-cant for very low conversion, when the separation unit hasto recover a small amount of polymer from a huge amountof monomer. The rigorous model correctly predicts this be-haviour.Fig. 11B shows di8erent molecular weight distributions

for several feed 6ow rates of monomer, corresponding to

A. A. Kiss et al. / Chemical Engineering Science 58 (2003) 2973–2984 2983

Recycle

Ifeed

1 2

3

4

LDPE

Ethylene

LPSHPS

0100 ˚C2025 bar

2025 bar

250 bar 2 bar

D = 0.05 mL = 2000 m

170 ˚C2025 bar

cI = 0.18 mol/m3

Tc = 170 ˚C

1100 m 1400m 1700m

Fig. 10. LDPE process 6owsheet.

0

0.05

0.1

0.15

0.2

0.25

50 75 100 125 150 175

Da

X

Simplified model

AspenPolymersPlus z I1= 1·10-5

B = 3.595H = 5.87·10-3

θ c= 0α = 17.737β = 2.05·105

γ = 9.35·104

1

3

4

2

0.0

0.1

0.2

0.3

0.4

0.5

0 75000 150000 225000 300000

Molecular weight

Wei

gh

t fr

acti

on

Case 1: Da=92.9; X=0.2Case 2: Da=92.4; X=0.125Case 3: Da=102.07; X=0.17Case 4: Da=102.86; X=0.215

(A)

(B)

Fig. 11. LDPE process: Da–X bifurcation diagram. (A) Several fold pointsexist in the practical range of conversion. The simpli7ed model agreeswith the rigorous simulation (Aspen Polymer PlusTM). (B) Molecularweight distribution for four di8erent operating points.

operating points 1–4 in diagram A. Transitions 1 → 2 and3 → 4 are expected when the production rate is changed(i.e. Da). It can be observed that the quality of the polymeris greatly a8ected. In the 7rst case (1 → 2), a small increase

of the feed 6ow rate reduces the conversion from 0.2 to0.125, and the polydispersity index (PDI) from 5.21 to 4.23.In the second case (3 → 4), for small decrease of the feed6ow rate, the conversion and polydispersity index increasefrom 0.17 to 0.215, and from 4.94 to 5.27, respectively. Inpractice, these changes might be unacceptable. Therefore,the design of recycle system should take into account thestate multiplicity and instability.

6. Conclusions

Six reactions with di8erent stoichiometry taking placein PFR–separator–recycle systems were investigated. Fora 7xed feed 6ow rate and a given kinetics, feasible steadystates are possible only if the reactor volume exceedsa critical value. For simple stoichiometry (S1, S2) andisothermal operation, the critical point represents a trans-critical bifurcation. When heat e8ects are included or forconsecutive, stoichiometry involving one reactant (S3, S5),one fold point may enter the feasible range of positive con-version leading to state multiplicity. If two reactants areinvolved in consecutive-autocatalytic reactions (S4, S6),state multiplicity is a generic feature.This behaviour is identical to the one previously found in

similar isothermal systems involving a CSTR. The agree-ment refers not only to qualitative features, but also to theparameter values at which di8erent bifurcation phenomenaoccur. This suggests that the behaviour is determined by re-action stoichiometry, recycle policy and control structure,and not by the reactor type.The multiplicity of states is important because it is accom-

panied by the instability of the low-conversion branch thatsets a lower limit on the achievable conversion. For isother-mal reactor–separator–recycle polymerization systems, thisbehaviour has practical importance (i.e. the conversion

2984 A. A. Kiss et al. / Chemical Engineering Science 58 (2003) 2973–2984

on the unstable branch is signi7cant) when the radicals’quasi-steady-state approximation is not valid (slow termina-tion, gel-e8ect). For non-isothermal systems the large heate8ect of polymerization reactions renders multiplicity tobe very probable. When designing PFR–separator–recyclesystems one must be aware of the non-linear behaviour andits implications on plant operability.

Notation

A heat transfer area, m2

c concentration, mol=m3

cp speci7c heat, J=kg KF 6ow rate, mol/sRH heat of reaction, J=mol Kk reaction rate constant, (mol=m3)n−1=sT temperature, KTA Arrhenius temperature, KU heat transfer coeIcient, W=m2 KV reactor volume, m3

X conversion, dimensionless

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