Steady-State Nonlinear Bifurcation Analysis of a High-Impact

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1972

Ind. Eng. Chem. Res. 2000, 39, 1972-1979

Steady-State Nonlinear Bifurcation Analysis of a High-Impact Polystyrene Continuous Stirred Tank Reactor Juan Carlos Verazaluce-Garcı´a and Antonio Flores-Tlacuahuac* Departamento de Ciencias, Universidad Iberoamericana, Prolongacio´ n Paseo de la Reforma 880, Me´ xico DF 01210, Me´ xico

Enrique Saldı´var-Guerra Centro de Investigacion y Desarrollo Tecnolo´ gico (CID), Avenida de los Sauces 87-6, Lerma, Edo de Me´ xico 52000, Me´ xico

In this work the open-loop nonlinear bifurcation analysis of a continuous stirred tank reactor where the high-impact styrene polymerization takes place is performed. The effect of potential manipulated and disturbance variables is addressed. The effect of nonlinearities on reactor startup is also discussed. One of the multiplicity sources is identified, and the implications for operability are stressed. Using the bifurcation maps, some ways of removing potential undesired nonlinearities are also mentioned. 1. Introduction For a long time it has been recognized that the nonlinear behavior (i.e., input/output multiplicities, limit cycles, etc.) of chemical reactors might have an important effect on the operation difficulty of such processes.1 Bifurcation theory has been recognized as a very useful tool to address the nonlinear pattern behavior of processing systems subject to the variation of some meaningful parameters.2 Using such bifurcation and continuation methods, it becomes feasible to numerically detect input multiplicities, output multiplicities, isolas, limit cycles (Hopf bifurcations), etc. This information could be used to remove undesired nonlinearities because a common point of view is that sometimes the presence of such kinds of nonlinearities might affect adversely the performance of closed-loop systems and that nonlinear plants require nonlinear control laws for satisfactory closed-loop control. However, there are in the academic literature evidences that highly nonlinear processes might be controlled using linear controllers3 and that open-loop nonlinearity does not necessarily imply the use of nonlinear control laws for acceptable control of nonlinear processes. The degree of nonlinearity in the control law for nonlinear process control is still a research topic.4,5 In any case the selection of a control law depends not only on the openloop nonlinearity characteristics but also on some closedloop performance objective.4 In this work we adopt the more classical viewpoint that open-loop nonlinearity might be an indicator of closed-loop control difficulties. It is clear that such an open-loop nonlinearity depends also upon the design operating region. In particular, because of complex kinetic behavior, mass- and heat-transfer limitations, etc., complicated steady-state nonlinear bifurcation behavior has been * Author to whom correspondence should be addressed. E-mail: [email protected]. Phone/fax: +52 5 267 42 79. Website: http://kaos.dci.uia.mx/∼aflores.

reported in several types of polymerization reactors.6 In bulk polymerization of methyl methacrylate (MMA), the emergence of steady-state multiplicity has been associated with the so-called gel effect.7 This effect appears in high-conversion regions in which there is a high concentration of polymer and highly viscous polymer mixtures. The effect of such a viscosity increase is to decrease the frequency of the termination reactions compared to that of the propagation reactions because the termination step becomes diffusion-controlled and the molecular weight increases. Using bifurcation analysis, Russo and Bequette8 studied the impact of process design on the multiplicity behavior of a continuous stirred tank reactor (CSTR) for the case of an irreversible first-order reaction. Later on, the same authors9 analyzed the effect of changes in process design on the open-loop behavior of the reaction system previously studied. Russo and Bequette10 also conducted a multiplicity and operability analysis in a styrene polymerization reactor. More recently, SilvaBeard and Flores-Tlacuahuac11 performed a similar analysis for an MMA bulk polymerization reactor. A more complete literature review on the nonlinear behavior of polymerization reactors can be found in a recent paper of our research group.11 In this work we perform the steady-state nonlinear bifurcation analysis of a CSTR for the production of high-impact polystyrene (HIPS). HIPS is an important commodity material which combines the ease of processing of polystyrene with increased mechanical resistance. It is used in a variety of applications: packaging, household appliances, etc. Although there are still a few industrial facilities operating the batch mass-suspension process for the production of HIPS, most of this material is presently produced via the more profitable bulk continuous process. There are several variations of this process,12,13 but most of them use a CSTR in series with some kind of tubular reactor(s). The residence time distribution of the reactors, as well as the

10.1021/ie990560a CCC: $19.00 © 2000 American Chemical Society Published on Web 04/27/2000

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reaction conditions (mixing effectiveness, temperature, viscosity), plays an important role in determining the size and morphology of the rubber particles and the final properties of the product. The polymerization can be carried out by thermal initiation or chemical initiation with monofunctional or bifunctional initiators.14,15 Elementary catastrophe theory might be used in order to detect analytical conditions under which input/output multiplicities and isola behavior could emerge. However, one of the major problems related to the use of catastrophe theory is that it requires collapsing all of the mathematical model into a single algebraic equation. This procedure is totally impractical for large-scale models. Besides, because of the presence of transcendental functions, it may be difficult to combine all of the model equations. Even if one could end up with a single algebraic equation, the resulting expression might be quite complicated to manipulate because theoretical conditions from catastrophe theory require evaluation of first and higher order derivatives (a difficult task even using an algebraic manipulation program). Because of the complexity of our HIPS model, we decided to try in the first place a purely numerical continuation procedure as a way to characterize the HIPS reactor input/output multiplicity behavior. The aim of this work is to provide a first look into the operability problems faced by HIPS reactors. The steady-state operability problem is addressed by using nonlinear bifurcation techniques. In the open literature there are only a few recent works on the modeling of the HIPS process.16-18 There are not published works on the nonlinear analysis of the HIPS manufacturing process. Even when the industrial HIPS process involves several series connected reactors, we have selected only a CSTR to perform the bifurcation analysis in order to keep things as simple as possible and to avoid the effect of interactions among the reactors on the nonlinear behavior. Also a simple case of the HIPS production process is considered, therefore avoiding mathematical modeling complexities. All of the nonlinear bifurcation diagrams were generated using XPPAUTO software.19

Table 1. HIPS Kinetic Mechanism Initiation Reactions thermal ki0

3MS 98 2R1S chemical f1kd

I 98 2R ki1

R + MS 98 R1S ki2

R + B098 BR ki3

BR + MS 98 B1RS Propagation Reactions kp

RjS + MS 98 Rj+1 S kp

BjRS + MS 98Bj+1 RS Definite Termination Reactions homopolymer kt

j+m RjS + Rm S 98 P

grafting kt

RjS + BR 98 BjP kt

j+m RjS + Bm RS 98 BP

cross-linking kt

BR + BR 98 BEB kt

BjRS + BR 98 BjPB kt

j+m BjRS + Bm RS 98 BPB

Transfer Reactions monomer kfs

RjS + MS 98 Pj + R1S kfs

BjRS + MS 98 BjP + R1S grafting sites kfb

RjS + B0 98 Pj + BR

2. Mathematical Modeling

kfb

BjRS + B0 98 BjP + BR

In this part the mathematical model of the freeradical bulk polymerization of the system styrene/ polybutadiene, using a monofunctional initiator, is described. The set of polymerization reactions is carried out in a nonisothermal CSTR assuming perfect mixing, constant physical properties, quasi-steady state, and the long-chain hypothesis. Constant volume in the reactor has also been assumed. Changes in the density of the monomer-polymer mixture have been neglected. The kinetic mechanism involves initiation, propagation, transfer, and termination reactions. Polybutadiene is added in order to guarantee desired mechanical properties by promoting grafting reactions. Network formation reactions are also modeled because they may lead to an undesirable excess of cross-linking in the rubber particles. There is some debate about the reactions causing cross-linking. Keskkula20 has proposed as the main reaction responsible for cross-linking one occurring between two just-activated polybutadiene radicals. Other

Table 2. Rate Constants Information kd ) 9.1 × 1013e-29508/RgT ki0 ) 1.1 × 105e-27340/RgT ki1 ) 1.0 × 107e-7067/RgT ki2 ) 2.0 × 106e-7067/RgT ki3 ) 1.0 × 107e-7067/RgT kp ) 1.0 × 107e-7067/RgT kfs ) 6.6 × 107e-14400/RgT kfb ) 2.3 × 109e-18000/RgT kt ) 1.7 × 109e-843.0/T-2(C1xs+C2xs2+C3xs3) c1 ) 2.57 - 0.00505T c2 ) 9.56 - 0.0176T c3 ) -3.03 + 0.00785T

1/s L2/(mol2 s) L/(mol s) L/(mol s) L/(mol s) L/(mol s) L/(mol s) L/(mol s) L/(mol s)

authors14,16 assume that coupling between two radicals in the active ends of graft chains is the main contributor to cross-linking. In this model both possibilities are included. The kinetic mechanism is summarized in Table 1, while Table 2 contains information about rate

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constants; design parameters are shown in Table 3.

dI Q ) (I - I) - kdI dt V 0

(1)

dMS Q ) (MS0 - MS) - kpMS(µ0R + µ0B) dt V

(2)

dB0 Q ) (B00 - B0) - (ki2R + kfsµ0R + kfbµ0B)B0 dt V

(3)

dR Q ) (Ro - R) + 2f1kdI - R(ki1MS + ki2B0) dt V

(4)

dBR Q ) (BR0 - BR) + (B0(ki2R + kfb(µ0R + µ0B)) dt V BR(ki3MS + kt(µ0R + µ0B + BR))) (5) kt dBEB Q ) BR2 - BEB dt 2 V

( )

(6)

dS ) Q(So - S) dt

(7)

dT Q ) (To - T) + dt V

∆HrkpMS(µ0R

+

µ0B)

FsCps

UA(T - Tj) FsCpsV (8)

Table 3. Design Information for a Typical Industrial HIPS CSTR volume jacket volume initiator volumetric flow rate feedstream volumetric flow rate cooling water volumetric flow rate monomer feedstream concentration polybutadiene feedstream concentration initiator feedstream concentration feedstream temperature cooling water temperature heat of reaction heat-transfer area global heat-transfer coefficient feedstream density heat capacity cooling water density cooling water heat capacity initiator efficiency ideal gas constant

9450 2000 0.0015 1.14 1 8.63 1.05 0.981 333 294 -69919.56 19.5 80 0.915 1647.27 1 4045.7 0.57 1.987

L L L/s L/s L/s mol/L mol/L mol/L K K J/mol m2 J/(s K m2) kg/L J/(kg K) kg/L J/(kg K) cal/(mol K)

where

R1 ) 2ki0M3S + ki1RMS + kfsMSµ0B

(20)

R2 ) -(kpMS + kt(µ0R + µ0B + BR) + kfbB0)

(21)

R3 ) kpMS

(22)

B1 ) ki3BRMS

(23)

UA(T - Tj) dTj Qw ) (Tow - Tj) + dt Vc FwCpwVj

(9)

dµ0R Q ) (R1 + (R2 + R3)µ0R) - µ0R dt V

(10)

B3 ) kpMS

dµ1R Q ) (R2µ1R + R3(µ1R + µ0R)) - µ1R dt V

(11)

The above set of equations, derived using the method of moments, allow us to compute the average of the molecular weight distribution:

dµ0B Q ) (B1 + (B2 + B3)µ0B) - µ0B dt V

(12)

dµ1B

Q 1 µ V B

(13)

dλ0P kt 2 Q ) µ0R + (kfsMS + kfbB0)µ0R - λ0P dt 2 V

(14)

dλ1P Q ) (ktµ1Rµ0R + (kfsMS + kfbB0)µ1R) - λ1P dt V

(15)

dt

) (B1 + B2µ1B + B3(µ1B + µ0B)) -

(

dλB0 P dt

)

) (ktµ0R(µ0B + BR) + (kfsMS + kfbB0)µ0B) -

dλB1 P dt

) (kt(µ1RBR + (µ1Rµ0B + µ0Rµ1B)) + (kfsMS + kfbB0)µ1B) -

dλB0 PB dt dλB1 PB dt

Q 0 λ (16) V BP

(

) ktµ0BBR +

)

kt 0 2 Q µ - λB0 PB 2 B V

) (ktµ1BBR + ktµ1Bµ0B) -

Q 1 λ V BPB

Q 1 λ (17) V BP (18)

(19)

B2 ) -(kpMS + kt(µ0R + µ0B + BR) + kfsMS + kfbB0) (24) (25)

Mn )

λ1P + µ1R λ0P + µ0R

(26)

3. Results and Discussion (i) Effect of Manipulated Variables. In Figure 1 bifurcation diagrams of the reactor temperature and number-average molecular weight using the cooling water flow rate as the main continuation parameter and the initiator volumetric flow rate as the secondary parameter are shown. Under nominal operating conditions (denoted by the symbol O; as usual, stable steady states are denoted by a continuous line while unstable steady states are denoted by a dashed line) the reactor displays output multiplicities. This comes as no surprise as many systems exhibiting even simple kinetic schemes may show multiple steady states at different operating conditions. Output multiplicities might be a potential source of control problems. The nominal upper temperature steady state is unstable while the lower steady state is stable. A third upper branch in the continuation curve could be expected at higher temperatures giving rise to ignition and extinction phenomena; this branch was not found because the continuation parameter was kept in a realistic range. Therefore, open-loop control of the HIPS reactor under nominal design conditions is not possible because any arbitrary size disturbance would move the reactor away from the desired operating point.

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Figure 1. Continuation diagrams using the cooling water volumetric flow rate as the continuation parameter.

Figure 2. Continuation diagrams using the initiator feedstream volumetric flow rate as the continuation parameter.

This situation is easily analyzed from the bifurcation diagram. For instance, any small variation in the initiator flow rate would change the reactor operation to the lower temperature steady state. This implies around a 500% increase in Mn, a clearly undesired situation because HIPS properties are highly dependent upon the Mn. If bifurcation diagrams are obtained for larger initiator flow rate values, then input multiplicities for the Mn can be observed in Figure 2. In fact under nominal operating conditions, up to three different values of the initiator flow-rate would give rise to the same Mn. We should note that input multiplicities are not observed for the reactor temperature, a valid situation because input multiplicities are the result of nonmonotonic variation of a system state. From this bifurcation diagram we observe that multiplicities cannot be removed by manipulating either the cooling water flow rate or the initiator flow rate. Because such variables might be used for closed-loop control, the emergence of input and output multiplicities is very interesting. From a control point of view, input multiplicities seem to lead to a harder control problem because, under certain conditions, they can be related to inverse response21 (i.e., zeros crossing the imaginary axis). In situations like this one, because of changes in the plant gain sign, the use of proportional-integralderivative (PID) controllers is not recommended. It is important to stress that the presence of right-half plane zeros limits the achievable closed-loop performance, regardless of the control law used. Once input multiplicities are detected, they might be easily handled by imposing constraints on the manipulated variables that display this sort of undesired behavior. Hence, constrained nonlinear predictive control might be used for closed-loop control purposes because this control strat-

egy naturally takes into account constraints in the manipulated variables. The emergence of output multiplicities is not a very difficult control problem because the upper temperature unstable steady state might be stabilized by feedback control. (ii) Effect of Disturbances. We now analyze the open-loop behavior of the HIPS reactor in the presence of variables that might be considered as process disturbances. Figure 3 shows bifurcation diagrams using the monomer feedstream concentration as the main continuation parameter and the feedstream temperature as the secondary parameter. We observe that at the nominal operating point input and output multiplicities are exhibited by Mn. This result is interesting because this means that the same Mn might be achieved by using a smaller amount of monomer and operating at a lower reactor temperature; however, a lower productivity would be expected. A similar behavior is observed when the effect of the initiator feedstream concentration changes is analyzed as shown in Figure 4 where the feedstream flow rate is used as the secondary parameter. Again input and output multiplicities are observed in Mn. It should be noted that the upper limit of the initiator concentration (in order to get the same Mn) is rather high for real operation, but it allows us to observe interesting nonlinear phenomena. Input/output multiplicities can not be removed by changes in the initiator concentration and feedstream volumetric flow rate. In Figure 5 the bifurcation diagrams using the cooling water temperature as the main continuation parameter and the feedstream volumetric flow rate are shown. Again, the range of feasible cooling water temperature is extended beyond realistic values; this, however, shows the form of the complete curves and gives an idea of

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Figure 3. (a) Reactor temperature and (b) molecular weight using the feedstream monomer molar concentration as the continuation parameter.

Figure 4. Continuation diagrams using the initiator feedstream concentration as the continuation parameter.

the phenomena that can occur should other cooling media be used. In this diagram we clearly notice that

Figure 5. Continuation diagrams using the cooling water feedstream temperature as the continuation parameter.

the reactor has an additional stable open-loop hightemperature steady state, which was not observed previously. In this case only output multiplicities were observed for both the reactor temperature and Mn. The temperature bifurcation diagram shows that output multiplicities might be removed by increasing the feedstream volumetric flow rate 90%. However, an undesired effect would emerge: the reactor temperature would be reduced, leading to a drastic increase in MWD failing to meet the desired polymer properties. Figure 6 shows the bifurcation diagrams using the feedstream volumetric flow rate as the main continuation parameter and the monomer concentration as the secondary parameter. Even when at the nominal operating point the HIPS reactor only displays output multiplicities, a 25% decrease in the monomer feedstream concentration would give rise to an operating situation where both input and output multiplicities will be present. A similar behavior is observed when the feedstream temperature is used as the secondary parameter, as shown in Figure 7. (iii) Startup of the Reactor. If an analysis approach of sequential quasi steady states is taken, the bifurcation maps obtained in this work might also be used to analyze some of the problems to be presented during reactor startup. In order to illustrate this point, let us take the situation displayed in Figure 3. In order to arrive at the desired steady-state operating point, one might start feeding solvent and then initiator, followed by feeding monomer and polybutadiene at the nominal feedstream temperature. This is equivalent to operate the reactor along the number 1 operating curve. As this curve shows, extremely high monomer concentrations are required to reach the turning point (around 32 mol/ L); in a practical situation such large monomer concen-

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Figure 6. Continuation diagrams using the feedstream flow rate as the continuation parameter.

Figure 7. Continuation diagrams using the feedstream flow rate as the continuation parameter.

trations are not possible. However, if such infeasible monomer concentration values could be used, then a

Figure 8. Continuation diagrams using the residence time as the continuation parameter.

closed-loop control system should be used (before reaching the turning point) to arrive at the nominal operating point. A more practical way to startup the reactor can be seen from this same figure. By increasing the reactor feedstream temperature, so we reach operating curve number 2 or 3, the transition to the nominal steady state can be done using feasible monomer concentration values. In any case, a closed-loop control system would be needed to reach the desired nominal operating point. (iv) Source of Output Multiplicities. From a nonlinear analysis point of view, it becomes important to understand how multiplicities emerge. This knowledge might be used to identify potential ways of removing them. In polymerization reaction engineering, there have been some evidences that relate the presence of multiplicity behavior to the existence of the gel effect. The gel effect is the term used to denote a fast polymer viscosity increase leading to higher molecular weights and faster overall rate of reaction because termination reactions become more unlikely compared to propagation reactions. Multiplicity effects due to the gel effect might be reduced by adding a solvent, as Figure 8 shows. In the limit output multiplicities might all be removed by adding enough solvent (i.e., around 60% of the monomer volumetric flow rate). However, the implications for reactor productivity are clear because a smaller amount of monomer would be transformed into polymer chains. It is well-known22 that for polymerization reactors, and because of the gel effect, output multiplicities may arise even in the isothermal case. We decided not to analyze this case in order to have a more realistic approach to modeling industrial polymerization reactors. (v) Operability Implications. Because of the fact that the nominal operating point is unstable, since

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the open-loop system exhibits a single right-half plane (RHP) pole (p ) 4.6 × 10-4 rad/s), feedback control is necessary to stabilize the HIPS reactor. Therefore, manual operation is not recommended for such a reactor because operation at the nominal design point would be difficult. Manual operation of the reactor around the unstable design point is almost impossible because of the hysteresis behavior exhibited by the reactor. The presence of a RHP pole might impose some limitations on closed-loop control performance23 if RHP zeros happen to be located at frequencies similar to that of the RHP pole. RHP zeros might be introduced by the reactor design or by measurement delays. Hence, for good control performance, controlled variables should be available with negligible delays. The unstable HIPS reactor can be stabilized by manipulating the cooling water flow rate to control the reactor temperature. Because normally temperature measurements are readily available, this control loop should render an acceptable closed-loop control performance. If the temperature measurement delay were approximated by a first-order Pade´ approximation, then the closed-loop system might tolerate a maximum delay of θd < 2/p ≈ 72 min without stability worries. The situation is different if composition analyzers are used for feedback control because normally large measurement delays are implied. Youla et al.24 have shown that for a system with a single RHP pole and a single RHP zero, system stabilization will be impossible using a stable linear controller if p > z (where z stands for the zero location). Because PID controllers are the most commonly used industrial controllers, this result has an important implication. A way to stabilize reactor operation might be to use a composition estimation technique to eliminate measurement delays and perhaps to use a nonlinear control law to cope with reactor nonlinearities (even when a linear controller might be used sometimes to control nonlinear plants4). With respect to the measurement of other variables, weight cells can be used to measure with no delay changes in the reactor holdup. Molecular weight via online gel permeation chromatographic analysis would introduce a significant delay in the measurements.25 On the other hand, it should be possible to implement online measurement of viscosity with virtually no delay coupled with state estimation techniques in order to infer molecular weight. (vi) Removing Multiplicities. Bifurcation maps can also be used to analyze process condition changes that might lead to removal or reduction of the impact of multiplicities. From Figure 3 a form to remove input multiplicities can be observed. If the reactor might be operated in the low monomer concentration region, then, for practical purposes, the displayed input multiplicities are removed because very high monomer concentration (around 33 mol/L and possible beyond design values) is required to enter into the high temperature/low molecular weight region. Besides, the low monomer concentration region is open-loop stable which makes the closed-loop control problem less difficult. However, in this case the reactor might be operating in a low productivity region, so a compromise between operability and productivity should be pursued. (vii) Comparison against Experimental Plant Data. We have developed a steady-state process simulator to predict the operation of the HIPS process. The performance of such a simulator has been checked

Figure 9. Continuation diagram generated using experimentally verified kinetic parameters (experimental point denoted by the asterisk symbol).

against experimental plant data; mainly kinetic rate constants have been fitted to meet experimental information. Figure 9 shows one experimental point obtained in a commercial polystyrene plant (denoted by the asterisk symbol) and the corresponding bifurcation diagram generated by using the fitted kinetic parameters. Because both the model and the kinetic information had been experimentally verified, we strongly believe that the bifurcation diagram will be close to the real industrial one. Because of the problem of getting industrial plant data, we have included only one experimental point. Also, because of confidentiality reasons, we are not able to provide details on the operating conditions and kinetic parameters used to obtain Figure 9. 4. Conclusions Nonlinear bifurcation analysis of a HIPS CSTR has been carried out. The analysis showed that sometimes it might be difficult to remove input/output multiplicities and that a potential way to remove them would be to increase the ratio solvent/monomer volumetric flow rate. However, this situation might lead to reduced polymer productivity. Because the reactor displays both input and output multiplicities, the corresponding closedloop control problem might be difficult to address. This is an open research topic because some nonlinear plants had been successfully controlled by linear controllers.4 Presently we are addressing the closed-loop control problem and extending the HIPS model to include bifunctional initiator and transfer agents. Acknowledgment A.F.T. thanks the Centro de Investigacio´n y Desarrollo (CID) staff for their generous hospitality during his writing this paper. Thanks go to Grisel Ramı´rez for her help in reviewing the mathematical model and in providing experimental plant data. Thanks also go to Andrea Silva for her help in obtaining the bifurcation calculations shown in Figure 9. Financial support from Universidad Iberoamericana and CID is gratefully acknowledged. Nomenclature A ) heat-transfer area, m2 B ) grafted polymer, mol/L Bo ) polybutadiene unit, mol/L BEB ) cross-linked polybutadiene, mol/L

Ind. Eng. Chem. Res., Vol. 39, No. 6, 2000 1979 Bp ) grafted polymer, mol/L BPB ) cross-linked polymer, mol/L BR ) activated polybutadiene unit, mol/L BRS ) grafted radical with a styrene end group, mol/L Cp ) heat capacity, J/(kg K) I ) initiator concentration, mol/L M ) monomer concentration, mol/L Mn ) molecular weight distribution P ) homopolymer, mol/L Q ) volumetric flow rate, L/s R ) radical concentration, mol/L Rg ) ideal gas constant, cal/(mol K) S ) solvent concentration, mol/L T ) reactor temperature, K U ) global heat-transfer coefficient, J/(s K m2) V ) volume, L f1 ) initiator efficiency kd ) chemical initiation kinetic constant, L/s kfb ) polybutadiene transfer kinetic constant, L/(mol s) kfs ) styrene monomer transfer kinetic constant, L/(mol s) ki0 ) thermal initiation kinetic constant, L2/(mol2 s) kil ) chemical initiation kinetic constant (size 1), L/(mol s) ki2 ) polybutadiene activation chemical initiation kinetic constant, L/(mol s) ki3 ) grafted chemical initiation kinetic constant (size 1), L/(mol s) kp ) propagation kinetic constant, L/(mol s) kt ) coupling termination kinetic constant, L/(mol s) p ) pole location, rad/s z ) zero location, rad/s xs ) monomer conversion fraction ()(M0 - M)/M0) Superscripts 0 ) zeroth moment 1 ) first moment j, m ) polymer length Subscripts w ) cooling water j ) reactor jacket o ) feedstream condition s ) styrene Greek Symbols ∆Hr ) heat of reaction, J/mol λ ) moment of a dead species µ ) moment of an active species F ) density, kg/L θd ) time delay, m

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Received for review July 28, 1999 Revised manuscript received January 20, 2000 Accepted January 26, 2000 IE990560A