Extended Two-Step Method for Molecular Weight ... - ACS Publications

Aug 11, 2001 - While the conventional two-step method uses a quadratic form for the instantaneous average chain length (IACL), it is proved here that,...
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Ind. Eng. Chem. Res. 2001, 40, 3929-3935

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PROCESS DESIGN AND CONTROL Extended Two-Step Method for Molecular Weight Control in a Batch Polymerization Reactor Hyung-Jun Rho† and Hyun-Ku Rhee* School of Chemical Engineering and Institute of Chemical Processes, Seoul National University, Kwanak-ku, Seoul 151-742, Korea

This paper presents an extended two-step method for the control of the molecular weight distribution in a batch polymerization reactor. While the conventional two-step method uses a quadratic form for the instantaneous average chain length (IACL), it is proved here that, starting from an arbitrary form for the IACL, one can generate a temperature trajectory that would produce polymer having desired molecular weight averages at a specified level of conversion. Furthermore, the form chosen for the IACL exercises influence not only on the temperature trajectory but also on the reaction time required. These observations are readily incorporated to develop an extended version of the two-step method in such a way that constraints imposed on the reactor temperature may be taken into account and also the reaction time may be reduced. Simulation results for the methyl methacrylate polymerization reactor demonstrate that the extended two-step method is very useful for constructing the reactor temperature trajectories bounded by both upper and lower bounds. It also shows that the reaction time can be reduced effectively by the extended two-step method without changing the desired molecular weight averages and monomer conversion. The trajectory tracking control experiment proves that the polymer product having desired molecular weight averages is indeed obtained with the specified level of monomer conversion at a reduced reaction time. 1. Introduction The batch polymerization reactor has been widely used in chemical industries for manufacturing lowvolume and high-quality polymer materials. Such product specifications require a reactor operation strategy to meet the desired objectives such as productivity, molecular weight distribution, and process requirements at the end of the reactor operation. Many researchers have devoted their effort to generate the constrained temperature trajectories or the trajectories having shorter reaction time. However, their endeavor encountered a complex optimization problem due to the time dependency of the mathematical model used.1-9 Kwon and Evans10 proposed a coordinate transformation method to eliminate the time dependency of state variables. The two-step method is an excellent scheme for generating the reactor temperature trajectory for a batch polymerization reactor. Because it changes the optimization process into a simple algebraic scheme, the computing time can be considerably reduced and the convergence check routine can be avoided.11-13 This feature becomes particularly important when the twostep method is implemented online.14 However, not only does the conventional two-step method have limitations in the range of attainable * To whom correspondence should be addressed. E-mail: [email protected]. Tel: (+82) 2-880-7405. Fax: (+82) 2-8887295. † Current address: Hyundai Information Technology Co., Seoul 140-2, Korea.

properties of polymer15 but also it suffers from difficulties in the treatment of process constraints or the reduction of the reaction time. The conventional twostep method cannot admit all of the feasible trajectories, because it fixes the shape of the instantaneous average chain length (IACL) as a quadratic form whereas the shape of IACL is independent of the desired molecular weight averages. The present paper deals with an extension of the conventional two-step method that can handle the process constraints and reduce the reaction time required by iteratively updating the shape for the IACL. The present scheme does not have the limitation arising from the feature of the feasible regions for the molecular weight averages. Here, this scheme is proven effective by simulation studies. Control experiments are also conducted to show that polymers with desired properties are obtained with the prescribed conversion at a reduced reaction time by tracking the reactor temperature trajectory. 2. Polymerization Reactor Model The system considered in this study is the batch solution polymerization process of methyl methacrylate (MMA) using benzoyl peroxide (BPO) as the initiator and ethyl acetate (EA) as the solvent. The fundamental model includes initiation, propagation, termination, and chain-transfer reactions. Termination reactions by both combination and disproportionation as well as chaintransfer reactions to monomer and solvent are taken into account. For the fundamental model derivation and notations, one is referred to Yoo et al.14 and Yoo and Rhee.15

10.1021/ie000699h CCC: $20.00 © 2001 American Chemical Society Published on Web 08/11/2001

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To begin with, the kth moment of the chain concentrations of living polymers and the total moles of polymers are defined respectively as ∞

Gk ≡

jkRj, ∑ j)1

k ) 0, 1

(1)

∞

Hk ≡

jk(Rj + Pj)V, ∑ j)1

k ) 0, 1, 2

(2)

in which Rj and Pj denote the concentrations of living and dead polymers of chain length j. Then the balance equations for Hk, k ) 0, 1, 2, are derived in the form

1 dH0 ) 2fkdI + (ktrmM + ktrsS - 0.5ktdG0)G0 V dt

(3)

1 dH1 ) 2fkdI + (kpM + ktrmM + ktrsS)G0 ) V dt 1 d(MV) d(SV) + (4) V dt dt

(

)

1 dH2 ) 2fkdI + kpM(G0 + 2G1) + V dt (ktrmM + ktrsS)G0 + ktcG12 (5) where I, M, and S represent the concentrations of initiator, monomer, and solvent, respectively, and V denotes the volume of the reaction mixture. In fact, the first moment of the total polymer produced may be readily determined without solving the differential equations by using the following identity:

one for the zeroth moment of total moles of polymers produced, H0. Because the zeroth moment H0 is a monotonically increasing function of the reaction time, it may be regarded as an independent variable replacing time t. The equations for the first and second moments of total moles of polymers are rearranged to give

{ {

dH1 0 ) y, H1 ) H * dH0 1 dH2 0 ) py2, H2 ) H * dH0 2

at H0 ) 0 at H0 ) H0* at H0 ) 0 at H0 ) H0*

(10)

(11)

where y and p are defined by the IACL and the instantaneous polydispersity index, respectively. The boundary values, H0*, H1*, and H2*, can be obtained from the desired values of monomer conversion and number- and weight-average molecular weights in eqs 7-9. The IACL may be expressed by using eqs 3 and 4 as y) 2fkd(IV)V + kp(MV)(G0V) + ktrm(MV)(G0V) + ktrs(SV)(G0V) 2fkd(IV)V - 0.5ktc(G0V)2 + ktrm(MV)(G0V) + ktrs(SV)(G0V)

(12) This implies that y > 1. The equality y ) 1 is excluded because it requires G0 ) 0. Because the instantaneous polydispersity index p remains fairly constant during the course of free-radical polymerization, it may be assumed to be constant at pc. This assumption is required in the two-step method.11,12 Equations 10 and 11 are then integrated to give

(6)

H 1* )

∫0H *y dH0

(13)

Then, the monomer conversion, the molecular weight averages, and the polydispersity index are determined as follows:

H2* ) pc

∫0H *y2 dH0

(14)

H1 ) (MV)0 - MV + (SV)0 - SV

Mn )

X ) H1/(MV)0

(7)

H1 H2 Wm, Mw ) Wm H0 H1

(8)

P ) Mw/Mn

(9)

Here, Wm is the molecular weight of the monomer and the ratio H1/H0 is known as the average chain length of the polymer product. The volume of the reaction mixture is determined by the sum of the volumes of monomer, solvent, and polymer. In this study, free-volume correlations suggested by Schmidt and Ray16 are used to take into account the gel effect. The kinetic rate constants were determined from the experimental data by using the parameter estimation technique,17,18 and the physical properties were taken from the literature.19

0

There exist a whole variety of y profiles that satisfy eqs 13 and 14. The first step in the extended two-step method is to determine an appropriate y profile that satisfies eqs 13 and 14. 3.1. First Step: Determination of the y* Profile. Let y* denote the desired profile of y that is a function of H0. It should have two parameters to satisfy both eqs 13 and 14. Takamatsu et al.11 and Chang and Lai12 proposed three special types of y* profiles in their twostep method. For example, they employed a quadratic form of the y* profile

y*(H0) ) y(0+) + c1H0 + c2H02

(15)

where y(0+) is the limiting value to be determined from the initial reactor conditions. The extended two-step method proposed in this study does not specify the form of the y* profile a priori. Instead, it starts with an arbitrary shape function v(H0) for the y* profile; i.e.,

3. Extended Two-Step Method Let us transform the mass balance equations for initiator, monomer, solvent, moments of living polymer concentrations, and moments of total moles of polymers, respectively, by dividing each of the equations by the

0

y*(H0) )

v(H0) - c2 > 1, c1 > 0 c1

(16)

where c1 and c2 are the parameters that play the roles of rescaling and shifting, respectively. Equation 16 can

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represent all of the feasible solutions satisfying eqs 13 and 14 because v(H0) is arbitrary. Substituting eq 16 into eqs 13 and 14, we find

x

pc(H0*A2 - A12) P* - pc

1 c1 ) H1*

c2 )

(A1 - c1H1*) H0*

(17)

(18)

where A1 and A2 are defined as

A1 ≡

∫0H *v(H0) dH0

(19)

A2 ≡

∫0H *v2(H0) dH0

(20)

0

0

Any shape function v(H0) may be employed in this extended two-step method and will lead to the same desired number- and weight-average molecular weights at a specified monomer conversion. Therefore, various optimization policies may be devised by properly designing the shape function. It should be pointed out, however, that the reaction time required will be all different depending on the shape function chosen. According to the Cauchy-Schwarz inequality for integral and the natural physical limitation for freeradical polymerization reaction, we find

H0*A2 > A12

(21)

P* > pc

(22)

Here, H0*A2 * A12 and P* * pc because 0 < c1 < ∞. 3.2. Second Step: Determination of the T* Profile. Now, let us consider the coordinate-transformed mass balance equations. At a fixed value of H0, the ICAL depends on the reactor temperature only. When the root-finding algorithm is used, the desired reactor temperature T*(H0) is searched at every fixed value of H0 until the condition y(H0;T*) ) y*(H0) is satisfied. Here, y(H0;T*) represents the value of y in eq 12 with parameter T*. While y*(H0) is determined in the first step, the trajectory for the reaction temperature is constructed in the second step by making a sequence of these temperatures from H0 ) 0 to H0 ) H0*. The coordinate-transformed equation for the zeroth moment of total moles of polymers is used for mapping T*(H0) into the corresponding time profile T*(t); i.e.,

V dt ) dH0 2fkdI + (kpM + ktrmM + ktrsS - 0.5ktcG0)G0 (23) The coordinate-transformed equations of the fundamental model need to be integrated simultaneously for the state variables. 3.3. Illustrations. Let us illustrate the computational flow of the extended two-step method. Suppose that, starting from the initial charges in a 2 L batch reactor of 8 g for the initiator, 800 mL for the monomer, and 800 mL for the solvent, it is desired to produce polymer with Mn* ) 80 000 g‚mol-1 and Mw* ) 144 000 g‚mol-1 at X* ) 0.7 as listed in Table 1. Figure 1a shows the reversed sine curve chosen as the shape function v(H0). In this case, it turns out that

Figure 1. Computational steps in the extended two-step method. An example of the shape function and the desired profiles of y, T, X, Mn, and Mw obtained by the shape function. Table 1. Initial Charges of Reactants and Desired Properties of the Polymer Product Initial Charge monomer (MMA) solvent (EA) initiator (BPO)

800 mL 800 mL 8g

Desired Properties monomer conversion, X* number-average molecular weight, Mn* weight-average molecular weight, Mw*

0.7 80 000 g/mol 144 000 g/mol

c1 ) 0.365 mol-1 and c2 ) - 291.4 mol-1 (cf. eqs 1720). The first step procedure then gives y*(H0) as depicted in Figure 1b, while T*(H0) is computed in the second step to give Figure 1c. The profile of the reactor temperature is then mapped into the time domain as shown in Figure 1d by using eq 23. Time evolution of the monomer conversion and the molecular weight averages are computed in the second step by using eqs 7-9 and are presented in parts e and f of Figure 1, respectively. It is observed that the two molecular weight averages reach their respective target values at the desired level of monomer conversion. In this case, the reactor starts to operate at a rather high temperature of 70.8 °C and the polymerization reaction proceeds for 178.1 min to reach the target. Three different shape functions are tested, and the results are presented in Figure 2. In all of the three diagrams the solid curves correspond to the case of Figure 1, while the dashed curves and the dash-dotted curves represent the cases of using a sine curve and a linearly decreasing function, respectively, as the shape function. Although the temperature trajectories and the reaction times are different from one another, the final product indeed obtains X*, Mw*, and Mn* in all of the three cases. The reaction time with the first temperature trajectory (solid curve) is substantially shorter than those of the other two cases. It is to be noticed that the ranges of the temperature variation along the trajectories are all different from one another. Also, the temperature trajectory for the case of a linearly decreasing function takes a much lower initial temperature but the reaction time becomes longer. Another aspect to be taken into account is that the variation in temperature be mild and smooth so that the controller may be able to track the trajectory rather smoothly. Because it employs a quadratic form for the IACL, the conventional two-step method may be applied only

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Figure 2. Temperature trajectories based on three different shape functions and the time profiles of the monomer conversion and the molecular weight averages corresponding to each of the shape functions.

for the values of Mn* and Mw* in the feasible regions.15 With the present scheme, however, there would be no restrictions on the values of Mn* and Mw* other than those originating from the polymerization kinetics and the process constraints. In the second step, one can consider the possibility of using the initiator concentration instead of the reactor temperature or using both of them simultaneously. In such cases, however, it would be much more difficult to track the initiator concentration trajectory (cf. Jeong and Rhee20).

Figure 3. Flowchart for the extended two-step method when the constraints are imposed on the reactor temperature.

4. Treatment of Temperature Constraints Suppose that the reactor is operated within a certain temperature range because of the process operability or safety. For example, the initiator may be decomposed at a reactor temperature higher than a certain level and the energy can be saved when the process is operated under a certain level of reactor temperature. We have just noticed that the desired values of polymer properties can be obtained irrespective of the shape function v(H0), so one may apply the following shape update rule for the shape function to treat the constraints imposed on the reactor temperature.

{

c1y(H0;Tmax) + c2 v(H0) ) c1y(H0;T) + c2 c1y(H0;Tmin) + c2

if T g Tmax if Tmin < T < Tmax (24) if T e Tmin

Figure 4. Evolution of the profiles of y*, T*, X, Mn, and Mw starting from the shape function of Figure 1a when the updated rule is applied with the constraints 67 °C < T < 75 °C imposed on the reactor temperature.

where the expression y(H0;Tmax) or y(H0;Tmin) is the value of IACL when the kinetic rate constants at Tmax or Tmin are used, respectively, instead of those at T in eq 12. Figure 3 presents the flowchart of the extended twostep method when the temperature constraints are taken into account. At first, guess the initial shape function v(H0) among the appropriate candidates. In the first step, the coefficients c1 and c2 are calculated based on the shape function and the desired profile of IACL is determined. In the second step, the time profile of the desired reactor temperature T*(H0) is determined through an efficient tracking of the y*(H0) profile in such a way that y(H0;T*) ) y*(H0) for all H0. At the same time, the mass balance equations are integrated to give

the IACL y(H0) and the function v(H0) is updated according to the above update rule. The function v(H0) is used in the next iteration as a new shape function. The iteration is terminated when the following criterion is satisfied:

Tmin e T*(H0) e Tmax

for 0 e H0 e H0* (25)

The simulation results of the extended two-step method are presented in Figure 4 when the reactor temperature is constrained within the range from 67 to 75 °C. Parts a and b of Figure 4 show the desired trajectories of the IACL y* and the reactor temperature T*, respectively. Part c of Figure 4 presents the monomer conversion and the number- and weight-average

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molecular weights. It is observed that, starting from a reversed sine curve as the initial shape function and applying the constraint update rule of eq 24, the temperature trajectory obtained after five times of iteration satisfies the constraints. Further iteration shows that the trajectory tends to converge to an ultimate one as may be noticed in Figure 4. The desired values for X, Mn and Mw are always reached at the final stage of the reaction course while the temperature trajectory is updated. The updated scheme for the temperature constraints is expected to assume an important role when the extended two-step method is implemented online. 5. Reduction of the Reaction Time Because the reaction temperature is closely related to both the average molecular weights and the reaction time, it is difficult to reduce the reaction time without changing the molecular weight averages. However, it is observed that the reaction time varies depending on the shape function v(H0) chosen for the IACL when the desired molecular weight averages are fixed. This implies that the reaction time can be reduced by properly choosing the shape function. We suggest an update rule among the possible candidates to reduce the reaction time; i.e.,

y ∂F y ∂F ∂T vi+1 ) vi + a ) vi + a F ∂y F ∂T ∂y

Figure 5. Flowchart for the extended two-step method when the reduction scheme for the reaction time is applied.

(26)

where the subscript i and the parameter a denote the iteration number and the proportionality constant, respectively, and the function F is defined as dt/dH0 given in eq 23. Because the reaction rate constants depend on the reaction temperature, eq 23 indicates that F is a function of T and thus eq 26 holds. The factor (∂F/∂y)/(F/y) represents the sensitivity of F ) dt/dH0 with respect to y. Therefore, the update scheme (26) indicates that the greater the sensitivity is, the more the shape function is updated. Because it changes in the opposite direction to y, the temperature T tends to become lower when the sensitivity is high. Figure 5 presents the schematic diagram of the extended two-step method when the reduction scheme for the reaction time is applied. Initially, an arbitrary shape function v(H0) is guessed. In the first step, the coefficients c1 and c2 are calculated and the desired profile of IACL is determined. In the second step, the time evolution of the desired reactor temperature is determined through an efficient tracking of the IACL. Simultaneously, the mass balance equations are integrated and the shape function is updated for the next iteration. The iteration proceeds until the present profile becomes sufficiently close to the previous one within the specified criterion. When the reduction scheme for the reaction time is used, the simulation results obtained by the extended two-step method are shown in Figure 6. Parts a-c of Figure 6 represent the profiles of IACL, reactor temperature, and reaction time, respectively, as a function of H0. In part c of Figure 6, the reaction time is really decreased as the iteration proceeds. The ultimate curve shows the reaction time of 174.5 min. The ultimate curves for y* and T* include a sharp change in the middle of the reaction course due to the occurrence of the gel effect.

Figure 6. Desired profiles of IACL, reactor temperature, and reaction time when the reduction scheme for the reaction time is applied.

In this case, the same trajectory is obtained whatever the initial shape function that may be used. According to our experience based on simulation studies, the ultimate curve appears to be unique, but the uniqueness has not been proved yet. For this reason, one may not claim that the reaction time is minimized. However, the reaction time is substantially reduced. Irrespective of the value of a, the IACL curve always converges to the ultimate curve; however, the rate of approach to the ultimate curve is found to depend on the magnitude of the proportionality constant a in eq 26. On the other hand, the appropriate value of a will have to be determined empirically. Obviously, it is better to avoid extremely large or small values. The updated scheme of eq 26 certainly bears further investigation. Figure 7 shows the simulation results for the reactor temperature, the monomer conversion, and the numberand weight-average molecular weights obtained in the case of Figure 6. All of the variables X, Mn, and Mw reach

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Figure 7. Simulation results of monomer conversion and numberand weight-average molecular weights corresponding to the case of Figure 6.

Figure 8. Time profiles of the ultimate curves in Figures 6 and 7: (a) reactor temperature trajectory and conversion; (b) numberand weight-average molecular weights; (c) polydispersity index.

their respective target values satisfactorily at the final stage of the reaction. In Figure 8, the time profiles of the ultimate curves in Figures 7 and 8 are presented. Part a of Figure 8 shows the reactor temperature trajectory and the time profile of conversion, whereas part b of Figure 8 gives the time profiles of Mn and Mw. The polydispersity index shown in part c of Figure 8 clearly demonstrates the occurrence of the gel effect at the time of the sharp decrease in temperature. 6. Experimental Results The set-point tracking experiment was conducted under the same conditions as those for the illustrative example of subsection 3.3 (cf. Table 1) with the updated scheme applied for the reaction time. Therefore, the temperature trajectory corresponds to the ultimate curve in part b of Figure 6, which is reproduced in part a of Figure 8. The experimental results are presented in Figure 9. Parts a-c show the reactor temperature, monomer conversion, and number- and weight-average

Figure 9. Experimental and simulation results for the reactor temperature, monomer conversion, and molecular weight averages when the reaction temperature trajectory with reduced reaction time is tracked by using a conventional PID controller. Both the constraints, 67 °C < T < 75 °C, imposed on the reactor temperature and the reduction scheme for the reaction time are taken into account.

molecular weights, respectively. The curves denote the simulation result, while the filled keys represent the experimental data. The stainless steel reactor had a jacket volume of 1.1 L, and two conventional PID controllers were used in the cascade control system. The proportional band, the reset time, and the derivative time were 100.0, 3.5, and 0.001 s for the master loop and 55.0, 4.0, and 0.0 s for the slave loop. The reactor temperature was raised to the desired starting temperature, and then the purified initiator dissolved in the solvent was fed to start the polymerization. The monomer conversion was measured by the gravimetric method, while the molecular weight averages were measured by gel permeation chromatography. Not only the monomer conversion but also the molecular weight averages are found to be in good agreement with the simulation results and to reach their respective target values at the reduced reaction time of 174.5 min. The tracking control experiments were conducted with other temperature trajectories, too. In these cases, the target values were approached only with longer reaction times. This indicates that the reactor temperature trajectory with the reduction scheme for the reaction time may be applied to actual polymerization processes. 7. Conclusions While the conventional two-step method employs a quadratic form for the instantaneous average chain length, it is found that an arbitrary form can give rise to a temperature trajectory that would produce polymer having desired molecular weight averages at a prescribed level of conversion. Naturally, the temperature trajectory depends on the form chosen for the IACL and so does the reaction time required. These features are combined to develop an extended version of the twostep method that can handle the constraints imposed on the reactor temperature and reduce the reaction time. For these purposes, two different updated schemes are proposed: one to accommodate the constraints on the reactor temperature and the other to reduce the

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reaction time. Simulation results demonstrate that both schemes are useful and valid. The present method is also corroborated by a trajectory tracking control experiment for the MMA polymerization reactor system. The control experiment shows that the polymer product having desired polymer properties is indeed obtained with the specified level of monomer conversion at a reduced reaction time. This indicates that the extended two-step method may be readily applied to actual polymerization processes. However, the updated scheme for the reaction time demands a further study for its theoretical basis. Nomenclature a ) proportionality constant, eq 26 A1, A2 ) integral operators, eqs 19 and 20 c1, c2 ) parameters, eqs 17 and 18 f ) initiator efficiency Gk ) kth moment of living polymer concentration, eq 1, k ) 0, 1, mol‚L-1 Hk ) kth moment of total polymer produced, eq 2, k ) 0, 1, 2, mol I ) concentration of the initiator, mol‚L-1 ki ) rate constant of reaction i, s-1 or cm3‚mol-1‚s-1 M ) concentration of the monomer, mol‚L-1 Mn ) number-average molecular weight, g‚mol-1 Mw ) weight-average molecular weight, g‚mol-1 p ) instantaneous polydispersity index, eq 11 pc ) instantaneous polydispersity index constant, eq 14 P ) polydispersity index, eq 9 Pj ) concentration of dead polymer of chain length j, mol‚L-1 Rj ) concentration of living polymer of chain length j, mol‚L-1 S ) concentration of the solvent, mol‚L-1 t ) reaction time, s T ) temperature, °C v ) shape function, eq 16 V ) volume of the reaction mixture, L Wm ) molecular weight of the monomer, g‚mol-1 X ) monomer conversion, eq 7 y ) IACL, eq 10 Superscripts max ) maximum value min ) minimum value * ) desired value Subscripts d ) initiator decomposition i ) initiation m ) monomer p ) propagation or polymer s ) solvent t ) termination tc ) termination by combination trm ) chain transfer to monomer trs ) chain transfer to solvent

Literature Cited (1) Chen, S.-A.; Jeng, W.-F. Minimum End Time Policies for Batchwise Radical Chain Polymerization. Chem. Eng. Sci. 1978, 33, 735. (2) Chen, S.-A.; Lin, K.-F. Minimum End Time Policies for Batchwise Radical PolymerizationsII: A Two-Stage Process for Styrene Polymerization. Chem. Eng. Sci. 1980, 35, 2325. (3) Chen, S.-A.; Huang, N.-W. Minimum End Time Policies for Batchwise Radical Chain PolymerizationsIII: The Initiator Addition Policies. Chem. Eng. Sci. 1981, 36, 1295. (4) Rao, B. B.; Mhaskar, R. D. Minimum End Time Policies in Batch Chain Addition Polymerizations. Polymer 1981, 22, 1593. (5) Farber, J. N. Optimization of Polymerization Reactor. Ph.D. Thesis, University of Massachusetts, Amherst, MA, 1983. (6) Chen, S.-A.; Hsu, K.-Y. Minimum End Time Policies for Batchwise Radical Chain PolymerizationsIV: Consideration of Chain Transfer Effect in Isothermal Operation with One-Stage of Initiator. Chem. Eng. Sci. 1984, 39, 177. (7) Ponnuswamy, S. R.; Shah, S. L.; Kiparissides, C. A. Computer Optimal Control of Batch Polymerization Reactors. Ind. Eng. Chem. Res. 1987, 26, 2229. (8) Secchi, A. R.; Lima, E. L.; Pinto, J. C. Constrained Optimal Batch Polymerization Reactor Control. Polym. Eng. Sci. 1990, 30, 1209. (9) Scali, C.; Ciari, R.; Bello, T.; Maschio, G. Optimal Temperature for the Control of the Product Quality in Batch Polymerization: Simulation and Experimental Results. J. Appl. Polym. Sci. 1995, 55, 945. (10) Kwon, Y. D.; Evans, L. B. A Coordinate-Transform Method for the Numerical Solution of Nonlinear Minimum-Time Control Problems. AIChE J. 1975, 21, 1158. (11) Takamatsu, T.; Shioya, S.; Okada, Y. Molecular Weight Distribution Control in a Batch Polymerization Reactor. Ind. Eng. Chem. Res. 1988, 27, 93. (12) Chang, J.-S.; Lai, J.-L. Computation of Optimal Temperature Policy for Molecular Weight Control in a Batch Polymerization Reactor. Ind. Eng. Chem. Res. 1992, 31, 861. (13) Chang, J.-S.; Liao, P.-H. Molecular Weight Control of a Batch Polymerization Reactor: Experimental Study. Ind. Eng. Chem. Res. 1999, 38, 144. (14) Yoo, K.-Y.; Jeong, B.-G.; Rhee, H.-K. Molecular Weight Distribution Control in a Batch Polymerization Reactor Using the On-line Two-Step Method. Ind. Eng. Chem. Res. 1999, 38, 4805. (15) Yoo, K.-Y.; Rhee, H.-K. Molecular Weight Distribution Attainable in a Batch Free-Radical Polymerization. AIChE J. 1999, 45, 1298. (16) Schmidt, A. D.; Ray, W. H. The dynamics Behavior of Continuous Polymerization ReactorssI. Isothermal Solution Polymerization in a CSTR. Chem. Eng. Sci. 1981, 36, 1410. (17) Ahn, S.-M.; Chang, S.-C.; Rhee, H.-K. Application of Optimal Temperature Trajectory to Batch PMMA Polymerization Reactor. J. Appl. Polym. Sci. 1998, 69, 59. (18) Rho, H.-J.; Huh, Y.-J.; Rhee, H.-K. Application of Adaptive Model-Predictive Control to a Batch Polymerization Reactor. Chem. Eng. Sci. 1998, 53, 3729. (19) Brandrup, J.; Immergut, E. H. Polymer Handbook, 3rd ed.; John Wiley & Sons: New York, 1989. (20) Jeong, B.-G.; Rhee, H.-K. Experimental Validation of the Optimal Trajectory of Initiator Concentration in a Batch MMA Polymerization Reactor. J. Appl. Polym. Sci. 2000, 78, 1256.

Received for review July 28, 2000 Revised manuscript received February 7, 2001 Accepted June 15, 2001 IE000699H