Molecular Weight Distribution Control in a Batch Polymerization

Ind. Eng. Chem. Res. 1999, 38, 4805-4814

4805

Molecular Weight Distribution Control in a Batch Polymerization Reactor Using the On-Line Two-Step Method Kee-Youn Yoo, Boong-Goon Jeong, and Hyun-Ku Rhee* School of Chemical Engineering and Institute of Chemical Processes, Seoul National University, Kwanak-ku, Seoul 151-742, Korea

This paper applies the two-step method to analyze the feasibility of the desired molecular weights and develops the on-line two-step method to obtain the polymer product with the desired molecular weight distribution (MWD) under process disturbances. If adequate sensors and process model are available to predict the effects of disturbances, then midcourse correction policies are obtained by using the on-line two-step method. For an illustrative example of the on-line twostep method, we conduct the styrene solution polymerization in a batch reactor system and demonstrate the excellent performance for MWD control under the measured process disturbances. 1. Introduction The batch mode of reactor operation is very popular in the manufacture of low-volume, high-value polymers. In general, the major objective is not to keep the system at a set point but to optimize an objective function at the end of the batch cycle. In a batch free radical polymerization process, a typical operation task involves manipulation of reactor temperature, initiator concentration, or other variables to achieve the desired objectives such as productivity, specified molecular weight, and residual impurities at the end of the reactor operation. It is well-known that molecular weight (MW) and molecular weight distribution (MWD) significantly affect the mechanical and rheological properties of the polymer.1 Hence, operating a batch reactor to achieve a specified average molecular weight and molecular weight distribution is highly desired. Usually, the polymerization process is operated according to a predetermined recipe (such as reactor temperature trajectory and initiator supply policy) which has been found to produce the polymer with desired properties. In industrial practice, the recipe is usually determined by trial and error experience. A less common industrial approach for determining the recipe is to use a polymerization reactor model. Many researchers have used Pontryagin’s maximum principle and the standard numerical optimization techniques to determine the recipe that minimizes the reaction time and produces the polymer with desired average molecular weights. For instance, Thomas and Kiparissides2 calculated the optimal open-loop temperature and initiator policies in a batch polymerization reactor by using a discretization control method along with a first-order gradient technique. Ahn et al.3 implemented the optimal temperature policy on a fully automated experimental reactor system. Experimental values of conversion and average molecular weights turned out to be in good agreement with their respective desired values. The classical approach to the end point optimization leads to a two-point boundary value problem. The numerical solution of this two-point boundary value problem * To whom correspondence should be addressed. Tel.: +822-880-7405. Fax: +82-2-888-7295. E-mail: [email protected].

provides the time profile of the manipulated input, which has to be implemented in an open-loop fashion. However, the derived optimization problem is nonconvex and leads to many difficulties for numerical calculation, i.e., feasibility, optimality, and computation. Also, because of batch-to-batch variations in the system parameters and initial conditions, the implementation of this open-loop profile can lead to suboptimal performance. Takamatsu et al.4 proposed a two-step method based on the instantaneous degree of polymerization to determine the operation policy that produces the desired average molecular weights at the end of the batch cycle. In this approach, one separates the MWD control problem into two steps: the first step is to determine the profile for the instantaneous degree of polymerization giving the desired number- and weight-average molecular weights at the final conversion level; and the second step is to explore the operating condition forcing the instantaneous degree of polymerization to the value determined in the first step. This method formulates the average molecular weight control problem into a one-parameter searching problem. Hence, the timeconsuming optimization procedure can be avoided and the criterion to judge whether the specified molecular weight objectives are achievable can be derived.5 The main disadvantage with a predetermined recipe (using Pontryagin’s principle or the two-step method) is that poor control of MWD may arise as a result of batch-to-batch variation in the process conditions and process disturbances. Krothapally and Palanki6 and Krothapally et al.7 developed and implemented a neural network method to calculate the recipe on-line under variable initial loading conditions. This approach used the initial conditions of the batch process as inputs to the neural network and thus could deal with the batchto-batch variation for a variety of initial conditions. However, it would be difficult to apply this method when process disturbances are introduced during the process operation. For this reason, it is necessary to develop closed-loop schemes for MWD control in a batch polymerization reactor. One approach for developing closedloop methods is to repeatedly use a numerical algorithm developed for open-loop schemes at each time step. In this study, we develop the on-line two-step method to obtain the polymer product with the desired MWD

10.1021/ie980799b CCC: $18.00 © 1999 American Chemical Society Published on Web 11/16/1999

4806

Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999

Table 1. Free-Radical Polymerization Mechanism kd

I 98 2φ

initiation

ki

φ + M 98 R1

to take into account the volume change of the reaction mixture; i.e.,

Wm Ws Wm + (SV) + (M(0)V(0) - (MV)) Fm Fs Fm

V ) (MV)

kp

Rj + M 98 Rj+1

propagation

ktc

termination by combination

Ri + Rj 98 Ri+j

chain transfer to monomer

Ri + M 98 Pi + R1

chain transfer to solvent

Ri + S 98 Pi + S

ktrm

ktrs

To describe MWD, it is not necessary to solve the infinite number of equations for the concentrations of polymer chains of each length. Instead, the method of moments is adopted to calculate the number-average degree of polymerization, Y, and the polydispersity, P. The definitions of the moments are as follows: ∞

Table 2. Balance Equations for a Batch Polymerization Reactor

1 d(IV) ) -kdI V dt

Gk )

∞

∑ nkRn n)1

and Fk )

∑ nkPn n)1

k ) 0, 1, 2

(2)

For the sake of simplicity, we use the moments of total polymer formed in a batch reactor defined as follows:

1 d(MV) ) -2fkdI - kpMG0 - ktrmMG0 V dt 1 d(SV) ) -ktrsSG0 V dt

Hk ) (Fk + Gk)V

k ) 0, 1, 2

(3)

Here, one can derive the balance equations for G1 and Hk, k ) 1, 2, 3, as presented in the lower part of Table 2. On the basis of simple mass-balance consideration, the conversion, X, is calculated as

1 d(G0V) ) 2fkdI - ktG02 V dt 1 1 dH0 ) 2fkdI - ktcG02 + (ktrmM + ktrsS)G0 V dt 2

(

(1)

)

1 dH1 1 d(MV) d(SV) )+ V dt V dt dt dH 1 2 ) 2fkdI + kpM(G0 + 2G1) + ktcG02 + (ktrmM + ktrsS)G0 V dt 1 d(G1V) ) 2fkdI + kpMG0 - ktG0G1 + V dt (ktrmM + ktrsS)(G0 - G1)

under process disturbances. If adequate on-line sensors are available to detect the presence of disturbances and reactor models can be developed to predict the effects of these detected disturbances on the polymer quality, then midcourse correction policies are calculated by using the on-line two-step method. As an illustrative example of the on-line two-step method, we perform the styrene solution polymerization in a batch reactor system and verify the superior performance for MWD control under the measured reactor temperature disturbances.

X)

(4)

whereas Y and P are expressed as

Y)

H0H2 H1 and P ) H0 H2

(5)

1

In this work, we apply this batch polymerization reactor model to the styrene polymerization system. As the conversion of monomer increases, the reaction mixture becomes more and more viscous and the rate of the termination reaction decreases. These phenomena in reaction rates are termed the “gel” (or “Trommsdorff”) effect. Although the gel effect in styrene polymerization is not as strong as that in MMA polymerization, this has to be considered in describing the polymerization kinetics at high conversion or with low solvent volume fraction. Here, the empirical gel effect correlation suggested by Hamer et al.8 is used:

gt(X,T) )

2. Batch Polymerization Reactor Model In this section, we consider a batch reactor model for the solution polymerization. The reaction kinetics is assumed to follow the free radical polymerization mechanism including chain-transfer reactions to both the solvent and monomer. The free radical polymerization mechanism is summarized in Table 1. With the assumption of perfect mixing of the reaction mixture, the dynamic behavior of the reactor is described by the equations listed in the upper part of Table 2. As the monomer is converted to a polymer, the density of the reaction mixture increases and thus the volume, V, of the reactor contents shrinks as the reaction proceeds. Therefore, we use the appropriate density correlation

H1 M(0)V(0) - MV ) M(0)V(0) M(0)V(0)

kt ) exp(-2(Ax + Bx2 + Cx3)) kt0 x ) X(1 - fs) (6)

where

A ) 2.57 - (5.05 × 10-3)T B ) 9.56 - (1.76 × 10-2)T C ) -3.03 + (7.85 × 10-3)T The physical properties and kinetic parameters were taken from the literature and are listed in Table 3. However, the chain-transfer rate constants were deter-

Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999 4807

following equations:

Table 3. Parameters Used in the Reactor Model for Styrene Polymerization Fm (g/L) Fs (g/L) Fp (g/L)

Physical Properties 924.0-0.981×(T-273.15) 885.5-0.955×(T-273.15) 1084.0-0.605×(T-273.15)

Schuler and Schuler and Suzhen9 Takamatsu et al.4

kd (s-1) kp (L/mol/s) kt0 (L/mol/s) ktrm (L/mol/s) ktrs (L/mol/s)

Rate Constants 1.58 × 1015 exp(-30780/RT) 1.051 × 107 exp(-7064/RT) 1.255 × 109 exp(-1680/RT) 1.186 × 107 exp(-11767/RT) 3.148 × 109 exp(-16264/RT)

Duerksen et al.10 Duerksen et al.10 Duerksen et al.10 estimated estimated

mined by applying the parameter estimation technique to the experimental data.

In general, MWD is fairly well-described by the number-average degree of polymerization, Y, and the polydispersity, P. Therefore, the MWD control problem in a batch reactor boils down to that of searching an operating condition to produce the polymer with the prescribed number-average degree of polymerization, Y*, and the prescribed polydispersity, P*, at the desired conversion level, X*. For a given X*, the desired value of H1 may be given by

(7)

If Y and P in eq 5 are replaced by Y* and P*, respectively, the desired values for H0 and H2 at final conversion level are obtained as follows:

H/0 ) H/1/Y* and H/2 ) H/1Y*P*

(8)

Therefore, given the control objectives X*, Y*, and P* and the initial monomer amount, the MWD control problem is equivalent to that of searching an operating condition to produce the polymer with H0*, H1*, and H2*. It is then necessary to monitor the relations among H0, H1, and H2 during a batch reactor operation. For this purpose, the instantaneous degree of polymerization, y, and the instantaneous polydispersity, p, are defined as

y)

dH1 dH0 dt dt

/

(9)

( )( )/( )

dH0 dH2 p) dt dt

dH1 dt

(13)

∫0H p(H0) y2(H0) dH0

(14)

/ 0

H/2 )

/ 0

These equations must be satisfied to obtain the polymer product with desired properties at the final conversion level. In free radical polymerization p(H0) remains fairly constant during the course of polymerization, so this will be assumed constant during a batch. Under this condition, eq 14 may be rearranged to yield

H/2/p )

3. Two-Step Method for MWD Control in a Batch Reactor

H/1 ) M(0)V(0)X*

∫0H y(H0) dH0

H/1 )

Suzhen9

∫0H y2(H0) dH0 / 0

(15)

Furthermore, eqs 13 and 15 may be put into dimensionless form:

1) P* ) p

( )( ) ∫( ) ( )

y ∫01 Y* 1

0

y Y*

d

H0

(16)

H/0

2

d

H0

(17)

H/0

Here, we can separate the MWD control problem into two steps: the first step is to determine y*(H0) satisfying eqs 16 and 27 whereas the second step is to explore the operating conditions forcing y(H0) to y*(H0) determined in the first step. 3.1. First Step Based on the Quadratic Expression for y. Takamastu et al.4 used three different types of expression for y(H0); i.e., (1) a rectangular type, (2) a quadratic type, and (3) a mixed type of zero and firstorder polynomials. In this study, we will consider the quadratic expression for y(H0) to execute the two-step method; i.e.,

() ()

H0 H0 Y(H0) y(0) ) + a1 / + a2 / Y* Y* H0 H0

2

(18)

where y(0) is calculated by using the initial reactor condition. In general, the quadratic expression is preferred to other ones because of the fact that the variation of y is smooth and the attainable MWD is sufficiently extensive. Various expressions for y(H0), however, may be chosen considering the reactor operability. The unknown coefficients, a1 and a2, can be determined by inserting eq 18 into eqs 16 and 17. This yields

3 15 a1 ) - (3β + 5D) and a2 ) (β + 3D) (19) 2 4

2

(10)

where

Using eqs 9 and 10, the following pair of equations are obtained:

dH1 )y dt

(11)

dH2 ) py2 dt

(12)

Integration of eqs 11 and 12 from 0 to H0* gives the

R* )

y(0) P* - 1, β ) - 1, and p Y*

x151 (8R - β)

D)(

2

(20)

For easy reference, we shall call eq 18 the quadratic constitutive equation (QCE). We note that, for given values of R and β, there exist two different trajectories: one with D g 0 and the other with D < 0. The desired polymer product is to be obtained if the instantaneous

4808


degree of polymerization y tracks one of the two trajectories given by QCEs during the course of polymerization. 3.2. Second Step: Calculation of the Recipe for the Reactor Operation. In the second step, the reactor operating condition that realizes the y* profile obtained in the first step should be calculated on the basis of polymerization reactor model. Here, we assume that the reactor temperature is primarily used to control the MWD for the given amount of initiator and solvent fraction. By using the conservation equations for the moments of polymer formed, we can calculate the righthand side of eq 9 for a given temperature and indeed determine the temperature trajectory that makes the y profile of eq 9 equal to the y*(H0) profile obtained in the first step. In actual calculation, the reactor model equations need to be integrated for I, M, S, and G0 simultaneously and the balance equation for H0 is used to map the y*(H0) profile into the corresponding time profile y*(t). This procedure will lead to the real operating policy required.

4. MWD Attainable by the Two-Step Method Based on QCEs In practice, the reactor often needs to be operated under the process constraints because of process safety or operability. In general, the instantaneous degree of polymerization y varies monotonically along with the operating variables such as the reactor temperature, the initiator concentration, and the solvent fraction. Thus, the process constraints for these operating variables may be replaced by the appropriate constraints on y (see Yoo and Rhee5). The transformed constraints, s1 e y e s2, can be readily identified by performing the reactor model simulation under the operating conditions close to the process constraints. We shall put s1 and s2 into dimensionless form as follows:

δ1 )

s1 s2 - 1 and δ2 ) -1 Y* Y*

(21)

and thus

δ2 )

( )

s2 s2 δ1 + -1 , s1 s1

-1 < δ1 e 0 and δ2 g 0 (22)

It is now evident that the attainable polydispersity is restricted by the values of δ1 and δ2. In this case, the attainable MWD by the two-step method based on QCEs is approximated by the physically realizable set of (R, β) satisfying the inequalities

()

()

H0 H0 3 15 δ1 e β - (3β + 5D) / + (β + 3D) / 2 4 H0 H0

2

e δ2

for 0 e H0 e H/0 (23)

in which the maximum and minimum values of QCEs are required to exist in the constrained range of y, respectively. The plane of the constraint parameters (δ1, δ2) is divided into seven subregions based on the set of

Figure 1. Feasible regions for the pairs of (R, β) corresponding to the subregion of (δ1, δ2) at s2/s1 ) 2.0 in the case of the QCE for D g 0.

the boundary inequalities enclosing the feasible (R, β) region.5 To each of these subregions, there corresponds a particular shape of the feasible (R, β) region as illustrated in Figure 1, which reveals the information about the attainable values of R and β within a specific subregion of constraint parameters. Using this information about the attainable average molecular weights, one can efficiently develop the MWD control strategies for producing the desired polymer product.5 Also, this information is crucial when on-line modification of the recipe is implemented by using the two-step method. 5. On-Line Two-Step Method Because of the external disturbances and limited control performance, the predetermined operating policy may not be realized satisfactorily and the final polymer property may deviate from the desired polymer property. For this reason, it is necessary to develop a closedloop scheme for MWD control. One approach is to repeatedly use the numerical method for the open-loop scheme with available process information at each time step. 5.1. First Step Based on QCEs and Feasibility Consideration for MWD. The base recipe is calculated by using the two-step method based on QCEs. In the


Figure 2. MWD control structure based on the on-line two-step method.

on-line approach, eqs 16 and 17 for the two-step method are modified as

1)

( )( ∫( )(

y ∫01 Yh *(t )

d

i

P h *(ti) ) p

1

0

y Y h *(ti)

)

H0 - H0(ti)

H/0 - H0(ti)

2

d

(24)

)

H0 - H0(ti)

H/0 - H0(ti)

(25)

where ti is the update instant and

Y h *(ti) )

H/1 - H1(ti) H/0

- H0(ti)

R j (ti) )

(H/0 - H0(ti))(H/2 - H2(ti)) (H/1

(

)

H0 - H0(ti) 3 h (ti)) / + )β h (ti) - (3β h (ti) + 5D 2 Y h *(ti) H0 - H0(ti) H0 - H0(ti) 2 15 h (ti)) / (27) (β h (ti) + 3D 4 H - H (t ) y(H0)

(

0

0

i

)

where

and

P h *(ti)

the update instant ti and the final reaction time, respectively. As a result, the updated QCEs are derived in the dimensionless form

2

- H1(ti))

(26)

h *(ti) correspond to the degree of In practice, Y h *(ti) and P polymerization and the polydispersity of the polymer that the updated operating policy must produce between

P h *(ti) y(ti) - 1, - 1, β h (ti) ) p Y h *(ti)

x151 (8Rj(t ) - βh (t )) (28)

and D h (ti) ) (

2

i

i

Here, the original and updated QCEs are equivalent if the predetermined operating policy is perfectly realized. In the on-line approach, the feasibility of the desired MWD must be checked at every update instant ti. This

4810


Figure 3. Schematic diagram of the batch polymerization system used in this study.

task on the two-step method is easily accomplished by inspecting the feasible (R, β) region. When the desired MWD specification becomes infeasible because of the midcourse disturbances, the specific strategies are required to obtain the feasible operating policy. If the h (ti)) is infeasible and R j (ti) is smaller than set of (R j (ti), β the attainable maximum value of R, one can always find the desired operating policy by simply adjusting y(ti) (see Figure 1). This strategy results in the step change of the reactor operating condition to meet the desired j (ti) is larger than the MWD at the update instant ti. If R attainable maximum value of R, the following optimization strategy can be used to resolve the infeasible problem:

min

(η,µ)∈feasible(R j ,β h)

(η - R j (ti))2 + F(µ - β h (ti))2

(29)

where F is a weighting factor. The optimal value of (η, µ) is easily obtained at the boundary of the feasible h (ti)). Replacing (R j (ti), β h (ti)) in eq 27 by region of (R j (ti), β (η, µ) we can obtain the operating policy that adequately minimizes the deviation from the desired MWD. In this case, the MWD specification can be modified as follows:

Y/m )

H/1 H/0,m

and P/m )

H/0,mH/2,m H1*2

(30)

where the subscript m indicates the modified MWD specification and

H/0,m )

( )

µ+1 (H/1 - H1(ti)) + H0(ti) y(ti)

H/2,m ) y(ti)p

η+1 (µ+1 )(H

/ 1

- H1(ti)) + H2(ti)

(31) (32)

Here, the specification of the first moment of total polymer formed is preserved to keep the conversion level to the desired value.

5.2 Second Step Based on the On-Line Process Information. The second step in the on-line approach parallels that in the off-line approach. However, the reactor model equations or the state estimator are computed, keeping pace with the progress of the reaction to monitor the state variables. That is, the required operating condition is calculated on-line by using the available process information during a batch. Then, the on-line two-step method translates the cumulative errors under process disturbances to the modified set point about the reactor operating condition. The controller selects a control effort within the constraints to make the process output follow the set-point generated by the on-line two-step routine as closely as possible. In such a circumstance, if we use a controller, for which the present set point value is only needed, i.e., PID controller, it is sufficient to generate the present set point value in the second step. However, if a model-predictive controller is implemented, for which the future set point sequences are needed at every control optimization step, the second step must generate the set point sequences up to the prediction horizon. 5.3. MWD Control Structure of the On-Line TwoStep Method. We present the MWD control structure based on the on-line two-step method in Figure 2. The structure permits use of all process information (i.e., process measurements and process models) to accomplish the MWD control objective. Frequently, all available process input and output measurements are needed to estimate the effects of the unmeasured disturbances. The on-line two-step routine is tunable via a single, physically meaningful parameter, i.e., the update interval of the first step, t1, in Figure 2b. The tuning parameter determines the speed of response of the closed-loop system when the model is perfect. For a shorter update interval, the control system provides rapid disturbance rejection. The speed of system response may be adjusted to provide stable control. In general, because the dynamics of MWD is slower than


Figure 5. Results of the temperature-tracking experiments for both off-line and on-line two-step methods (nominal case).

Figure 4. Experimental results (filled keys) compared to the model predictions (curves) under various isothermal conditions (styrene: 0.9 L, toluene: 0.3 L, AIBN: 5.91 g).

that of a manipulated variable such as the reactor temperature, the update interval of the first step, t1, must be set to a larger value than that of the second step, t2. One of the remaining obstacles in MWD control is the lack of adequate and frequent measurements. For example, the reactor temperature is available directly in the reactor system whereas MWD is measured infrequently and associated with large time delay. Hence, the recent trend has been to use the wellestablished closed-loop state estimation techniques such as Kalman filter. Ellis et al.11 achieved on-line estimation of MWD in a batch reactor for MMA solution polymerization using the two-time scale Kalman filter based on on-line GPC (gel permeation chromatography) data. Further, they used the estimator as the molecular weight measuring device for feedback control of the weight-average molecular weight. Mutha et al.12 developed a fixed-lag smoothing-based extended Kalman filter to deal with multirate measurements and implemented it on the CSTR for MMA solution polymerization, in which the conversion and the weight-average molecular weight are estimated and controlled using the estimator and model-predictive controller. When these on-line closed-loop estimation schemes are combined with the first step of the on-line approach, the performance of MWD control will be remarkably improved under process disturbances. Moreover, the number- and weight-average molecular weights can be simultaneously controlled in a batch polymerization reactor by the on-line two-step method. 6. Experimental Verification 6.1. Reactor Control System and Experimental Procedure. Figure 3 shows the schematic diagram of

the batch polymerization system used in this study. A 2-L stainless steel reactor with a 1.07-L jacket is used for the reaction with a 4-blade turbine-type agitator. Some coolant portion is circulated to reduce disturbances in the jacket temperature and to save heating energy. The reactor temperature control is achieved by manipulating the valve stem positions of the hot and cold water lines using a cascade PID control configuration. The outer discrete-time PID controller generates a set point for the jacket temperature to control the reactor temperature and the inner discrete-time PI controller calculates the valve stem positions of hot and cold water lines to control the jacket inlet temperature. The controller tuning is conducted by the rule of thumb and the parameter values are PB (proportional band) ) 125, τI ) 0.70 min, and τD ) 0.025 min for the outer loop and PB ) 60.0 and τI ) 1.25 min for the inner loop, respectively. In the on-line two-step method, the temperature trajectory is to be modified if the temperature disturbance and the resulting variation of the moments of total polymer formed in the batch reactor are detected. The controller performance may deteriorate under the variable operating condition. In this case, it would be necessary to employ an adaptive or nonlinear control strategy to deal with the variable operating condition. In our experimental study, however, the conventional cascade PID controller showed the satisfactory performance to track the modified temperature trajectory under the presence of disturbances. An IBM 486 personal computer and PLC (TI505) are employed for data acquisition and implementation of the on-line two-step routine and controller. Passing the packed column with activated Al2O3 purifies the styrene monomer. The solvent is toluene and the initiator is R,R′-azobis(isobutyronitrile) (AIBN). The amounts of reactants used are 0.9 L of monomer, 0.3 L of solvent, and 5.91 g of initiator. The purified styrene monomer, 0.9 L, and the toluene, 1.5 L, are initially charged in the reactor and heated to the desired initial reactor temperature. The initiator, 5.91 g, is dissolved in toluene, 1.5 L. The experiment is conducted

4812


Figure 6. Conversion histories obtained by the temperaturetracking experiments (nominal case): off-line measurements (filled keys) and model predictions (curves).

by pouring the initiator solution after the purging of nitrogen in the reactor. The reactants are sampled every 30 min and the reaction is quenched with cold methanol. The conversion is determined by the gravimetric method while the average molecular weights are measured by Waters GPC system equipped with a RI detector and an Ultrastyragel column (Styragel HR4). 6.2. Model Validation. The batch reactor model is tested for its validity by conducting polymerization experiments under the isothermal conditions and comparing the results with the model predictions. Figure 4 shows the comparison between the experimental and simulation results at three different temperatures. In all cases, the two are in good agreement. Therefore, it may be concluded that the model used in this study is adequate to the reactor system under consideration. 6.3. MWD Control Using the On-Line Two-Step Method. The MWD control experiment based on the two-step method was conducted in both on-line and offline manners. To make a better simulation of actual reactor operation, we take into account the constraints imposed on the reactor temperature for the experiments of MWD control. Here the reactor temperature is constrained to vary in the range between 50 and 90 °C. Using the results of reactor model simulation at 50 and 90 °C, we set s1 and s2, the constraints on y, to 190 and 590, respectively. Thus, the temperature constraints are translated to the constraints on y and the attainable MWD specification can be easily evaluated in the twostep method based on QCEs. The target is to obtain a product having desired monomer conversion (X* ) 0.5), degree of polymerization (Y* ) 250), and polydispersity (P* ) 1.8). The temperature set point for the off-line experiment was calculated a priori. For the on-line experiment, the temperature trajectory was generated on-line by the online two-step routine, in which the update interval of the first step is 10 min and that of the second step is 2 min. In the nominal case, both on-line and off-line methods show similar performances with respect to MWD control (see Figures 5-7), although one may

Figure 7. Profiles of the number- and weight-average molecular weights obtained by the temperature-tracking experiments (nominal case); off-line measurements (filled keys), model predictions (curves).

recognize the improved performance of the on-line method with respect to the average molecular weights in Figure 7. This improvement may be attributed to the feed-forward compensation effect using the measured temperature deviation. Next, we introduced an intentional disturbance during the reactor operation to examine the performance for MWD control under the measured temperature disturbances. Here, the coolant pump stopped to function at 35 min after the start-up and was restored after 25 min; i.e., the coolant medium was not supplied to the jacket for 25 min. Figure 8 presents the results of the temperature-tracking experiments with the temperature disturbance. The solid line represents the time profile of the set point while the dashed line represents the profile of the controlled reactor temperature. In the on-line experiment, the temperature trajectory, after the introduction of disturbance, turned out to be located in


Figure 8. Results of the temperature-tracking experiments for both off-line and on-line two-step methods under the external disturbances.

Figure 10. Profiles of the number- and weight-average molecular weights obtained by the temperature-tracking experiments under the external disturbances: off-line measurements (filled keys) and model predictions (curves). Table 4. Comparison of the Results at the End of the Batch Cycle between the On-Line and Off-Line Two-Step Methods under the Presence of External Disturbances

Figure 9. Conversion histories obtained by the temperaturetracking experiments under the external disturbances: off-line measurements (filled keys) and model predictions (curves).

the higher temperature range compared to the off-line temperature trajectory to make up for the measured temperature deviation. In addition, the on-line scheme made the total reaction time slightly increased to obtain the desired conversion level. Presented in Figure 9 is the conversion history whereas Figure 10 shows the average molecular weight profiles for both off-line and on-line experiments, respectively, with the temperature disturbance. For the on-line experiment, the state of reactants such as the moments of polymer concentration is estimated by integrating the reactor model using the measured reactor temperature. On the basis of the estimated state, the temperature trajectory is updated by the on-line two-step method. As one can see in Table 4, the results

desired value on-line off-line

conversion

Mn

Mw

P

0.50 0.51 0.45

26 100 26 137 27 046

46 980 44 930 44 745

1.80 1.72 1.65

of the on-line experiment at the final reaction time showed much less deviation from the desired conversion and the target average molecular weights in comparison to the results of the off-line experiment. In other words, the temperature trajectory update compensates for the measured temperature deviation during the course of reaction. The deviation of the weight-average molecular weight appears to occur as a result of both the model uncertainty and the unmeasured disturbances that were not captured by the reactor temperature measurement. If the on-line closed-loop estimation schemes for the conversion and weight-average molecular weight are combined with the first step of the on-line approach, the performance of MWD control will be remarkably improved under these unmeasured disturbances as well as the model uncertainty.

4814


7. Conclusions In this work, we developed the on-line two-step method in which the operating policy is updated by taking into account the available process information and the feasibility of MWD. The applicability of the online two-step method was verified by control experiments on a batch reactor system for the styrene solution polymerization. The on-line experiment gave much better results than that of the off-line experiment under the measured temperature disturbances; i.e., the results of the on-line experiment were much closer to the target values than those of the off-line experiment. This is attributed to the excellent disturbance rejection performance of the on-line two-step method. Acknowledgment This work was supported by the Korea Science and Engineering Foundation (KOSEF) through the Automation Research Center at Pohang University of Science and Technology. Nomenclature f ) initiator efficiency Fk ) kth moment of dead polymer concentration (k ) 0, 1, 2) (mol/L) Gk ) kth moment of living polymer concentration (k ) 0, 1, 2) (mol/L) Hk ) kth moment of total polymer formed in a batch reactor (k ) 0, 1, 2) (mol/L) I ) initiator concentration (mol/L) k ) reaction rate constant (s-1) or (L/(mol‚s)) M ) monomer concentration (mol/L) Mn ) number-average molecular weight (g/mol) Mw ) weight-average molecular weight (g/mol) Pn ) dead polymer of chain length n or its concentration (mol/L) Rn ) living polymer of chain length n or its concentration (mol/L) S ) solvent concentration (mol/L) V ) volume of reaction mixture (L) W ) molecular weight (g/mol) x ) fractional monomer conversion F ) density (g/L) Subscripts d ) initiator decomposition m ) monomer

p ) propagation reaction or polymer s ) solvent (toluene) t ) termination tc ) termination by combination trm ) chain transfer to monomer trs ) chain transfer to solvent

Literature Cited (1) Nunes, R. W.; Martin, J. R.; John, J. F. Influence of Molecular Weight and Molecular Weight Distribution on Mechanical Properties of Polymers. Polym. Eng. Sci. 1982, 22, 205. (2) Thomas, I. M.; Kiparissides, C. Computation of the Nearoptimal Policies for a Batch Polymerization Reactor. Can. J. Chem. Eng. 1984, 62, 284. (3) 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. (4) Takamatsu, T.; Shioya, S.; Okada, Y. Molecular Weight Distribution Control in a Batch Polymerization. Ind. Eng. Chem. Res. 1988, 27, 93. (5) Yoo, K.-Y.; Rhee, H.-K. Molecular-Weight Distribution Attainable in a Batch Free Radical Polymerization. AIChE J. 1999, 45, 1298. (6) Krothapally, M.; Palanki, S. A Neural Network Strategy for Batch Process Optimization. Comput. Chem. Eng. 1997, 21(S), S463. (7) Krothapally, M.; Bennett, B.; Finney, W. C.; Palanki, S. Experimental Implementation of an On-line Optimization Scheme to Batch PMMA Synthesis. ISA Trans. 1999, 7, 1. (8) Hamer, J. W.; Akarmov, T. A.; Ray, W. H. The Dynamic Behavior of Continuous PolymerizationsII. Nonisothermal Solution Homopolymerization and Copolymerization in a CSTR. Chem. Eng. Sci. 1981, 36, 1897. (9) Schuler, H.; Suzhen, Z. Real-Time Estimation of the Chain Length Distribution in a Polymerization Reactor. Chem. Eng. Sci. 1985, 40, 1891. (10) Duerksen, J. H.; Hamielec, A. E.; Hodgins, J. W. Polymer Reactors and Molecular Weight Distribution: Part I. Free Readical Polymerization in a Continuous Stirred-Tank Reactor. AIChE J. 1967, 13, 1081. (11) Ellis, M. F.; Taylor, T. W.; Jensen, K. On-line Molecular Weight Distribution Estimation and Control in Batch Polymerization. AIChE J. 1994, 40, 445. (12) Mutha, R. K.; Cluett, W. R.; Penlidis, A. On-line Nonlinear Model-Based Estimation and Control of a Polymer Reactor. AIChE J. 1997, 43, 3042.

Received for review December 30, 1998 Revised manuscript received September 8, 1999 Accepted September 29, 1999 IE980799B

Molecular Weight Distribution Control in a Batch Polymerization

Recommend Documents