(PDF) Multiobjective dynamic optimization of a semibatch free-radical copolymerization process with interactive cad tools

0098-I 354188 $3.00 + 0.00 Copyright 0 1988 Pergamon Press plc

MULTIOBJECTIVE DYNAMIC OPTIMIZATION OF A SEMIBATCH FREE-RADICAL COPOLYMERIZATION

PROCESS WITH INTERACTIVE CAD TOOLS

D. BUTALA, K. Y. CHOIR and M. K. H. FANS Department of Chemical and Nuclear Engineering and Systems Research Center,

University of Maryland, College Park, MD 20742, U.S.A.

(Received 20 August 1987; final revision received 8 January 1988; received for publicarion 26 January 1988)

Abstract-Optimal open-loop control strategies are developed for batch and semibatch free-radical copolymerization of styrene and acrylonitrile using the interactive computer-aided design (CAD) tool, CONSOLE, which was developed at the University of Maryland in 1987. CONSOLE emphasizes control engineer’s intuition and man-machine interaction and includes a classification of various design specifications. Conflict resolution among the different objectives is made by scaling of specification space based on designer’s knowledge of the polymerization process. A feasible direction algorithm is used to obtain optimal monomer addition and reactor temperature policies. Both the monomer addition policy and the reactor temperature program have been computed to produce a copolymer of uniform molecular weight and copolymer composition.

INTRODUCTION

A precise control of polymer properties has become one of the major issues in optimizing industrial polymerization processes recently. Many polymerization processes involve batch or semibatch reactors to produce small-to-medium volume engineering and specialty polymers. Although continuous processes, in general, produce polymers of more consistent quality in large volume through process automation than batch processes, the latter are still industrially very important and particularly well-suited for the production of polymers of varying grades in a rapidly changing market environment.

In free-radical copolymerization processes, con- trolling both copolymer composition and molecular weight or molecular weight distribution (MWD) is of primary importance. For instance, two styrene- acrylonitrile (SAN) copolymers differing more than 4% in acrylonitrile level are incompatible, resulting in poor physical and mechanical properties (Molau, 1965). It is also well-known that significant batch-to- batch variation in product polymer properties is quite common in industrial processes and resulting off-specification products are often wasted.

Many of the batch or semibatch polymerization reactor control problems stem from the lack of accurate on-line sensors for the measurement of polymer properties such as molecular weight, MWD and composition. Although some promising progress has been made in recent years in developing on-line sensors for certain polymer&ion systems (Pollock, 1983; Guyot et al., 1981; Alonso et al., 1986; Schork

tTo whom correspondence should be addressed. $Electrical Engineering Department and Systems Research

Center.

and Ray, 1983; Jo, 1975), on-line sensors for the measurement of many important polymer properties are not readily available at this time. Therefore, batch or semibatch copolymerization processes pose challenging open-loop control problems (task level control). Inevitably, accurate process models are re- quired in order to synthesize open-loop time-varying control policies which minimize properly defined objective functions.

The semibatch free-radical copolymerization process is a typical example of a multivariable open-loop optimal control design problem. Here, one has to manipulate at least two control variables (e.g. monomer addition rate, reactor temperature) in order to obtain copolymers of desired composition and molecular weight or its MWD in the shortest possible time under the operating constraints. If only one polymer quality parameter is controlled by manipu- lating one control variable, uncontrolled property parameters may deviate from their desired values as the reaction proceeds. In other words, performance objectives compete against each other and a compro- mise has to be considered. One simple technique used quite often for handling such problems is the weighted sum approach, in which the single cost function is a weighted sum of various performance objectives. However, this has the disadvantage of hiding the physical significance of these objectives during the course of optimization. In other words, this technique gives a solution to the overall problem but does not provide insight into the conflicts among the competing objectives. Furthermore, it requires a priori knowledge of the weights to vary the emphasis given to each objective, and adjusting them is usually a cumbersome process. In recent years, several publi- cations dealing with single-cost function optimization

1115

1116 D. BUTALA et al.

in hom*opolymerization have appeared (Farber and Laurence, 1986; Wu et al., 1982; Thomas and Kiparissides, 1984; Ponnuswamy, 1984). Recently, Chen and Lee (1987) used the single objective function to design an optimal copolymerization process with initiator addition rate as a single control variable. For such a multiobjective optimization problem, as copolymerization process optimal solutions gener- ally do not exist and one must be satisfied with obtaining a noninferlor solution or Pareto optimum. The noninferior solution is one in which no decrease can be obtained in any of the objectives without causing a simultaneous increase in at least one of the other objectives. Tsoukas et al. (1982) were the first to apply the dynamic two-objective ~-constraint algorithm developed by Haimes et al. (1975) to a semibatch copolymerization process. Farber (1986) reported the use of a similar multiobjective optimization technique to determine the noninferior sets for continuous copolymerization of styrene-acrylonitrile and methyl methacrylate-vinyl acetate systems.

With a rapid progress in computer technology, there is a growing interest in developing interactive optimization tools to solve a variety of engineering problems. In this paper, we apply a new methodology to the open-loop optimal control problem for semibatch copolymerization processes. This methodology was developed by Nye and tit* (1986) and it has been applied successfully to numerous design problems in various branches of engineering such as integrated circuits (Nye et al., 1983), control systems (Baras et al., 1984; Fan et al., 1985a, b, c; Chen, 1987; Wang, 1987) or earthquake-resistant structures (Austin, 1985). The central idea in this CAD methodology is to emphasize designer intuition and man-machine interaction in an optimization-based approach to engineering design in which the designer and the computer are complementary as they work together to optimize the performance of designs. Furthermore, it handles constrained multiobjective design problems where some of the objectives or the constraints could be “functional”. This means that the specification could also be a function of an independent variable and that the specification needs to he optimized or satisfied for all values of the independent variable in a given set. The delivery of designer intuition and knowledge pertaining to the design problem is through an application-oriented problem formulation. We summarize the procedure of the problem formulation below. For more details, see Nye and tit* (1986), Fan et al. (1987) and Nye (1983).

1. Partition of the various specifications into three categories:

(a) hard (functional) constraint-a specification whose satisfaction is considered essential and hence achieving it should proceed with the utmost priority. For our design problem of semibatch copolymerization processes in which monomers are added to the reactor, good

examples for hard constraints are the requirements of total reaction mass and operating temperature. It is clear that the resulting design has no engineering value if the total reaction mass volume exceeds the designed reactor volume.

(b) soft (functional) constraint-a specification which involves a desired or target value that the designer should try to approach and reach if possible, but such that no further gain would be obtained if the specification overachieved its target value. A soft constraint could be the product specifications. A slight violation of the constraint would probably not jeopardize the value of the design, even though a design satisfying the constraint would be preferable.

(c) (functional) objective-a specification for which some quantity should be minimized or maximized. Minimization in deviation of product specifications from their set values and minimization of reaction time will be our objectives.

Based on the above definition of constraints and objectives, the algorithm proceeds in three phases. In phase I, only the hard constraints are considered. When all the hard constraints are satisfied, the second phase starts. In phase II, objectives and soft constraints compete simultaneously while hard constraints remain satisfied. The third phase starts only when all the soft constraints are satisfied and all the objectives have reached at least their good values (see below) and hard constraints are still maintained satisfied. In phase III, effort is made to improve the objectives while all hard and soft constraints are still kept satisfied.

2. Choose a good value (curve) and a bad value (curve) for each specification by the uniform satisfaction/dissatisfaction rule: having all of the various (functional) objectives and soft (functional) constraints achieving their corresponding good values (curves) should provide the same level of satisfaction to the designer for each, while achieving the bad values (curves) should provide the same level of dissatisfaction. Having all of the various hard (functional) constraints achieving their corresponding good values (curves) should provide the same level of satisfaction to the designer for each, while achieving the bad values (curves) should provide the same level of dissatisfaction. The use of good and bad values in this way provides a very simple way to do tradeoff analysis; if the designer is dissatisfied with the performance level achieved by a particular objective or constraint, he simply changes what he considers to be satisfactory (good value) or unsatisfactory (bad value).

With this problem formulation, comparison between various scaled specifications becomes meaning- ful. Namely, under the assumption that the levels of satisfaction are affine functions of the specifications,

Dynamic optimization with interactive CAD tools 1117

we can say that the ith specification is more satisfied than the jth, if fi > f< and uice uersa. Hence the maximal scaled value of all specifications represents an index of the quality of the design. Therefore we consider the following nonliner programming problem:

min max f i(x) X i (1)

subject to g’,(x)GO, j= 1,2,...

where x is the design parameter vector,/: and g< are the ith “scaled” (functional) objective and the ith (functional) constraint respectively. The “scaled” (functional) objective is defined by:

where fi is the ith (functional) objective, XL,_, and .ft are its good value (curve) and bad value (curve) of the ith objective, respectively. With this trans- formation, 0 and 1 correspond respectively to the specified good and bad values. A similar definition applies to the “scaled” (functional) constraint. In the case of the functional objective, f f stands for the maximal value of all possible values of the independent variable, and similarly, for the functional constraint, gi < 0 means that the constraint must be satisfied for all possible values of the independent variable.

The methodology mentioned above has been im- plemented in the CAD package CONSOLE (Fan et al., 1987). CONSOLE is an interactive optimization- based design tool applicabie to a very broad range of engineering systems provided that system simulators are given. It was developed by Fan et al. (1987) at the University of Maryland. CONSOLE meets many of

the very specific requirements that design engineers demand of the packages, such as: (1) the description of design problem is closely related to the character of design; (2) the optimization algorithms can be linked with arbitrary simulators in an extremely easy way; and (3) graphic display of the results during the optimization, etc. CONSOLE provides a user- interactive interface which allows the user to visualize various outputs related to the performance of the design at a specified iteration. Such outputs include displaying of design parameters, performance comb, active specifications and plots of functional objectives or functional constraints. See subsequent sections for more details.

KINETICS AND REACTOR MODELING

The polymerization reactor optimization problem was investigated with solution copolymerization of styrene and acrylonitrile. The solvent and initiator used were xylene and AIBN, respectively. The ma- sons for selecting this system were: SAN copolymer is commercially important and optimal control is necessary if copolymer composition has to be maintained at points other than the azeotropic point. The details of reaction kinetics and reactor modeling equations are described below and the numerical values of kinetic constants are listed in Table 1.

The following kinetic model is used to describe the hom*ogeneous solution free-radical copolymerization of styrene with acrylonitrile. At high solvent volume fraction, the effect of diffusion-controlled termination (gel effect) is not significant. Penultimate effect is also assumed negligible (these assumptions do not restrict the applicability of the proposed controller design technique, however).

Table 1. Numerical values of kinetic parameters and reaction conditions for styrene acrvlonitrile cooolvmerization

Initiator AIBN

Activation Pre-exponential energy

Parameter factor (Cal gmo1- ’ ) Ref.

t 6.02 x lOI s-’ 31,730 f 0.6 -

Monomer (1 mol-s-l) styrene k #II 1.06 x 10’ 7067

k 11 I 1.25 x 109 1677 k/n, 2.31 x IO6 12,670

Acrylonitrile k c22 3.0 x IO’ 4100 k dl 3.3 x 10’2 5400 kw 6.93 x lo6 5837

rl 2.56 (-) 1190 J.2 6.67 x 1O-5(-) -4340

k/u 30 x k,,, 12,670 ‘c/21 5 x $z 5837 +, 23 (--) -

(a)

AH,= l.OZ{l -[I.393 F,(l - F,)p5} + 16.03 F, + 16.73(1 -F,)

Reactor parameters

f = 0.25, tMx = 5 (h), I’,,, = 4.0 (I), Jo - 0.05 (mol I-‘)

kc = 6.5 x lo-’ (kcal s ‘cm-‘Km’), pC.=O.332 (kcall-‘K-‘)

(b)

(a) Tsoukas er al. (1982): (b) Miyama and Fujimoto (1961).

1118 D. BUTALA et a[.

Initiation

I-2R

kt I

R+M, - PI0 kiz

R fM,-Qo,.

Propagation

P kp 12 ..m+Mz- Q Il.*+ I

Q,., + M?-k+ ,,m

Q,.m + M$+Qn.m+,-

Combination termination

k,c I I P”,?n + P,, -M,+,.,+r

*,c 12 P n. m + Q,.,-- n+r.m+y

km 22 Q.., + Q,,--n+r,m+q.

Disproportionation termination

*rdll P tl,l?l+ P,.,- Mm., + Mr.4

krd12 P n. m + Q, q- Mn,, + M..c,

bf22 Qn. m + Q, q- M,, + Mr,q.

Chain transfer

P n,m+M,- k”’ M,,, + P,,

P,, m + MI- ‘y/12 Mu, + Qo,

Qn,, + WIfl*Km + J’,,

Q.,m + M,- km Mn., + Qo,

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

Vi=,<4.0 (I) (44c)

where N is the number of mesh points, the subscript i represents the i th mesh points,& represents the k th objective at the ith mesh point and +_P and MNr are the desired monomer mol ratio and the number average molecular weight. The desired value for r#~~ was selected as 1 .O and the same for MN, was chosen as 30,000, whereas uJ, represents the k th control at the ith mesh point.

In the case of molecular weight control, the initial deviation from the desired specification is very high. This is because of the fact that at the initial stage of the reaction MW suddenly rises to the Beak value and then tends to decrease with time. Thus the objectives at the initial mesh points will be always the worst. This is not desirable for our problem as the final properties are of more importance than the initial values. To avoid this drawback, two modifications are made in defining the objective for the molecular weight control: (1) a first few initial mesh points are not considered; and (2) different weights are used for the different mesh points.

Finally, the objective function definition is modi- fied to achieve the desired properties in the minimum possible time. For this case, objectives for the properties were transformed to soft constraints and the soft constraints for the final conversion for the volume of the final reaction mass are defined as shown below.

Soft constraints:

[q5(i)-+s]z=0.0; i=l,...,N (45a)

[M,,,(i) - MN,)’ = 0.0; i = 1, . . . , N (45b)

x , _ ,%J = 0.75 (45c)

3.0(l) c VIE/$ ,( 4.0(l). (45d)

Dynamic optimization with interactive CAD tools I121

One more design parameter (i.e. final reaction time) may bc included in the problem definition and all the other hard constraints are kept as before. Here, optimization ends when the point is reached where the scaled value of time cannot be improved any further without deteriorating the scaled values of the properties by the same amount.

SYNTHESIS OF OPTIMAL CONTROL P0LICIF.S

The algorithm used in CONSOLE is a method of feasible direction. The basic operation of such a method can be described in the following procedure:

Step 1. Step 2.

Step 3.

Step 4.

Step 5.

Initialization. Stop if a certain criterion for the current design parameters is satisfied. Find a search direction such that it is a descent direction for all c-active specifications. A specification is said to be c-active if the sum of its scaled value and c is larger than the scaled values of all specifications. If the search direction is large enough in magnitude, proceed to the next step. Otherwise, reduce L and go to Step 2. Along the search direction, perform a line search such that the maximum of the sealed values of all specifications is sufficiently decreased. Update the design parameters and go to Step 2.

A detailed description of the actual optimization algorithm used is given in Nye and tit* (1986) and Nye (1983). The following description in the rest of this section explains how the Pcomb performance chart and other graphical representation of CONSOLE can be used to obtain the desired optimal control policies for semibatch copolymerization processes. The complex information that needs to be conveyed to a designer requires graphical feedback showing algorithm and problem performance. Figure la illustrates the displays of initial parameter values in polynomials and the initial performance for the optimization problem of maintaining desired polymer properties (molecular weight and copolymer composition) during the entire span of reaction, when the final time is fixed.

The performance of CONSOLE is displayed as a chart in the lower half of Fig. la and it is called the Pcomb performance comb. This allows a designer using the multiple-objective formulation quickly to grasp the performance tradeoffs of a design. Figure la clearly shows that our initial choice of parameters is unsatisfactory. The interpretation of the Pcomb diagram is as follows:

On the Pcomb display, the first row displays the current iteration number, phase number (I _ III), the value of c and the worst (largest) scaled value. The second row is the heading. Following that, there is a

row associated with each specification and it is identi- fied by both a symbol and its name. For instance, in the third row of the Pcomb display in Fig. la, “FOl ” identifies the first functional objective (MN - MNs)z (minimization of squared error for the desired MW). The number under PRESENT is its current (un- scaled) value. Its good and bad values are indicated under GOOD and BAD, respectively. When displaying a functional specification, its current good and bad values are displayed at the independent variable where the specification has the maximal “scaled” value. The scaled value of the specification is graphi- cally displayed between good value column and bad value column. The position of the tip of each line represents the scaled value. If the scaled value is between - 1 and 2, the corresponding tip position is shown in the display window and it is marked by a ‘&*“. Therefore, the tip position lying under the heading G represents the scaled value 0, and similarly, under the heading B stands for the scaled value 1. If the scaled value is less than - 1, it is displayed by an arrow towards the left. If the scaled value is more than 2, it is then displayed by an arrow towards the right. For example, in Fig. la, the scaled value of F02 is 0.75 and its tip position is about three quarters away from G to B. As FCl and FC2 have scaled values less than - 1, they are indicated by left- pointing arrows on the left. Whereas Cl and FC3 have scaled values more than 2 and are indicated by right-pointing arrows on the right. Notice that some lines in Fig. la are drawn from the left whereas some are drawn from the right. Lines drawn from the left stand for specifications which are to be minimized or constrained for upper bounds and similar arguments apply to the lines drawn from the right. Notice also that the line corresponding to FOI and F02 are drawn = while others are drawn -. Lines drawn = represent either objectives or soft constraints, and lines drawn - represent hard constraints.

The Pcomb may be output automatically during each optimization iteration or manually after, say, adjusting the good or bad values for a particular objective or constraint. Tradeoffs between competing objectives or constraints are explored by adjusting good and bad values after best or near-best performance of the system being designed and have been achieved following several iterations of optimization. Basically, after several optimization algorithms have been carried out with a set of good and bad values, the designer displays a performance comb and de- cides whether the present values of objectives and constraints are satisfying. A designer who is not happy with the present performance can adjust good and/or bad values to reflect these feelings and then resume optimization.

In the example Pcomb shown, it is clear that many of the specifications are definitely not satisfying. The final volume is almost 2 1 above the target value, the product molecular weight specification is not met, and the temperature limit is not observed.

1122 D. BUT&A e; al.

Design Parametera:

Name al a2 a3

:: b2

Value 3.33OOOe+O2 1 .OOoOOe+Ol 1 .OQQOOe-01 1 .OOoOOe-01 2.OOOOOe+Ol

1; *Z=‘Z -1 :OOOOOe-02

Variation wrt 0 Prev Iter=O 6.Oe+OO 1 . Oe+OO 1 .oe-02 1 .Oe-02 1. oe+oo 1. oe-01 1 .Oe-02 1 .Oe-03

Pcomb (Iter- 0) <Phase 1) (epa- 1 .OOO) CMnr_HARD- 36)

SPECIFICATION

:z: :7- -7:;:: Cl final vol FCl upper flow PC2 1ov.c flow FC3 upper flow FC4 lower flow

PBESBNT GOOD 1.67e+O8 o.OOe+OO 4.768-02 O.OOe+OO 6.Ole+OO 4.OOe+OO 2.00s-02 7.OOe-02 l .lPe-02 0. oo~*oo 9.98e+O2 3.63e*OP 3_33e+O2 3_28e+O2

G B BAD 111-11111-111*-11111------------> 2.6Oe+O7 rl*l*rrrrrrrrrlrlrlr*I 6.06e-02 ‘_‘_‘____*_-_-__--_----_________, 4.1oe+oO <-- l I 7.600-02 <-~-~-~---~-~~---------~-~~~~--~~ -6 OOe-03 ‘_‘_____________________________> 3:64e+O2 *~~-~-~~----~-~~-~~~~~~~-~~_~_-_~ 3.23e+02

Fig. la. Open-loop control of SAN copolymerization reactor: initial design parameters and their performance comb as displayed by CONSOLE (semibatch nonisothermal reaction with fixed finat time

Case 3).

.-•-._

. 1 : Presant

. 2:Good 03:Bod

Mesh point Fig. 1 b. Open-loop control of SAN copolymerization reactor: performance plots of functional objectives with initial design parameters as displayed by CONSOLE (Case 3): (A) molecular weight; (B) bulk phase

composition.

Plots of the functional specifications are given in Fig. I b. Figure 1 b illustrates the graphical representation of two functional objectives and their current values at each mesh point. The plots also represent the good curve (desired leve) as curve 2 and the bad cun~e as curve 3. The aim is to keep the constraint curve I below curve 2 and curve 3 (in the case of minimizing). It can be observed that for the case of molecular weight, the initial eight mesh points are neglected, whereas for the case of copolymer composition, the initial four mesh points are neglected. In

both cases, one can observe that curve 1 is going away from the good curve and for molecular weight it is even going out of bounds. Not only is the final product far too low and the molecular weight far off-spec, but the control is not even feasible.

After running CONSOLE for 22 iterations, the results obtained are shown in Figs ?.a and b. Figure 2a shows the Pcomb display for this set of parameters. It can be observed that the performance has been improved significantly for both the objectives and for all the constraints. As shown in Fig. 2b, the

Dynamic optimization with interactive CAD tools 1123

Design Paramrtrrcl :

Name al

Value Variation vrt 0 PrW? Iter-22 3.69188e+02 6.Oe+OO 6Y ox

a2 a3 a4 bl b2 b3 b4

(Note:

-3.iQ240a+OO l.Oa+OO -68Y Q.l3616e-02 l.Oe-02 ox 6.9134Oe-02 l.Os-02 0.69793e+OQ l.Oe+oO "61:

-3.7488&-01 l.Oe-01 63% -4.073140-02 1.00-02 60% -6.405OOe-03 l.Oe-03 38%

0% change indicates change less then

Pcomb <1ter- 22) (Phase 2) (eps- 6.104E-6)

1% or no change at all)

<MAX_COST_SOFT= 0.0786327)

SPECIFICATION FOl <MN-&We)-2

final vol FCl upper ilow FC2 lower flow FC3 upper temp FC4 lower temp

PRESENT GOOD l.Q2e+OB o.OOe+OO 3.88e-03 O.OOe*OO 3.4re+OO 4.000+00 9.?Oe-03 7.oDe-02 6.OOe-03 o.OOe+OO 3.63e+02 3.63%+02 3.46e+02 3.28e+O2

G B 11111111111* I *==========* c-- c-- I !

BAD 2.6Oe+O7 6.06e-02 4.106+00 7.5Oe-02 - __ _~ <_~__-~~-~~~--~-~~-~~------~-~~-- -6 (JfJe_c)3

<-- i I 3:64e+02 <____-__-__-----_--__--_-________ 3_23e+02

Fig. 2a. Open-loop control of SAN copolymerization reactor: final design parameters and their performance comb as displayed by CONSOLE (Case 3).

x10’

1. . 1 : Prasent .- Iz

l 2 : Good 1.0 0 3 : Bad

”

0 hhuLubLutruuLuLuub~!~ x10’ 0.9 1.1 1.3 1.5 1.7 1.9 2.1

x lo+

6.0

Mesh point Fig. 2b. Open-loop control of SAN copolymerization reactor: performance plots of functional objectives with final design parameters as displayed by CONSOLE (Case 3): (A) molecular weight; (B) bulk phase

composition.

property objectives are well within the bound and very close to the good curoes. This could be taken as a feasible solution or it could be improved further, depending upon the requirements of the user.

RESULTS AND DlSCLlSSlON

Optimal control policies for several cases were developed using this methodology and the results are shown in Figs 3-7. The desired value for copolymer composition (F,) was selected such that the wt% of

acrylonitrile in the copolymer is in the range 25--35%, and the desired value of number average molecular weight was chosen as 30,000. Initial reactor volume was 1.0 1 and other initial design parameters were f, = 0.25 (mol fraction), I, = 0.05 (mol I-‘), 4, = &, = 1 .O. Solvent mol fraction and initiator concentration in the feed were the same as their initial values; the monomer mol ratio in the feed, M,/M,, was 1.5.

The first plot in Fig. 3 shows the optimal reactor temperature profile when only molecular weight is

D. BUTALA er al. 1124

Time (mid Fig. 3. Open-loop copolymer molecular weight control in a batch reactor with control profile obtained by the use of CONSOLE (batch nonisothermal reaction with fixed final

time, Case I).

controlled. Response of MW with this optimal temperature profile is also shown in the same plot. Note that initial deviation in MW from the desired value is very low and the reaction of the desired value of MW is achieved accurately at the end of the reaction. It is interesting to observe from the polydispersity plot that when good MW control is obtained, narrow molecular weight distribution is also observed. How- ever, the second plot showing the copolymer com-

x10= 1.501 (360.0 2

Time (min) Fig. 4. Open-loop copolymer composition control in a semibatch reactor with control profile obtained by the use of CONSOLE (semibatch isothermal reaction with fixed

final time, Case 2).

x10=

1.50 .380.0 ^y

> d

; 0.75 = u2 - 350.0 5 MN $

t 0.800 1~1~1111~11~,~ 5 - - 320_Om, 50.0 g

Fl E

rr-’ 0.60 -, 25.0 i

0 2.00 - MW - 1.0

- 0.5

Time (min)

2 Yl t E ”

Fig. 5. Open-loop copolymer composition and molecular weight control in a semibatch reactor with control profiles

obtained by the use of CONSOLE (Case 3).

position indicates that when only MW is controlled by varying the reaction temperature, the copolymer composition deviates considerably from its desired value. Figure 4 illustrates the results of optimal control when only the copolymer composition is controlled by adding the monomer to the reactor during the course of copolymerization. For this case, reactor temperature was kept constant at 333.0 K. As shown in the figure, copolymer composition is main-

xl05

l.501, 380.0 ;

i2 li

Fl 0.60 25.0 ;

O0-O 300 Time (min)

Fig. 6. Open-loop copolymer composition and molecular weight control at final time in a semibatch reactor with control profiles obtained by the use of CONSOLE (semibatch isothermal reaction with fixed final time and

properties control only at the final time, Case 4).

Dynamic optimization with interactive CAD tools I125

x10= 1.50,

t I’ 0.75 UP

0.40 2.00

2% 1.00 > P OK----L-

O 100 200 3 lime (min)

Fig. 7. Minimum end time problem for open-loop copolymer composition and molecular weight control in a semibatch reactor (semibatch nonisothermal reaction with

final time minimization, Case 5).

tained well at its desired level. Although the polydispersity is maintained almost constant, the MW is far away from its desired value. Also note that the use of low temperature leds to low final monomer conversion (below 50%). It should be noted that the isothermal semibatch copolymerization process poses a singular control problem when the monomer feedrate is the only control variable. By using the prespecified form of control policy (i.e. polynomials), we can approximate the excact singular arc, which is extremely difficult to compute analytically with the maximum principle. Figure 4 shows that no dis- continuity in control policy is apparent and that the polynomial approximation in the monomer feedrate profile is adequate to describe the control policy. In the third case, a combination of two controls were used and the results are shown in Fig. 5. Here MW and copolymer composition were controlled simultaneously. Very good results were achieved for both MW control and copolymer composition control. Copolymer composition is maintained at its desired level throughout the reaction and the initial deviation of MW from the desired value decreases with time; eventually the desired value is obtained with very little error. Polydispersity remains at a good level and monomer conversion of 80% is achieved. For all the three cases shown in Figs 3-5, MW and copolymer composition were controlled at several mesh points.

In the next case, the aim was set to achieve desired MW and copolymer composition only at the final time, keeping all the other variables the same as before. The results obtained are shown in Fig. 6. Because of the fact that MW is controlled only at the final time, a very high initial deviation from the

desired value is obvious; however, the desired value of MW at the final time is reached very accurately. It can also be observed that although the copolymer composition is controlled only at the final time, the desired value is maintained throughout the reaction span. This leads to a conclusion that it is important to control MW at an early stage, whereas copolymer composition should be controlled at the final stage of the reaction. Note that very desirable polydispersity and conversion values are obtained.

Finally, the commercially-important case of minimizing batch polymerization time was studied. Desired MW, copolymer composition, monomer conversion and desired minimum and maximum limit of reaction mass at final time were set as soft constraints, and the constraints on temperature and flowrate were kept as before. The final time is an additional design parameter and was included in the problem definition and optimization was performed again with the design parameters obtained in Case 4 (cf. Fig. 6). The objective was to minimize the reaction time while still maintaining the desired specifications at the final time. The results obtained are illustrated in Fig. 7. Note that the optimal control profile is almost the same as that for Case 4 and that the same desired properties are obtained in the shorter reaction span. In other words, there is no need to continue the reaction for 5 h. Instead, the same level of satisfaction for all the objectives and the constraints can be obtained within 4.58 h. This indicates that there is no other optimal path which gives the same level of satisfaction for the desired copolymer properties and simultaneously minimizes the reaction time.

CONCLUSIONS

In this paper, we have introduced a user-interactive CAD package for optimization-CONSOLE- developed at the University of Maryland, to solve the optimal control problem in semibatch CO- polymerization reactors. The multivariable dynamic optimization approach is used for this problem, where the control variables are parameterized in third-order polynomials in time. The coefficients of these polynomials are considered as the design parameters of the multivariable optimization problem. For our study, reactor temperature and monomer feed rate are used as the control variables. The dynamic optimization problem associated with the batch or the semibatch processes is approximated by the use of a mesh type structure, where the infinite dimension is reduced to the finite dimension equiv- alent to the number of mesh points. The time- dependent objectives and constraints are defined as functional quantities and their specifications are observed only at the definite number of time instants corresponding to the mesh points. Constraints on the time-varying variables and the static variables are also included in the problem definition.

1126 D. BUTALA et al

It has been shown that CONSOLE provides a very good approach for resolving the tradeoffs between the competing objectives by introducing the concept of good values and bad values. The criteria of selecting the good values and the bad values are related with the designer’s intuition and his knowledge of the problem. The whole concept of objectives, soft constraints, hard constraints and their related good and bad values is easy to understand and provides the platform for design engineers’ intuition. One definite advantage gained over the single-cost function approach is the elimination of the cumbersome process of selecting the weights for the indi- vidual objective. Five different cases based on the different objectives are studied. For all cases, numerical simulations show excellent open-loop controls of copolymer composition and molecular weight. The control profiles obtained are easy to implement in any computer control system. The successful use of this approach for the multivariable dynamic optimization problem, such as copolymerization reactor optimization studied in this work, opens the door for many other dynamic chemical reactor systems.

Acknowledgement-This work was supported by the Systems Research Center at the University of Maryland, College Park.

NOMENCLATURE

A, = Heat transfer area reactor-’ volume (cm2 1-i) C, = Specific heat of reaction mixture (Cal g-i K-i) F, = Mol fraction of ith monomer in copolymer, i = I, 2 f = Initiator efficiency f; = Mzll fr2action of ith monomer in reaction mixture;

h, = Overall heat transfer coefficient (cal cm-* s-’ K-‘) I = Initiator concentration in reaction mixture

(mol I-‘) Z,= Initiator concentration in feed (mol 1-l)

k, = Initiator decomposition rate constant (s-l) k,+ = Chain transfer rate constant, i,j=1,2

(1 mol-’ s-‘) kqY = Propagation rate constant, i, j = I,2 (1 mol-’ s-‘) k,,, = Combination termination rate constant, i,j = I,2

(1 mol-’ s-i) k,dU = Disproportionation termination rate constant, i, j

(1 mol-' s-‘) M, = ith monomer concentration in reaction mixture,

i = I,2 (mol 1-i) M,,= ith monomer concentration in feed, i = 1,2

(mall-‘) MN = Total number average molecular weight

MNs = Desired value of total number average molecular weight

M, = Total weight average molecular weight P = TotaI growing polymer concentration of type-l

(mol I-‘) Pi = ith moment of the total number MWD of radicals

of type-1 P n. m = Concentration of growing polymer containing n

units of monomer-I and m units of monomer-2 (mall-‘)

PD = Polydispersity Q = I$t;\_~wing polymer concentration of type-2

pi = ith moment of the total number MWD of radicals of type-2

Q.,, = Concentration of growing polymer containing n units of monomer-I and m units of monomer-2 and ending in monomer-2 (mol 1-l)

R, = Copolymerization reaction rate (moi 1-i s-‘) rij = Monomer reactivity ratio T = Reactor temperature (K) T, = Jacket media temperature (K) T,= Monomer feed temperature (K) I = Reaction time (min) u = Monomer feed rate (I min-’ ) u, = ith manipulated variable, i = 1; feed rate, i = 2,

reactor temperature I’ = Reactor volume (1) w, = Molecular weight of ith monomer, i = 1, 2

(g mol-‘)

Greek Ietters AH, = Heat of copolymerization (cal mol-‘)

,li = kth moment of the dead copolymer total number MWD, k =O, I,2

p = Density of reaction mixture (g Ii’) & = Molar ratio of monomers in reaction mixture &1= Monomer mol ration in feed stream 4, = Desired value of molar ratio of monomers in

reaction mixture @, = Cross termination factor

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