dynamic optimization of semibatch vinyl acetate/acrylic acid suspension copolymerizations

Dynamic Optimization of Semibatch VinylAcetate/Acrylic Acid Suspension Copolymerizations

Fabricio Machado,1 Enrique Luis Lima,2 Jose Carlos Pinto2

1 Instituto de Quımica, Universidade de Brasılia, Campus Universitario Darcy Ribeiro, CP 04478,70910-900 Brasılia, DF, Brazil

2 Programa de Engenharia Quımica/COPPE, Universidade Federal do Rio de Janeiro, Cidade Universitaria,CP 68502, Rio de Janeiro 21945-970, RJ, Brazil

This work performs the dynamic optimization of semi-batch vinyl acetate (VAc)/acrylic acid (AA) suspensioncopolymerizations. The proposed dynamic optimizationstrategy is based on a direct search Complex algo-rithm and is used to control the copolymer composi-tion along the batch. First, a sequential optimizationprocedure is used to determine the optimum AA con-centration and feed rate profiles, required to providethe specified copolymer composition. In the secondstep, a sequential optimization procedure is coupledwith a predictive controller to guarantee that themanipulation of feed flow rates can allow for attain-ment of the desired copolymer compositions. The opti-mization strategy is validated through simulation, byassuming that reactions are subject to perturbations ofthe reaction temperature, initiator, and VAc concentra-tions. It is shown that the proposed optimization strat-egy can be used successfully both for design of mono-mer feed rate profiles and removal of processdisturbances during semibatch suspension copolymer-izations, to keep the copolymer composition constantthroughout the batch. POLYM. ENG. SCI., 50:697–708, 2010.ª 2009 Society of Plastics Engineers

INTRODUCTION

In most chemical fields, there is a growing interest for

development of optimization and control techniques

intended to maintain the process operation at optimum

and safe conditions. The definition of optimal condition

may depend on several technical and economical factors,

certainly varying from case to case; however, to keep the

operation at the optimum condition, it is necessary to con-

trol the process, as the process operation is subject to

uncontrolled and undesired disturbances [1].

Typical polymerization processes are characterized by

the simultaneous occurrence of several complex nonlinear

phenomena [2, 3]. The complex nonlinear behavior of

most polymerization processes motivates the development

of nonlinear monitoring, control, and optimization strat-

egies, to be used for prediction and control of the end-use

properties of the final polymer products [4]. Optimization

procedures are normally used for determination of the

process operation conditions that allow for minimization

(maximization) of specified performance criteria [5]. The

sets of controlled and manipulated variables are selected

based on both the impact on final polymer quality and the

possibility of manipulation in real time. Typical con-

trolled/manipulated variables are reactor temperature,

reactant concentrations, and reactant feed flow rates, used

to control copolymer compositions and average molecular

weights [6].

According to Marjanovic et al. [7], the combination of

nonlinear process dynamics and large batch-to-batch var-

iations of process variables, associated with poor real-time

measurements of process performance, often leads to

inconsistent operations in the polymerization field. To

overcome this common industrial problem, it is necessary

to keep the most important input process variables con-

stant and/or modify these variables within very strict

operation limits, leading to safer process operation, cost

reduction, and production of materials that satisfy speci-

fied product quality indexes [8, 9].

According to Gentric et al. [10], the optimization of a

polymerization process requires the definition of an appro-

priate objective function and specification of the process

constraints that must be satisfied, which are usually

expressed in terms of the reaction time and/or molecular

characteristics of the final product. Frequently, it is

desired to minimize the batch time and/or to minimize the

difference between specified and obtained properties (mo-

lecular weight distributions, copolymer composition, and

Correspondence to: Fabricio Machado; e-mail: [email protected]

Contract grant sponsors: Coordenacao de Aperfeicoamento de Pessoal de

Nıvel Superior (CAPES); contract grant sponsor: Conselho Nacional de

Desenvolvimento Cientıfico e Tecnologico (CNPq).

DOI 10.1002/pen.21572

Published online in Wiley InterScience (www.interscience.wiley.com).

VVC 2009 Society of Plastics Engineers

POLYMER ENGINEERING AND SCIENCE—-2010

monomer conversion, among others) of the final product

[11–14].

To solve the proposed optimization problem, a large

number of distinct analytical and numerical techniques

have been used [7–9, 15–20]. For instance, methods based

on the maximum principle of Pontryagin and on the opti-

mal control theory have been used to provide closed ana-

lytical solutions for the optimum conditions; methods

based on the use of Lagrange multipliers have been used

to provide closed analytical and numerical solutions to

constrained problems; methods based on orthogonal collo-

cation procedures have been combined with successive

quadratic programing to provide polynomial approxima-

tions of the optimal dynamic trajectories; direct search

optimization techniques have been used to provide opti-

mal solutions over a discretized time domain; among

many other possibilities.

The objective function used for optimization usually

depends on both the final properties of the system and the

dynamic trajectory that describes the polymerization pro-

cess. As a consequence, the dynamic optimization prob-

lem can be written as follows:

minu tð Þ;tf

= t; x tð Þ; u tð Þð Þ ¼ G x tfð Þð Þ þZ tf

t0

F t; x tð Þ; u tð Þð Þdt

(1)

subject to:

x ¼ fP x; uð Þ (2)

y ¼ gP xð Þ (3)

ul � u � uu (4)

xl � x � xu (5)

yl � y � yu (6)

where fP and gP are continuous and differentiable func-

tions, x is the vector of state variables, u is the vector of

decision variables, and y is the vector of output variables.

The indices l and u correspond to the lower and upper

bounds of the state, input and output variables. G is a dif-

ferentiable nonlinear function of the state variables at the

end of the batch and F is a differentiable nonlinear func-

tion of the state and input variables along the operation

trajectory [21, 22]. Equations 2 and 3 represent the pro-

cess model, whereas Eqs. 4–6 represent the process and

safety constraints.

Dynamic optimization problems frequently require that

sets of ordinary differential equations (ODEs) or differen-

tial algebraic equations (DAEs) be solved iteratively. A

possible alternative to solve these problems involves the

discretization of the time variable and the use of colloca-

tion methods to transform the set of ODE or DAE into a

system of algebraic equations that can be solved with the

help of sequential quadratic programing (simultaneous

optimization) [15, 22–25]. As the dynamic trajectories are

not known a priori, this approach may lead to very large

sets of algebraic equations, which may be difficult to

solve numerically. A second alternative involves the itera-

tive numerical integration of the differential equations (se-

quential optimization) [26, 27]. In this case, the numerical

integrator provides the dynamic trajectories, while the

optimizer manipulates process inputs and initial condi-

tions to optimize the designed performance criterion. As a

consequence, the optimization problem is usually much

smaller, although the iterative solution of the differential

equations may be time-consuming.

Vinyl acetate (VAc)/acrylic acid (AA) copolymer beads

produced in suspension reactors are used for immobiliza-

tion of enzymes and cells in biotechnological applications

[28–30]. The VAc/AA copolymerization is a typical exam-

ple of reaction where monomers present very different reac-

tivity ratios, so that copolymer composition drifts can be

very significant along the batch time. However, homogene-

ous copolymer composition is fundamental for develop-

ment of successful applications. For this reason, VAc/AA

polymerizations are usually performed in semibatch mode

[31–34]. To avoid the continuous drift of copolymer com-

position, operation policies must be designed, based on the

manipulation of reactor temperatures and/or monomer feed

flow rates. Manipulation of feed flow rates must be per-

formed with care in suspension reactors, as formation of

new polymer beads should be avoided during the batch. For

this reason, semibatch reactions are performed much more

frequently in solution and emulsion polymerizations than in

suspension polymerizations [14, 35, 36]. This also explains

why semibatch suspension reactions have not received

much attention in the literature. Particularly, publications

about the dynamic optimization of semibatch suspension

reactors are not available in the open literature.

In previous publications, it was shown that optimum

monomer feed policies might be developed for the produc-

tion of VAc/AA copolymers with homogeneous composi-

tion [31–33]. However, process operation is subject to per-

turbations; therefore, process optimization should be car-

ried out in-line iteratively. For this reason, this article

presents optimum operation policies for semibatch VAc/

AA copolymerizations performed in suspension for the first

time. The operation policies are updated in-line and in real

time, based on the availability of the current process states.

The proposed dynamic optimization strategy is based on a

direct search Complex algorithm and is used to control the

copolymer composition along the batch. First, a sequential

optimization procedure is used to determine the optimum

AA concentration profiles, required to provide a specified

copolymer composition. In the second step, a sequential

optimization procedure is coupled with a predictive control-

ler (corrective action based on reoptimization, where a new

feed flow rate profile is determined to maintain the compo-

sition constant along the polymerization time) to guarantee

that the manipulation of feed flow rates can allow for attain-

698 POLYMER ENGINEERING AND SCIENCE—-2010 DOI 10.1002/pen

ment of desired monomer concentrations. The optimization

strategy is validated through simulation, by assuming that

reactions are subject to perturbations of the temperature,

initiator, and VAc concentrations.

PROBLEM FORMULATION

The kinetic mechanism used to describe the VAc/AA

suspension copolymerization is based on the classical

free-radical mechanism and includes initiation, propaga-

tion, chain transfer to monomer, and termination by dis-

proportionation. The kinetic mechanism and the model

equations are presented in the Appendix. Detailed descrip-

tion of the polymerization model can be found elsewhere

[32], and the reader is referred to this publication for

additional details. The polymerization model presented in

the Appendix was validated with real experimental data

obtained at different reaction conditions for VAc/AA sus-

pension copolymerizations performed in both batch and

semibatch operation modes. Additionally, an open-loop

control strategy was also implemented successfully and

validated with real experimental data for semibatch reac-

tions [32, 33], showing that the model and the proposed

control strategy are very robust. The set of DAEs (see

Appendix) that describes the system was solved numeri-

cally with the numerical integrator DASSL [37].

VAc/AA copolymerizations are performed in suspen-

sion to produce beads with average size ranging from 200

to 500 lm. It is very important to emphasize that the

semibatch VAc/AA copolymerization performed in sus-

pension reactors is very different from reactions per-

formed in solution or bulk reactors because AA is soluble

both in the aqueous and organic phases. As a conse-

quence, AA is partitioned among both phases, which

affects the dynamic behavior of the polymerization and

the characteristics of the final polymer product [31–33]. It

is also very important to emphasize that thermodynamic

partition coefficients depend very strongly and nonlinearly

on the AA aqueous concentrations. As a consequence,

feed rate policies developed for the suspension reactors

cannot be compared to solutions eventually provided for

solution or bulk processes. It is also important to empha-

size that previous experimental studies [31–33] showed

that mass transfer rates between the aqueous and organic

phases are not significant, which means that the system is

always very close to thermodynamic equilibrium and that

feed rate policies can be implemented with confidence.

The optimization problem was solved numerically with

the direct search Complex algorithm [38, 39]. The standard

numerical procedure DBCPOL, obtained from the IMSLTM

library, was used for implementation of the computer code

[40]. Direct search methods are very helpful because they

require the computation of derivatives of neither the objec-

tive function nor the process constraints [41, 42]. Particu-

larly, the use of the complex technique is recommended

when the optimization is performed with a large system of

equations and/or when the manipulated variables present

distinct ranges of variation, so that the user does not need

to rescale the optimization problem [39, 43, 44]. To avoid

the possible existence of local optima, distinct starting

points were generated at random inside the search region.

Equations 1–6 present a general framework for dynamic

optimization problems, which is sufficiently general to

accommodate different optimization problems. In the par-

ticular case analyzed here, F (Eq. 1) is obtained as a sum-

mation over a discretized time domain (sampling times).

This can be justified in terms of the simpler numerical

implementation and also based on the fact that real meas-

urements are only available at the specified sampling times.

Additionally, G (Eq. 1) is not used because all variables are

controlled not only at the end of the batch but also through-

out the dynamic trajectory. It is also important to emphasize

that the final batch time was not controlled, as control of

polymer properties is much more important than the control

of polymer productivity in the present example. However,

inclusion of the final batch time in the objective function

can be made in a straightforward manner. Equation 2 repre-

sents the process model (see the Appendix) and Eq. 3 repre-

sents the process outputs (see Eqs. A.6 and A.7 in Appen-

dix). Equations 4–6 account for process and safety con-

straints for state variables (e.g., reaction temperature and

initiator concentration), output variables (e.g., monomers

conversion and copolymer composition), and manipulated

variables (e.g., feed flow rates).

RESULTS AND DISCUSSION

Optimization of Feed Rate Policies

The main objective pursued during the process operation

is to keep the copolymer composition constant at the

desired setpoint value along the whole reaction time

through manipulation of the feed flow rate of AA (F2). Feed

rate profiles required to keep the copolymer composition

constant were determined for discretized time intervals, so

that feed rate profiles can be described as a sequence of

pulses. Machado et al. [33] had computed optimum AA

feed rate profiles for continuous feed rate profiles, although

previously obtained solutions could not be used efficiently

for in-line optimizations because they do not consider the

inherent uncertainties of the process states and inputs.

Sampling intervals of 10 min were defined for reaction

times of �3 hr. As observed from our simulations, the nu-

merical optimization problem can always be performed to

obtain the required F2 values in less than 10 min on a

desktop computer, which means that the operation can be

carried out on-line in real time. It is assumed that compo-

sitions can be obtained in real time with the help of near

infrared spectroscopy and conductimetry, based on previ-

ous experimental studies performed in our group [31, 34,

45]. Therefore, it is assumed that the current states

(including compositions and temperatures) are known.

DOI 10.1002/pen POLYMER ENGINEERING AND SCIENCE—-2010 699

Optimum AA feed rate profiles were determined assum-

ing that both pure AA and AA solution (0.025 M) are avail-

able. It was also assumed that the copolymer composition

must be kept constant and equal to 20% in molar basis, as

required by real applications [32]. According to Machado

et al. [33], when solutions containing different concentra-

tions of AA are added into the reaction medium, the com-

plementary amount of water may exert a strong positive

influence on the stabilization of the reaction system, keep-

ing the suspension viscosity under control. The addition of

water avoids the increase of the polymer concentration and

maintains the polymer holdup at safe levels, avoiding parti-

cle agglomeration. Additionally, feeding of AA solutions

allows for adequate homogenization of the suspended drop-

lets during the semibatch operation, preventing the forma-

tion of new organic droplets in the medium.

The optimization problem can be defined as follows:

minF2

= ¼XNIi¼1

<i � <di

<di

� �2(7)

where Ri is the cumulative copolymer composition at

each discretized sampling time i, Rdi is the desired copol-

ymer composition at each discretized sampling time i,and NI is the number of discretized sampling times in the

considered control window. Calculations were performed

on an Intel1 Pentium1 4 processor with Intel1 Hyper-

Threading technology (used to improve parallelism), CPU

3.2 GHz and 512 Mb memory RAM. The sequential opti-

mization procedure is presented in Fig. 1.

In this article, Eq. 7 is subject to the set of differentialalgebraic Eqs. A.1–A.21, presented in the Appendix (cor-responding to Eq. 2 of the proposed optimization frame-work), and to Eqs. 8 and 9, presented later (correspondingto Eqs. 4–6 of the proposed optimization framework).

To perform computations for a 1-L lab-scale reactor, a

typical polymerization recipe is presented in Table 1. The

following feed rate constraints were defined, in accord-

ance with the real operation constraints (limitation of

available pumps and reactor capacity), as analyzed else-

where [33, 46]. In the case of feeding pure AA, the lower

and upper bounds were defined as follows:

0 � F2 � 0:00012 (8)

When AA solutions (0.025 M) were used, the lower and

upper bounds were defined as follows:

0 � F2 � 0:005; (9)

where F2 corresponds to the feed flow rate in mol/s. The

feed flow rates required to keep the AA copolymer com-

FIG. 1. Proposed scheme for optimization of acrylic acid feed rate policies.


position constant during the reaction are depend on the

AA molar concentration of the feed solution. Conse-

quently, the feed flow rate increases as the AA molar con-

centration in the feed solution is decreased.

Figure 2 shows the optimum feed rate profiles obtained

when pure AA is used to keep the copolymer composition

constant and equal to 20 mol % of AA. The individual

and global conversions are also presented, as predicted by

the model. It can be observed that it is possible to obtain

polymer resins with homogeneous chemical composition

with the help of the proposed optimization procedure. It

is important to notice that the discontinuous AA feed rate

profiles are very different from the previously obtained

continuous feed rate policies. Particularly, one should

observe that the series of feed rate pulses is oscillatory,

while the optimum continuous profiles are smooth and

present a single point of maximum [33]. This happens

because of the unavoidable accumulation (and depletion)

of monomer during a finite time window, while the feed

rate is kept constant. Therefore, it seems clear that the

continuous optimum feed rate profile should not be used

as a nominal reference for the discontinuous implementa-

tion of the feed rate policy. On the other hand, conversion

trajectories (and consequently reaction rates and batch

times) are not affected significantly during the discontinu-

ous feed operation. This occurs because the characteristic

reaction time (measured in hours and controlled by the

consumption rate of the VAc monomer) is much longer

than the characteristic sampling time (10 min). Despite

that as the consumption rates of the AA monomer are

high, strict control of the AA concentrations must be per-

formed to keep copolymer composition under control.

Figure 3 shows the optimum feed rate profiles obtained

when AA solution (0.025 M) is used to keep the copoly-

mer composition constant and equal to 20 molar % of

AA. The individual and global conversions are also pre-

FIG. 2. Dynamic behavior of VAc/AA suspension copolymerization.

(A) Optimum AA feed rate and cumulative AA copolymer composition

profile; (B) conversion profile. [Color figure can be viewed in the online

issue, which is available at www.interscience.wiley.com.]

TABLE 1. Typical polymerization recipe.

Species Weight (g)

Water 420

Vinyl acetate 144

Acrylic acid 36

Benzoyl peroxide 1.5

Poly(vinyl alcohol) 0.5

FIG. 3. Dynamic behavior of VAc/AA suspension copolymerization. (A)

Optimum 0.025 M AA solution feed rate and cumulative AA copolymer

composition profile; (B) conversions profile. [Color figure can be viewed

in the online issue, which is available at www.interscience.wiley.com.]


sented, as predicted by the model. It can be observed

once more that it is possible to obtain polymer resins with

homogeneous chemical composition with the help of the

proposed optimization procedure. Again, the discontinu-

ous feed rate policies are oscillatory and conversion tra-

jectories (and consequently reaction rates and batch times)

are not affected significantly during the discontinuous

feed operation, as described previously.

The results presented in Figs. 2 and 3 show that there

are no relevant differences between the evolutions of

monomer conversion and copolymer composition when

AA is fed as a pure component or in a diluted aqueous

solution. This finding encourages the use of diluted AA

feed streams to perform semibatch VAc/AA copolymer-

izations, as problems related to mixing, homogenization,

and stabilization of the reaction medium can be mini-

mized without significant loss of productivity.

Figures 4 and 5 show how the initial guesses used to per-

form the numerical computations can affect the obtained

results when pure AA and AA solution (0.025 M) are used

as feeds. Depending on the initial guess, distinct feed rate

profiles can be obtained. This is a consequence of the dis-

tinct reaction rates of AA (very high) and VAc (very low).

Despite that, it is very important to emphasize that the cu-

mulative copolymer composition is kept very close (around

61.0%) to the desired setpoint value throughout the entire

polymerization trajectory in all cases. (As one might al-

ready expect, the instantaneous copolymer compositions

follow very similar dynamic trajectories in these cases.)

The model performance may be considered as very good as

model deviations are always smaller than the observed ex-

perimental errors of the analytical methods normally used

to determine the copolymer composition.

As one can observe in Figs. 4 and 5, the copolymer

composition oscillates slightly during the polymerization

reaction. This oscillatory behavior of the predicted copol-

ymer composition is more pronounced at each discretized

sampling time i, when the feed flow rate is changed. This

is due to the very high reaction rates of the AA monomer.

In spite of that, the copolymer composition is maintained

within a very narrow range around the desired setpoint

value.

Besides, monomer conversions and reaction times are

always very similar in all cases, indicating that global

properties are not very sensitive to fast fluctuation of the

operation conditions. One should also observe the charac-

teristic oscillatory series of pulses obtained in all cases.

The sensitivity of the final solutions to modification of

the initial guesses can be attributed to the nonlinear char-

acteristics of the model equations used to describe the

VAc/AA copolymerization. As a consequence, multiple

local solutions may exist, encouraging the analyses of sol-

utions obtained with distinct initial guesses. To minimize

the risk of finding local optima, multiple starting points

(at least five) are generated at random within the search

region for initialization of the numerical code.

FIG. 4. Influence of the initial guess on the optimum feed rate profile of pure AA. [Color figure can be

viewed in the online issue, which is available at www.interscience.wiley.com.]


FIG. 5. Influence of the initial guess on the optimum feed rate profile of AA solution (0.025 M). [Color fig-

ure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

FIG. 6. Proposed sequential optimization scheme with predictive action for composition control in VAc/AA

suspension copolymerizations. [Color figure can be viewed in the online issue, which is available at www.

interscience.wiley.com.]


Sequential Optimization of Feed Rate Policies

To compensate for unavoidable perturbations of reac-

tion temperature, initiator concentrations and efficiencies

(which cannot be measured in-line), and monomer con-

centrations, a predictive control action can be included

into the optimization scheme (corrective action based on

reoptimization, where new feed flow rate profiles are

determined to maintain the composition constant along

the polymerization time). In this case, predicted and

measured performances can be compared, allowing for

indirect estimation of the reaction activity. This provides

an implicit integral correction scheme, as the accumula-

tion (or depletion) of AA in the medium leads necessar-

ily to the decrease (or increase) of the AA feed flow

rates in the subsequent control intervals, leading to cor-

rection of copolymer compositions even when process

disturbances are present or when there is significant

model mismatch, as shown in the following examples.

Figure 6 illustrates the proposed sequential optimiza-

tion scheme after introduction of the predictive control

action. In this case, after determination of the current

states at sampling time i, NI 2 i optimum feed rate

pulses are reoptimized to maintain the copolymer com-

position constant and equal to the desired values. This

can also compensate for model uncertainties during real

implementations and for possible inhibition of the poly-

merization caused by impurities. Simulations were per-

formed as described in the previous section. At each se-

quential optimization step, the new initial guesses for

the remainder of the trajectory were the converged solu-

tions from the previous optimization step.

Simulations were performed by assuming that changes

of the initiator concentration (activity) may lead to large

copolymer composition drifts and should be avoided. To

illustrate this effect, after 40 min of reaction, the amount

of initiator used to perform the simulations was

increased 25% in respect to the initial value. Figure 7A

and B shows the cumulative copolymer compositions

obtained when the predictive controller is not used (in

the particular case of Fig. 7A, the optimization scheme

was not updated, leading to significant model mismatch)

and when it is used to update the feed rate profiles. It is

clear that the control action can remove the undesired

perturbation and keep the copolymer composition con-

stant throughout the batch. As the initiator concentration

increases, reflecting the unexpected increase of the reac-

tion activity, the AA copolymer content decreases, as

VAc is consumed faster. This leads to significant bias of

the final copolymer composition. On the other hand,

when the predictive control action is used, the AA feed

rate policy is changed to accommodate for the increased

reactivity of the VAc monomer.

Simulations were also performed to evaluate the si-

multaneous variation of initiator concentration and VAc

concentration in the reaction medium. The main objec-

tive was to evaluate the performance of the proposed

optimization scheme when both the overall reaction

activity and the copolymerization kinetics are disturbed.

To illustrate this effect, after 40 min of reaction, the

amount of initiator used to perform the simulations

was increased 25% in respect to the initial value. After

70 min, the amount of monomer was also increased

25% in respect to the initial value. As in the previous

case, it is important to say that the optimization

scheme was not updated, leading to significant model

mismatch.

It can be observed in Fig. 8B that the proposed optimi-

zation scheme is able to determine new feed profiles and

simultaneously maintain the copolymer composition at the

desired setpoint value. The small oscillatory behavior can

be observed after introduction of the process disturbances;

however, obtained results can be regarded as excellent.

Simulations were also performed to evaluate the im-

portance of temperature disturbances on the performance

of the proposed optimization scheme. To illustrate this

effect, it was assumed that the actual reactor temperature

was different from the value used to perform the optimi-

zations. Figure 9 illustrates the importance of tempera-

ture uncertainties on the obtained copolymer composi-

tions, when differences between the actual reactor tem-

FIG. 7. Sequential optimization after perturbation of the initiator con-

centration: (A) without predictive action and (B) with predictive action.

[Color figure can be viewed in the online issue, which is available at

www.interscience.wiley.com.]


peratures and the values used for optimization were

equal to 108C. As one can observe, temperature exerts a

significant effect on the AA composition, favoring the

preferential incorporation of AA and leading to continu-

ous drift of copolymer composition. However, when the

optimization is carried out considering the predictive

action, copolymer composition drift is not observed. As

can be noted, in this case, the AA copolymer composi-

tion is kept constant at the desired setpoint value during

all stages of the polymerization, indicating that the

adopted optimization strategy is able to remove the tem-

perature disturbance efficiently.

CONCLUSIONS

The dynamic optimization of semibatch VAc/AA sus-

pension copolymerizations was carried out, based on a

direct search Complex algorithm. The dynamic optimiza-

tion scheme was used to control the copolymer composi-

tion along the batch. A sequential optimization proce-

dure was coupled with a predictive controller to guaran-

tee that the manipulation of feed flow rates could allow

for attainment of the desired copolymer compositions.

The optimization strategy was validated through simula-

tion, by assuming that reactions were subject to pertur-

bations of the reaction temperature, initiator, and VAc

concentrations. It was shown that the proposed optimiza-

tion strategy can be used successfully both for design of

monomer feed rate profiles and removal of process dis-

turbances during semibatch suspension copolymeriza-

tions, to keep the copolymer composition constant

throughout the batch.

Particularly, it was shown that the computed opti-

mum feed rate profiles present oscillatory behavior,

because of the proposed feed rate discretization

scheme. Besides, it was shown that computed optimum

feed rate profiles are sensitive to the initial guesses

provided for optimization, because of existence of mul-

tiple local optima. Despite that, copolymer composition

could be controlled efficiently in all cases, with negli-

gible effects on the overall reaction rates and batch

time. As required CPU computer times in standard

desktop computers are smaller than a few minutes, the

proposed sequential optimization procedure can be used

in real time in actual semibatch suspension polymeriza-

tion applications.

FIG. 8. Sequential optimization after perturbation of the initiator and

monomer concentrations: (A) without predictive action and (B) with pre-

dictive action. [Color figure can be viewed in the online issue, which is

available at www.interscience.wiley.com.]

FIG. 9. Sequential optimization after perturbation of the reaction tem-

perature: (A) 108C increase and (B) 108C decrease. (The optimal feed

flow rate profile was determined with the updating optimization proce-

dure). [Color figure can be viewed in the online issue, which is available

at www.interscience.wiley.com.]


APPENDIX: KINETIC MECHANISMAND MODEL EQUATIONS

The kinetic mechanism proposed for VAc/AA copoly-

merization comprises the following steps:

Initiation Transfer to monomer

I�!kD 2R

RþM1 �!kl P1;0

RþM2 �!kl Q0;1

Pi;j þM1 �!ktrM11 Ci;j þ P1;0

Pi;j þM2 �!ktrM12 Ci;j þ Q0;1

Qi;j þM1 �!ktrM21 Ci;j þ P1;0

Qi;j þM2 �!ktrM22 Ci;j þ Q0;1

Propagation Termination by disproportionation

Pi;j þM1 �!kP11

Piþ1;j

Pi;j þM2 �!kP12

Qi;jþ1

Qi;j þM1 �!kP21

Piþ1;i

Qi;j þM2 �!kP22

Qi;jþ1

Pi;j þ Pm;n �!kT11 Ci:j þ Cm;n

Pi;j þ Qm;n �!kT12 Ci:j þ Cm;n

Qi;j þ Qm;n �!kT22 Ci:j þ Cm;n

Assuming that the long chain and quasi steady-state

hypotheses are valid for the polymer radicals, it is possi-

ble to develop the following set of mass balance equa-

tions for the VAc/AA copolymerization process.

� Initiator

dI

dt¼ �kDI (A:1)

� Monomer 1 (vinyl acetate)

dM1

dt¼ � kP11P

I� �

M1 þ 1

jþ 1

� �M2

r1

� �(A:2)

� Monomer 2 (acrylic acid)

dM2

dt¼ � kP11

PI� � 1

jþ 1

� �M2

r1þ r2r1

1

jþ 1

� �2M22

M1

" #

þ F2 ðA:3Þ� Incorporated monomer 1 (vinyl acetate)

d}1

dt¼ kP11

PI� �

M1 þ 1

jþ 1

� �M2

r1

� �(A:4)

� Incorporated monomer 2 (acrylic acid)

d}2

dt¼ kP11

PI� � 1

jþ 1

� �M2

r1þ r2r1

1

jþ 1

� �2M22

M1

" #(A:5)

� Global conversion

xM ¼ MW1}1 þMW2}2

MW1 M1 þ }1ð Þ þMW2 M2 þ }2ð Þ (A:6)

� Cumulative copolymer composition

< ¼ }2

}1 þ }2

(A:7)where

r1 ¼ kP11kP12

(A:8)

r2 ¼ kP22kP21

(A:9)

kP11PI

� �¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2fkDI

1c1þ 2z 1

c1c2ð Þ1=2r2r1

1jþ1

� M2

M1þ 1

c2

1jþ1

� 2r2r1

M2

M1

� 2

vuutðA9Þ

z ¼ 107:84 exp �63:64M2

M1 1þ jð Þ þM2

� �(A:11)

ci ¼kPii

2

kTii

¼ kPii02

kTii0

gðT; xMÞ i ¼ 1; 2 (A:12)

The correlation for the combined gel and glass effects is

given by:

gðT; xMÞ ¼ exp �146:8 vf � vf0ð Þ � 1076:3 vf � vf0ð Þ2n o

þ 92:9 vf � vf0ð Þ ðA:13Þ

The system free volume can be given as the sum of the

individual contributions of each of the system compo-

nents, as follows [47]:

vf ¼X4i¼1

vfifi (A:14)

The individual free volume contributions and the volume

fractions of the components can be given as [48]:

vfi ¼ 0:025þ ai T � Tgi� �

(A:15)

fi ¼ri=miP4i¼1 ri

mi

(A:16)

where ri is the pure density of the component i, mi is the

mass of the component i; ai and Tgi are, respectively, the


expansion coefficients and the glass transition tempera-

tures required for calculation of the free volume.

The global partition coefficient is given by:

j ¼ KVII

VI(A:17)

where Vi is the volume of the phase i. The superscripts I

and II stand for the organic and aqueous phases, respec-

tively. The following functional form can be proposed for

the partition coefficient [31].

K T; M2½ �II�

¼ Aþ B M2½ �IIþ C

M2½ �II� 2

264

375�1

(A:18)

where the coefficients A, B, and C are temperature-de-

pendent adjustable parameters in the form.

A ¼ �16:67þ 0:455 T � 273:15ð Þ� 2:92 � 10�3 T � 273:15ð Þ2 ðA:19Þ

B ¼ 23:02� 0:594 T � 273:15ð Þþ 3:96 � 10�3 T � 273:15ð Þ2 ðA:20Þ

C ¼ 0:317� 9:02 � 10�3 T � 273:15ð Þþ 6:04 � 10�5 T � 273:15ð Þ2 ðA:21Þ

The polymerization kinetic constants and model parame-

ters used for simulation were similar to the one described

by [32], and the reader is referred to this publication for

detailed information.

NOMENCLATURE

Parameters

F initiator efficiency

Fi feed flow rate of species iG combined gel and glass effect correlation

I concentration of initiator

K partition coefficient

kD kinetic constant for initiator decomposition

kPi,j kinetic constant for propagation of the radical iwith the monomer j

kTi,j kinetic constant for termination of radicals i and jMi concentration of monomer iMWi molecular weight of monomer iPi,j radical chain containing i mers of the species 1

and j mers of the species 2 in the chain and the

species 1 at the active site (radical 1)

PAA poly(acrylic acid)

PVAc poly(vinyl acetate)

Qi,j radical chain containing i mers of the species 2

and j mers of the species 1 in the chain and the

species 2 at the active site (radical 2)

ri reactivity ratio of monomer iT reaction time

T reaction temperature

Tgi glass transition temperatures of the species iVi volume of phase ivf free volume of the reaction system

vf0 free volume of the reaction system at zero con-

version

vfi free volume of species ivfM free volume of the reaction system at zero con-

version

xM monomer conversion

Greek symbols

fi volume fraction of the species i in the reactor

qi pure density of the component iqw density of the water

}i polymer iai expansion coefficients of species iu global partition coefficient

f cross termination constant

Gi,j dead polymer chain containing i mers of the spe-

cies 1 and j mers of the species 2

Superscripts

I organic phase

II aqueous phase

Subscripts

1 vinyl acetate

2 acrylic acid

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dynamic optimization of semibatch vinyl acetate/acrylic acid suspension copolymerizations

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