dynamic optimization of emulsion polymerization

9th International Scientific – Technical Conference – PROCESS CONTROL 2010 June 7 – 10, 2010, Kouty nad Desnou, Czech Republic

DYNAMIC OPTIMIZATION OF EMULSION POLYMERIZATION PROCESS PAULEN RADOSLAV, FIKAR MIROSLAV, LATIFI M. A.† FCFT, STU in Bratislava, Slovakia, e-mail: {radoslav.paulen, miroslav.fikar}@stuba.sk † LRPG, CNRS–ENSIC–INPL, Nancy, France, e-mail: [email protected] Abstract: In this work we study emulsion polymerization reaction in a batch reactor. Overall process model is described by seven first-order non-linear ordinary differential equations (ODEs) which right hand sides are varying according to one of the three stages of the process. Our goal is to produce polymer of prescribed terminal quantity and quality in minimum time. The process can thus be described as a hybrid system with both continuous and discrete behaviour. We obtain optimal trajectories of control variable using control vector parameterization (CVP) method. This method translates infinite dynamic optimization problem into finite non–linear programming (NLP) problem. The NLP computation requires gradient information which we provide using method of adjoint variables. Keywords: Dynamic Optimization, Hybrid Systems, Emulsion Polymerization

1 INTRODUCTION In this study we treat emulsion copolymerization of styrene and α–methylstyrene taking place in batch reactor. Emulsion polymerization refers to a unique process employed for some radical chain polymerizations. It involves the polymerization of monomers in the form of emulsions. This process has several distinct advantages from the process control point of view. The physical state of the emulsion (colloidal) system makes it easy to control the process. The product of an emulsion polymerization, referred to as a latex, can in many instances be used directly without further separations [Odian, 2004]. This process model can be described as a hybrid system with ordinary differential equations (continuous character) but with differing right-hand sides (logical conditions) according to the stage of the process. An approach used to find an optimal control of a plant or process is termed dynamic optimization [Chm´urny et al., 1988; Mikleˇs and Fikar, 2007]. It encompasses several techniques, which can be divided in two broad frameworks, variational methods and discretization methods. In variational methods [Luus and Cormack, 1972], a classical calculus of variations together with Pontryagin’s maximum principle [Pontryagin et al., 1962] is applied. These methods address the dynamic optimization problem in its original infinite dimensional form. A big advantage of this is that we are looking for an exact solution to problem without any transformations. But it is disadvantageous if we want to solve a problem for not quite simple systems. Then a discretization plays important role. The original infinite dimensional problem is approximated by a simpler one and transformed to a non-linear programming (NLP) problem. Sequential discretization, usually achieved via control parametrization [Brusch and Schappelle, 1973], is an discretization approach in which the control variable profiles are approximated by a sum of basis functions in terms of a finite set of real parameters. These parameters then become the decision variables in a dynamic embedded NLP. Function evaluations are provided to this NLP via numerical solution of a fully determined initial value problem (IVP), which is given by fixing the control profiles. This method has the advantages of yielding a relatively small NLP and exploiting the robustness and efficiency of modern IVP and sensitivity solvers [Chachuat et al., 2006]. Dynamic modeling and optimization of this process was previously studied in Gentric [1997] and Salhi et al. [2004]. In next sections we introduce the emulsion polymerization reaction model, give hybrid representation of this model and then we describe dynamic optimization approach used to compute optimal control profiles for this process. Finally we give and discuss obtained results.

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9th International Scientific – Technical Conference – PROCESS CONTROL 2010 June 7 – 10, 2010, Kouty nad Desnou, Czech Republic

2 PROCESS MODEL The overall process is described by a kinetic model, a molecular weight distribution model, and heat balance equations in the reactor and in the jacket. 2.1 Kinetic Mechanism The copolymerization of styrene and α-methylstyrene has been previously studied both in the bulk and in solution. Similarities with the homopolymerization behavior of styrene or α-methylstyrene are often emphasized. The kinetic model of an emulsion polymerization usually divided into three stages. The experimental validation was carried out by Gentric [1997]. The three stages of the model are as follows: 1. In the first stage, free radicals are produced in the aqueous phase by initiator decomposition. They are captured by the micelles swollen with monomer. The polymerization begins in these micelles. This stage, corresponding to particle heterogeneous nucleation (the polymerization rate is increasing), stops when all of the micelles have disappeared. 2. Particle growth occurs during the second stage. Monomer diffuses rapidly from monomer droplets toward the particles, which are saturated with monomer as long as monomer droplets exist. This stage ceases when all of the monomer droplets have disappeared (the polymerization rate is constant). 3. The third and final stage is characterized by the decrease of the monomer concentration in particles, also the polymerization rate is decreasing. The following assumptions are used in the modeling of the kinetic process: Styrene and α-methylstyrene are both hydrophobic monomers; thus, only micellar nucleation is taken into account. Because of this hydrophobic character, the reactions of propagation, transfer to monomer, termination in the aqueous phase, and radical desorption are neglected. Termination in particles is considered to be very rapid compared to radical entry into particles; thus, it can be assumed that no more than one radical is present in each polymer particle (zero-one system). This allows us to write Pj• = N • , where variable Pj• represents number of activated polymers of chain length j and N • represents number of active particles. The maximum conversion rate considered is generally around 65%, so that gelation has not yet occurred. Thus, the gel effect is not included in the model to avoid unnecessary complexity. Coagulation is assumed to be negligible. The kinetic mechanism is then written as follows: • Initiator decomposition A → 2R•

Ra = 2f kd A

• Particle formation R• + m → N•

Rn = k1 mR•

• Initiation N + R• → N•

Ri = k2 N R•

• Termination N• + R• → N Rt = k2 N • R• C023a – 2

9th International Scientific – Technical Conference – PROCESS CONTROL 2010 June 7 – 10, 2010, Kouty nad Desnou, Czech Republic

• Propagation P•j + M → P•j+1

Rp = kp Mp N •

• Transfer to monomer P•j + M → M• + P•j

RtrM = ktrM Mp N •

Here A means initiator concentration, R• stands for initiator radical concentration, m represents number of micelles per unit volume, M is global monomer concentration and finally f is initiator efficiency. Variables R and k differing by index of partial reaction of kinetic mechanism denote reaction rates and reaction kinetic coefficients. 2.2 Kinetic Model Only the global monomer concentration is described. The depropagation is taken into account by including in the propagation rate constant a factor depending on the α-methylstyrene molar fraction in the initial charge kp = kp′ exp(−a.fMS )

(1)

where a is a constant, fMS is molar fraction of α-methylstyrene at the beginning of a reaction and kp′ is the propagation rate constant for styrene homopolymerization. The quasi-stationary state approximation allows to calculate the initiator radicals concentration R• =

Ra k1 m + k2 Np

(2)

where Np = N + N • is overall number of polymer particles. Then the rate of particles formation may be written1 N˙ p = k1 m

Ra NA k1 m + k2 Np

(3)

Here NA stands for Avogadro’s number. Like in Harada et al. [1972], we introduce the factor which is similar to an effectiveness for the particles relative to the micelles in collecting an initiator radical ε=

k2 SNA k1 m

(4)

The rate of particle formation is now expressed by Ra N A N˙ p = k1 m εNp 1+ SNA

(5)

With the classical hypothesis that the emulsifier molecules are adsorbed in a monomolecular layer on the polymer particles surface, the emulsifier concentration may be written 1/3

S = So − kv (XMo )2/3 Np

(6)

where So denotes initial emulsifier concentration, Mo means initial monomer concentration, X signifies monomer conversion and 1/3  2 36πMM (7) kv = ωP2 (as NA )3 ρ2P 1

through the whole paper a dot over a variable represents its time derivative

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9th International Scientific – Technical Conference – PROCESS CONTROL 2010 June 7 – 10, 2010, Kouty nad Desnou, Czech Republic

where MM denotes monomer molecular weight, ωP means polymer weight fraction in particles, variable as represents surface area occupied by an emulsifier molecule and ρP stands here for polymer particle density. The rate of monomer consumption is expressed according to Np M˙ = −Rp = −kp Mp n ¯ NA

(8)

with average number of radicals per particle n ¯ =0.5, and monomer concentration in particles Mp which can be expressed as (1 − Xc )ρM X ≤ Xc [(1 − Xc ) + Xc ρM /ρP ] MM (1 − X)ρM X > Xc Mp = [(1 − X) + XρM /ρP ] MM

Mp = Mpc =

(9a) (9b)

there constant Mpc denotes critical monomer concentration in particles, Xc stands for critical monomer conversion and ρM represents monomer density. 2.3 Molecular Weight Distribution Model Besides, it is important to evaluate the polymer properties. They are linked to the polymer structure and can be described in terms of average molecular characteristics (number and weight average molecular weights, polydispersity index, . . . ). The used approach is an extension of the tendency model [Villermaux and Blavier, 1984] to emulsion polymerization. According to the kinetic mechanism, the rates of production of the zeroth, first and second order moments of the molecular weight distribution may be written Q˙ 0 =Rt + RtrM Q˙ 1 =L(Rt + RtrM ) Q˙ 2 =2L2 (Rt + RtrM )

(10)

where Rt =

Ra n ¯ Np Np + Sε

RtrM =ktrM Mp L=

Np n ¯ NA

(11)

Rp Rt + RtrM

Here variable L denotes kinetic chain length. Once these moments are known, the number-average ¯ n ), weight-average molecular weight (M ¯ w ) and polydispersity index (Ip ) can be molecular weight (M calculated according to ¯ n = MM Q1 M Q0

¯ w = MM Q2 M Q1

¯w M Ip = ¯ Mn

(12)

2.4 Reactor Temperature Dynamics Model In order to achieve temperature control, it is necessary to describe the reactor temperature dynamics. The reaction takes place in a glass stirred tank reactor. Heat control is realized thanks to a cooling fluid circulating in a jacket at constant flow rate. The inlet temperature of the cooling fluid is controllable. Energy balances on the jacket and the reactor contents give the following equations for the

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9th International Scientific – Technical Conference – PROCESS CONTROL 2010 June 7 – 10, 2010, Kouty nad Desnou, Czech Republic

reactor and jacket temperature dynamics V ∆H UA T˙ = − Rp + (Tj − T ) m r Cp mCp Fj UA (Tj − T ) T˙j = (Tj,in − Tj ) − Vj ρj Vj Cp,j

(13) (14)

where V and Vj are reactor contents volume and reactor cooling jacket volume respectively, ∆H means polymerization reaction enthalpy, term mr Cp represents reactor total heat capacity, constant U means heat-transfer coefficient, A is cooling surface, variables Fj , ρj and Cp,j signify cooling fluid flow rate, cooling fluid density and cooling fluid heat capacity respectively. All the kinetic coefficients are calculated according to Arrhenius law ki = ki,0 exp(−Ej /RT )

(15)

2.5 Hybrid Representation of the Process Model Majority of this section uses notation introduced by Feehery [1998]. We consider system described by ODEs of the form mode Si

x˙

(i)

=f

(i)

(i)

(i)

(x , u , p, t) ∀t ∈

(i) (i) [t0 , tf ],

Si ∈

nk [

Sk

(16)

k=1 (i)

(i)

(i)

(i)

where x(i) ∈ Rnx , u(i) ∈ Rnu and p ∈ Rnp . Function f (i) is such that f (i) : Rnx × Rnu × Rnp × (i) R → Rnx . Switching from one stage (mode Sk ) to another (mode Sj ) occurs when a unique transition (k) condition Lj (x˙ (k) , x(k) , u(k) , p, t), Sj ∈ P (k) is satisfied. Set P (k) contains all possible successive (k)

stages of mode Sk . Transition function Tj (x˙ (k) , x(k) , u(k) , x˙ (j) , x(j) , u(j) , p, t) determines possible discontinuity in variables participating in model. There is a special case of transition function defined at (k) mode initial time t = t0 where (k)

(0)

(k)

Tk (x˙ (k) , x(k) , u(k) , p, t) = 0 ⇒ x(k) (t0 ) = x0

(17)

Hence, model represented by (5), (8), (10), (13), and (14) is strongly non-linear and hybrid as well (see [Salhi et al., 2004]). Its hybrid character can be expressed as follows mode Si

x˙ (i) = f (i) (x(i) , u, p)

(i)

(i)

∀t ∈ [t0 , tf ], Si ∈ {S1 , S2 , S3 }

(18)

where state vector is defined as x = (M, Np , Q0 , Q1 , Q2 , T, Tj )T

(19)

In our case, switching between stages passes in order known in advance. Thus, we can define transition conditions as (1)

(1)

L2 (x(1) , t) = 0 ⇒ Np (t) = Npst

(20)

(2)

L3 (x(2) , t) = 0 ⇒ X (2) (t) = Xc

(21)

where constant Npst represents number of particles which depends on temperature and initial load in reactor. Variable X(t) means monomer conversion which depends on actual monomer concentration M (t) and initial load in reactor. Note that due to known sequence of stages we can write (3) (3) (2) (2) (1) (1) t0 = t0 tf = t0 tf = t0 tf = tf

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9th International Scientific – Technical Conference – PROCESS CONTROL 2010 June 7 – 10, 2010, Kouty nad Desnou, Czech Republic

Transition functions can be defined as (0)

(1)

(1)

T1 (x(1) , p, t) = 0 ⇒ x(1) (t0 ) = x0 (1)

(2)

(1)

(2)

(3)

(2)

T2 (x(1) , x(2) , t) = 0 ⇒ x(2) (t0 ) = x(1) (tf )

(23)

T3 (x(2) , x(3) , t) = 0 ⇒ x(3) (t0 ) = x(2) (tf ) The rest of this section forms conditions for particular stages of the process. As it was mentioned in (3) (2) Section 2.1, the number of particles stays constant in the second and third stage. Thus, N˙ p = N˙ p = 0. In the third stage, calculation of variable Mp changes (see (9)). This changes also right hand sides of (8), (10), and (13) where Mp is present. 3 PROCESS OPTIMIZATION 3.1 Problem Definition Dynamic optimization problems can in general be defined for continuous process models as Z tf min G(x(tf )) + F (x(t), u(t), p, t)dt u(t),p

t0

s.t. x˙ = f (x(t), u(t), p, t)

∀t ∈ [t0 , tf ]

x(t0 ) = x0 h(t, x(t), u(t), p) = 0

(24)

g(t, x(t), u(t), p) ≤ 0 u(t) ∈ [uL (t), uU (t)] p ∈ [pL , pU ] where function G(·) is real-valued function and function F (·) is such that F : Rnx ×Rnu ×Rnp ×R → R. Functions h(·) and g(·) are vector functions of appropriate dimensions that define process constraints. Our objective is to compute such a control u(t) of the system that will drive system to desired terminal state (some state values are prescribed at final time) at minimum time possible. Based on this and Section 2.5 we can form our optimization problem as Z t(3) f (1) (3) dt = tf − t0 min u(t),p

(1)

t0

(i)

s.t. x˙ (i) = f (i) (x(i) , u(i) , p) (1)

(i)

∀t ∈ [t0 , tf ], i ∈ {1, 2, 3}

(1)

x(1) (t0 ) = x0 (3)

(25)

(3)

h(x(3) (tf ), tf ) = 0 u(t) ∈ [uL (t), uU (t)] p ∈ [pL , pU ] where initial state vector is defined as (1)

x0 = (M0 , 0, 0, 0, 0, T0 , Tj,0 )T

(26)

where T0 and Tj,0 are variables which represent initial reactor and cooling jacket temperature respectively. These are parameters p to be optimized. Terminal state of the reaction is characterized both by quantity and quality terminal conditions. Quantity terminal condition means that we desire final conversion of monomer to be (3)

(3)

Xf

(3)

= X (3) (tf ) = 1 −

M (3) (tf )

(27)

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9th International Scientific – Technical Conference – PROCESS CONTROL 2010 June 7 – 10, 2010, Kouty nad Desnou, Czech Republic

Quality terminal condition is represented by final value of number-average molecular weight, weightaverage molecular weight or polydispersity index of polymer and can be computed by (12). The optimized control variable u(t) is reactor cooling jacket inlet temperature Tj,in . 3.2 Optimization Method From application of Pontryagin’s maximum principle [Pontryagin et al., 1962] one can obtain necessary conditions of optimality for dynamic optimization problem. These determine definition of adjoint variables and their corresponding boundary conditions (specified at final time). Definition of gradients with respect to controls and parameters can be derived also using this principle. Necessary conditions of optimality for hybrid processes were studied in Ruban [1997] where possible discontinuity in process variables was taken in account. This kind of discontinuity is not present in our case. Detailed analysis of the optimized system showed that discontinuity in adjoint variables is also not present. Therefore, adjoint variable definition is expressed as ∂H (i) λ˙ (i) = − ∂(x(i) )T ∂G (3) (3) λ (tf ) = ∂(x(3) )T

i ∈ {1, 2, 3}

(28) (29)

(3)

t=tf

where

H (i) (x, λ, u, p, t) = F (·) + (λ(i) )T f (i) (·)

(30)

is Hamiltonian function and λ is vector of adjoint variables. We employ control vector parametrization (CVP) method to solve dynamic optimization problem. In first step, control trajectory u(t) is discretized to final number of intervals with constant control uj , where j ∈ {1, . . . , Nu } indicates control interval number. Hence, an infinite-dimensional optimization problem in continuous control variables is transformed into a finite-dimensional non-linear programming problem that can be solved by any gradientbased method (e.g., a sequential quadratic programming (SQP) method). The performance index evaluation is carried out by solving an initial value problem (IVP) of the original ODE system, and gradients of the performance index as well as the constraints with respect to the parameters (p and approximations of u(t)) can be evaluated by solving the adjoint equations. 3.2.1 Computing Gradients Accurate gradients calculation is one of the most challenging step in solving the NLP problem which was derived from original problem (25) using piecewise-constant control uj applied on specified instant of time ∆tj = tj − tj−1 . Gradients of objective function J can be computed such that ∂J (3) ∂tf

∂J (i) ∂tj

(3)

=H (3) (tf ) +

∂G (3)

∂tf

(i) + =H (i) (t− j ) − H (tj ) ∀j ∈ {1, . . . , Nu − 1} (1)

∂G ∂J (1) (1) ∂x = T − Jp (t0 ) + (λ(1) )T (t0 ) 0T T ∂p ∂p ∂p ∂J (i) (i) =Ju (tj−1 ) − Ju (tj ) ∀j ∈ {1, . . . , Nu } ∂uj

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9th International Scientific – Technical Conference – PROCESS CONTROL 2010 June 7 – 10, 2010, Kouty nad Desnou, Czech Republic

where i ∈ {1, 2, 3} and (i)

∂H J˙u = ∂u (i) ∂H J˙p = ∂pT

(32) (33)

Any constraint function can be expressed in form of optimization criterion [Salhi et al., 2004], so that constraint gradients can be found in same manner like presented in (31). 3.2.2 Optimization Algorithm We employ the procedure which can be described by following steps: 1. Make initial guess for values of ui , ∆ti and p. 2. Integrate system forward in time. 3. Evaluate objective and constraint functions. 4. Integrate system of adjoint equations backward in time. 5. Calculate gradients. 6. Use NLP solver to get new values of variables ui , ∆ti and p. 7. If the optimality conditions are satisfied then quit. Else, go to Step 2. 3.3 Results and Discussion Optimization algorithm proposed in Section 3.2 was implemented using MATLAB and its NLP solver fmincon which solves constrained NLP problems. Integration was performed using ode45 integrator where 4th order Runge–Kutta numerical integration method is implemented. Optimization problem was solved for different control discretization scenarios concerning piece-wise constant control. Obtained results are summarized in following table where variable Nu represents the number of control ¯ n,f = Table 1 – Comparison of different control strategies for desired terminal state: Xf = 0.7, M 6 2 × 10 Nu 3 4 5 6 7 8 9 10

tf [s] 4950 4859 4673 4609 4550 4539 4502 4445

Xf 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7

¯ n,f × 10−6 M 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0

¯ w,f × 10−6 M 4.8 5.7 5.8 5.8 6.0 6.3 6.3 6.4

Ip 2.38 2.87 2.87 2.87 3.01 3.13 3.13 3.18

segments considered. It can be seen that by the rising number of control segments not only the optimization criterion is lowering its value (as expected) but also an overall quality of polymer produced rises (this can be seen in last two columns of Table 1). Using of multistart method (i.e. starting algorithm

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9th International Scientific – Technical Conference – PROCESS CONTROL 2010 June 7 – 10, 2010, Kouty nad Desnou, Czech Republic

from multiple different starting points) showed that optimization problem appears to have multiple local minima. Graphically, the results can be represented by following Fig. 1, where the control trajectories as well as reactor temperature trajectories are depicted for selected control discretization scenarios. 345

345 T Tj,in

340

j,in

T,T

T,Tj,in [K]

335

[K]

335 330

325

320

320 315 0

0.2

0.4

0.6

0.8

1

0

0.6

(a) Nu = 3

(b) Nu = 4

0.8

1

345 T Tj,in

T Tj,in

340

T,Tj,in [K]

335

[K]

335 j,in

0.4 time/tf

340

T,T

0.2

time/tf

345

330

330

325

325

320

320

315

315 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

time/tf

time/tf

(c) Nu = 5

(d) Nu = 6

345

0.8

1

345 T Tj,in

340

T Tj,in

340

T,Tj,in [K]

335

[K]

335 j,in

330

325

315

T,T

T Tj,in

340

330

330

325

325

320

320

315

315 0

0.2

0.4

0.6

0.8

1

time/t

0

0.2

0.4

0.6

0.8

1

time/t

f

f

(e) Nu = 7

(f) Nu = 8

Figure 1 – Trajectories of control variable and reactor temperature for different numbers of control segments (Nu ).

4 ACKNOWLEDGMENTS The authors gratefully acknowledge the contribution of the Scientific Grant Agency of the Slovak Republic under the grants 1/0071/09, 1/0537/10 and the Slovak Research and Development Agency under the project APVV-0029-07. This work is also supported by a grant No. NIL-I-007-d from Iceland, Liechtenstein and Norway through the EEA Financial Mechanism and the Norwegian Financial Mechanism. This project is also co-financed from the state budget of the Slovak Republic.

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345

345 T Tj,in

340

j,in

T,Tj,in [K]

335

[K]

335

T,T

T Tj,in

340

330

330

325

325

320

320

315

315 0

0.2

0.4

0.6

0.8

1

time/t

0

0.2

0.4

0.6

0.8

1

time/t

f

f

(g) Nu = 9

(h) Nu = 10

Figure 1 – Trajectories of control variable and reactor temperature for different numbers of control segments (Nu ). (Continued.)

5 CONCLUSIONS In this paper we studied dynamic optimization of a polymerization process with emulsion copolymerization of styrene and α-methylstyrene. We presented process model and its hybrid representation. Following this representation we derived necessary conditions of optimality of this process. According to these we were able to compute gradients of objective and constraints functions. These gradients were passed to NLP solver which solved NLP obtained by control discretization of original dynamic optimization problem. As the computations suggested there might exist more than one solution to our problem because when the algorithm was launched from different initial points (a multistart method) a different solution was reported at the end of an algorithm run. This suggests that our optimization problem possesses multiple (local) minima. Further this tells that to find an optimal process operation we should utilize another approach (global optimization) to achieve guarantee of global optimality of control variable profiles. References BRUSCH, R. G.; SCHAPPELLE, R. H. 1973. Solution of highly constrained optimal control problems using nonlinear programming. AIAA Journal, 11, 2, 135–136. CHACHUAT, B.; SINGER, A. B.; BARTON, P. I. 2006. Global methods for dynamic optimization and mixed-integer dynamic optimization. Ind. Eng. Chem. Res., 45, 25, 8373–8392. ˇ ´ ´ M. 1988. Automatic Control of Technological Processes. CHMURNY, D.; PROKOP, R.; BAKOSOV A, Bratislava: Alfa. (in Slovak). FEEHERY, W. F. 1998. Dynamic Optimization with Path Constraints. Ph.D. thesis. GENTRIC, C. 1997. Optimisation Dynamique et Commande non Lineaire d’un Reacteur de Polymerisation en Emulsion. Ph.D. thesis, INPL, Nancy, France. HARADA, M.; NOMURA, M.; KOJIMA, H.; EGUCHI, W. 1972. Rate of emulsion polymerization of styrene. J. Appl. Polym. Sci, 16, 811–833. LUUS, R.; CORMACK, D. 1972. Multiplicity of solutions resulting from the use of variational methods in optimal control problems. Canadian Journal of Chemical Engineering, 50, 2, 309–311. ˇ J.; FIKAR, M. 2007. Process Modelling, Identification, and Control. Berlin: Springer Verlag. MIKLES, ODIAN, G. 2004. Principles of Polymerization. John Wiley & Sons, Inc., 4th edition. PONTRYAGIN, L. S.; BOLTYANSKII, V. G.; GAMKRELIDZE, R. V.; MISHCHENKO, E. F. 1962. The Mathematical Theory of Optimal Processes. New York: John Wiley & Sons, Inc. RUBAN, A. I. 1997. Sensitivity coefficients for discontinuous dynamic systems. Journal of Computer and Systems Sciences International, 36, 4, 536–542.

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SALHI, D.; DAROUX, M.; GENTRIC, C.; CORRIOU, J. P.; PLA, F.; LATIFI, M. A. 2004. Optimal temperature-time programming in a batch copolymerization reactor. Ind. Eng. Chem. Res., 43, 7392– 7400. VILLERMAUX, J.; BLAVIER, L. 1984. Free radical polymerization engineering – I A new method for modeling free radical homogeneous polymerization reactions. Chem. Eng. Sci., 39, 87–99.

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