Optimal Operating Policies for the Nitroxide-Mediated Radical

Ind. Eng. Chem. Res. 2006, 45, 4637-4652

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Optimal Operating Policies for the Nitroxide-Mediated Radical Polymerization of Styrene in a Semibatch Reactor Roberto Lemoine-Nava and Antonio Flores-Tlacuahuac* Departmento de Ingenierı´a y Ciencias Quı´micas, UniVersidad Iberoamericana, Prol. Paseo de la Reforma 880, Me´ xico DF 01210, Me´ xico

Enrique Saldı´var-Guerra Centro de InVestigacio´ n en Quı´mica Aplicada, BlVd. Enrique Reyna 140, Saltillo, Coahuila 25100, Me´ xico

In this work, optimal operating policies for the industrial-level semibatch living free radical polymerization of styrene are proposed. Several operating scenarios are addressed, and one of the most promising ones is identified. The determination of the time-dependent optimal operating policies is posed as a dynamic optimization problem. The simultaneous dynamic optimization approach, involving the full discretization of the states and manipulated variables, was used to solve the resulting optimization problem. Moreover, the optimization problem also involved the determination of design parameters such as the initial holdup, feed stream temperature, and concentrations resulting in a large-scale nonlinear program. The resulting optimal operating conditions lead to a polymer featuring living characteristics matching monomer conversion, molecularweight distribution, and polydispersity index target values. 1. Introduction Living/“free” radical polymerization (LFRP) constitutes a field which has undergone a very high and rapid development, and this could be attributed to its potential applications in important fields, such as biotechnology, microelectronics, molecular biology, and nanotechnology.1 However, although research concerning the kinetic and mechanistic aspects of these processes has been made, still process systems engineering (PSE)-oriented research will be necessary in order to make LFRP economically viable.2 Most publications concerning this kind of polymerization analyze batch isothermal systems. Nevertheless, for large-scale applications, additional aspects such as process start-ups, heat-transfer requirements and limitations, agitation regimes, and adequate dosage of the reactants must be addressed due to the operational complications that they might bring. The importance of LFRP processes lies in the fact that through them it is possible to manufacture polymers with well-defined structures. Moreover, in LFRP processes, polymers with narrow molecular-weight distributions and low polydispersities can be achieved. Polymers with narrow molecular-weight distributions featuring low polydispersities may not be a goal by themselves, because a narrow polydispersity is not necessarily more attractive in the final product. However, if the product resulting from an LFRP is going to be used in a second process step for the growth of a second block of a different monomer, then narrow polydispersities are desirable to ensure the same global monomer composition in all the final block copolymer chains. Among all the different types of reactors that could be used for manufacturing polymers featuring living characteristics, the semibatch operating scheme offers a viable commercial alternative for manufacturing products with desired living properties.3 Moreover, it would be worthwhile to find the adequate operating policies that could lead to the synthesis of more monodisperse polymers than the ones that could be obtained by means of * To whom correspondence should be addressed. E-mail: [email protected]. Tel./Fax: +52(55)59504074. Web: http:// 200.13.98.241/∼antonio.

traditional free radical polymerizations. Although, in most industrial environments, the semibatch operating policies stem from process operators’ experience,4 it is likely to operate under suboptimal conditions due to the lack of a systematic methodology for the determination of some of the best operating policies. Dynamic optimization of batch and semibatch systems has proved to be a useful tool for the enhancement of the performance of such processes, as it can help to reduce the processing time for manufacturing products of a given quality or to achieve certain properties of them that would be difficult to obtain without optimal operating policies. There is a wide list of studies concerning the dynamic optimization of polymerization reactions. For these particular processes, in fact, dynamic optimization can provide an additional benefit, because although some properties such as compositions can be efficiently measured online, it is still necessary to develop reliable and relatively fast sensors for other quality parameters, such as the molecularweight distributions, which could be particularly difficult to measure in systems such as emulsion polymerizations.5 Thus, if a reliable mathematical model for the process is available, the dynamic optimization approach can provide a way to find the most adequate operating regime to achieve a desired product quality, to minimize the processing time, or to achieve a desired objective value for a given set of process parameters. Although this approach, as stated in this work, does not take into account the effect of disturbances or modeling errors hitting the process, it can provide a systematic methodology for the determination of the most proper open-loop operating policies for such purposes. Open-loop optimal trajectories determined in this way could be closed-loop tracked using a feedback control system featuring an estimation scheme to take care of both modeling errors and unavailable measurements, as proposed in ref 6. Among the numerous studies that have been published concerning the dynamic optimization of polymerization reactions, the following ones are the more relevant for the purposes of our work. Chen and Jeng7,8 worked on the determination of the minimum end time policies for the polymerization of styrene. In a first publication, they considered the traditional case of generation of radicals by means of an initiator, using the reaction

10.1021/ie050849u CCC: $33.50 © 2006 American Chemical Society Published on Web 05/19/2006

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initiator feeding rate and the reaction temperature (for which both optimal value and optimal policy where calculated) as the manipulated variables. In a second work, they extended this study to the case in which thermal polymerization of this monomer takes place. In both publications, experimental verification of their results was made. Jang and Yang9 studied the processing time minimization problem of the emulsion polymerization of vinyl acetate, by means of optimal initiator feeding policies using the method of mixed integration-collocation optimization and compared their results with experimental ones. Choi and Butala10 studied the optimal control problem for the copolymerization of styrene and acrylonitrile. In this work, the optimal operation policies for the temperature and the monomer addition, to control the copolymer composition and molecular weight, were studied. One of the main contributions of this study lies in the fact that such operation policies were determined by inverse feedback control, a technique in which the optimal profiles of the manipulated variables are obtained by the assumption of the presence of a fictitious controller, which is designed and used for the calculation of the time-dependent trajectories for the decision variables, which are tracked and recorded for their application in the open-loop operation of the system. Cawthon and Knaebel3 studied the copolymerization of styrene and acrylonitrile in a semibatch reactor. They attempted to minimize both the processing time and polydispersity, maintaining predefined values for the copolymer composition and the molecular weight. To meet the specified requirements, the reaction time, the process temperature, and the monomer feeding rates were considered as manipulated variables. Maschio et al.11 addressed the control of the molecular weight and the polydispersity for the bulk, suspension, and solution polymerization of methyl methacrylate. For such a purpose, the values of concentrations and the optimal temperature and feeding rates of initiator, monomer, solvent, and chain transfer agent profiles were evaluated. In this work, a comparison between actual and predicted data was made, and a good agreement between them was observed. Vicente et al.5 manipulated the monomer and chain transfer agent feeding rates in the emulsion copolymerization of methyl methacrylate and n-butylacrylate to control the copolymer composition and its molecular-weight distribution. The authors compared their theoretical results against experimental data. Silva and Biscaia12 determined the optimal operation policies for the batch polymerization of styrene using a genetic algorithm, attempting to maximize the monomer conversion up to a desired value and to minimize the initiator content in the product. The authors selected the initiator flow rate and the reaction temperature as the manipulated variables and imposed constraints to the values of the molecular weight and polydispersity. Doyle et al.13 applied a hybrid model for the semibatch emulsion polymerization of styrene to control the particle-size distribution. For such a purpose, a fundamental mechanistic model is complemented with a statistical one, using data history from past batch operations to refine the results provided by the fundamental model and to achieve the desired polymer properties and manipulating the surfactant and initiator feeding rates. Joly and Pinto14 studied the optimal control problem of the batch production of nylon-6,6 in autoclaves. In this paper, predefined values of molecular weight and amine end concentration were achieved through the manipulation of the heat input and the reactor pressure. To our knowledge, the only PSE-oriented publication concerning LFRP processes has been made by Faliks et al.15 (however, several researchers16,17 had previously advanced the idea of using semibatch operating mode to reduce polymeriza-

tion time, keeping the living character of the polymerization reaction system, and the subject can be further exploited18), in which the authors attempted to minimize the reaction time and the polydispersity of the final product in a tubular reactor for the TEMPO-mediated polymerization of styrene by means of the manipulation of the TEMPO flux along the reactor. Although they improved the performance of the process in comparison with a base case in which no optimization was employed, they achieved values for the polydispersity greater than 1.3 for all the cases. Although this value is adequate for the LFRP standards, it would be desirable to find an alternative for the synthesis of a more monodisperse polystyrene. Regarding actual industrial processes based on the analyzed polymerization scheme addressed in this work, there is at least one commercial process based on nitroxide mediated polymerization (NMP) revealed by EFKA Additives in The Netherlands in recent years, and it is aimed at the production of pigment dispersants for coatings. They use a sequential polymerization in a semibatch reactor for building acrylic block copolymers (ref 19 and references therein). Also ARKEMA, a company based in France, has presented a family of acrylic block copolymers made by a sequential process using NMP. The process can be regarded as semibatch (see ref 20 and references therein). Finally, one of us (E. Saldı´var-Guerra) has industrial experience, and while in industry, he developed semibatch and continuous processes for the production of block copolymers containing functional groups based on NMP chemistry. In this paper, the first published industrial-scale dynamic optimization study for a LFRP process under batchwise operation conditions is developed. There have been works21 dealing with reactor design studies in LFRP systems, but without the dynamic optimization detail addressed in the present work. Morever, in the present work, due to strong interactions among the process variables, design parameters, such as initial reactor holdup, feed stream concentration, and temperature, are also simultaneously determined together with the optimal time-dependent values of the chosen manipulated variables, leading to an optimization problem that is harder to solve. The paper is structured as follows. In Section 2, the reaction mechanism and the mathematical model of the process are detailed. In Section 3, the formulation of the dynamic optimization problem is presented. In Section 4, all the cases of study for the optimization are detailed, and the general analysis of the resulting optimal policies is made. In Section 5, the final conclusions are presented. 2. Reaction Kinetics and Mathematical Model Before the mathematical model of the process is discussed, it is convenient to give a brief description and the modeling assumptions behind it. A flowsheet of the polymerization reactor and its heat exchange system is presented in Figure 1. The reaction kinetic scheme used in this study is the same as in ref 2 and is reported in Table 1. In that reference, the diffusional effects (which become significant at high conversions) are neglected. This assumption also holds for this study. Following the monomolecular reaction mechanism described in ref 22, a nitroxyl ether (NOE) is used to provide primary radicals (for the propagation step) and nitroxyl radicals (for the formation of dormant species) as a product of its cleavage. An additional amount of radicals are generated from the thermal autopolymerization of the styrene. The kinetic coefficients are also the same as in ref 2 and are shown in Table 2. The reactor consists of a perfectly mixed tank in which, for all the operating schemes (which will be detailed later), initial amounts of styrene and nitroxyl ether are prefed. Although it

Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006 4639

Figure 1. Flowsheet of the polymerization process. Table 1. NMRP Living Polymerization Kinetic Scheme nitroxyl ether decomposition

ka2

8 R• + NOx• NOE 79 k d2

mayo dimerization

kdim

M + M 98 D ki

thermal initiation

D + M 98 M• + D•

first propagation (primary radicals)

R• + MT98 P1•

kp

kp

first propagation (monomeric radicals)

M• + M 98 P1•

first propagation (dimeric radicals)

D• + M 98 P1•

propagation

Pn• + M 98 Pn+1•

dormant-living exchange (monomeric alkoxyamine) dormant-living exchange (polymeric alkoxyamine) alkoxyamine decomposition

M• + NOx• 79 8 MONx k

kp

kp

rate enhancement reaction

kd a

kd

Pn• + NOx• 79 8 PnONx k a

kdecomp

MONx 98 M + HONx kh

3

D + NOx• 98 D• + HONx ktc

termination by combination

Pn• + Pm• 98 Dn+m

termination by disproportionation

Pn• + Pm• 98 Dn + Dm

ktd

ktrm

transfer to monomer

Pn• + M 98 M• + Dn

Transfer to dimer

Pn• + D 98 D• + Dn

ktrd

Table 2. NRMP Living Polymerization Kinetic Information; T Is in K and R ) 8.314 × 10-3 kJ/(gmol‚K) ki kdim kp ktc ktd ktrm ktrd kdecomp kh3 kd ka kd2 ka2

e7.0233 e-7616.7/T 23 104.4 e-93.5/RT 24 107.63 e-32.51/RT 25 1.7 × 109 e-843/T 26 0 22 0 22 0 22 5.7 × 1014 e-153/RT 25 0.1 22,27 4.7 × 109 e-9.6296/RT 28 3 × 1013 e-124/RT 25 kd ka

L/(mol s) L/(mol s) L/(mol s) L/(mol s) L/(mol s) L/(mol s) L/(mol s) 1/s L/(mol s) L/(mol s) 1/s L/(mol s) 1/s

has been shown that important parameters such as the viscosity and the heat-transfer coefficient show significant changes as the polymerization progresses,29 for the sake of simplicity, constant thermodynamic properties and the worst-case scenario heat-transfer global coefficient (i.e., the smaller heat-

transfer coefficient at high viscosity) are assumed. Also, a temperature-independent heat of polymerization has been considered. The heat exchange requirements of the process are provided by a heating-cooling system similar to the one described in ref 30. It consists of a loop in which water or steam are driven by a split-range controller. When heating is required, the steam (which has been assumed here to be saturated and at 100 °C) is allowed to flow to the loop, and the entrance of cooling water is restricted. When it becomes necessary to cool the system, a given amount of water leaving the cooling jacket (at a relatively high temperature) is replaced by the same amount of water, which is at atmospheric conditions, thus increasing the cooling system capacity to receive the heat released by the reaction. For the cases in which reactants are fed (i.e., semibatch cases), any increase of their temperature above environmental conditions is assumed to be achieved through previously available heat exchangers, whose residence time is assumed to be negligible, and no prepolymerization in them takes place. According to the previous description, the mass balances, moment balances, and volume variations can be described according to the next set of equations (refer to the nomenclature section for variables definition):

d[M] [M0]QPM + [M0]QMM [M] dV ) - 2kdim[M]2 dt V V dt ki[D][M] - kp[M]{[D•] + [M•] + [R•]} -kp[M][Y0] ktrm[M][Y0] + kdecomp[MONx] (1) d[D] [D] dV )+ kdim[M]2 - ki[D][M] dt V dt ktrd[D][Y0] - kh3[D][NOx•] (2) d[NOE] [NOE,0]QMM [NOE] dV ) dt V V dt ka2[NOE] + kd2[NOx•][R•] (3)

4640


d[M•] [M•] dV )+ ki[D][M] - kp[M][M•] dt V dt kd[NOx•][M•] + ka[MONx] + ktrm[M][Y0] (4) d[R•] [R•] dV )- kp[R•][M] + ka2[NOE] - kd2[R•][NOx•] dt V dt (5) •

•

d[D ] [D ] dV )+ ki[D][M] - kp[D•][M] + dt V dt ktrd[D][Y0] + kh3[D][NOx•] (6) [NOx•] dV d[NOx•] )- kd[NOx•][Y0] + ka[Z0] + dt V dt ka2[NOE] - kd[NOx•][M•] + ka[MONx] kd2[NOx•][R•] - kh3[D][NOx•] (7) d[MONx] dt

)-

[MONx] dV V

dt

+ kd[NOx•][M•] ka[MONx] - kdecomp[MONx] (8)

[Y0] dV d[Y0] )+ kp[M]{[D•] + [M•] + [R•]} + dt V dt ka[Z0] - [Y0]{kd[NOx•] + ktrm[M] + ktrd[D] + (ktc + ktd)[Y0]} (9) [Y1] dV d[Y1] )+ kp[M]{[D•] + [M•] + [R•]} + dt V dt ka[Z1] + kp[M][Y0] - [Y1]{kd[NOx•] + ktrm[M] + ktrd[D] + (ktc + ktd)[Y0]} (10) [Y2] dV d[Y2] )+ kp[M]{[D•] + [M•] + [R•] + dt V dt [Y0] + 2[Y1]} + ka[Z2] - [Y2]{kd[NOx•] + ktrm[M] + ktrd[D] + (ktc + ktd)[Y0]} (11)

adapted from ref 31, neglecting the contribution of the nitroxyl ether in the system, which can be assumed to be true because it is present in trace quantities. The energy balances for the former (which describe the variations of the temperature of the cooling jacket before and after interacting with the reactor) have been based on ref 30.

dT ) dt (-∆HR)Vkp[M]([M•] + [R•] + [D•] + [Y0]) - UA(T - Tj) NM,0Cp,M {QPM[M0]Cp,M(T - TPM,0) + QMM[M0]Cp,M(T - TMM,0)} NM,0Cp,M (19)

dTj,in Qcw(Tj,out - Tj,in) - Qnew(Tj,out - Tnew) ) + dt Vloop λFs (20) FwCp,wVloop dTj,out Qcw(Tj,in - Tj,out) UA(T - Tj) ) + dt Vj FwCp,wVj

(21)

The average cooling jacket temperature is given by the following expression:

Tj )

Tj,in + Tj,out 2

(22)

The initial moles of monomer initially loaded in the reactor are calculated with the following formula:

NM,0 ) V0[M0]

(23)

The heat-transfer area can be calculated as a function of the reactor volume as follows:

[Z0] dV d[Z0] )+ kd[NOx•][Y0] - ka[Z0] dt V dt

(12)

4 π A ) Dr2 + V 4 Dr

d[Z1] [Z1] dV )+ kd[NOx•][Y1] - ka[Z1] dt V dt

(13)

The molecular weights and the polydispersity index can be evaluated using the next equations:

[Z2] dV d[Z2] )+ kd[NOx•][Y2] - ka[Z2] dt V dt

(14)

[Z1] + [Q1] + [Z1] Mn ) MWM [Z0] + [Q0] + [Z0]

(25)

[Q0] dV d[Q0] 1 )+ ktc + ktd [Y0]2 + dt V dt 2 [Y0](ktrm[M] + ktrd[D]) (15)

[Z2] + [Q2] + [Z2] Mw ) MWM [Z1] + [Q1] + [Z1]

(26)

(

)

[Q1] dV d[Q1] )+ (ktc + ktd)[Y0][Y1] + dt V dt [Y1](ktrm[M] + ktrd[D]) (16) [Q2] dV d[Q2] )+ (ktc + ktd)[Y0][Y2] + dt V dt ktc[Y1]2 + [Y2](ktrm[M] + ktrd[D]) (17) dV ) QPM + QMM dt

(18)

Now, the energy balances for both the reactor and the cooling jacket will be shown. The one describing the latter has been

Polydispersity )

Mw Mn

(24)

(27)

The monomer fractional conversion can be evaluated using the next equation, adapted from ref 21:

X)

[Y1] + [Z1] + [Q1] [Y1] + [Z1] + [Q1] + [M]

(28)

The set of design parameters at the nominal point, as well as thermodynamical properties, are shown in Table 3. This table contains two types of information: design parameters and ther-


Initial conditions:

Table 3. Design Parameters and Thermodynamic Information monomer concentration in the feed streams loop cooling water flow rate loop inlet cooling water temperature cooling jacket volume loop volume heat-transfer coefficient heat of reaction reaction mixture heat capacity cooling water heat capacity steam latent heat cooling water density styrene molecular weight reactor diameter

8.7 mol/L32 3.33 L/s 293.15 K32 2000 L32 2500 L 80 J/(m2 s K)32 -73 000 J/mol32 1 647.27 J/(kg K)32 4 045.7 J/(kg K)32 2.26 × 106 J/kg35 1 kg/L32 104 g/gmol32 2m

modynamic constants. All the thermodynamic information was obtained from a previous paper of our research work.32 Regarding the design information, it is almost the same as that in the above-mentioned paper. We have only changed the following: (a) cooling water flowrate (from 1 up to 3.33 L/s) and (b) jacket volume (from 2 000 up to 2 500 L). Those values were adjusted to improve the cooling capabilities of the semibatch reactor. Previous values shown in ref 32 were for a CSTR; therefore, they needed to be changed for a semibatch reactor. Moreover, reactor diameter was fixed at 2 m. The kinetics of the NMP process has been extensively studied in the last 10 years. It is true that there are still open questions about some aspects of the kinetics, but most of the basic aspects are reasonably well-understood (see refs 33 and 34). Temperature effects can be accounted for by the activation energies of the capping-decapping steps of the nitroxide equilibrium, which have been reported for different nitroxide structures. Our group is at present actively working in fine details of the basic aspects of this chemistry. Therefore, we believe that our calculations and predictions have a reasonably strong foundation and are close to reality. Moreover, in a previous paper of our group,22 we show comparisons of our model vs experimental kinetic data for a monomolecular and a bimolecular styrene NMP, and the fitting is quite good. This gives us confidence in the predictions for the system of this study. 3. Formulation of the Dynamic Optimization Problem For the solution of the dynamic optimization problem, the so-called simultaneous approach36,37 is used. In this method, trough proper model discretization, the differential and algebraic equation (DAE) system is transformed into a set of algebraic equations, giving rise to a normally large-scale optimization problem. This method is advantageous for the resolution of systems which might exhibit instabilities37 and for problems where constraints for the state variables are to be imposed. To find the optimal values and trajectories of the manipulated variables under a batch time-minimization framework,30 in this work the next objective function is used:

Min

∫0θ {ω1||PI(t) - PIdes||2 + ω2||X(t) - Xdes||2 +

u ω3||Va(t) - Va,des||2 + ω4||Qnew(t)/Qnew - Qnew,des/Qunew||2 +

ω5||QPM(t) - QPM,des||2 + ω6||QMM(t) - QMM,des||2 + ω7||Fs(t) - Fs,des||2} dt (29) s. t. DAE model:

z(0) ) z0

(32)

Bounds: zL e z(t) e zU

(33)

yL e y(t) e yU

(34)

uL e u(t) e uU

(35)

pL e p(t) e pU

(36)

tLf e tf(t) e tUf

(37)

In the objective function, θ is the transition horizon, which corresponds to the nominal batch time; PI stands for the polydispersity index; and Va is the dimensionless reaction mixture volume, defined by the equation

V - Vl , 0 e Va e 1 Va ) u V - Vl

(38)

where Vl and Vu are the lower and upper bounds for the reaction mixture volume, respectively. Qunew is the upper bound for the loop inlet cooling water flow rate; this quantity included in the objective function works as a scaling factor for this manipulated variable. The subscript des stands for the desired values of the variables at the end of the process. The aim of the objective function is to maximize the monomer conversion (X), as well as the volume of the reaction mixture (V), at the end of the batch period and to minimize the polydispersity index (PI). Thus, the desired values of these variables have been declared as Xdes ) 1, Va,des ) 1, and PIdes ) 1. Because of the facts that it would be desirable that, at the end of the batch period, all the services were shut down and that no reactants were fed to the reactor because of security reasons, the desired values of the manipulated variables at the end of the batch period have been set to Qnew,des ) 0 L/s, QPM,des ) 0 L/s, QMM,des ) 0 L/s, and Fs,des ) 0 kg/s. In the constraints set, F is the vector of right-hand sides of the differential equations which describe the mass, moments, and energy balances; G is the vector of algebraic equations, which are assumed to be index one; z is the vector of differential states; z0 contains the initial values of z; y is the vector of algebraic states; u is the vector of control profiles; and p is a time-independent parameter vector. Although in living radical polymerizations the polydispersity is one of the most critical variables to control, the molecular weight is also an important parameter because it determines the applications that the polymer can be used for in practice. However, this variable has not been incorporated in the objective function to avoid tradeoffs that there could exist among it and the polydispersity or the monomer conversion. ωi are weighting factors for the approximation of the decision variables to the desired values and to attain smooth profiles for the manipulated variables. To transform the original DAE optimization problem into a nonlinear programming one, the profiles are approximated by a family of polynomials on finite elements. The details of the solution of the dynamic optimization problem can be consulted elsewhere.30,37

dz(t) ) F(z(t), y(t), u(t), t, p) ) 0 dt

(30)

4. Results and Discussion

G(z(t), y(t), u(t), t, p) ) 0

(31)

In this paper, four main cases for the nitroxide mediated radical polymerization (NMRP) of styrene have been analyzed,

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Table 4. Weight Factors Used for Each Case of Study case

A

B

C

D

ω1 ω2 ω3 ω4 ω5 ω6 ω7

3 × 104 7 × 104 1 × 105 5 × 102 0 0 1 × 104

3 × 104 7 × 104 1 × 105 5 × 102 1 × 107 0 1 × 104

3 × 104 7 × 104 1 × 105 2 × 103 0 3 × 104 2 × 104

3 × 104 7 × 104 1 × 105 5 × 102 1 × 107 1 × 104 1 × 104

Table 5. NMRP Living Polymerization Kinetic Scheme case NOE,0 TPM,0 TMM,0 V0

A

B

TIDV X X X X

X

C X X X

D X X X X

case Fs QPM QMM Qnew

A

B

TDMV X X X X

X

C

D

X

X X X X

X X

with each one characterized by its feeding policy. It is important to mention that, for all the analyzed cases, the initial temperature of the reaction mixture has been fixed to 20 °C in order to also find the best operating policies for the start-up of the reactor under a nominal batch run time of 8 h. To evaluate the performance of the process for the control of polydispersity, the results obtained in each case of study will be compared with the ones reported in ref 22, under isothermal conditions at 130 °C for the so-called monomolecular process. Such a study has been selected for the mentioned comparison because the kinetic model used in this work is an adaptation of the one reported in ref 22. It is important to stress some facts about the results published by the authors. They worked with two different values for the nitroxyl ether initial concentration: 0.0087 and 0.087 mol/L. Employing the former, they could achieve number-average molecular weights as high as 4 × 104 g/gmol, but only polydispersities equal to or greater than 1.3 were recorded. With the latter, polydispersities close to 1.2 were obtained, but the highest number-average molecular weight reported was 1 × 104. The tradeoff between the control of the polydispersity and the molecular weight, pointed out in the formulation of the dynamic optimization problem section, now becomes evident. Nevertheless, the highest value reported in the mentioned reference, i.e., 4 × 104, will be taken as a target number-average molecular weight in order to evaluate the performance of the optimal operating policies. As the main criterion used for the evaluation of the livingness of the reaction, the highest value that the polydispersity index should achieve is 1.4, a limiting value for living radical polymerization standards. Moreover, it is important to stress that no end-point constraints have been imposed either for the molecular weight or for the polydispersity index (which is supposed to be minimized through its inclusion in the objective function), and thus, these targets are only used for comparison with the results obtained in this work. Although the decision variables considered in each case study will be explained in detail later, they are presented explicitly in Table 5 for a better understanding of the individual cases. The weight factors used in the cases of study are presented in Table 4. Case A: Pure Batch Operation. In this case, no reactants are continuously added to the reaction mixture; only the temperature profile is varied through the manipulation of cooling water and steam flow rates. The decision variables are the latter, as well as the volume of styrene/NOE loaded to the reactor (which is assumed to remain constant all along the process due to the lack of addition of reactants) and the initial nitroxyl ether concentration of this mixture. For the reaction mixture volume, both minimum and maximum values have been chosen as

bounds to maintain at least a minimum productivity; these values are 5 000 and 10 000 L, respectively. It was found that the optimal volume of styrene/NOE mixture that should be loaded into the reactor is 5 000 L, and the concentration of NOE in this mixture must be 1 × 10-3 mol/L. From Figure 2a, we see that, working under a purely batch operating scheme, the monomer conversion that can be achieved is poor for practical purposes, being 0.21, its maximal value reached at the end of the processing period. We also see in this figure that the monomer conversion achieves significant values just after 4.5-5 h of operation. At this point, a temperature of ∼88 °C is reached, where both thermal and nitroxyl ether initiation have just started. The increments of the monomer conversion can be attributed to the thermal autopolymerization reaction, which gives rise to the formation of monomeric radicals. However, the population of radicals in the reaction mixture increases due to the cleavage of the nitroxyl ether, which, besides yielding primary radicals, gives place to the formation of nitroxyl radicals. It could not be expected to have a proper control of the polymerization because it is necessary to let the propagating radicals react with the nitroxyl radicals to yield polymeric alkoxyamines, therefore establishing the characteristic equilibrium of the dormant-living exchange reaction. Thus, at this stage, it could be inferred that the propagation step due to the presence of the previously mentioned radicals becomes predominant, and no effective control of the polymerization is achieved yet. This situation changes after 7 h of operation, when a temperature of 110 °C is reached. When such a condition prevails, the equilibrium in the dormant-living exchange is achieved, and the control of the polymerization becomes noticeable. This fact can be corroborated by looking at Figure 2 parts b and c. Although some irregularities might be a consequence of numerical inaccuracy, shown in Figure 2b, we see that the molecular weight starts to increase almost linearly, just as the monomer conversion does in this period. From Figure 2c, it can be seen that, in fact, from this point on the polydispersity begins to decrease. Both facts provide evidence of the “livingness” of the polymerization at this stage. However, the control of the polymerization is reached just 1 h before the nominal batch period ends, and thus, the overall control of the reaction can be taken as unacceptable. Although the final molecular weight is relatively high (1.6 × 105 g/gmol), the polydispersity index is greater than 1.7, and the usage of the nitroxyl ether for the control of the reaction becomes unjustified. We could expect better results if longer batch periods were allowed. From Figure 3 parts a and b, we see that, although some noise is present in these plots, the main heat requirements are provided by the steam at the first 2 h of operation. Smaller amounts of heat are provided to the reaction mixture until 7 h are reached. The reduction of the steam flow rate occurs due to the propagation step; the reaction provides its own heat, and thus, external heating sources become less required. Beyond this period, significant amounts of renewal cooling water are required in order to counteract the heat released by the reaction. It is important to notice that, during the first 4.5 h, the heat released by the steam is used to raise the temperature of the reaction mixture from 20 °C up to conditions in which the formation of radicals becomes feasible. This large initial period, which could be seen as an induction stage, is responsible for the low performance of the process, and by shortening it, we would expect to achieve improved control of the polymerization. A potential solution could be to load smaller quantities of monomer/NOE mixture in the reactor. This reduction of the initial holdup would have, as a consequence, a faster heating


Figure 2. Operation policies under batch scheme: (a) monomer conversion, (b) Mn, (c) polydispersity index, (d) reactor temperature, (e) cooling jacket inlet temperature, and (f) reactor volume.

with the same amounts of steam, and thus, the reaction could begin faster. In fact, it can be observed that the optimal volume for the reactor corresponds to the specified lower bound for this parameter; thus, it could be inferred that a lower holdup can lead to a better performance of the process. Nevertheless, the tradeoff between control of the polymerization and productivity of the process would become evident. Although the results of this section could be questioned, they represent the simulation of a real situation where the polymerization process might not be economically feasible beyond a certain operating time. Case B: Addition of Pure Monomer. In this case, all the decision variables mentioned in Case A are included. However, the option of feeding a stream of pure monomer is also

addressed. The problem is formulated to obtain a dynamic profile of the monomer feed streamflow rate, as well as the temperature of this stream, and the initial volume of styrene/ NOE that should be loaded into the reactor. The monomer concentration of the entering stream has been set to 8.7 mol/L (which is the styrene molar density at ambient temperature). By solving the optimization problem, it was found that the monomer stream should be fed at a temperature of 130 °C; the reactor should be loaded with 500 L, and initially the reaction mixture should have a nitroxyl ether concentration of 0.087 mol/L. As can be seen from Figure 4a, under this semibatch operating scheme, we can achieve conversions ∼0.9, enhancing the monomer conversion over the specified batch time. Also, it is possible to achieve better polymer properties; the molecular

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Figure 3. Operation policies under batch scheme: (a) renewal cooling water flowrate and (b) steam flowrate.

weight achieved at the end of the processing period is ∼2 × 104 (see Figure 4b). Moreover, as can be seen from Figure 4c, a less polydisperse polymer is obtained, featuring a polydispersity index lower than 1.14 at the end of the operation period. Although it could be considered that it is absolutely advantageous to operate under this regime, Figure 4f shows that there is a dramatic decrease of the reactor productivity (when comparing this case with the purely batch operation mode). At the end of the operating period, the reactor volume is only 1150 L. The small increment from the original volume to the final one is congruent with the low pure-monomer flow rates that can be observed from Figure 5b. This fact could be adverse for the large-scale production of polystyrene. Figure 4d shows that the heating of the reaction mixture is much faster than in the pure batch case (for example, in the current case, temperatures ∼80 °C are reached in 4 h to reach such condition). This happens obviously due to the much lower holdup at the beginning and during the semibatch operation, which helps not only to ease the preheating of the system but also to initiate the propagation by autopolymerization in a shorter period. Although feeding the monomer stream increases the reactor holdup, thus bringing heat-transfer difficulties, it must be kept in mind that this stream is supposed to enter the reactor at a temperature of 130 °C, and due to the fact that it gets directly in contact with the reaction mixture, it also helps to increase the reactor temperature. It is also interesting to take a simultaneous look at Figure 4d and Figure 5 parts a-c. We see that the steam provides the necessary heat for the temperature increase for the first hour of operation; the maximum value of the pure monomer flow rate is reached at such time. After such period, there is a change in the slope in Figure 4d, which indicates that the reactor temperature increases at a lower rate. The coincidence of this change in the temperature slope and the decrease in the monomer flow rate indicates that, in fact, monomer flow rate has a deep impact on the polymerization heat of reaction. After 2 h of operation, the reaction temperature reaches 110 °C. As mentioned before, at this temperature, the cleavage of the nitroxyl ether becomes important, and thus the propagation rate increases from this point on, in which the cooling water flow rate begins to increase, thus helping to control the temperature increments that arise as a consequence of the heat generated by the propagation reaction step. However, the cooling water flow rate does not avoid that the reactor hits its temperature upper bound, that is, 160 °C. This temperature is reached when the cooling water flow is switched off, and the steam is allowed to enter to the heating/cooling cycle. Thus, the temperature rise

could be attributed to this fact, rather than to the heat generated by the propagation reaction step. Although small quantities of cooling water are supplied again past 5.4 h (when the reactor temperature has reached its maximum and is about to fall), the temperature decrements can be attributed to the low monomer concentration. This fact can be confirmed by looking at Figure 4a, where we can see that the conversion after 5.4 h is ∼0.85 and remains almost constant. Under this operating scheme, an important fact must be observed. Although the maximum reaction mixture volume that the equipment can contain has been set to 10 000 L, the volume of the cooling jacket remains constant and has a value of 2 000 L. However, at the end of the batch period, the actual volume of the reaction mixture about a half of that of the cooling jacket. Now, as will be shown in cases C and D, under this operating scheme, the maximum molecular weight is achieved at the end of the nominal batch period, under proper conditions of polydispersity index control. This could be evidence of the fact that the cooling jacket must have a relatively high volume in comparison with the one of the reaction mixture in order to achieve higher molecular weights, keeping the living characteristics of the polymerization. Although it has been mentioned that the productivity using this operating scheme is quite poor due to the small reactor volume, the product quality is considerably better than the one obtained under the purely batch operating mode. Thus, the present semibatch operating scheme could be considered advantageous if the monodisperse polystyrene is to be taken as a specialty product. Case C: Addition of a Monomer/Nitroxyl Ether Mixture. In this case, instead of feeding a stream of pure monomer, a mixture of styrene and nitroxyl ether is used. The main decision variables are the renewal cooling water and steam flow rate profiles, the monomer/NOE mixture flow rate profiles, the temperature at which this stream should be fed, the volume of monomer/NOE mixture that should be initially loaded into the reactor, and the initial NOE concentration that this stream should have. To keep the problem as simple as possible, the concentrations of NOE in both the initially loaded reaction mixture and the styrene/NOE stream have been assumed to be equal. It was found that the optimal initial volume of styrene/NOE that should be loaded is 500 L; this mixture should have a NOE concentration of 0.087 mol/L. The monomer/NOE stream must enter into the reactor at a temperature of 108 °C, and due to the problem specifications, its nitroxyl ether concentration should be equal to that of the reaction mixture before the batch period begins. As can be seen from Figure 6a, at the end of the


Figure 4. Operation policies under semibatch scheme with pure monomer addition: (a) monomer conversion, (b) Mn, (c) polydispersity index, (d) reactor temperature, (e) cooling jacket inlet temperature, and (f) reactor volume.

batch period, a monomer conversion of 0.71 is achieved. Although this value is slightly lower than the one obtained in Case B, it is satisfactory for practical purposes. Interestingly, under this operating scheme, the preheating period is shorter, compared to case B (∼0.5 h, see Figure 6d), and a temperature of 125 °C is reached in such a time. Both the steam flow rate entering the heating/cooling cycle and the relatively hot styrene/ NOE mixture flow rate fed into the reactor help in shortening the initial heating period of the process. An interesting analysis can be made if the reactor temperature plot (Figure 6b) is partitioned in three main sections, according to the elapsed time. The first section is the already mentioned preheating stage, which begins at t ) 0 and ends at t ) 0.5 h. Taking a look at Figure 7 parts b and c, it can be seen that both the steam and

the monomer/NOE mixture flow rates contribute to such a temperature increment. The second stage goes from 0.5 to 3 h. A monotonically decreasing temperature tendency can be observed in this section. Looking at Figure 7b, it can be seen that the monomer/NOE stream is fed into the reactor, but there is no steam supply for the heating/cooling cycle. The reactor temperature, however, reaches a minimum temperature of ∼110 °C, getting closer to that of the entering stream. Thus, it could be thought that the feed stream acts as a sort of coolant, if only its thermal effects are considered. However, the reaction mixture does not match the entering stream temperature because of the heat generated by the propagation reaction step. Finally, the third stage of the temperature behavior can be identified, going from 3 to 8 h. In this final period, the temperature continuously

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Figure 5. Operation policies under semibatch scheme with pure monomer addition: (a) renewal cooling water flow rate, (b) pure monomer flow rate, and (c) steam flow rate.

increases. It can be seen that no steam or styrene/NOE mixture enters the reactor any more. Thus, this temperature rise must be exclusively due to the exothermicity of the reaction. From Figure 7a, we see that with exception of the process startup and ending periods, the cooling water flow rate remains almost constant, with a value of 3.3 L/s, helping to avoid reaction thermal runaway conditions, which could be specially dangerous in the third stage of the reactor temperature behavior where the temperature increases are only due to the heat of reaction. From Figure 6c, we see that, after 0.3 h from the start-up, the polydispersity index continuously decreases. Taking a look at Figure 6d, at such time, the reactor temperature reaches a value of ∼100-110 °C. Under these conditions, the nitroxyl ether is likely to undergo cleavage, yielding nitroxyl radicals and thus allowing control of the reaction and gradually reducing the polydispersity index. A value of ∼1.17 is achieved for this variable, thus showing that this operating scheme allows the reaction to have a true living-like behavior. The productivity under the present operating conditions is satisfactory when compared against the purely batch operating mode, because a final reactor volume of ∼8500 L is obtained at the end of the batch period. However, the molecular weight still remains low with respect to the target. Case D: Addition of Pure and Mixed Monomer. In this case, the option of adding both a pure monomer and a monomer/ NOE stream is addressed. As can be anticipated, the decision variables are the combination of the ones of the individual cases. Thus, summarizing, the cooling water, steam, pure monomer, and monomer/NOE mixture flow rates, as well as the initial

volume of styrene/NOE mixture loaded into the reactor, the temperatures of the styrene and styrene/NOE streams (which are not assumed to be equal), and the initial nitroxyl ether concentrations of the reaction mixture and the mixed monomer stream are the decision variables. According to the optimal solution of the problem, the reactor should be initially loaded with 500 L of monomer/NOE mixture, with a concentration of NOE of 0.087 mol/L. The pure monomer stream should be fed at 130 oC, and the monomer/NOE stream should be fed at 42.2 °C. Looking at Figure 8a, we see that the induction period is quite short (

Optimal Operating Policies for the Nitroxide-Mediated Radical

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