A Composition-Temperature Control Strategy for Semibatch Emulsion Copolymer Reactors

Copyright © IFAC Dynamics and Control of Process Svstcms. Corfu. Greece, 1998

A COMPOSmON-TEMPERATURE CONTROL STRATEGY FOR SEMIBATCH EMUlSION COPOLYMER REACTORS Fernando Zaldo', and JesUs A1varez 2

Universidad Autonoma Metropolitana-Iztapalapa Depto. de Ingenieria de Procesos e Hidraulica Apdo. 55534, 09340 Mexico D.F

MExIco [email protected]

Abstract: Emulsion copolymerization in semibatch reactors is an industrial process where important commodity and engineering plastics are manufactured, and the design and control of the semibatch operation is a relevant problem which has not yet been adequately understood and solved. Acknowledging that the semibatch copolymer reactor means a complex nonlinear plant that has a strong and asymmetric input-output coupling, and that operates in a sustained transient, in this work a nonlinear control framework is used to address the problem of obtaining a homogeneous emulsion copolymer, at constant temperature; and in a minimum semibatch period. In particular, two basic issues are treated: the existence, and the construction of a robust nominal operation. The proposed methodology is illustrated with the emulsion copolymerization of vinyl acetate (V AM) and n-butyl acrylate (BA). Copyright © 1998 IFAC Keywords: Process control, nonlinear control systems, optimal control.

1.

INTRODUCTION

condition, which consists in feeding a monomer mixture with a composition which is the same as the one of the product copolymer, at a sufficiently slow dosage rate such that the comonomers polymerize as they are added into the reactor. The main drawback of this copolymerization strategy is the excessive reaction time. Accordingly, the research efforts have been aimed to obtain homogeneous copolymers at faster polymerization rates. At the cost of some copolymer heterogeneity, Hanna (1957) proposed to initially load the less reactive monomer and to feed the more reactive one. This operation mode has been implemented and verified experimentally (Guyot et al., 1981; Arzamendi and Asua, 1989). Closed-loop realizations of this operation strategy have been proposed and implemented using conventional-type control tools (Chiang, 1977; Guyot et al., 1981); as well as model-based non linear-type operation-control strategies: using an extended Kalman filter approach (Dimitratos et al., 1989a, b, 1991), a non linear

Emulsion polymerization is one of the most important industrial processes for the production of commodity and engineering plastics. In particular, the copolymerization in a semibatch process offers the advantage of modifying homopolymer properties into tailor-made materials with higher aggregated value. These reactors have complex and nonlinear dynamics with strong and asymmetric input-output coupling. The simplest, oldest, and still widely used industrial method to produce a homogeneous polymer is an approximation to the so-called starved operation 'Permanent address: Centro de InvestigaciOn en Polimcros (COMEX), M. Acbar Lobaton 2, 55855 Tepexpan, EdoMex., MEXICO. [email protected], 2To whom correspondence must be addressed.

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adaptive-type scheme (Urretabizkaia et al. , 1994), and an optimization-based design with conversion instead of time as the evolution variable (Canu et al.. 1994). These studies demonstrate the importance and benefits of using model-based control and exhibit the lack of a systematic approach to the operation-control problem. An additional drawback of the existing copolymer composition control studies is that they have overlooked the thermal dynamics, which are present in industrial reactors (Arzamendi et al., 1991).

M\·

= q\ (t)p\

M 2• =q2(t)P2

T = _ I_

[q\ (t)p\ (Cp\I; -CpT)

pVeCp

+ q2 (t)P2 (C p2 T2 -

CpT)+ (- Mf p\ )Rp"

+ (-Mf p2 )RPT2 +q/(t)Pw(CpwT/ -CpT) +UA/TJ -T)]

re =q\(t)+q2(t)+q/(t)-S rw =q/(t)

In the present work, a nonlinear control framework is used to rigorously address the operation policy design of a semibatch emulsion copolyrnerization reactor that must produce a homogeneous copolymer, nm at constant temperature, and be minimum-time. Following A1varez et al. 's (1995) equipmentoperation-control design strategy, two basic issues, which underlie the robust solvability of any open or closed-loop reactor operation-control strategy, are addressed: the existence, and the construction of a robust nominal operation; in the understanding that the associated dynamic invertibility (Hirschom, 1979) property constitutes a necessary existence condition and a construction pre-requisite to design in the future the closed-loop feedback version, either conventional (padilla and Alvarez, 1997) or advanced (Alvarez, 1996), of the proposed open-loop control design operation. The proposed methodology is illustrated with the emulsion copolymerization of vinyl acetate (VAM) and n-butyl acrylate (BA).

This model is a standard one, and its detailed description can be found elsewhere (Hamielec et al., 1987). The states of the reactor are: M j (mass of unreacted monomer i; i=I,2), I (initiator molar concentration), Vw (volume of water), Ve (volume of emulsion), M j • (mass of added monomer i; i=I,2), N (number of latex particles per unit volume of water phase), and T (reactor temperature). The control inputs are: q;(t) (feedrate of monomer i; i=I,2) and Tc (jacket temperature or equivalently, heat exchange rate). To build a homogeneous copolymer at maximum through output rate, the following outputs must be kept constant during the entire semibatch operation: Y I = Rp/(Rpl + Rp2) (instantaneous copolymer composition), M (mass of monomer droplets, if they exist), and T (reactor temperature). In compact notation the reactor can be written as x = f(x, U, r), y =h(x, r)

(I)

XI = MJ, X2= M2, X3 = I, X. = N, Xs = M I·, ~ = M2·, X7 = T , X8 = Ve. ~ = Vw UI = qJ, U2 = q2, U3 = Tc YI = Y I , Y2 = M, Y3 = T r: model parameters

2. THE COPOLYMER REACTOR AND ITS CONTROL PROBLEM 2. J Copolymer reactor

In an emulsion copolymer reactor, two monomers are converted into polymer, producing a significant amount of heat (Penlidis et al., 1985; Harnielec et al., 1987). Water, emulsifying agent, and initiator are loaded into the reactor. First, heat is added to start the reaction. Then, heat must be removed to prevent a reactor thermal run-away. The monomer feedrates and the heat exchange policy must be such that the reaction takes place as fast as possible, having adequate safety margins, and yielding a product within certain quality specifications. The reactor is described by the following set of nine non linear ordinary differential equations:

2.2 Control problem The control problem consists in determining the two monomer feedrates, ql(t) and q2(t), as well as the coolant temperature policy T c(t) (or equivalently, the heat exchange policy)

such that the reactor yields a homogeneous copolymer, at a constant temperature, in a minimum time. The idea is to use a nonlinear dynamics and control framework to rigorously cast and solve the operation policy design of a semibatch emulsion copolymerization reactor. The problem of designing the reactor operation will be treated as the existence and construction of a robustly stable motion (see Alvarez et al., 1995; and references therein) that satisfies the dynamic inverse for the semibatch reactor process. In principle, the dynamic inverse can

220

be used to design the nominal operation policy and to design a feedback control scheme either conventional (Padilla and Alvarez, 1997) or advanced (Alvarez, \996).

iii) The map cl> is Rx-invertible,

3. DYNAMIC INVERSION In this section we recall the nonlinear control tools that will be used to address our reactor control problem: a robust version of the nominal inversion of a nonlinear dynamic plant (Hirschom, 1979; Isidori, 1995). An application oriented approach of the dynamic inverse can be seen in Alvarez et al. 's (1995) treatment of a homopolymer reactor.

iv) The map cp is Ru-invertible, kl km cp(x, u, r) = [L f hI' .. . , L f h m ]'

Definition 1. Let us consider a nonlinear plant of form (1) with n states, m control inputs and m control outputs. The plant (1) is said to be RE(robustly exponentially)-dynamically invertible for a given (output trajectory-initial condition) pair y(t)-x if o there is a unique and RE-stable (Alvarez, 1997) input trajectory u(t) which, applied to the plant, produces the prescribed output yet) . •

v) x(t) is a unique and RE-stable motion ofthe (n-K)dimensional I(inverse)-dynamics: (2a)

XI(t)= {x E Xwl ~(x, r) = y(t)} eX

(2b)

Where (dim XI=n-K) _ (Kl-l). . (Km-I) , y(t)-[Yl' .. . 'YI , .. . 'Ym' · ··'Ym ] (t)

The map q,(x, U. r) is x-invertible if there is an inverse map cl>-I such that cI>-1 [(x, u. r), u, r] = X,

. -1 x = f{x, cp [x,v(t),r],r}, xEXI(t)

_ (K\) (Km) , v(t)-[Yl '···'Ym ](t) . • (x, U. r)

E

S

I

Corollary 1. Let the plant (I) meet the RE-stable dynamic invertibility conditions of Theorem 1. Then, the nominal RE-stable input trajectory u(t) = eu[t, to'

are L(Lipschitz)-continuous in S. Let L if a denote

x , yet), r] is given by the output of the following

the recursive directional derivative of the timevarying scalar field a along the time-varying vector field f. This is (a = aaJax, at=aaJat): x

input-state-output system:

The map q,(x, u, r) is said to be R(robustly)x-

invertible in S if it is x-invertible in S, and cl> and

f

o

.

(n-K)-dimensional

1

x = f{x, cp- [x, vet), r], r}, x(O) = Xo u=cp-I[x,v(t),r]

E

dynamic

XI (t)

(3a) (3b)

where the manifold X\[y(t)], the input vet) and the input y(t) are given in eq. (2).•

The following theorem is a straightforward robust tracking version (see Alvarez et al., 1994) of the nominal solvability conditions (Alvarez, 1996) of the regulation state-feedback control problem.

4. DESIGN OF THE REACTOR NOMINAL OPERATION

Theorem 1. The nonlinear plant (1) is REdynamically invertible if there are m strictly positive integers (invertibility indices, or output relative degrees), KI' ... , Km' and n-K scalar maps cl>K+l(X,

According to the control systems theory (Morari and Zafiriou, 1989), the dynamic inverse of a plant is the limit of what can be accomplished with a perfect controller. In this section, the RE-dynamic invertibility of the semibatch copolymer reactor is addressed to obtain the nominal operation.

r), ... , cl>n(x, r), such that, in a given state-input set S=XxUxR: i) The relative degrees meet the following condition

4.1 The choice ofoutputs To obtain a homogeneous copolymer with a prespecified composition Y i (i=l,2), the rate quotient Y I = Rp1 / (Rpl + Rp2) must be kept constant throughout

ii) The map ~ is independent of u,

221

,

the course of the polymerization. According to Mayo's Equation (Odian, 1991) and the thermodynamic equilibriwn, a homogeneous copolymer implies that the monomer ratios in the particle and droplet phases are constant. Specifically, ~ = M Ip / M2p = M Id / M2d , ~ = { (K - 1) + [ (K - 1 + 4rlr2K ]1/2 }l2rl where rl and r2, are the monomer reactivity ratios, Y I is the mole fraction of monomer 1 in the copolymer, and K is the constant quotient of monomer polymerization rates: K = Rpl / Rn = Y I / (1 - YI)· To make simpler the mathematical manipulations associated to the dynamic inversion of the reactor, let us replace the original composition output YI = Y I by the equivalent output YI = Mid - ~ M2d = c, which is a linear version of the preceding equation for the composition quotient ~. As stated in the formulation of the control problem, the second output Y2 = Mid + M2d = M if t < t., is chosen to obtain the maxirnwn polymerization rate, where the time tl (to be determined) corresponds to the end of the period of operation with maximwn particle swelling. The temperature is an output

q,n(x, r) leads

=

to

[I, N, M I·, M 2• , Ve> V w the

,

Rx-invertible

;(x,r)=[hl,~,h3';II'] (x,r)

i

]

map

implying

that

conditions (i), (ii) and (iii) of Theorem 1 are met. On the other hand, we have that the map

0. Since det A> 0, and A and b are L-continuous, the map

.

, (4a)

x 1= fj(x.,r), XI =[1, N, M I• ,M2• ,Ve , V w ],

-1-1 -1fI(xl' r) = f.{xI,h (Y ,XI)'

y = h(xc' XI) f. = [f3,

,

4, f5,f6'£8,f9 ] (xc' XI)' Xc = [Mp ~, T]

with nwnerical integration and sensitivity analysis, it was found that the I-dynamics motion xit) was REstable, implying that condition (v) of Theorem 1 is

to be controlled in order to meet product properties related to the molecular architecture, as well as to ensure a safe operation.

met. Summarizing, there is a RE-stable dynamic inverse for the semibatch copolymer reactor,

For the design of the nominal semibatch reactor operation, two periods must be considered:

implying the existence of a RE-stable nominal operation for period I.

Period I (0 < t < t 1 ): from the beginning of the In period n, the reactor is a 2-input 2-output control system, and therefore the choice

operation at t = 0, up to the time tl where the prespecified load of one of the monomers (say the less reactive) has entered the reactor. In this period, the reactor will have three inputs and three outputs:

, q,n(x, r) = [M., I, N, M I·, M 2• , Ye, Vw

]

, leads to the Rx-invertible map cp(x, r) = [h, ~I] implying that conditions (i), (ii) and (iii) of Theorem 1 are met. On the other hand, we have that the map

0, and A and b are L-continuous, the map

Period II ( t 1S t < (0): from time t I up to the end of the reaction. In practice, this period is stopped at some sufficiently large finite time t > tl such that an f arbitrarily small amount of unreacted monomers is left. In this period, only the most reactive monomer is fed, and therefore, the reactor has two inputs and Y = [c, T]' two outputs: u = [q2' T

.

cl',

,

X I = fix.,r), XI = [M.,I,N,M I·,M2·,Ve ,Vw]

For robustness purposes the output relative degrees are required to be the unity (Alvarez et al., 1995). This is, lC = 1, i = I,m (m = 3 ift:::;t., or 2 ift> tl). i

,

where

4.2 RE-stable dynamic inversion

, f. (xl' Xc ,r) = [fh f3,

In period I, the reactor is a 3-input 3-output control system, and therefore, ~ = h in Theorem 1 and the choice

4, f5, 4, f8, f9] (xc' XI)

Xc = [~ , T]

222

(4b)

with numerical integration and sensitivity analysis, it was found that the I-dynamics motion XI(t) was REstable, implying that condition (v) of Theorem I is met. Summarizing, there is a RE-stable dynam ic inverse for the semibatch copolymer reactor, implying the existence of a RE-stable nominal operation for period H. According to the preceding section and to Corollary I , the nominal RE-stable input trajectory

and there is maximum swelling in the particle phase, meaning that the increase in the amount of unreacted VAM is due to the increase of the volume of the particle phase. In period II (once the VAM has been fed and only BA is being fed, as it can be seen in Fig. I d), the feedrate of BA gradually decreases until it vanishes asymptotically, following the rate of consumption of the remaining VAM swollen in the particle phase.

5. CONCLUSIONS is given by the output of the following variable-

The problem of obtaining a homogeneous emulsion copolymer in a minimum-time semibatch reactor at constant temperature has been formulated and solved within a nonlinear dynamical systems framework. Using a dynamic inverse approach, the existence of a robust nominal operation was established for any copolymer system, and a systematic procedure to design the nominal operation was given. The present work yields the nominal open-loop strategy, as well as the basic solvability conditions and constructive steps to design in the future the feedback control version of the proposed operation design. The approach can be extended to include the regulation of the molecular weight distribution .

structure input-state-output dynamic system:

(5a)

(5b) dim X I

= 6 if t < t I '

or 7

if

t ~ tI

Thus, given the prescribed output y and the nominal initial condition xo , the (numeric) integration of equation (5) yields the nominal RE-stable state evolution

x{t)

=

[x'I (t), h-\y , XI )]' and the

corresponding nominal RE-stable input

u{t) .

BIBLIOGRAPHY Alvarez, 1., R. Suarez and A. Saochez (1994). Semiglobal nonlinear control based on complete input-output linearization and its application to the start-up of a continuous polymerization reactor. Chem. Eng. Sci., 49 (21), 3617. Alvarez, 1., F. Zaldo and S. Padilla (1995). Integration of process and control designs by nonlinear geometric methods. Proc. DYCORD + '95, 363-368. Alvarez, 1. (1996). Output-feedback control of nonlinear plants. AIChE J., 42 (9), 2540-2554. Alvarez, 1. (1997). A robust estimator design for nonlinear plants. Proc. Am. Contr. Con/., Vol 4, 3058-3062. Arzamendi, G. and 1.M. Asua (1989). Monomer addition policies for copolymer composition control in semicontinuous emulsion copolymerization. J. Appl Polym. Sci. , 38, 20192036. Arzamendi, G. and J.M. Asua (1991). Copolymer composition control of emulsion copolymers in reactors with limited capacity for heat removal. Ind. Eng. Chem. Res., 30, 1342-1350. Canu, P., S. Canegallo, M. Morbidelli and G. Storti in emulsion (1994). Composition control copolymerization. 1. Optimal monomer feed policies. J. of Appl. Polym. Science, Vol. 54, 18991917. Chiang, T.C., C. Graillat, 1. Guillot, Y.T. Pham and A. Guyot (1977). Copolymerisation radicalaire du methacrylate de methyle et du chlorure de

4.3 Application to a VAM-BA copolymer Let us consider a 3000 Gal. industrial reactor where -10 Tons. of emulsion copolymer must be produced. Surfactant, initiator, and water are initially loaded. The polymerization should be carried out at 60°C. The goal is to produce a 82% mol vinyl acetate - 18% mol butyl acrylate homogeneous copolymer. The nominal operation of the semibatch reactor was obtained from the integration of eq. (5), and the results are presented in Figure 1. Period I lasts up to tl ~ 40 min. (see Fig. Id), the entire semibatch (i.e., period I and II) takes tf ~ 330 minutes to reach a 99.5% global conversion (see Fig. la), and there is no droplet phase. As expected, from the dynamic inverse approach, the instantaneous composition (see Fig. le), and the temperature (see Fig. lc) are kept constant at their prescribed values, and the reaction rate is set at its maximum value. The corresponding monomer feedrate and jacket temperature timeevolutions are shown in Figures Id and lc, respectively. In period I, the ratio ql /q2 is constant; and q 1 and q2 exhibit an exponential-type response with a settling time of about 20 minutes. In the last -20 min. of period I, qI and q2 reach constant values. According to Figure I b, the VAM monomer is fed in period I such that there is no droplet phase

223

vinylidene avec an sans derive de composition. J. Polym. Sci. Polym. Chem., 15,2961-2970. Dimitratos, J., C. Georgakis, M.S. EI-Aasser and A. Klein (1989a). Control of product composition in emulsion copolymerization. In: Polymer Reaction Engineering (K.H. Reichert and W. Geiseler, Eds.). pp. 33-42. Dimitratos, J., C. Georgakis, M.S. EI-Aasser and A. Klein (1989b). Dynamic modeling and state estimation for an emulsion copolymerization reactor. Comp. Chem. Eng., 13,21-33. Dimitratos, J., C. Georgakis, M.S. EI-Aasser and A, Klein (1991). An experimental study of adaptive Kalman filtering in emulsion copolymerization. Chem. Eng. Sei., 46 (12), 3203-3218. Guyot, A., J. Guillot, C. Pichot and L. Rios-Guerrero (1981). New design for producing constant in emulsion composition copolymers polymerization. In: Emulsion Polymers and Emulsion Polymerization (Eds. D.R Basset, and A. E. Hamielec), ACS Symposium Series 165, pp. 415-436. Hamielec, A.E., J.F. MacGregor and A. Penlidis (1987). Multicomponent free radical polymerization in batch, semibatch and continuous reactors. Malcromol. Chem., Macromol. Symp. 10111, 521-570. Hanna, R.J. (1957). Synthesis of chemically uniform copolymers. Ind. Eng. Chem., 49, 208-209. Hirschorn, RM. (1979). Invertibility of multivariable non Iin ear control systems. IEEE Trans. Automat. Contr., AC-24, 855-865. Isidori, A. (1995). Nonlinear Control Systems. Springer Verlag, N.Y. Morari, M. and E. Zafiriou (1989). Robust Process Control. Prentice Hall, New Jersey. Odian, G. (1991), Principles of Polymerization, Wiley-Interscience, N.Y. Padilla, S. and J. Alvarez (1997). Control of continuous copolymerization reactors. AIChE J., 43 (2),448-463 . Penlidis, A., J.F. MacGregor and A.E. Hamielec (1985). A theoretical and experimental investigation of the batch emulsion polymerization of vinyl acetate. Polym. Proc. Eng., 3 (3), 185-218. Urretabizkaia, A., J.R Leiza and J.M. Asua (1994). On-line terpolymer comPOSItion control in semicontinuous emulsion polymerization. AlChE J.,40,1850-1864.

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