Optimal Grade Transitions in the High-Impact Polystyrene

Ind. Eng. Chem. Res. 2006, 45, 6175-6189

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PROCESS DESIGN AND CONTROL Optimal Grade Transitions in the High-Impact Polystyrene Polymerization Process Antonio Flores-Tlacuahuac* and Lorenz T. Biegler Department of Chemical Engineering, Carnegie-Mellon UniVersity, 5000 Forbes AVenue, Pittsburgh, PennsylVania 15213

Enrique Saldı´var-Guerra

Downloaded by TEXAS A&M INTL UNIV on August 24, 2015 | http://pubs.acs.org Publication Date (Web): August 3, 2006 | doi: 10.1021/ie051140q

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

We address the dynamic optimal grade transition problem in large scale high-impact polystyrene (HIPS) plants. The problem of computing the dynamic optimal transition trajectories is cast as a nonlinear programming problem whose solution provides the optimal time dependent values of the manipulated variables. The optimal transition trajectories are computed off-line requiring a feedback control system for their closed-loop tracking and to take care of modeling errors. The optimal dynamic transition trajectories were computed between unstable open-loop operating points. When nonlinear bifurcation diagrams were used, it was found that the simultaneous manipulation of both monomer and initiator flow rates leads to a decrease in monomer conversion to achieve higher monomer conversions. To get smoother grade transition trajectories, it was also found that grade trajectories performed in a step by step manner are advantageous over transitions done with a single overall step. 1. Introduction To meet polymer market demands, chemical plants should be able to manufacture required polymers featuring desired end use properties. This means that a given plant that manufactures a certain type of polymer (for instance, for food processing) might also produce polymers with totally different end use properties (for instance, for car components). In both cases, the manufactured polymers are often the same (from a chemical composition point of view). However, it is clear that there is a remarkable difference between both polymers, because they might feature different mechanical and impact properties. Grades are polymer products that are made from the same polymer but that exhibit different end use properties. Among such properties, the following ones can be mentioned: brightness, color, texture, mechanical strength, molecular weight, and so forth. Therefore, different grades are manufactured by running a given process under different processing (temperatures, pressures, flow rates) conditions. In this work we define as grade transitions the set of process operations required to change from producing an initial grade to producing a final desired grade. In industrial practice, grade transition operations are often based on heuristic rules derived from the knowledge of how the process operates, and there are a few commercial products for implementing closed-loop grade transition policies.1 Grade transition operation is a dynamic operation, that is, during process transition all properties associated with the polymer change with respect to time. From an operational point of view, the transition policy (i.e., the way the transition is * To whom correspondence should be addressed. On leave from Universidad Iberoamericana. Address: Departamento de Ingenieria y Ciencias Quı´micas, Universidad Iberoamericana, Prol. Paseo de la Reforma 880, 01280 Me´xico DF, Me´xico. E-mail: antonio.flores@ uia.mx. Phone/fax: +52(55) 59504074. http://200.13.98.241/∼antonio.

carried out) must satisfy the following requisites. (1) Safety: Industrially one of the most widely employed ways of carrying out grade transition consists of using open-loop step changes in the manipulated variables. This means that the value of the manipulated variable is changed to the maximum or minimum values that such a variable can take. Because most polymerization reactions are highly exothermic, there is always a risk of thermal runaway. (2) Minimization of the out of specification product: During grade transition certain amounts of undesired out of specification product are manufactured. Normally, this product cannot be placed on the market. Hence, the grade transition operation should be done as quickly as possible, leading to minimization of off-spec product. However, due to residence time constraints, grade transition operations cannot be done in an arbitrarily short time. Thus, grade transition policies must take into account both points discussed above. It seems obvious to us that any grade transition optimization is well worth it if it results in minimization of an off-spec lesservalued product, as in this case. However, we can provide some estimation of the savings. Assuming a medium-size high-impact polystyrene (HIPS) plant, with a capacity of 100 000 ton/year and a process with a residence time of 4 h (typical for HIPS continuous processes), the material produced during a transition performed with simple (nonoptimized) feed-condition step changes corresponds conservatively to at least 3 residence times (12 h) or roughly 137 ton/transition. At a price of about 0.80 USD/lb, this amounts to about 240 000 USD. Not necessarily all of this will be lost because the off-spec material can probably be diluted or sold at a lower price and an optimal transition will not eliminate all the off-spec material; however, considering that many transitions are common in a year (perhaps around 20), it is no exaggeration to estimate that the losses for nonoptimized transitions are on the order of several hundred thousand dollars per year in a typical HIPS plant.

10.1021/ie051140q CCC: $33.50 © 2006 American Chemical Society Published on Web 08/03/2006


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Because grade transitions often deal with proprietary information, there are few industrial studies in this area. McAuley and McGregor2 analyzed the grade transition problem in polymerization reactors. The authors used an open-loop optimal control approach. To analyze the quality of the end product, they used commonly applied and easily obtained density and melt index measurements. The authors attempted to quantify the effect of modeling errors on the computed trajectories and showed that the optimal transition trajectories exhibited a modeling error dependence. Such a dependence might be attenuated by using a feedback control strategy, as in ref 3. Working with the polyolefin production process, Debling et al.4 analyzed the open-loop dynamic process response using different transition policies. Some of the operating policies were just heuristic rules extracted from process experience, while some others were proposed by the authors. All this work was made in an open-loop manner; this means that the difficulty for the closed-loop implementation of the transition trajectories was not addressed. From the work reported by Debling et al.4 it is difficult to quantify how close the heuristic rules are to the optimal operating region. Moreover, for different polymerization systems the derivation of heuristic rules might not be simple. Sirohi and Choi5 computed off-line optimal grade and startup transitions and tracked them using an on-line optimization approach based on a model predictive control (MPC) scheme. The authors tested several heuristic transition policies in a polyolefin reactor. Takeda and Ray6 formulated the problem of grade transition trajectories determination as an open-loop dynamic optimization problem. To test their grade transition strategy the authors used a polyolefin polymerization reactor. The authors analyzed two kinds of grade transitions: (1) the easy transition (where few manipulated variables should change from their nominal values) and (2) the difficult one (where all manipulated variables should change from their nominal values). In the easy case, the authors concluded that the observed transition using simple manipulated variable step changes was similar to that obtained by over- or undershooting the same set of manipulated variables. In this case, the optimal transition profiles are not much better than with the over- or undershooting techniques. The situation was different when the difficult case was analyzed, because the authors found that both transition times and out of specification product were minimized when optimal transition profiles were used. In addition, sometimes the optimal transition policies might not be intuitive, making it difficult to obtain them from process experience. Hillestad and Andersen7 proposed an on-line optimization scheme for polypropylene grade transition. The model used for the on-line optimization was a first principles on-line model, updated to predict the reactor behavior; some variables were estimated on-line using an ad hoc estimation technique. Although no details are given on how the tracking of the control signals was implemented, the authors mention that their online optimization grade transition control approach was implemented on a real polypropylene reactor. Wang et al.8,9 proposed tracking a trajectory, determined by off-line dynamic optimization, and a closed-loop MPC approach for polyolefin grade transition operations. In their approach, the optimal profile of manipulated and controlled variables is computed off-line by solving a dynamic optimization problem. To implement the sequence of optimal input/output signals, the controlled variables form the set point signals to be tracked, while the manipulated variables are employed in a feedforward form. A linear MPC strategy, whose model representation is obtained by linearization of the nonlinear first principles model along the

optimal solution trajectory, is used to take care of disturbances and modeling errors. The resulting MPC is cast in terms of sequential quadratic programming (SQP). The approach was applied to a mathematical model of an olefin polymerization reactor. Seki et al.10 proposed an on-line grade transition and MPC feedback strategy. In their approach, at each sampling time, the authors computed the optimal tracking trajectory by solving a linear quadratic problem. Then a linear feedback MPC was employed for the tracking of the optimal reference input/output signals. An ad hoc estimation scheme was also used. The online optimal grade transition and feedback MPC scheme was applied on two polymerization systems. Mahadevan et al.11 analyzed the interplay between optimal scheduling and grade transitions. The authors proposed an optimal scheduling based on the transition cost measured in terms of the amount of off-specification products under closedloop PI control. They also proposed the use of robust control theory concepts to identify grade transitions that are “difficult” in the presence of model uncertainty. The approach used by the authors to solve the simultaneous optimal scheduling and grade transition problem should be considered as approximate, because the rigorous problem should be addressed as a mixedinteger nonlinear programming (MINLP) problem. Van Brempt et al.1 discussed the use of linear MPC for the grade transition tracking control problem. The optimal transition trajectory is computed off-line and passed to a linear MPC control scheme. The authors proposed ranking the set of process constraints. Their approach was applied to a model of a polystyrene process. Unfortunately, the authors did not address state estimation nor comment on the specific polymer properties that were tracked. Recently, BenAmor et al.12 employed nonlinear model predictive control (NLMPC) for the closed-loop tracking control of reference trajectories. Although the authors did not address the optimal computation of the input/output signals, the work highlights the implementation of NLMPC strategies into realtime commercial optimization software for grade transition tracking purposes. The outline of this paper is as follows. In section 2, the polymerization dynamic mathematical model is described. In section 3, we consider steady-state states and parameters of the entire HIPS reaction section and then briefly describe the simultaneous dynamic optimization (SDO) strategy used to compute the optimal transition trajectories. In section 4, we describe the set of polymer grades and transitions for this study. Section 5 contains the computation of the optimal transition trajectories as well as the discussion of the results. Finally, the conclusions of this work are presented in section 6. 2. Mathematical Model In the open literature there are only a few studies on the modeling of the HIPS process.13,14 In this section, we describe the mathematical model of the free-radical bulk polymerization of the system styrene/polybutadiene, using a monofunctional initiator. In Figure 1a, a typical industrial process15 for HIPS manufacturing is shown. The set of polymerization reactions are carried out in a continuous stirred tank reactor (R1) followed by a tubular reactor (R2) and finally by a heat exchanger (R3). No recycle streams are present in the process. The tubular reactor consists of several compartments in a rotating cylinder slightly tilted downward downstream. The compartments are separated by solid disks of nearly the same diameter as the internal diameter of the cylinder. The material flows from one compartment to the next one by gravity through clears in sections of

Ind. Eng. Chem. Res., Vol. 45, No. 18, 2006 6177 Table 1. Rate Constant Informationa Kdj ) 9.1 × 1013e-29508/RgTj (1/s) Ki0j ) 1.1 × 105e-27340/RgTj (L2/mol2‚s) Ki1j ) 1.0 × 107e-7067/RgTj (L/mol‚s) Ki2j ) 2.0 × 106e-7067/RgTj (L/mol‚s) Ki3j ) 1.0 × 107e-7067/RgTj (L/mol‚s) Kpj ) 1.0 × 107e-7067/RgTj (L/mol‚s) Kfsj ) 6.6 × 107e-14400/RgTj (L/mol‚s) Kfbj ) 2.3 × 109e-18000/RgTj (L/mol‚s) Ktj ) 1.7 × 109e-843.0/T-2(C1xsj+C2xsj2+C3xsj3) (L/mol‚s) c1 ) 2.57 - 0.00505Tj c2 ) 9.56 - 0.0176Tj c3 ) -3.03 + 0.00785Tj a

Tj is in K, Rg ) 1.987 cal/(mol‚K), and xsj ) (C°j,m - Cj,m)/C°j,m.


Table 2. Optimal Values of the Decision Variables

Figure 1. (a) Flowsheet of the HIPS plant and (b) approximation of the HIPS plant by a set of seven series-conneced CSTRs. The dashed box stands for the five CSTRs employed for approximating the steady-state behavior of the R2 tubular reactor.

the disks left between the disks and the cylinder and having slightly smaller diameter than the rest of the disk. The stirring is provided by the rotation of the cylinder, and it does not represent a perfect mixing; however, isothermal operation at each compartment is approximated. The reactor is not full of material and leaves a space which is filled by vapors of styrene and/or nitrogen. The reactor operates isobarically with the help of a split-range temperature control system which injects nitrogen when it is desired to raise the temperature or allows vapors of styrene to go to a condenser on top of the reactor so liquid styrene returns to the reactor (at one of the compartments). Owing to the complexity of the mathematical model of the tubular reactor, the steady-state behavior of such a reactor was approximated by a set of five continuously stirred tank reactor (CSTRs) connected in series, thus giving rise to a system of seven series-connected CSTRs as shown in Figure 1b. The steady-state behavior of the reaction train has been matched against actual experimental plant data. The dynamic behavior of the HIPS process during grade transition has been reported elsewhere.16 The model used in this work for each reactor is a slight extension of the one previously used in ref 16. The main differences between the model used in this work and the one reported in ref 16 are the computation of the first moment of the dead species and the computation of the zeroth and first moments of the living species. The extension of the model was needed to compute the molecular weight distribution as given in eq 24 of the present work. The polymerization reactions are carried out in a nonisothermal CSTR assuming perfect mixing, constant physical properties, quasi-steady state, and the long chain hypothesis. Constant volume in the reactor has also been assumed. Changes in density of the monomer-polymer mixture have been neglected, and we assume that the system is homogeneous. (In practice, rubber particles form a separate phase, but this is usually small in weight percentage (around 4-6%); therefore, this assumption is a reasonable approximation). The kinetic mechanism involves initiation, propagation, transfer, and termination reactions. Polybutadiene is added to guarantee desired mechanical properties by promoting grafting reactions. Network formation reac-

reactor

volume (L)

jacket volume (L)

Qcw (L/s)

heat-transfer area (m2)

1 2 3 4 5 6 7

6000 900 1000 650 1000 1000 5000

1200 180 200 130 200 200 1000

0.1311 1.0 1.0 1.0 1.0 1.0 1.0

11.718 1.7578 1.9531 1.2695 1.9531 1.9531 9.5676

Table 3. Nominal Steady State of the HIPS Reaction Train reactor

Cm (mol/L)

Tr (K)

Tj (K)

To (K)

1 2 3 4 5 6 7

7.318 5.388 3.860 3.578 3.375 3.240 3.158

377.460 454.356 464.848 445.925 434.152 428.635 394.283

347.304 299.387 300.353 297.720 299.212 299.006 310.231

333.000 377.460 408.920 441.605 437.006 434.132 428.635

tions are also modeled because they may lead to an undesirable excess of cross-linking in the rubber particles. There is still some debate about the reactions causing cross-linking. H. Keskkula (personal communication, 1998) assumed that the main reaction responsible for cross-linking occurs between two just-activated polybutadiene radicals. On the other hand, others17,18 assume that coupling between two radicals in the active ends of graft chains is the main contributor to cross-linking. In this model both possibilities are included. These modeling considerations have been shown to reproduce the performance of an industrial HIPS continuous stirred tank reactor.19 In addition, the gel effect is modeled through modification of the termination rate constant (see Table 2 in ref 19). Because of wide variations in temperature during the transitions, one would expect that the heat transfer coefficients would vary considerably during this transient period. However, to keep the computations more manageable, we used an average value for the heat transfer coefficient averaging over the conditions of all the reactors in the train. Notice that, as a consequence of the design of the process, the first reactors in the train, showing lower conversion values, are operated at lower temperatures (productivity, control, and product quality reasons, as higher molecular weight fractions are produced at this stage, are design issues). As the conversion is increased along the train, the temperature is increased to reduce the viscosity and promote higher conversions (and also to produce fractions of lower molecular weight). As a result, the increase in viscosity due to the increase in conversion downstream the train is somewhat compensated by the higher temperatures employed at higher conversions. Consequently, the variations in the heat transfer

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coefficient are also attenuated. However, a more precise calculation would require the inclusion of the heat transfer coefficient variation with conversion and temperature. This is expected to make even more nonlinear the reactor behavior. When the above modeling assumptions are used, the mathematical model for each CSTR is described by the following set of differential equations.

dCj,i Qj,iC°j,i - QjCj,i - KdjCj,i ) dt Vj

(1)

dCj,m Qj(C°j,m - Cj,m) ) - KpjCj,m(µrj0 + µbj0) dt Vj

(2)

dCj,b Qj(C°j,b - Cj,b) ) - Cj,b(Ki2jCj,r + Kfsjµrj0 + Kfbjµbj0) (3) dt Vj


dCj,r ) 2f*KdjCj,i - Cj,r(Ki1jCj,m + Ki2jCj,b) dt

(4)

dCj,br ) Cj,b(Ki2jCj,r + Kfbj(µrj0 + µbj0)) dt Cj,br(Ki3jCj,m + Ktj(µrj0 + µbj0 + Cj,br)) (5) dµrj0 ) 2Ki0jCj,m3 + Ki1jCj,rCj,m + Cj,mKfsj(µrj0 + µbj0) dt

(6)

dµrj1

(KpjCj,m + Ktj(µrj0 + µbj0 + Cj,br) + KfsjCj,m + KfbjCj,b)µrj0 + KpjCj,mµrj0 (7)

dt

) Ki3jCj,brCj,m - (KpjCj,m + Ktj(µrj0 + µbj0 + Cj,br) + KfsjCj,m + KfbjCj,b)µbj0 + KpjCj,mµbj0 (8) dµbj1

Qj 1 µ V j bj

(9)

Ktj Qj (µ 0)2 + (KfsjCj,m + KfbjCj,b)µrj0 - λpj0 2 rj Vj

(10)

dt dλpj0 dt dλpj1 dt dλbpj0 dt

)

) (B1 + B2µbj1 + B3(µbj1 + µbj0)) -

(

)

) (Ktjµrj1µrj0 + (KfsjCj,m + KfbjCj,b)µrj1) -

dt

Qj 1 λ (11) Vj pj

dTj,c Qj,c(T°j - Tj,c) UjAj(Tj - Tj,c) ) + dt Vj FjCpjVj

(17)

R1 ) 2Ki0jCj,m3 + Ki1jRCj,m + Kfsjµbj0

(18)

where

R2 ) -(KpjCj,m + Ktj(µrj0 + µbj0 + Cj,br) + KfbjCj,b) R3 ) KpjCj,m

(19) (20)

B1 ) Ki3jCj,brCj,m

(21)

B2 ) -(KpjCj,m + Ktj(µrj0 + µbj0 + Cj,br) + KfsjCj,m + KfbjCj,b) (22) B3 ) KpjCj,m

(23)

The average molecular weight distribution (Mn) is given by

λpj1 + µrj1 λpj0 + µrj0

(24)

where C stands for concentration; T is the reactor temperature; the subscripts j, i, m, b, and r stand for reactor number, initiator, monomer, butadiene, and radical, respectively; and the subscripts o and c mean feed stream and jacket related variables, respectively. Here µ0 and µ1 are the zeroth and first moments of the death species, while λ0 and λ1 are the zeroth and first moments of the living species, Q is the feed stream volumetric flow rate, V is the volume, f* is the initiator efficiency, ∆H is the reaction heat, F is the density, and Cp is the heat capacity. K stands for the kinetic rate constants, and its subindexes d, p, i0, i1, i2, i3, t, fs, and fb represent the rate constants of the different polymerization steps. The numerical values of the rate constants are shown in Table 1 while design parameters are contained in ref 19. Notice that each reactor has its own cooling system that is not shared with the rest of the reactors. This means that the coolant is added at different positions along the reaction system. 3. Dynamic Optimization Formulation

)

(Ktjµrj0(µbj0 + Cj,br) + (KfsjCj,m + KfbjCj,b)µbj0) dλbpj1

0 0 Qj(T°j - Tj) ∆HKpjCj,m(µrj + µbj ) UjAj(Tj - Tj,c) (16) Vj FsCps FsCpsVj

Mj,n )

Fj ) (R2µrj1 + R3(µrj1 + µrj0)) - µrj1 dt Vj

dµbj0

dTj ) dt

Qj 0 λ (12) Vj bpj

) (Ktj(µrj1Cj,br + (µrj1µbj0 + µrj0µbj1)) + (KfsjCj,m + KfbjCj,b)µbj1) dλbbpj0 dt dλbbpj1 dt

(

) Ktjµbj0Cj,br +

)

Qj 1 λ (13) Vj bpj

Kt j Qj (µbj0)2 - λbbpj0 2 Vj

(14)

Qj λ 1 Vj bbpj

(15)

) (Ktjµbj1Cj,br + Ktjµbj1µbj0) -

This section deals with two components for determination of optimal grade transition policies. First, we describe the computation of optimal steady states and parameters that describe the desired grades. Next, we briefly describe the SDO strategy based on orthogonal collocation on finite elements, which was used to compute optimal transitions between grades. 3.1. Optimal Process Parameters. Optimal steady-state operating states and parameters were determined to match the ranges of industrial monomer conversion. In a typical industrial process,16 a feed stream consisting of styrene monomer, polybutadiene, and initiator is fed to the reactor train (see Figure 1a). The first reactor (R1) is operated at around 120-130 °C, and a conversion of 20-30% is achieved there. The product stream is sent to a second reactor (R2) which, in the particular process analyzed here, is an isobaric tubular reactor operating

Ind. Eng. Chem. Res., Vol. 45, No. 18, 2006 6179


at around 160-170 °C where a 60 overall conversion is normally achieved. Next, the product stream from R2 is fed to a heat exchanger operating at around 200 °C where the polymerization reaches around 70% overall conversion. As mentioned above, the complete HIPS reaction train was approximated by a set of seven series-connected nonisothermal CSTRs as shown in Figure 1b. In this figure the dashed box represents the five CSTRs employed to approximate the tubular reactor behavior. Our industrial experience with the HIPS process suggests that five CSTRs suffice to approximate the tubular reactor steady-state behavior.19 The optimization problem decision variables considered are typical reactor design parameters such as volume, jacket dimension, heat transfer area, and cooling water flow rate which lead to a suitable conversion profile such as the one described above. Therefore, for each reactor, the steady-state optimal design of each CSTR was cast in terms of the following nonlinear program (NLP).

min (Xjd - Xj)2 x,u

s.t. eqs 1-17 with

dx )0 dt

0 e θj e θju jacket

293 K e Tj

e 373 K

x, u g 0 ∀ {j ∈ 1,..., 7}

(25)

where j stands for the reactor number in the cascade, u is the set of design parameters [V, Vc, A, Qcw] for every reactor j, and x are the system states. θ is the residence time, and θu ) [1.5, 0.1, 0.1, 0.1, 0.1, 0.1, 1.2] is the residence time upper bound. Xjd ) [0.15, 0.35, 0.5, 0.60, 0.7, 0.7, 0.75] is the desired overall conversion reached in each reactor, and Xj is the actual optimum overall fraction conversion in the same reactor. Here the overall monomer conversion is calculated from the feed stream to the sequence up to the jth reactor outlet. Optimal design information is presented in Table 2, while nominal steady-state conditions for the process are presented in Table 3. The optimization problem stated previously was solved using the mathematical modeling software AMPL20 and IPOPT21 for solving the NLP problem. With this optimization framework, we found that conversion levels in the tubular reactor (reactors 2-6) were extremely sensitive to small variations in the operating temperature. This phenomenon is influenced by the onset of the gel effect, included in our model. Under this situation the viscosity of the polymeric mixture tends to rise sharply, therefore leading to a decrease in mass diffusion. Because of this, the released heat of reaction cannot be efficiently removed, giving rise to large temperature increases. In fact, the onset of the gel effect has been recognized as a major source of nonlinearity in polymerization systems. It was found that employing reactor intercooling could overcome thermal runaway conditions, making it feasible to operate the process around the industrial desired monomer conversion range. Note from the conversion profile for the process that most of the polymer chains are produced in reactors 1-3 (55%), while in the final reactors 4-7 only an additional 8% conversion is reached. This was only possible by setting intercooling stages between reactors 2-3 and 3-4 with a fixed cooling capacity at 20 °C. Once the desired conversion level is achieved in the tubular reactor, common industrial practice is to send this product stream

into a final heat exchanger aimed to consume an additional amount of styrene. However, as a result of the fact that the rate of reaction in the heat exchanger tends to be low, leading to small amounts of released heat, the feed stream might require extra heating, although in this work no preheating was employed. Therefore, the final overall conversion obtained is around 65%. This optimal design will be taken as the base case for later application of bifurcation analysis. 3.2. SDO. The operating points in this study are open-loop unstable, and for this particular process no open-loop stable steady states yield the desired conversion levels. Although the steady-state optimization approaches are often constrained to keep the processes away from operating near or within regimes characterized by steady-state multiplicity, sustained periodic oscillations, and other nonlinear phenomena, this situation may prevent design and operation near more profitable steady-state economic optima.22 Furthermore, with new nonlinear programming strategies being developed to permit reliable and robust model predictive control near or within complex nonlinear operating regimes,23 the instability open-loop problem might not be hard to deal with. Moreover, as described in ref 16, SDO provides a way to compute optimal dynamic policies such as grade transitions, even in the presence of challenging nonlinear behavior. These include transitions to unstable points, optimization with chaotic systems,24,25 and systems with limit cycles and bifurcations. In SDO, computation of optimal transition policies reduces to the solution of a NLP26 and provides values of the decision variables (i.e., the manipulated variables) that drive the system toward minimum transition time or off-spec product. A common requirement in polymerization reactors is that grade transitions feature a minimum transition time, waste material, or utility consumption. The minimum time transition policy requires setting the following optimization problem (where we have assumed that the manipulated variables are the monomer volumetric feed flow rate Q and the initiator volumetric feed flow rate Qi):

min

∫0t {w1||z1(t) - zˆ 1||2 + wf|Q(t) - Qˆ ||2 + wi||Qi(t) f

Q ˆ i|2} dt (26) s.t. semi-explicit DAE model: dz(t) ) F(z(t), y(t), u(t), t, p) dt 0 ) G(z(t), y(t), u(t), t, p) Initial conditions: z(0) ) z0

(27) (28) (29)

Bounds: zL e z(t) e zU yL e y(t) e yU uL e u(t) e uU pL e p e pU

(30)

where F is the vector of the right-hand sides of the differential equations in the DAE model, G is the vector of algebraic equations, assumed to be index one, z is the differential state vector, z0 are the initial values of z, zˆ is the new desired transition state, y is the algebraic state vector, u ) [QQi]T is the control profile vector, uˆ ) [Q ˆ Q ˆ i]T is the reference control vector, p is a time-independent parameter vector, and tf is the


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transition horizon. Notice that Q ˆ and Q ˆ i represent the values of the decision variables at the end of the desired transition. Such values are normally available from steady-state calculations because, to compute a polymer grade transition, we need to know the values of the controlled and manipulated variables at the initial and final steady states (i.e., for the initial and final grades). In the above formulation, z1 stands for a vector which contains those states that are part of the objective function, while zˆ 1 means the desired final values of the same states. In our case z1 ) [Cj,i, Cj,m, Cj,r, Cj,br, µrj0,µbj0, Tj, Tj,c]′ for j ) 1, ..., 7. So, z1 stands for the value of the states that we desire to track at the outlet of all the reactors and not only at the outlet of the last reactor of the reaction sequence. The objective function represents the deviation of the states and the inputs from the desired grade. The objective function terms represent lost product as well as deviations from desired levels of monomer and initiator flow. Also, to get smooth control actions, weight factors, denoted by w1, wf, and wi, are used on states, monomer Q(t), and initiator Qi(t) flow rates, respectively. After several trials, it was found that by setting w1 ) 1, wf ) wi ) 1000 will allow smooth input/output reactor responses to be computed. The measurement units of all the weighting factors are the square of the inverse of the variable to which they are applied, so the terms in the objective function can be added. In the SDO approach, the DAE optimization problem is converted into a NLP by approximating both the state and the control profiles by a family of polynomials on finite elements. Here a Runge-Kutta discretization with Radau collocation points is used, as it allows constraints to be set easily at the end of each element and stabilizes the system more efficiently if high index DAEs are present. In addition, the integral objective function is approximated with Radau quadrature with ne finite elements and ncol quadrature points in each element. As shown in ref 16, substitution of this discretization into eqs 26-30 applied at the collocation points leads to the following NLP.

min f(x)

(31)

s.t. c(x) ) 0

(32)

x L e x e xU

(33)

x∈Rn

where x represents coefficients of the piecewise polynomials that make up the control and state profiles. More details of this approach can be found in refs 16 and 26. The dynamic optimization formulation given by eqs 31-33 was implemented using the AMPL mathematical programming language27 and solved using the IPOPT algorithm21 for large-scale nonlinear programming. This algorithm follows a barrier approach, where the bound constraints are replaced by logarithmic barrier terms which are added to the objective function. Consequently, a strategy for solving the original NLP is to solve a sequence of barrier problems for decreasing barrier parameters. IPOPT follows a primal-dual approach and applies a Newton method to the resulting optimality conditions. Exact first and second derivatives for this method are provided automatically through the AMPL interface. More information on IPOPT can be found in the work of Wa¨chter and Biegler.21 Sometimes grade qualities are specified by lower and upper limits around a central nominal value. The imposition of upper and lower limits in the specification of grades would simply require the replacement of endpoint equations by endpoint inequality constraints in the dynamic optimization problem. Similarly, a more general objective function could be added to

minimize out of spec material or maximize productivity. One could then certainly apply the simultaneous dynamic strategy to an extended version of this problem. 4. Selection of Grades and Transitions To develop grade transition policies, we first generate bifurcation diagrams of the HIPS reaction system; these are displayed in Figures 2-4 and present a clear picture of the nonlinear behavior patterns embedded in the complete HIPS plant. Their utility will be more obvious for highly nonlinear system responses. While not routinely employed in industrial environments, the aim of using these bifurcation diagrams is to demonstrate the highly nonlinear behavior that arises at the intermediate reactors of the reaction train. Moreover, under certain conditions, the expected reactor behavior is opposite to what we normally would expect. As a result, one of the main points to highlight in this paper is that optimal grade transitions are difficult to compute due to nonlinearities in the form of multiple steady states and high parameter sensitivity. Therefore, bifurcation diagrams are the right tool to present and discuss these features. In this work the monomer and initiator feed stream flow rates are the two manipulated variables that steer the reaction train from a given initial grade to the desired final grade. Accordingly, two grade transition policy scenarios were considered. A first set of three polymer grades (A, B, C) are manufactured by manipulating only the monomer feed stream flow rate, while a second set of polymer grades (D, E, F) requires the simultaneous manipulation of both monomer and initiator feed stream flow rates. Figure 2 shows the bifurcation diagram employing the monomer feed stream flow rate as continuation parameter and monomer conversion as the reactor response (diagrams for the temperature response show qualitatively similar behavior). In this figure, the nominal operating point is denoted by the symbol “O”, and the continuous line represents stable open-loop steady states, while the dashed line stands for unstable open-loop steady states. Notice the highly nonlinear steady-state behavior. For simplicity, only those steady states close to the nominal operating region are shown. Under optimal design conditions, each reactor operates at an open-loop unstable steady state. It is interesting to realize that the response from the second reactor turns out to be against process intuition. Here, increasing the monomer flow rate should increase the monomer conversion. However, from Figure 2, it is clear that, due to the onset of nonlinearities, the output of the second reactor exhibits an inverse response. The situation does not look better for the third reactor, because it displays an almost zero gain between the input and the output. Process sensitivity starts increasing from reactor 5. Actually, for selecting the polymer grades, the response from the last reactor is the only one strictly needed. However, it is important to realize the nonlinear behavior response of the intermediate reactors. Table 4 shows the values of the polymer properties that characterize the A, B, and C grades. Such values are close to commercial grades commonly manufactured in typical HIPS plants. On the other hand, the manufacture of grades D, E, and F requires the manipulation of both the monomer and the initiator feed stream flow rates. Figure 3 displays the bifurcation diagrams of each one of the seven series-connected CSTRs. Note the highly nonlinear response exhibited for each reactor output. In particular, the response displayed by the first reactor is interesting, due to the fact that increased monomer conversions can be attained by reducing the initiator flow rate. Normally, increasing the monomer conversion requires increasing the



Figure 2. Monomer conversion bifurcation diagram using the monomer flow rate (Q) as the main continuation parameter. The “O” symbol stands for the nominal steady state. The continuous and dashed lines represent open-loop stable and unstable steady states, respectively.

Figure 3. Monomer conversion bifurcation diagram using the monomer flow rate (Q) as the main continuation parameter and the initiator flow rate (Qi) as the secondary continuation parameter. 1, Qi ) 15 × 10-4 L/s; 2, Qi ) 7.5 × 10-4 L/s; and 3, Qi ) 3 × 10-4 L/s.

initiator flow rate, and this tendency is observed in the rest of the reactor responses, except at the second reactor outlet. The processing conditions leading to the manufacture of the D, E, and F grades are shown in Table 4. From this table, notice that similar monomer conversion rates can be attained by reducing the monomer flow rate if the initiator flow rate is also manipulated. Of course, monomer conversion is not the only property one is usually interested in. In this case, reducing the monomer flow rate implies a reduction in the polymer produc-

tion rate. Figure 4 displays the locations, in terms of monomer conversion, of the selected grades. 5. Optimal Grade Transitions Table 5 summarizes all the grade transition cases addressed in this work. As mentioned previously, grades can be manufactured either by manipulating the monomer flow rate (A, B, C grades) or by the simultaneous manipulation of both the monomer and the initiator flow rates (D, E, F grades). Grade


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Figure 4. Location of the designed polymer grades in terms of the monomer conversion at the outlet of reactor 7: (a) manipulating only the monomer feed flow rate stream and (b) simultaneous manipulation of the monomer and initiator feed stream flow rates. 1, Qi ) 15 × 10-4 L/s; 2, Qi ) 7.5 × 10-4 L/s; and 3, Qi ) 3 × 10-4 L/s. Table 4. Grade Design Informationa grade N A B C D E F

conversion

T (K)

Qm (L/s)

Qi (L/s)

0.635 0.734 0.83 0.92 0.734 0.83 0.92

395 440 475 517 430 465 500

1.14 1.48 1.62 2 1.33 1.48 1.7

15 × 10-4 15 × 10-4 15 × 10-4 15 × 10-4 7.5 × 10-4 7.5 × 10-4 3 × 10-4

a Q is the monomer flow rate. Q stands for the initiator flow rate m i conversion, and temperature information refers to the last reactor of the reaction train sequence.

transitions were obtained by trying to match actual industrial HIPS grades. They may be classified as “easy” and “hard” grade transitions. Easy grade transitions mean transitions from the nominal operating point (N) to a close conversion region (A, D grades). They can be reached easily in terms of control effort and short transition times. Moreover, easy transitions feature mild nonlinearities and are also easy to compute in terms of

Table 5. Grade Transition Trajectories, Problem Statistics, and Objective Function (OF) Values transition

no. iterations

CPU (s)

OF

NfA AfN NfB BfN NfC CfN NfAfBfC CfBfAfN

183 209 228 377 398 290 1086 785

47.6 47.6 64.1 94.1 122.8 72.8 319.8 221.3

24.7 253.42 57.4 1019 153.9 3435.4 49.7 697.3

NfD DfN NfE EfN NfF FfN NfDfEfF FfEfDfN

179 258 244 129 736 246 1115 872

46.5 73.04 75.4 30.8 251.97 63.95 343.9 231.4

43.8 228.8 75.96 759.6 164.5 2032.7 61.95 550.3

computational requirements. On the other hand, hard grade transitions mean transitions featuring highly nonlinear regions,



Figure 5. Grade transitions using the monomer flow rate as the manipulated variable: (a, b) individual and (c, d) overall grade transitions. The displayed results refer to the output behavior of the last reactor of the polymerization reaction train sequence.

greater control effort, larger transition times, and greater computational requirements (C, F grades). Between easy and hard grade transitions, there are a set of grades (B, E grades) of intermediate complexity in terms of nonlinear behavior, control effort, and transition time. From Table 5, it can be noticed that all grade transitions were computed for forward and backward directions. The purpose of such transitions is twofold: to show that our dynamic optimization formulation is robust enough to cope with any transition direction and to highlight the strong nonlinear behavior embedded in the HIPS model. We would like to highlight that although monomer conversion was one of the main system parameters used for grade transition tracking, HIPS properties also depend on other parameters such as melt index and cross-linking data. However, this information is confidential, and, therefore, it is not provided. Molecular weight data were not included mainly to simplify the calculations; however, in ref 28 some molecular weight data are included. A final remark on the difficulty of the optimal grade transitions computation should be made. From Figure 4, it can be noticed that all grade transitions feature transitions between open-loop unstable steady states. Previously, the robust optimal transition policy between unstable steady states has been addressed.16 To the best of our knowledge, no other optimal dynamic transition strategies have been proposed that feature

transitions between unstable steady states. To keep the discussion of the results as simple as possible, only the output response from the last reactor 7 of the polymerization reaction train sequence is shown. Moreover, in ref 28 a complete and detailed study was undertaken on the HIPS process dynamic behavior during grade transition operations in terms of molecular weight distributions, polydispersity, and monomer conversion. We now discuss each optimal grade transition. Monomer Flow Rate Manipulation. N S A Transition. Figure 5a,b indicates that both grade transitions are relatively easy to compute. However, it should be noticed that the N f A transition is the easier one to carry out in terms of transition time. In fact, the transition time for A f N is almost twice the one carried out in the opposite direction. This behavior can be explained looking at Figure 6a. From this figure, for the N f A transition, the shape of the monomer feed flow rate features a step function, meaning that the manipulated variable quickly reaches its final value. However, for the transition in the opposite direction (A f N), the shape of the monomer feed flow rate is a first-order transfer function with a small initial overshoot. The fact that both the shape and settling time of the manipulated variable trajectory depend on the direction of the intended transition (i.e., they are not symmetric) highlights the nonlinear nature of the polymerization system.



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Figure 6. Monomer flow rate: (a) six individual cases and the (b) N f A f B f C (solid) and C f B f A f N (dashed) cases.

N S B Transition. From Figure 5a,b, it can be noticed that although the size of the intended transitions is larger than with N f A and A f N, both the N S A and the N S B transitions feature similar manipulated and controlled variables shapes. Surprisingly, even though the N f B case is 10% higher than the N f A transition in monomer conversion, both cases exhibit similar transition times. On the other hand, transition times for A f N and B f N are different, and this can be explained by looking at Figure 6a. For the B f N transition, it takes longer for the monomer feed flow rate to attain its final steady-state value, and this time the initial manipulated variable overshoot can be easily observed. N S C Transition. Again, from Figure 5a,b, the shape of the optimal manipulated and controlled variables is similar to

the previously addressed transitions. Nevertheless, it should be noticed that even though the N f C transition features an almost 30% monomer conversion increase, compared to the 10% conversion increase exhibited by the N f A transition, the transition times of both the N f C and N f A cases are practically the same. However, from Figure 6a, it can be noticed that in the C f N case the feed stream monomer flow rate features a more than 100 initial overshoot and the transition time for the C f N case turns out to be larger than for the N f A and N f B cases. It is evident that, as the imposed transition gets larger, the size of the initial monomer flow rate overshoot will also increase. N f A f B f C Transition. Instead of carrying out overall transitions (i.e., N S C transitions) for a single grade transition



Figure 7. Grade transitions using the monomer and initiator flow rates as the manipulated variables: (a, b) individual and (c, d) overall grade transitions. The displayed results refer to the output behavior of the last reactor of the polymerization reaction train sequence.

operation, it might be easier to perform these grade transitions in a step by step manner (i.e., taking the reactor operation around intermediate grades). Figure 5c,d displays the advantages of carrying out large monomer conversion transition grades by proceeding sequentially. It can be noticed that the optimal shapes of the manipulated and controlled variables are almost a superposition of the individual transition grade cases previously addressed (the set point of the grades was changed after 2 h, even though the corresponding grade transition was completed earlier). Regarding the shape and time values of the monomer flow rate, it can be noticed from Figure 6b that the N f A f B f C case is almost a collection of the individual responses. However, because the size of the imposed monomer conversion gets smaller, this time the C f B f A f N case features only a small initial overshoot; the large overshoot in monomer feed from the C f N case is avoided. Therefore, based on the amount of consumed monomer, it is beneficial to perform the C f N transition sequentially, rather than in a single transition grade operation. Monomer and Initiator Flow Rates Manipulation. In this part, we evaluate the advantages of carrying out grade transition computations that employ the initiator flow rate as an additional manipulated variable. Intuition dictates that incorporating more degrees of freedom should contribute to reducing the complexity

of grade transition computations, measured in terms of both grade transition time and reactants consumption. N S D Transition. When manipulating two input variables (N f D and D f N cases), Figure 7a,b shows that the shape of the optimal controlled variables is similar to the case in which only the monomer flow rate was manipulated (N f A and A f N cases). However, what is remarkable is the fact that, even when the D and A grades feature the same conversion, the D grade requires smaller monomer and initiator flow rates than the A grade. Again, increasing the monomer conversion would require increasing both the monomer and the initiator flow rates. Nevertheless, owing to the nonlinear behavior exhibited by the polymerization system, similar monomer conversions can be achieved using less reactant. Therefore, this is an indication that it might be better to manipulate both feed stream flow rates, instead of only the monomer flow rate, to achieve similar monomer conversions. Figures 8a and 9a display the optimal shape and values of the monomer and initiator flow rates, respectively. It can be noticed that the shape of the monomer flow rate is similar to previously addressed grade transitions. The initiator flow rate shape features a simple steplike function in either grade transition direction. It should be noticed that the N f D and D f N cases transition times are similar to those exhibited by the N f A and A f N cases.



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Figure 8. Monomer flow rate: (a) six individual cases and the (b) N f D f E f F (solid) and F f E f D f N (dashed) cases.

N S E Transition. From Figure 7a,b, it can be noticed again that the shape of the optimal profiles of the controlled variables is similar to that of the N f B and B f N cases. Of course, the numerical values of the controlled variables in both cases are different. Once again, it should be noted that the same monomer conversion is achieved employing a smaller amount of the reactants. Figure 8a displays essentially the same shape of the monomer flow rate optimal profile as, for instance, that obtained from the N f B and B f N cases. From Figure 9a, it can be noticed that the initiator flow rate optimal profile takes a simple steplike response. N S F Transition. Figure 7a,b displays the optimal profiles of the output controlled variables. The observed response pattern is similar to previously analyzed grade transitions (N f F and

F f N cases). However, once again due to the nonlinear behavior embedded in the polymerization reactor model, higher monomer conversions can be reached reducing the amount of both monomer and initiator flow rates. Figure 8a displays the optimal time profile of the initiator flow rate. Again, a large initial overshoot is observed for the F f N case. However, as displayed in Figure 9a, the optimal profile of the initiator flow rate is nonmonotonic. It shows an oscillatory response until it settles down to its final steady-state value. N S D S E S F Transition. Finally, from Figure 7c,d we note again that sequential grade transitions seem to be a better approach to carry out an overall transition, in terms of transition time and reactant consumption. The sequential grade transitions resemble the individual grade transitions previously discussed.



Figure 9. Initiator flow rate: (a) six individual cases and the (b) N f D f E f F (dashed) and F f E f D f N (solid) cases.

Nevertheless, sequential grade transitions feature smoother optimal profiles. From Figure 9b, it can be noticed that the initial overshoot of the monomer flow rate is reduced for the F f E f D f N case, due to the fact that the imposed grade transition requires smoother control actions. As shown in Figure 9b, the initial initiator flow rate oscillatory response previously observed (N f F case) is completely eliminated. As a matter of fact, the optimal profiles of both manipulated variables now consist of simple steplike functions. As an intermediate conclusion, the results show that sequential grade transitions feature smoother grade transitions leading to nonaggressive control actions. Moreover, the simultaneous manipulation of both monomer and initiator flow rates leads to a reduction in reactant consumption.

6. Conclusions We propose a robust and efficient dynamic optimization formulation for tackling the optimal grade transition computations. The dynamic optimization formulation was applied to a large scale industrial HIPS plant, featuring unstable nominal open-loop operating conditions. Even in the presence of those computational difficulties, the dynamic optimization formulation was able to efficiently compute all the transition trajectories among the required polymer grades. A summary of the computational performance is given in Table 5. It was found that the simultaneous manipulation of both monomer and initiator flow rates leads to the same polymer conversion with less initiator, in comparison to the case where


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the monomer flow rate was the only manipulated variable. Such behavior was detected before running the computations by using nonlinear bifurcation diagrams. Another major conclusion drawn from this work indicates that overall grade transitions should be carried out using a step by step grade transition approach, instead of a single overall grade transition. This observation can be explained by noting that this highly nonlinear system also has many local minima, characterized by large overshoots for overall grade transitions. These overshoots and aggressive values of the manipulated variables are largely avoided by carrying out grade transition trajectory determinations in a stepwise manner. The solution of the dynamic optimization problem leads to optimal profiles of flows instead of a series of steps. The main point is that an overall grade transition seems to be best executed by proceeding through the intermediate grades. To address the closed-loop implementation of the optimal grade transition trajectories, the grade trajectories can be used as the reference set point that the closed-loop control system should track. Because some potential controlled variables might not be measured on-line, an estimation scheme would be required. In a recent work29 we have provided details about the robust tracking of the type of optimal dynamic signals such as the ones calculated in this paper. It must be recognized that, if strong disturbances hit the system during grade transition, then the implementation of the optimal grade transition policies would be difficult to achieve. Finally, this study has a number of distinguishing features: (i) Although the numerical approach is based on our previous work, we note that, to our knowledge, there have not been previous grade transition optimization studies in the literature of this size and complexity. In contrast to our previous paper16 we now consider a system that is now almost an order of magnitude larger in size and with a proportional increase in the number of unstable modes. Nevertheless, we still are able to get efficient results for this particular HIPS system. (ii) Also, while handling grade changes in a stage-wise manner is often common practice, it has never been challenged nor validated with an optimization study. The optimization results in this study are, therefore, useful to practitioners. (iii) Optimal transition policies are often open-loop unstable, and these must be determined before the process is stabilized. Alternately, imposing a stabilizing controller first, even if it can be designed, may compromise the solution of the optimization problem. Our optimization approach avoids this problem altogether, and we have demonstrated this property on a system with seven unstable CSTRs. (iv) The model uses conversion and temperature as controlled variables, and the use of cooling water temperature as a manipulated variable. However, this approach does not preclude adding additional detail to the model. On the other hand, the process on which this study is based is proprietary, so some of the product information (regarding melt flow index and other product properties) cannot be shared. Nevertheless, the nonlinear characteristics and the optimal transition policies can still be captured in our model by modeling conversion and temperature only. This property allows us to validate the results in this study. Acknowledgment Antonio F.-T. wishes to acknowledge the financial support from the Fullbright Commission, Universidad IberoamericanaSanta Fe, and from the Department of Chemical Engineering at Carnegie-Mellon University.

Nomenclature c ) nonlinear equality constraints C ) reactor concentration Cp ) heat capacity f ) objective function f* ) initiator efficiency F ) DAE differential right-hand side G ) vector of algebraic equations K ) kinetic rate constants with subscripts Mn ) average molecular weight distribution p ) time-independent parameter vector Q ) feed stream volumetric flow rate T ) reactor temperature tf ) transition horizon u ) control profile vector uˆ ) reference control vector V ) volume x, xL, xU ) optimization variables and associated bounds y ) algebraic state vector z ) differential state vector zˆ ) state of desired grade z1 ) subvector of states in objective function zˆ 1 ) desired values of z1 Greek Letters ∆H ) heat of reaction λ0 ) zeroth moment of live species λ1 ) first moment of live species µ0 ) zeroth moment of dead species µ1 ) first moment of dead species F ) density Subscripts j ) reactor number i ) initiator m ) monomer b ) polybutadiene unit br ) activated polybutadiene unit pb ) cross-linked polybutadiene r ) radical o ) feed stream related variable c ) jacket related variable d ) chemical initiation step p ) propagation step i0 ) thermal initiation step i1 ) size 1 chemical initiation step i2 ) polybutadiene chemical initiation step i3 ) grafted chemical initiation step t ) coupling termination step fs ) styrene monomer transfer step fb ) polybutadiene transfer step Literature Cited (1) Van Brempt, W.; Van Overschee, P.; Backx, T.; Ludlage, J.; Hayot, P.; Ootsvogels, L.; Rahman, S. Grad-Change Control using INCA Model Predictive Controller: Application on a DOW Polystyrene Process Model. Am. Control Conf. 2003, 5411-5416. (2) McAuley, K. B.; McGregor, J. F. Optimal Grade Transitions in a Gas Phase Polyethylene Reactor. AIChE J. 1992, 38 (10), 1564-1576. (3) McAuley, K. B.; McGregor, J. F. Nonlinear Product Property Control in Industrial Gas-Phase Polyethylene Reactors. AIChE J. 1993, 39 (5), 855866. (4) Debling, J. A.; Han, G. C.; Kuijpers, F.; VerBurg, J.; Zacca, J.; Ray, W. H. Dynamic Modeling of product Grade Transitions for Oplefin Polymerization Processes. AIChE J. 1994, 40 (3), 506-520.


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ReceiVed for reView October 12, 2005 ReVised manuscript receiVed June 30, 2006 Accepted July 6, 2006 IE051140Q

Optimal Grade Transitions in the High-Impact Polystyrene

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