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Chemical Engineering Science 58 (2003) 3643 – 3658 www.elsevier.com/locate/ces Optimal grade transition and selection of closed-loop controllers in a gas-phase ole$n polymerization 'uidized bed reactor C. Chatzidoukasa; b , J. D. Perkinsb , E. N. Pistikopoulosb , C. Kiparissidesa;∗ a Department of Chemical Engineering, Chemical Process Engineering Research Institute, Aristotle University of Thessaloniki, P.O. Box 472, University City, Thessaloniki 54006, Greece b Department of Chemical Engineering, Centre for Process Systems Engineering, Imperial College, London SW7 2BY, UK Received 27 November 2002; received in revised form 21 March 2003; accepted 12 May 2003 Abstract To satisfy the diverse product quality speci$cations required by the broad range of polyole$n applications, polymerization plants are forced to operate under frequent grade transition policies. Commonly, the optimal solution to this problem is based on the minimization of a suitable objective function de$ned in terms of the changeover time, product quality speci$cations, process safety constraints and the amount of o9-spec polymer, using dynamic optimization methods. However, considering the great impact that a given control structure con$guration can have on the process operability and product quality optimization, the time optimal grade transition problem needs to be solved in parallel with the optimal selection of the closed-loop control pairings between the controlled and manipulated variables. In the present study, a mixed integer dynamic optimization approach is applied to a catalytic gas-phase ethylene-1-butene copolymerization 'uidized bed reactor (FBR) to calculate both the “best” closed-loop control con$guration and the time optimal grade transition policies. The gPROMS/gOPT computational tools for modelling and dynamic optimization, and the GAMS/CPLEX MILP solver are employed for the solution of the combined optimization problem. Simulation results are presented showing the signi$cant quality and economic bene$ts that can be achieved through the application of the proposed integrated approach to the optimal grade transition problem for a gas-phase polyole$n FBR. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Gas-phase ole$n polymerization; Optimal grade transition; Optimal control structure selection; Mixed integer dynamic optimization; gPROMSJ simulator 1. Introduction Present market needs combined with the broad range of polyole$n applications have forced the polyole$n industry to operate under frequent grade transition policies. This trend has led the polyole$n industry to move away from large continuous production of a single polymer grade to a more 'exible production scheme comprising a number of polymer grades of high quality but low volume. In fact, in a polyole$n plant as many as 30 – 40 polymer grades can be produced. Consequently, under such market-driven operating schedules, the minimization of o9-spec polymer production and grade changeover time ∗ Corresponding author. Tel.: +30-31-99-6211; fax: +30-31-99-6198. E-mail address: [email protected] (C. Kiparissides). 0009-2509/03/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0009-2509(03)00223-9 are prerequisite to any pro$tability analysis of the process. Commonly, the optimal solution to this problem is based on the minimization of a suitable objective function de$ned in terms of the grade changeover time, product-quality speci$cations, process safety constraints and the amount of o9-spec polymer. However, optimal operation of a polymerization plant in terms of higher yield and better product quality at reduced cost can only be achieved when the process is operated under well-controlled conditions. In fact, the optimal selection of feedforward and feedback controllers is an essential requirement for the faithful implementation of an optimal control policy in a polymer plant. Due to the signi$cant economic importance of productquality optimization, extensive research e9orts have been undertaken to develop optimal control policies for di9erent polymerization processes. Thus, a great number of studies on open-loop optimal control of polymer quality (e.g., number 3644 C. Chatzidoukas et al. / Chemical Engineering Science 58 (2003) 3643 – 3658 and weight average molecular weights, polydispersity index, copolymer composition, molecular weight distribution, etc.) have been reported for batch and semi-batch polymerization reactors (Thomas & Kiparissides, 1984; Cawthon & Knabel, 1989; Choi & Butala, 1991; Crowley & Choi, 1997). Moreover, in a number of published reports, the actual implementation of the calculated optimal control policies in laboratory and industrial polymerization reactors has been demonstrated (Chen & Huang, 1981; Ponnuswamy, Shah, & Kiparissides, 1987; MacGregor, Penlidis, & Hamilec, 1984; Kravaris, Wright, & Carrier, 1989; Kozub & MacGregor, 1992; McAuley & MacGregor, 1993; Ohshima & Tanigaki, 2000). The calculation of optimal grade transition policies in catalytic ole$n polymerization processes has been the subject of several publications. Cozewith (1988) studied the e9ect of step changes in chain transfer agent and monomer feed rates on the number average molecular weight (Mn ), polydispersity index (PD) and copolymer composition for a continuous 'ow stirred tank polyole$n reactor. He clearly demonstrated that the direction and magnitude of the transition greatly a9ected the transient responses of the polymer-quality variables (e.g., melt index, density). Moreover, he showed that a steady-state reactor reinstatement following a reactor start-up was substantially faster than the establishment of a new steady state during a grade transition operation. This was explained by the slow dynamic response of the polymer-quality variables during a grade transition due to the accumulated polymer in the reactor. Despite this observation, shutting down and restarting the reactor to produce a new polymer grade, is a much more expensive practice than the slower reactor transition from one grade to a new one. Thus, the calculation of time-optimal control policies to drive the reactor from one steady state to a new one in minimum time, is a problem of signi$cant economic importance to the polyole$n industry. McAuley and MacGregor (1992) investigated the optimal grade transition problem for a gas-phase polyole$n 'uidized bed reactor (FBR). A simple kinetic model was assumed to describe the molecular weight developments in the FBR and the control vector parameterization method was employed to calculate the optimal transition policies. No constraints on the state variables were imposed. They showed that the calculated optimal transition policies were strongly dependent on the functional form of the selected objective function and the presence of hard constraints on the optimization variables. Debling et al. (1994) applied a heuristic approach based on industrial practice to solve the optimal grade transition problem for solution, slurry and gas-phase ole$n polymerization processes. The POLYRED simulation package was used to assess the performance of di9erent grade transition strategies. Dabedo, Bell, McLellan, and McAuley (1997) and Ali, Abasaeed, and Al-Zahrani (1998) studied the stability and multiplicity of steady states in industrial gas-phase polyole$n FBRs in terms of the cata- lyst feed rate, super$cial gas velocity and temperature of the coolant water. They also compared the performance of di9erent types of non-linear model-based controllers (e.g., error trajectory and model predictive control) with that obtained under conventional PID control. Recently Takeda and Ray (1999) studied the optimal grade transition problem for a multistage polyole$n loop reactor, using the control vector parameterization method. They de$ned a product speci$cation band at the end of the changeover time and assumed that the reactor temperature was perfectly controlled, thereby bypassing a major issue for such polymerization systems. In the present study, the optimal grade transition problem is examined in relation to an industrial Ziegler–Natta catalytic gas-phase ethylene-1-butene polymerization FBR. To take into account the e9ect of the selected closed-loop control con$guration (e.g., control pairings among the available manipulated and controlled variables) a mixed integer dynamic optimization (MIDO) approach is adopted. The solution of the resulting optimization problem involves the optimal selection of a number of discrete variables (e.g., best control pairings), the optimal values of the tuning parameters (e.g., gain and integral time) of the regulatory feedback controllers and the optimal trajectories of the feedforward “polymer-quality” controllers. The gPROMS/gOPT (Process Systems Enterprise Ltd.) computational tools for modelling and dynamic process optimization purposes, and the GAMS/CPLEX MILP solver are employed for the calculation of the time optimal grade transition policies and the selection of the “best” multivariable control con$guration. The paper is organized into $ve sections. In the following section, a comprehensive dynamic model is developed to describe the copolymerization of ethylene with 1-butene in the presence of a multi-site Ziegler–Natta catalyst. Dynamic molar species and energy balances are derived to follow the concentrations of the two monomers, the reaction temperature and the average molecular and compositional properties of the copolymer (e.g., number and weight average molecular weights, overall copolymer composition) in the FBR. The resulting di9erential-algebraic equations are solved using the gPROMS simulator. In the third section, the general optimal grade transition problem is examined in relation to an industrial ole$n polymerization FBR. More speci$cally, process and “polymer-quality” control objectives are de$ned and the set of available manipulated variables is identi$ed. Then, the general dynamic optimization problem is stated with respect to the minimization of a general objective function. Subsequently, the theoretical background for the solution of the combined grade transition and control structure selection problem is developed. In section four, detailed simulation results are presented on the calculation of the time optimal grade transition policies for a $xed control structure and the solution of the combined optimal grade transition/control structure selection (MIDO) problem. In the last section of the paper, the main conclusions of this work are summarized. C. Chatzidoukas et al. / Chemical Engineering Science 58 (2003) 3643 – 3658 3645 Fig. 1. Schematic representation of a gas-phase ethylene polymerization FBR unit. 2. The gas-phase catalytic ethylene copolymerization process Gas-phase solid catalyzed ole$n polymerization has long been recognized as one of the most eNcient processes for producing polyole$ns. The moderate operating conditions, the absence of solvents, as well as the high catalyst activity are the main advantages of the gas-phase process. In a catalytic gas-phase ole$n polymerization FBR (see Fig. 1), catalyst particles are continuously fed into the reactor, at a point above the gas distributor, and react with the incoming 'uidizing reaction medium (e.g., monomers, H2 , N2 ) to produce a broad distribution of polymer particles. The particulate polyole$n product is continuously withdrawn from the reactor at a point, preferably, close to the bottom of the bed. The recycled and make-up monomer feed streams are continuously fed to the reactor. An external heat exchanger is employed for the removal of the polymerization heat from the recycle gas stream. Industrial polyole$n FBRs typically operate at temperatures of 75 –110◦ C and pressures of 20 –40 bar (Xie, McAuley, Hsu, & Bacon, 1994). The super$cial gas velocity in the reactor is of the order of 50 –70 cm=s. The single-pass monomer conversion in the FBR can vary from 2% to 5%, whereas the overall monomer conversion can be as high as 98% (McAuley, Talbot, & Harris, 1994). Table 1 Kinetic mechanism of ethylene-1-butene copolymerization over a Ziegler– Natta catalyst kk Activation by aluminium alkyl: aA k Spk + A→P 0 Chain initiation: k P0k + Mi →P1; i Propagation: k + M → Pk Pn; i i n+1 Spontaneous deactivation: P∗k → Cdk + Dnk Spontaneous chain transfer: k → P k + Dk Pn; n i 0 Chain transfer by hydrogen (H2 ): k + H → P k + Dk Pn; 2 n i 0 Chain transfer by monomer (Mi ): k + M → P k + Dk Pn; i n i 1 k k0; i k kp; ij k kdsp k ktsp; i k ktH; i k ktm; ij 2.1. Polymerization kinetics In the present study, a comprehensive mechanism was considered to describe the copolymerization kinetics of ethylene with 1-butene over a Ziegler–Natta catalyst (see Table 1). The kinetic mechanism comprises a series of elementary reactions including site activation, propagation, site deactivation and chain transfer reactions. The symbol Pn;k i denotes the concentration of “live” copolymer chains of total length ‘n’ ending in an ‘i’ monomer unit, formed at the ‘k’ catalyst active site. P0k and Dnk denote the 3646 C. Chatzidoukas et al. / Chemical Engineering Science 58 (2003) 3643 – 3658 numerical values of the kinetic rate constants are reported in Table 3. Table 2 Net production–consumption rates of the various molecular species Potential catalyst sites of type ‘k’: k [A] S k RkSp = −kaA p 2.2. The FBR model Vacant catalyst sites of type ‘k’: Nm k k k k k RkP0 = −kdsp P0 − k0;k i P0k [Mi ] − Rksp + (ktsp; T + ktH; T [H2 ])0 i=1 Pseudo-kinetic rate constant for chain transfer reactions: k k k rt;k T = ktsp; T + ktH; T [H ] + ktm; TT [MT ] Zero-order moment rate of ‘live’ polymer chains: k k − r k + [M ]k k k Rk = 0k −kdsp T tm; TT + k0; T P0 [MT ] t; T 0 First-order moment rate of ‘live’ polymer chains: k k − rk k k k k Rk = 1k −kdsp t; T + 0 [MT ] (ktm; TT + kp; TT ) + k0; T P0 [MT ] 1 Second-order moment rate of ‘live’ polymer chains: k − rk k k k k k k k Rk = 2k −kdsp t; T + ktm; TT 0 [MT ] + kp; TT [MT ] (0 + 21 ) + k0; T P0 [MT ] 2 Zero-order moment rate of ‘dead’ polymer chains: k + rk Rk0 = 0k kdsp t; T First-order moment rate of ‘dead’ polymer chains: k + rk Rk1 = 1k kdsp t; T Second-order moment rate of ‘dead’ polymer chains: k + rk Rk2 = 2k kdsp t; T Monomer consumption rate: k k k RkMi = [Mi ] k0;k i P0k + (ktm; Ti + kp; Ti )0 ] Hydrogen consumption rate: k k RkH2 = ktH; T 0 [H2 ] d[Mi ] Fin XMi ;in − Frec XMi Q0 [Mi ] − = dt MWi bed Vbed Vbed Overall copolymerization rate: Rp = Ns Nm One of the main assumptions in modelling the operation of a catalytic ole$n polymerization FBR regards the number of phases present in the bed as well as the mixing conditions in each phase. This has been the subject of several publications (Choi & Ray, 1985; Talbot, 1990; Shiau & Lin, 1993; McAuley et al., 1994; Hatzantonis et al., 2000). Based on the results of the previous investigators, the FBR was approximated by a single-phase continuous stirred tank reactor. Under normal operating conditions, the above assumption holds true for the majority of industrial FBRs (Jenkins, Jones, Jones, & Beret, 1986; Chinh, Filippelli, Newton, & Power, 1996). Since no separate bubble phase is included in the model, the bed voidage, bed , accounts for the overall gas volume fraction in the bed. The assumption of perfect mixing in the bed implies that the temperature and concentrations of the various molecular species will be independent of their position in the bed. Furthermore, it was assumed that mass and heat transfer resistances between the polymer particles and the gas phase were negligible and the catalyst contained two types of active sites. Based on the above assumptions, the following dynamic molar balances for the two monomers, hydrogen and nitrogen are derived. Monomer i: (RkMi MWi ) N i=1 k=1 − s (1 − bed ) [Mi ]A dh ; RkMi − bed Vbed dt (3) k=1 concentrations of the activated vacant catalyst sites of type ‘k’ and “dead” copolymer chains of length ‘n’ produced at the ‘k’ catalyst active site, respectively. All other symbols are explained in the nomenclature section. For multicomponent polymerizations, the use of pseudokinetic rate constants can considerably simplify the kinetic rate expressions (Carvalho de, Gloor, & Hamielec, 1989; McAuley, McGregor, & Hamielec, 1990; Hutchinson, Chen, & Ray, 1992; Hatzantonis, Yiannoulakis, Yiagopoulos, & Kiparissides, 2000). Based on the proposed kinetic mechanism (see Table 1) and the de$nition of the moments of the “live” ( ) and “dead” ( ) total number chain length distributions (TNCLDs), ∞ ∞ n [Pn;k 1 ] + n [Pn;k 2 ] (1) k = ;k 1 + ;k 2 = k = ∞ n=1 n [Dnk ] n=1 (2) n=2 the net production/consumption rates of the various molecular species in the FBR can be derived (see Table 2). The Hydrogen: Ns d[H2 ] FH2 + Fin XH2 ;in − Frec XH2 1 − bed − RkH2 = dt MWH2 Vbed bed bed k=1 − Q0 [H2 ] [H2 ]A dh − ; Vbed Vbed dt (4) Nitrogen: d[N2 ] FN2 + Fin XN2 ;in − Frec XN2 = dt MWN2 Vbed bed − Q0 [N2 ] [N2 ]A dh ; − Vbed Vbed dt (5) where RkMi , RkH2 are the monomer and hydrogen consumption rates at the catalyst active site of type ‘k’. Xi and Xi; in are the mass fractions of species ‘i’ in the recycle and input streams, respectively. Similarly, the mass balances for the potential catalyst sites, Spk , and all other molecular species C. Chatzidoukas et al. / Chemical Engineering Science 58 (2003) 3643 – 3658 3647 Table 3 Numerical values of the kinetic rate constants for a two-site Ziegler–Natta catalyst Pre-exponential factor (cm3 = mol= s) Site type Activation energy 1 2 (kcal/mol) Site type Activation energy 1 2 (kcal/mol) Activation Propagation k kaA 102 k EaA k kp; 11 102 9 9 Initiation k0;k 1 E0;k 1 k0;k 2 E0;k 2 1× 103 9 1× 103 9 0:14 × 103 9 0:14 × 103 9 Chain transfer k a ktsp; i k Etsp; i k ktH; i k EtH; i k ktm; 11 k ktm; 12 k ktm; 21 k ktm; 22 k Etm; ij 1× a Units Pre-exponential factor (cm3 = mol= s) 10−4 1× 8 8 88 370 8 8 2.1 2.1 6 110 2.1 1 6 110 8 8 10−4 85 × 103 85 × 103 k Ep; 11 9 9 k kp; 12 2 × 103 15 × 103 k Ep; 12 k kp; 21 k Ep; 21 k kp; 22 9 9 64 × 103 64 × 103 9 9 1:5 × 103 6:2 × 103 k Ep; 22 9 9 1 × 10−4 1 × 10−4 8 8 Deactivation k a kdsp k Edsp in s−1 . Y k , (Y k : P0k ; 0k ; 1k ; 2k ; 0k ; 1k ; 2k ) can be derived: k k Fcat = [%cat (1 − bed )] Sp; d Spk in − Q0 Sp = dt Vbed k Sp A dh +RkSp − ; Vbed dt k k k Q0 Y Y A dh dY − = RkY − ; dt Vbed Vbed dt The dynamic energy balance for the reaction mixture in the bed is written as (6) (7) where RkY denotes the net formation rate of the molecular species Y k (see Table 2). The symbols h, Vbed , and A denote the bed height, the volume and the cross-sectional area of the bed, respectively. Accordingly, one can derive the unsteady-state mass balance for the polymer in the bed N Ns s Vbed dh = RkM1 MW1 + RkM2 MW2 dt %A k=1 (Hgas; in − Hgas; out − Hprod; out + Hgenr ) TA dh dT − = ; dt Haccum Vbed dt (9) where the terms Hgas; in , Hgas; out , Hprod; out , and Hgenr denote the enthalpies of the input, output and product removal streams and the heat of polymerization, respectively. Assuming that the dynamic behaviour of the external heat exchanger (see Fig. 1) can be approximated by a series of Nz well-stirred zones for the recycle stream and a single well-stirred zone for the coolant, the following energy balances can be written Mgas; j Cp; mean j = 1; 2; : : : ; Nz ; k=1 Fcat − Q0 (1 − bed )% + ; (1 − bed )%A (8) where % and %cat are the corresponding densities of polymer and catalyst. dTj = Frec Cp; mean (Tj−1 − Tj ) − Qj ; dt Q= Nz j=1 Mc Cp; w Qj = Nz {Uj Aj (Tj − Tw )}; (10) (11) j=1 dTw = Fw Cp; w (Tw; in − Tw ) + Q: dt (12) 3648 C. Chatzidoukas et al. / Chemical Engineering Science 58 (2003) 3643 – 3658 Table 4 Reactor and heat exchanger design parameters Bed diameter Bed voidage Catalyst density Overall heat exchanger area Overall heat transfer coeNcient Coolant inlet temperature Dbed = 2:5 m bed = 0:5 %cat = 2840 kg=m3 A = 255 m2 Uj = 1000 J=K=m2 =s Tw; in = 288:15 K In Table 4, the numerical values of the reactor and heat exchanger design parameters are reported. The average polymer properties of interest (i.e., number and weight average molecular weights, polydispersity index and copolymer composition) can be calculated in terms of the “bulk” moments of the TNCLDs and the monomer consumption rates at the various catalyst sites (Hatzantonist et al., 2000). Thus, the “instantaneous” copolymer composition, ’i , for a catalyst having Ns active sites, will be given by the consumption rate of the ‘i’ monomer over the total consumption rate of the Nm monomers: N Ns Ns m k ’i = RM i RkMi : (13) i=1 k=1 d(Mp +i ) = Fp; in +i; in − Fp; out +i dt + Vbed (1 − bed )Rp ’i ; (14) where Mp , Fp; in and Fp; out denote the total polymer mass in the bed and the input and output polymer mass 'ow rates, respectively. Accordingly, the number and weight average molecular weights of the copolymer will be given by the following equations: N Ns s Mn = MW (1k + 1k ) (0k + 0k ); (15) k=1 Mw = MW k=1 (2k + k=1 2k ) N s Nm +i MWi : + 1k ); (16) k=1 (17) i=1 Finally, the PD will be given by the ratio of the weight average over the number average molecular weight. PD = Mw =Mn : (19) MI = aMwb : (20) The values of the c0 , c1 , c2 , a and b parameters were calculated by $tting proprietary industrial measurements on MI and % to o9-line measurements of Mw and +2 (i.e., the cumulative copolymer composition of butene). 3. The optimal grade transition problem Commonly, the numerical solution to the optimal grade transition problem is based on the minimization of a suitable objective function de$ned in terms of the changeover time, product-quality speci$cations, process safety constraints and the amount of o9-spec polymer, using dynamic optimization methods. However, an essential requirement for the application of the calculated optimal control trajectories to the process is the selection of the closed-loop feedforward and feedback controllers and the estimation of the feedback controllers’ tuning parameters (i.e., proportional gain and integral time). This means that the time optimal grade transition problem for the FBR needs to be solved simultaneously with the selection of the “best” control pairings between the available manipulated variables and the speci$ed process and “polymer-quality” objectives (i.e., controlled variables). 3.1. Control structure selection (1k where MW is the average molecular weight of the repeating unit in the copolymer chains. MW = % = c0 + c1 exp(−+2 =c2 ); k=1 To calculate the cumulative copolymer composition, +i , in the reactor, during a transient operation, the following dynamic mass balance equation needs to be solved: Ns It is well known that the direct on-line measurement of the polymer molecular properties is not practically feasible (Kammona, Chatzi, & Kiparissides, 1999). Thus, easily available on-line measurements of melt index (MI) and density (%) are often utilized to control the “polymer quality” in a polymerization reactor. In the literature, several correlations (McAuley & MacGregor, 1991; Kozub & MacGregor, 1992; Xie et al., 1994; Ogunnaike, 1994) have been proposed to relate MI and % with the weight average molecular weight and copolymer composition, respectively. In the present study, the following semi-empirical equations were employed: (18) The identi$cation of the appropriate control variables that mostly a9ect the “polymer-quality” variables (i.e., %, MI, etc.) is imperative to the calculation of an economically feasible grade transition policy so that relatively small changes in the manipulated variables will be suNcient to realize the grade transition objectives. In addition, one needs to specify the necessary control loops (i.e., pairings of controlled and manipulated variables) to maintain the process within a safe operating envelope and to ensure closed-loop process stability and the rejection of the various process disturbances (e.g., time-varying catalyst activity) (Choi & Ray, 1985; McAuley & MacGregor, 1993; Dabedo et al., 1997; C. Chatzidoukas et al. / Chemical Engineering Science 58 (2003) 3643 – 3658 Ali, Abasaeed, & Al-Zahrani, 1999). For example, the selected control con$guration must keep the temperature in the bed below the polymer softening temperature to prevent catastrophic particle agglomeration. For process safety and operability, the reactor temperature (T ) and pressure (P) as well as the bed height (h) need to be maintained at speci$ed operating points. Regarding the product quality, the polymer density (%) and the melt index (MI) are usually controlled. Finally, the polymer production rate (Rp ) is often controlled to maintain the reactor productivity at a desired level. A schematic representation of a gas-phase catalytic ole$n polymerization FBR is depicted in Fig. 1. One can identify nine possible manipulated variables, namely, the monomer and comonomer mass 'ow rates (Fmon1 , Fmon2 ) in the make-up stream; the hydrogen, nitrogen and catalyst mass 'ow rates (FH2 , FN2 , Fcat ); the mass 'ow rate of the bleed stream (Fbleed ); the mass 'ow rate of the product removal stream (Fout , including both polymer and gases), the recycle stream (Frec ), and the mass 'ow rate of the coolant water stream to the heat exchanger (Fw ). In practice, instead of manipulating the comonomer mass 'ow rate, Fmon2 , the ratio of the comonomer to the monomer in'ow rate in the make-up stream (Ratio = Fmon2 =Fmon1 ) is selected as a manipulated variable. From the above analysis, it is apparent that there are six controlled variables and nine possible manipulated variables. Actually, the mass 'ow rate of the recycle stream will depend on the super$cial gas velocity of the inlet gas stream. Therefore, the remaining eight manipulated variables can be employed to control the six output variables (i.e., the three process variables (T , P and h), the two “polymer-quality” variables (% and MI) and the polymer production rate, Rp ). There is a great number of publications dealing with the problem of optimal controller synthesis for a chemical plant. In the past, several controller synthesis criteria, including process controllability, process economics, etc., have been employed for the selection of the “best” control structure con$guration (Kravaris & Kantor, 1990; Narraway & Perkins, 1993; Cao & Rossiter, 1997; Heath, Kookos, & Perkins, 2000; Kookos & Perkins, 2002). In polymerization, the combined problem of optimal controller selection and process optimization during a grade transition has been addressed by several investigators (Kravaris et al., 1989; Kozub & MacGregor, 1992; Ogunnaike, 1994; Meziou, Deshpande, Cozewith, Silverman, & Morisson, 1996; Dabedo et al., 1997; Ali et al., 1998; Ohshima & Tanigaki, 2000). However, in all previous publications, the traditional sequential approach was employed. This means that a control system architecture is $rst identi$ed or/and assumed and then the optimal open- or/and closed-loop control problem is solved. In general, the selection of the control con$guration is based on both heuristic rules and classical methods, including the relative gain array (RGA), the singular value decomposition (SVD), etc. 3649 3.2. The time optimal control problem In a grade transition problem, the time optimal trajectories of the control variables are sought to drive the process from one set of operating conditions to a new one, while minimizing at the same time a certain objective function. In general, the dynamic optimization problem can be stated as Min J = G(x(tf ); y(tf ); tf ) uopt ; tf + tf t0 L(x(t); y(t); uopt (t); t) dt s:t: ẋ = f(x(t); uopt (t); d; t) y = h(x(t); uopt (t); d; t) 0 6 g(x(t); uopt (t); d; t); (I) where t0 and tf denote the initial and $nal transition times. G and L are scalar functionals of the state, x, control, u, and output, y, variables. The functions f, h and g comprise the modelling equality and inequality constraints that must be satis$ed at all times. The vector uopt (t) denotes a time optimal control trajectory that forces the process to follow an admissible state trajectory, while minimizing a certain objective function. The vector d denotes the values of the model parameters and process disturbances. It should be noted that the total transition time, tf , can be treated either as an additional decision variable or as a constant. The optimization problem (I), which generally involves a large number of di9erential and algebraic equations (DAEs), can be solved numerically using well-known discretization methods (e.g., simultaneous and sequential). In the simultaneous solution method (Tjoa & Biegler, 1991), both state and control vectors are discretized in time, leading to the transformation of the DAE system into a set of purely algebraic equations. Accordingly, the resulting system of non-linear algebraic equations is solved using any conventional non-linear programming method. In the sequential solution method (Vassiliadis, Sargent, & Pantelides, 1994), only the control vector is parameterized in time. The resulting system of DAEs is then solved for a given set of values of the discretized control vector using a DAE integrator. In this case, the integrator conveys sensitivity information to the optimizer and receives in response from the optimizer the calculated discrete optimal control values. 3.3. The combined optimization problem The main disadvantage of the sequential approach discussed in Sections 3.1 and 3.2 is that the control structure is selected independently of the optimal transition problem and, therefore, the optimality of the selected control 3650 C. Chatzidoukas et al. / Chemical Engineering Science 58 (2003) 3643 – 3658 structure cannot be guaranteed. To address this issue, an MIDO method can be employed to calculate simultaneously the optimal control con$guration and the optimal transition policy (Mohideen, Perkins, & Pistikopoulos, 1996; Schweiger & Floudas, 1997; Algor & Barton, 1999; Androulakis, 2000; Bansal, 2000). MIDO algorithms are based on decomposition principles (e.g., the generalized benders decomposition, GBD). In general, an iterative MIDO algorithm decomposes the overall problem into two interactive sub-problems, namely, the “primal” and the “master” sub-problem (Bansal, Perkins, & Pistikopoulos, 2002). In the “master” sub-problem, the “best” control pairings among the available manipulated and controlled variables are identi$ed using a mixed integer linear programming (MILP) method. The solution to the “master” sub-problem provides a lower bound to the $nal solution of the combined problem. In the “primal” level, the dynamic optimization problem is solved using the “current optimal” control structure con$guration. The latter provides an upper bound to the $nal optimal solution and dual information (Lagrange multipliers) to the “master” sub-problem. This iterative procedure continues until satisfactory convergence (e.g., within a speci$c tolerance) between the upper and lower bound solutions has been achieved. When the selection of the optimal control structure is coupled with the dynamic optimization problem, optimal continuous decision variables (corresponding to the optimal control trajectories and the tuning parameters of the feedback controllers) and optimal discrete decision variables (corresponding to the “best” control pairings) are identi$ed. As a result, the complexity of the combined (e.g., continuous and discrete) optimization problem considerably increases. In the present study, the two “product-quality” variables (i.e., % and MI) were held under optimal feedforward control, while the remaining four process variables (i.e., T; P; h, and Rp ) were kept under PI feedback control. A multivariable control con$guration among the six controlled variables (Yj : T; P; h; Rp ; %; MI) and the eight manipulated variables (Ui : Fmon1 ; Ratio = Fmon2 =Fmon1 ; FH2 ; FN2 ; Fcat ; Fout ; Fbleed and Fw ) was identi$ed using an MIDO approach. In the “master” sub-problem, the values of time invariant binary variables, bi; j , were calculated. The binary variable bi; j was set equal to 1 when the ‘i’ manipulated variable was coupled with the ‘j’ controlled variable. In any other case, the value of bi; j was set equal to zero. The GAMS/CPLEX MILP algorithm was employed for solving the “master” sub-problem. In the “primal” sub-problem the time optimal control trajectories of the two “product-quality” feedforward controllers and the tuning parameters (i.e., gains and integral times) of the four PI feedback process controllers were estimated using the gOPT optimizer of PSE Ltd. Accordingly, the overall control action for the ‘i’ manipulated variable was calculated by adding the contributions of both feedback and feedforward controllers: Ui; total (t) = Ui; feedback (t) + Ui; feedforward (t) (21) or Ui; total (t) = 4 bi; j Kc; ij Ej (t) + j=1 6 {bi; j Ui; opt } + 1 3I; ij o t Ej (t) dt (22) j=5 Kc; ij and 3I; ij denote the gain and integral time of the PI feedback controller for the (i; j) pair of manipulated-controlled variables. The di9erence term Ej (=Yj; sp − Yj ) is the error between the set point and measured values of the ‘j’ controlled variable. Ui; feedforward is the time optimal trajectory for the ‘i’ manipulated variable calculated by the gOPT optimizer, as a sequence of piecewise constant values. It is important to point out that the total number of possible “pairings” between the six controlled and the eight available manipulated variables is prohibitively large. Therefore, heuristic rules based on physical limitations on the possible control alternatives were employed to eliminate infeasible pairings (i.e., the pairing of bed height with the coolant 'ow rate) and reduce the binary search space in the “master” sub-problem. 4. Results and discussion In the present study, the time optimal control policies for the transition from grade A to grade B and back to grade A were determined for an ethylene-1-butene copolymerization FBR for both $xed and variable control con$gurations. It is well known that a number of important polymer end-use properties (e.g., sti9ness, transparency, hardness, etc.) as well as the rheological and processability characteristics of polyole$ns are directly linked with the values of % and MI. Thus, to minimize the transition time and the amount of o9-spec polymer produced during a grade changeover, an objective function expressed in terms of the time-dependent squared deviations of the polymer density and melt index from their corresponding desired values, was de$ned: 2 2 tf MI(t) − MIf %(t) − %f J= dt: (23) + MI0 − MIf %0 − % f to The subscripts 0 and f denote the values of the corresponding “polymer-quality” variables at the start and the $nal time of a grade transition. It should be noted that the selected form of the objective function ensures the satisfaction of the “product-quality” speci$cations while, at the same time, minimizes the transition time since the $nal time, tf , is treated as an additional control variable. Needless to say that additional terms (e.g., the polydispersity index, the amount of o9-spec polymer, etc.) could be included in the objective function resulting in alternative optimal transition policies. The selected product speci$cations for grades A and B as well as the corresponding operating conditions at steady C. Chatzidoukas et al. / Chemical Engineering Science 58 (2003) 3643 – 3658 Table 5 Operating conditions and product speci$cations for grades A and B 0.932 Operating conditions Grade A Grade B 0.930 Grade A hsp (m) Tsp (K) Psp (bar) Rp; sp (g/s) Fbleed (g/s) Frec (g/s) 6.0 360 21 2390 0.1 1:33 × 105 6.0 360 21 2390 0.1 1:33 × 105 0.928 Density-OP1 Density-OP2 Product speci?cations Mw (g/mol) +2 % (g=cm3 ) MI 3:8 × 105 0.024 0:9299 (±0:05%) 0:01376 (±4%) 2:9 × 105 0.046 0:91904 (±0:05%) 0:03604 (±4%) Table 6 Best pairings of manipulated and controlled variables based on the RGA analysis Manipulated variables Product withdrawal rate (Fout ) Cooling water feed rate (Fw ) Nitrogen feed rate (FN2 ) Monomer feed rate (Fmon1 ) Comonomer ratio (Ratio) Hydrogen feed rate (FH2 ) Controlled variables → → → → → → Bed height (h) Temperature (T ) Pressure (P) Production rate (Rp ) Density (%) Melt index (MI) state, are reported in Table 5. As can be seen, the transition from grades A to B results in a polyole$n having a higher melt index, MI, and a lower density, %. An opposite behaviour is obtained for the transition from grades B to A. In what follows, simulation results on the optimal grade transition and selection of control structure for a polyole$n FBR are presented. 4.1. Fixed control structure In this case, a multiple-input, multiple-output control con$guration was $rst identi$ed via the application of the RGA analysis to a linearized form of the FBR model. For the RGA analysis, six controlled variables (i.e., T; P; h; %; MI and Rp ) and eight manipulated variables (i.e., Fmon1 , Ratio, FH2 ; FN2 , Fcat , Fout , Fbleed and Fw ) were considered. Table 6 shows the “best” control pairings resulted from the application of the RGA analysis. The remaining two control variables (Fbleed , Fcat ) were held constant at their optimal values found via the solution of a static optimization problem that maximized the polymerization rate. According to the RGA results (see Table 6), the $rst four manipulated variables (i.e., Fout , Fw , FN2 , Fmon1 ) were employed for the feedback control of the bed height, temperature, pressure and production rate, while the remaining two control variables (i.e., Ratio, FH2 ) were chosen for the dynamic optimization of the “polymer-quality” variables (i.e., % and MI). 3651 3 Density (g/cm ) ±0.05% ~ 70 min 0.926 0.924 0.922 0.920 ±0.05% Grade B 0.918 0 100 200 300 400 500 600 700 800 Time (min) Fig. 2. Calculated optimal density pro$les for the transition from grades A to B and back to A for a $xed control structure obtained via the RGA analysis. The calculation of the time optimal trajectories for the comonomer to monomer ratio and the hydrogen feed rate and the optimal values of the tuning parameters (i.e., gain and integral time) of the four PI feedback controllers, were calculated by minimizing the objective function (23) subject to a set of equality constraints, Eqs. (1)–(20). In fact, the total number of model DAEs was equal to 343 (including 34 di9erential and 309 algebraic equations). On the other hand, the total number of calculated discrete control moves and single-value control parameters varied from 28– 68, depending on the number of selected time-varying control variables. It should be pointed out that no path constraints on the three process variables and the polymer production rate were imposed. On the other hand, upper and lower end-point constraints on T; P; h; Rp ; % and MI were set to ensure the satisfaction of the speci$ed values at the $nal time, tf . It is important to point out that, due to the slow dynamics of MI caused by the polymer mass accumulation, an end-point constraint on the time derivative of MI was also introduced to ensure the attainment of a steady state. In the present study, the gOPT sequential optimizer of PSE Ltd was employed for the solution of the time optimal control problem. In general, 30 – 40 iterations were required for the convergence of the sequential optimizer to the optimal solution. In Figs. 2 and 3, the calculated optimal trajectories for % and MI (marked by the broken lines, OP1) are plotted with respect to time. As can be seen, for the A to B grade transition, the time required for the “polymer-quality” variables to reach their corresponding end-point speci$cations is less than 300 min. However, the optimizer continues to update the set-point values of the feedforward “polymer-quality” controllers till the FBR reaches a steady state. Subsequently, the FBR undergoes a reverse transition from grades B to A. It is important to point out that the time required for a grade transition largely depends on the direction of change of polymer properties. More speci$cally, the transition time required for a decrease of a polymer property (i.e., %, MI) 3652 C. Chatzidoukas et al. / Chemical Engineering Science 58 (2003) 3643 – 3658 pro$les (marked by the solid lines, OP2) for %, and MI are plotted for the “best” alternative grade transition policy. As can be seen when the four control variables are used to minimize the objective function (23), the total transition time as well as the amount of o9-spec polymer are reduced by 5.4% and 6%, respectively. 0.040 0.038 0.036 Grade B ±4% Melt Index (MI) 0.034 0.032 MI-OP1 MI-OP2 0.030 ~ 50 mins 0.028 0.026 0.024 4.2. Variable control structure 0.022 ~ 20 min 0.020 0.018 0.016 0.014 Grade A ±4% 0.012 0 100 200 300 400 500 600 700 800 Time (min) Fig. 3. Calculated optimal MI pro$les for the transition from grade A to grade B and back to A for a $xed control structure obtained via the RGA analysis. from its current value to a lower one, is in general larger than that for a corresponding property increase. In fact, when the polymer density decreases during a grade transition (e.g., A → B), the butene composition in the bed increases, leading to an increase of the transition time due to the lower polymerization rate of butene. On the other hand, when the density increases (e.g., B → A), the transition time decreases because the polymerization rate of ethylene is higher than that of butene. The results of Fig. 2 are in full agreement with the previous kinetic justi$cation. In the case of MI, its increase or decrease is directly related to the weight average molecular weight of the polymer (see Eq. (20)), which is controlled by the hydrogen concentration in the bed. Thus, when the polymer melt index increases during a grade transition (e.g., from A to B), the hydrogen concentration in the bed increases, which lowers the value of Mw . On the other hand, a decrease in MI leads to a decrease of hydrogen concentration in the bed. However, due to the faster dynamics of hydrogen concentration repletion, the transition time for an increase in MI will be faster than that required for a corresponding decrease of MI (see Fig. 3). To further reduce the transition time, alternative heuristic approaches based on “best” industrial practice, were investigated. Thus, besides the two control variables identi$ed by the RGA analysis (FH2 , Ratio), additional control variables including the bleed 'ow rate, Fbleed , and the set points of the bed height, hsp , and production rate, Rp; sp , feedback controllers were considered for solving the optimal grade transition problem. Several combinations of the $ve control variables (i.e., FH2 , Ratio, Fbleed , hsp and Rp; sp ) were examined in order to minimize the total transition time and the amount of o9-spec polymer produced during the grade transition sequence from A to B and back to A. The best performance was obtained when, in addition to the time optimal trajectories of FH2 and Ratio, the set points of the bed height and production rate PI feedback controllers were optimally varied with respect to time. In Figs. 2 and 3, the optimal Subsequently, the time optimal grade transition problem was solved in combination with the optimal selection of the feedforward and feedback control loops, using an MIDO algorithm. In the “primal” sub-problem, an objective function similar to the one used in the $xed control-structure problem was employed (see Eq. (23)). Eqs. (1)–(22) comprised the system of equality DAEs. In addition, a set of continuous time invariant search variables, R, were introduced to represent the equivalent integer variables, b, identi$ed in the “master” sub-problem. Upper and lower end-point constraints on the process and “polymer-quality” variables were imposed as discussed in the $xed control-structure case. In the “master” sub-problem, a new objective function was formulated in terms of the objective function (23) and the Lagrange multipliers, , identi$ed at the “primal” sub-problem: L(b) = Jopt + T (b − R): (24) A set of equality and inequality constraints were imposed on the binary variables bi; j , to ensure that all the process variables were held under feedback control and a manipulated variable could be used for the feedforward control of only one “product-quality” variable. Note that the initial binary search space included 216 combinations. In Table 7, the imposed constraints on the binary variables are reported. As can be seen the coolant 'ow rate, Fw , and the product removal, Fout , are only used to control the temperature and the bed height, respectively. Moreover, the comonomer/monomer ratio and the hydrogen feed rate are always used in the respective feedforward controllers of the polymer density and melt index, which is consistent with the results of the RGA analysis. In Table 8, the initial control structure and the one calculated by the MIDO algorithm are reported for the transition from grade A to grade B. As can be seen the polymerization temperature is controlled by three manipulated variables (i.e., Fw , Ratio, Fcat ), while the polymer density and melt index are controlled by the set of variables (Ratio, Fbleed , Fcat ) and the FH2 , respectively. In general, 3– 4 iterations were required for the convergence of the MIDO algorithm to the optimal solution. It should be noted that the same optimal control structure was obtained for the transition from grade B to grade A. In Figs. 4 and 5, the calculated by the MIDO algorithm optimal pro$les for % and MI (marked by the broken lines, OP1) are plotted with respect to time. To further reduce the transition time, the set points of the bed height, hsp , C. Chatzidoukas et al. / Chemical Engineering Science 58 (2003) 3643 – 3658 3653 Table 7 Constraints imposed on the bi; j binary variables T P 1 Fw 0 FN 2 0 Fout 0 F mon1 (0; 1) Ratio 0 F H2 0 Fbleed F (0; 1) cat h Rp % MI 0 0 (0; 1) 0 0 0 0 1 (0; 1) 0 0 (0; 1) 0 0 0 0 (0; 1) 0 (0; 1) 0 0 0 0 0 (0; 1) (0; 1) (0; 1) (0; 1) 0 (0; 1) 0 0 0 0 0 (0; 1) 0 (0; 1) (0; 1) (0; 1) 6 b1; j 6 1 j=5 6 b2; j 6 1 j=5 6 b3; j 6 1 j=5 6 b4; j 6 1 j=5 6 b5; j 6 1 j=5 6 b6; j 6 1 j=5 6 b7; j 6 1 j=5 6 b8; j 6 1 j=5 bi; 1 ¿ 1 i bi; 2 ¿ 1 i bi; 3 ¿ 1 i bi; 4 ¿ 1 i bi; 5 ¿ 1 i bi; 6 ¿ 1 i Table 8 Calculated optimal control structure for the A to B grade transition {T Fw FN 2 Fout Fmon1 Ratio FH 2 Fbleed Fcat P h Rp % MI }{ T 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 Initial control structure and production rate, Rp; sp , feedback controllers were treated as additional control variables. The optimal pro$les for % and MI (marked by the solid line, OP2), derived under the second optimization policy, are also plotted in Figs. 4 and 5, respectively. It is apparent that the use of the two additional control variables signi$cantly improves the performance of the system, resulting in a decrease in the total transition time ! " " " " " " " " " " " " " " " " " " " " " " " " " P h Rp % MI } 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 0 ! Final control structure by 5%. At the same time, the amount of o9-spec polymer is further reduced by 7.7% (see also Fig. 6). Figs. 7–10 depict the calculated time optimal trajectories of the four optimization variables (i.e., Ratio, FH2 , hsp , Rp; sp ). Note that the calculated trajectories for the Ratio and FH2 represent the feedforward contributions of the two control variables to the multivariable controller given by 3654 C. Chatzidoukas et al. / Chemical Engineering Science 58 (2003) 3643 – 3658 0.932 Grade A Grade A Grade B Grade A ± 0.05% ~ 70 min 0.926 0.924 Density-OP1 Density-OP2 ~ 35 min 0.922 0.920 ± 0.05% 0.12 0.8 0.10 0.6 0.08 0.4 0.06 0.2 0.04 Grade B 0.918 0 100 200 300 400 500 600 700 0 800 100 200 300 400 500 600 0.0 800 700 Time (min) Time (min) Fig. 4. Calculated optimal density pro$les for the transition from grade A to grade B and back to A for a variable control structure obtained by an MIDO algorithm. Fig. 7. Time optimal control policies of the comonomer/monomer ratio and the hydrogen 'ow rate for the transition from grade A to grade B and back to A 0.040 0.036 Grade B 0.034 ± 4% Grade A 4.0 Bleed Flow Rate (g/s) Melt Index (MI) Grade B 5 0.032 0.030 0.028 0.026 0.024 0.022 ~ 70 min 0.020 4.5 Grade A MI-OP1 MI-OP2 0.038 4 3.5 3.0 3 2.5 2 2.0 0.018 0.016 1 0.014 1.5 Grade A ± 4% 0.012 0.010 0 0 100 200 300 400 500 600 700 800 0 100 200 300 Time (min) 400 500 600 1.0 800 700 Time (min) Fig. 5. Calculated optimal MI pro$les for the transition from grade A to grade B and back to A for a variable control structure obtained by an MIDO algorithm. Fig. 8. Time optimal control policies of the bleed and catalyst 'ow rates for the transition from grade A to grade B and back to A. 2.65 70 Grade A ~ 5 tn Grade A 50 40 ~ 2 tn 30 Grade B ~ 30 min 20 2.55 2.50 2.45 2.40 R p,sp Rp 2.35 Offspec-OP1 Offspec-OP2 10 Grade A Grade B 2.60 Polyme Production Rate (kg/s) 60 Off-spec Polymer (tn) Catalyst Feed Rate (g/s) Density (g/cm3 ) 0.928 1.0 Hydrogen Feed Rate(g/s) Comonomer to Monomer Feed Ratio 0.14 0.930 2.30 0 0 100 200 300 400 500 600 700 800 Time (min) Fig. 6. Amount of o9-spec polymer produced under di9erent optimization policies using an MIDO approach. 0 100 200 300 400 500 600 700 800 Time (min) Fig. 9. Time optimal set-point trajectory of the production rate feedback controller and time variation of the respective controlled variable. C. Chatzidoukas et al. / Chemical Engineering Science 58 (2003) 3643 – 3658 605 Grade B Grade A Grade A 70 600 ~ 10 tn 595 60 590 Off-spec Polymer (tn) Bed Level (cm) 3655 585 580 575 570 565 560 hsp h 555 50 40 ~ 6 tn 30 ~ 150 min 20 10 Fixed Control Structure MIDO 550 0 100 200 300 400 500 600 700 800 Time (min) Fig. 10. Time optimal set-point trajectory of the bed height feedback controller and time variation of the respective controlled variable. Eq. (22), while the calculated trajectories for hsp and Rp; sp are applied to the respective PI feedback controllers as piecewise set-point changes. In Fig. 7, the optimal trajectories for Ratio and FH2 are illustrated for the total transition from grade A to grade B and back to grade A. It is apparent that for the A to B transition (e.g., increase of MI and decrease of %), the comonomer to monomer feed ratio increases, leading to a decrease of the polymer density caused by the higher incorporation rate of 1-butene in the copolymer chains. Similarly, the hydrogen feed rate initially increases, which brings about an increase in the H2 concentration in the bed, resulting in a decrease of the molecular weight of the polymer and in an analogous increase of MI. On the other hand, for the transition from grades B to A, the two control variables are optimally reduced to their initial starting values. In Fig. 8, the optimal variations of the bleed and catalyst 'ow rates are plotted. It can be seen that for the A→B grade transition, the catalyst 'ow rate increases to compensate for the lower reactivity of 1-butene. Moreover, the bleed 'ow rate increases to accelerate the transition of the comonomer and monomer gaseous concentrations to the desired values. For the B→A grade transition, the catalyst 'ow rate is optimally reduced to its starting value, while the bleed 'ow rate initially operates at its upper limit to speed up the change of the comonomer/monomer composition in the gas phase to its optimal value. However, the end of the transition, it also returns to its initial operating value. Figs. 9 and 10 depict the calculated optimal set-point trajectories of the two process controllers as well as the actual values of the respective controlled variables. It should be noted that the polymer production rate PI controller closely follows the set-point changes. On the other hand, the bed height PI controller exhibits signi$cant overshooting when a decrease in the bed height is required. From the results of Figs. 9 and 10, it can be concluded that, independently of the direction of the grade transition, a decreasing policy with respect to the bed height and an increasing policy with 0 0 100 200 300 400 500 600 700 800 900 Time (min) Fig. 11. Amount of o9-spec polymer produced under a $xed (RGA) and a variable (MIDO) control structure. respect to the production rate are always required to reduce the changeover time. Finally, in Fig. 11, the o9-spec amount of polymer produced under both $xed and variable control structures is plotted with respect to the transition time. It is evident that a signi$cant improvement in the reactor performance is obtained when an MIDO approach is employed for the solution of the combined optimal grade transition problem and the selection of an optimal control structure. In this case, the total transition time as well as the amount of o9-spec polymer are reduced from their respective values obtained under a $xed control structure, by 17.7% and 15%, respectively. 5. Conclusions The optimal grade transition operation of a polymerization plant in terms of increased productivity and improved product quality can only be achieved when the process is e9ectively controlled. Since the closed-loop control system con$guration substantially a9ects the calculation of the time optimal control policies, it is of paramount importance the simultaneous solution of the combined optimal grade transition problem and the selection of the “best” closed-loop feedback/feedforward controllers for the plant. In the present study, an MIDO method was applied to a gas-phase ole$n copolymerization FBR to calculate the time optimal control policies for a sequence of grade transitions and identify the “best” closed-loop feedforward and feedback controllers in order to maintain the process within a safe operating envelope and ensure the faithful implementation of the calculated optimal control policies to the plant. Based on the results of the MIDO analysis, four feedback controllers were identi$ed to control the three process variables (i.e., temperature, pressure and bed height) and the polymer production rate. For the control of the “polymer 3656 C. Chatzidoukas et al. / Chemical Engineering Science 58 (2003) 3643 – 3658 quality” (i.e., polymer density and melt index), two feedforward controllers were identi$ed. The six output variables were controlled by the combined action of eight manipulated variables (i.e., Fw , FN2 , Fout , Fmon1 , Ratio, FH2 , Fbleed and Fcat ). To further reduce the transition time and the amount of o9-spec polymer produced during a grade transition sequence (i.e., A → B → A), the set points of the bed height and production rate feedback controllers were treated as additional control variables, which signi$cantly improved the overall process performance. Moreover, it was shown that the simultaneous solution to the combined problem of optimal grade transition and selection of closed-loop control structure resulted in a superior performance of the FBR, in terms of both reduced transition time and amount of o9-spec polymer, over that obtained under a $xed control structure derived by the RGA analysis. [Cdk ] Cp; mean Cp; w [Dnk ] Fbleed Fcat FH2 Fmon1 ; Fmon2 FN2 Fout Fp Frec [H2 ] Haccum Hgas; in Hgas; out Hgenr Hprod; out ka kdsp k0 kp ktsp [Mi ] Mc Mgas Mn Mp [MT ] MW Mw [N2 ] Nz Nm Ns [P0 ] Notation A [A] kt cross-sectional area, m2 aluminium alkyl cocatalyst concentration, mol=m3 concentration of deactivated catalyst sites of type ‘k’, mol=m3 speci$c heat capacity of the reaction mixture in the recycle stream, cal/g/K speci$c heat capacity of water, cal/g/K concentration of “dead” copolymer chains of length ‘n’ produced at ‘k’ catalyst active site, mol=m3 bleed 'ow rate, g/s catalyst feed rate, g/s hydrogen feed rate into the bed, g/s monomer and comonomer make-up feed rates, g/s nitrogen feed rate into the bed, g/s total product removal rate, g/s polymer 'ow rate, g/s recycle 'ow rate, g/s hydrogen concentration in the bed, mol=m3 accumulation enthalpy term, cal=K=m3 gas input enthalpy rate, cal=s=m3 gas output enthalpy rate, cal=s=m3 polymerization heat rate, cal=s=m3 product output enthalpy rate, cal=s=m3 kinetic rate constant of activation reaction, m3 =mol=s kinetic rate constant of spontaneous deactivation reaction, s−1 kinetic rate constant of initiation reaction, m3 =mol=s kinetic rate constant of propagation reaction, m3 =mol=s [Pn; i ] [P∗ ] Q Q0 [Sp ] T Trec Tw; in Tw U Vbed XM i XH2 XN2 kinetic rate constant of chain transfer reaction, m3 =mol=s kinetic rate constant of spontaneous chain transfer reaction, s−1 monomer concentration in the bed, mol=m3 coolant mass in the heat exchanger, g total mass of gases in the heat exchanger, g number average molecular weight of polymer, g/mol total polymer mass in the reactor, g total monomer concentration in the bed, mol=m3 component molecular weight, g/mol weight average molecular weight of polymer, g/mol concentration of nitrogen in the bed, mol=m3 number of well-stirred zones in the heat exchanger total number of monomers number of catalyst active sites concentration of vacant catalyst active sites, mol=m3 concentration of “live” copolymer chains of length ‘n’ ending in an ‘i’ monomer unit, mol=m3 total concentration of active sites (vacant and occupied by polymer chain), mol=m3 heat transfer rate, cal/s volumetric product removal rate, m3 =s concentration of potential catalyst active sites, mol=m3 temperature, K temperature of the recycle stream at the heat exchanger exit, K inlet water temperature to the heat exchanger, K water temperature in the heat exchanger, K overall heat transfer coeNcient, J=K=m2 =s bed volume, m3 mass fraction of monomer ‘i’ in the bed mass fraction of hydrogen in the bed mass fraction of nitrogen in the bed Greek letters XHr×n bed ‘ ‘ % ’i +i heat of reaction, cal/g bed void fraction “live” copolymer moment of ‘-order, mol=m3 “dead” copolymer moment of ‘-order, mol=m3 density, g=m3 instantaneous copolymer composition with respect to the ‘i’ monomer cumulative copolymer composition with respect to the ‘i’ monomer C. 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