- Email: [email protected]
Model predictive control of an open-loop unstable train of polymerization reactors
Computers chern. EIIgng, Vol. 18, Suppl., pp. SS25-S528, 1994 Printed in Great Britain
0098-13S4f94 $6.00+0.00 Pergamon Press Ltd
MODEL PREDICTIVE CONTROL OF AN OPEN-LOOP UNSTABLE TRAIN OF POLYMERIZATION REACTORS F. Gobin 1 , L. C. Zullo2 and J.-P. Calvet 2 Koninklijke/Shell-Laboratorium, Amsterdam (Shell Research B.V.) Badhuisweg 3, 1031 CM Amsterdam The Netherlands
ABSTRACT Model Predictive Control (MPC) schemes have the appealing property of combining powerful multivariate control algorithms with ease of implementation and conceptual simplicity. We have applied MPC to the model of a typical non-linear process of industrial interest, involving two Continuous Stirred-Tank Reactors (CSTR) in cascade. The reaction taking place in these reactors is the polymerization of styrene in the presence of a binary initiator mixture. Under certain conditions, such a system exhibits multiple steadystates and open-loop unstable behaviour. Because of their open-loop unstable behaviour, the reactors give rise to a difficult temperature control problem. Control is achieved by manipulating the flow rates of coolant fluid circulating inside the reactors' jackets. The present work aims at designing proper control strategies for this unstable train of reactors using a Dynamic Matrix Control (DMC) algorithm and to compare such a scheme with the use of conventional digital PID controllers. The design of the DMC controller was based on a linearization of the dynamic model around an open-loop stable operating point. The implementation of the DMC algorithm was carried out using a standard formulation.
INTRODUCTION Model-based controllers have the advantage over conventional types of controllers to be of a predictive nature. Moreover, a practical advantage in their implementation is that the algorithm computing the discrete control moves can cope not only with bounds on the manipulated variables reflecting real process constraints (e.g. bound on flow rates) but also with bounds between two consecutive moves. These represent limitations of the "speed" of control variables (e.g. rate of valve opening). The ability to handle input constraints and also (if feasible) output constraints made Model Predictive Control (MPC) very popular in industry. However, one of the drawbacks of MPC is its restriction to linear systems. Although most chemical processes have non-linear dynamics, MPC algorithms have yet been successfully used by approximating the dynamics of a process by a first or second-order linear system with delay, around the operating condition of interest. The model identification is usually carried out by perturbation of the real plant with step input changes while the output responses are recorded. This procedure is usually delicate and time-consuming. But if a reliable non-linear model is available, the identification on the plant can be avoided or reduced to a minimum. Instead, the linear approximation can be derived by linearizing the non-linear model at a given operation point. Moreover, the performance of a MPC algorithm can be tested and if necessary adapted by simulation of the plant via the rigorous non-linear model. This 'off-line' control performance analysis provides a wealth of information, such as better understanding of the process behaviour under different scenarios, e.g set point changes, disturbance rejections. These scenarios can also show if the linear approximation used for the MPC algorithms is valid and within what range. Finally, these control simulations can determine whether advanced control is really necessary or if traditional (PI, PID) type controllers are sufficient. This paper presents the proposed analysis applied to a process where nonlinearities are evident. In particular, the performance of a Dynamic Matrix Control algorighm (DMC) is compared with PIDs controllers when two polymerization reactors in series are driven from a stable to an unstable operating regime. All simulation results were obtained using the dynamic simulator SpeedUp [Pantelides, HISS]. 1 2
Ecole Nationale Su.,erieure des Industries Chimiques, 1 rue Grandville, BP 451, 54001 Nancy Cedex, France. Authors to whom correspondence should be addressed.
S525
European Symposium on Computer Aided Process Engineering-3
S526
PROCESS DESCRIPTION The process to be controlled is the homopolymerization of styrene in a cascade of two continuous stirred tank reactors (CSTRs, Figure 1). The free-radical polymerization is initiated by a mixture of two initiators having different thermal decomposition activities (tert-butyl perbenzoate and benzoyl peroxide). In previous studies of this system (Kim et al. 1990; 1991), both reactors have been found to exhibit quite complex behaviour: using the reactor residence time as a parameter, Kim found up to three steady states in the first reactor and five in the second one. In the previous studies, the reaction was carried out under adiabatic conditions. In the present work, however, we address the problem of controlling the temperature of the two-reactor system. Therefore, a detailed description of heat transfer to and from a coolant jacket has been added to the original model.
Polymerization kinetics The kinetics of the system are described in depth by Kim et al. (1991). Briefly, the kinetic assumptions made are: (i) the primary radicals generated by the decomposition of liable groups in both initiators are assumed indistinguishable, (ii) the chain transfer to solvent (ethyl benzene), the primary radical termination, and induced decomposition reactions of the initiators are assumed negligible, (iii) thermal initiation is not negligible, (iv) the gel effect is considered. The relevant kinetic scheme required for the model is given below.
k.. _ k..
+
Initiation by initiators:
fA
Thermal initiation:
3M~2PI k. Pn + M Pn + 1 (n ~ 1) Pn + M ~ Mn + PI (n ~ 1) Pn + S -CI. Mn + PI ( n ~ 1) Pn + Pm ~ Mn+m (n,m ~ 1)
Propagation : Chain transfer to monomer: Chain transfer to chain transfer agent: Chain termination:
--+
2R, fB -
2R, R
M
k.
----!. PI
where R is the primary radical, M the monomer, Pn the live polymer with n repeating units, Mn the dead polymer with n repeating units. fA and fB are the initiators having different thermal stabilities and S is the solvent.
The two-reactor model To simplify the model, we adopted a number of assumptions (see Kim et al. (1991) for details): (i) the quasi-steady-state approximation, meaning that the concentration of free radicals is constant, (ii) the longchain approximation, (iii) an empirical correlation for gel effect 3 . This last correlation implies that, for high conversion, increased viscosity leads to the termination reactions being diffusion controlled. Finally, perfect mixing and constant density are assumed. Now each reactor can be modelled as follows (both reactors have the same geometry). Mass balances for each reactant (rate constants k{.) are of Arrhenius type):
VdM/dt
= q(MJ -
M) - Vkp(T)M P
(1 )
(2) Vdf8/dt
=q(lBJ -
(3)
fB) - VkdB(T)f8
The energy balance for the reactor (heat of reaction is due to propagation only): pCpV(dT/dt)
= pCpq(TJ -
T)
+ V(-AH)kp(T)M P -
UA(T -1j)
(4)
The energy balance for the jacket : (5) Equation 5 is not present in the original model (Kim et al. (1991» and was added to design the control scheme described in this paper. Note that with the present model, polymer properties such as average chain 3 gl (X 1 ,T)
== k,jk: = exp[-2(Ax+ Bx 2 +Cx 3 )], from Hui and Hamiliec and modified by Hamer et at. (see
Kim et at. (1991».
European Symposium on Computer Aided Process Engineeriog-3
S527
length as well as polydispersity can be predicted. Also, by applying the quasi-steady-state approximation an expression for the concentration of growing polymer chain in the reactor is obtained (see Kim et al. (1991 »: (6)
Open-loop steady states This train of polymerization reactors is highly non-linear and can exhibit multiple steady states. In particular, for the residence time given in Table 1, each reactor will have three steady states, of which one is unstable. The numerical values of the feed conditions and the various process parameters (MJ' IAJ' IB/' TI and k{.},Il.H,p,Cp ) are the same as in used by Kim et al. (1991). The additional parameters used in the energy balance of the jacket are given in Table 2. From a practical point of view, the multiple steady states mean that for a given coolant flow rate (manipulated variable) the reactor can produce three different types of polymer. Although the unstable steady state can be regarded as an "exotic" phenomenon, it was noticed that this operating point yields better polymer properties. Moreover, one can see in Table 1 that, under the unstable condition, the conversion of the monomer is more evenly distributed between the reactors. Hence, rather than being "exotic", this particular unstable point can be considered the desired operating condition for the polymerization reactors.
CONTROL OPERATION AT THE UNSTABLE STEADY STATE Operating a process at an unstable operating condition is a challenging control problem. If the process is poorly controlled, even small disturbances (feed temperature and/or composition) may cause the reactor to drift away from the desired operating point. Under stable open-loop conditions, most controllers have similar performance, but if the dynamics of the process are highly non-linear or unstable, the use of advanced control techniques - in particular predictive ones - can be beneficial . It is thus desired to investigate the use of DMC when the reactors are driven from the stable to the unstable point. Since the DMC algorithm is based on a linear model, one must first linearize the non-linear model of the process. The identification of the real process can only be done at a stable operating point. The linearization of the process was derived at the steady state SS!. This task is rather tedious for complex and/or large non-linear systems. It was facilitated by means of the Control Design Interface (COl) of SpeedUp. A Fortran program of the DMC algorithm (see below) was interfaced to SpeedUp for the control simulations via the External Data Interface (EOI). A schematic DMC-SpeedUp coupling with the linearization step (via COl) is shown in Figure 2.
The DMC algorithm A complete description of the DMC algorithm (as used in this paper) can be found in Prett and Garcia (1988). For clarity, the philosophy of the predictive algorithm is sketched in Figure 3. In practice, all control algorithms are digital, i.e. the control actions are changed and held constant at every sampling time. The DMC algorithm uses the linear model to predict the trajectory of process response. Its task is to calculate, at every sampling time, the next m control moves that will minimize an objective function representing the discrepancy between the predicted trajectory and the set point over an horizon of p ~ m sampling times. The discrepancy is computed as a weighted Euclidean norm of an error vector 4 . These control moves, to be implemented, can also be included in the objective function as shown below.
(7)
Moreover, constraints can be added to this optimization problem. First, the control moves can be bounded so as to render the control action less aggressive (Il.umin(.) :5 Il.u :5 Il.u moz (.». Second, the operational constraints can be included by adding bounds on the control variables (Umin :5 u(.) :5 umoz). After some algebraic manipulations, this optimization problem leads to a classic constrained QP problem. Upon solution of the optimization, the first control move is implemented and the optimization is repeated at the next sampling time. 4\1x\l!T q = x T qT qx = x T Qx where x E R"'. In t.he DMC implementat.ion Q is usually diagonal (here it is identity).
S528
European Symposium on Computer Aided Process Engineering-3
SIMULATION RESULTS AND CONCLUSIONS Several control scenarios have been tested with the DMC algorithm. The influence offactors such as sampling time and weight on the error trajectory and control moves were also analysed. Here we only show a set point change simulation with input constraints (sampling time 10 s). Figure 4 represents the temperature profiles in both reactors with the PID and DMC implementation respectively. Comparison of performance can be seen in Figure 5, which gives a conversion - temperature plot for both reactors. It is known that for linear systems the unconstrained DMC algorithm will give the best performance. However, this does not apply to the polymerization reactors. The increase of the coolant flow rate must be done slowly (this can be achieved by imposing constraints on the control moves), otherwise reactor 1 will extinguish with no possibility to start it up again (even with the valve fully closed). This remarkable result reveals the difficulty of controlling non-linear systems away from the linearization point. It was found that the parameters of the PID controllers derived via a Cohen and Coon tuning (Stephanopoulos, 1984) were not satisfactory. It was only after repeated trials and errors that an appropriate tuning was found. As a measure of performance the time needed to reach ±O.FC and ±0.1% conv. of the desired set point was computed for the DMC and PID implementation. The results were 9000 sand 6000 s for reactors 1 and 2, respectively, with the PIDs and only 2700 sand 2900 s with the DMC implementation.
REFERENCES K. J. Kim,K. Y. Choi and J. C. Alexander, Polymer Engineering and Science, Vol.30, No.5, 1990. K. J. Kim,K. Y. Choi and J. C. Alexander, Polymer Engineering and Science, Vol. 31 , No.5, 1991.
D. M. Prett and C. E. Garcia, "Fundamental Process Control," Butterworth, 1988. G. Stephanopoulos, "Chemical Process Control: An Introduction to Theory and Practice," PrenticeHall, 1984. C. C. Pantelides, Compo and Chern. Eng., Vo1.12, 745-755, 1988.
Acknowledgement: The authors are indebted to Ms. Joanna Elder (Imperial College, London) who implemented the model in SpeedUp. Table 1: The multiple steady states in both reactors (XF,. is the cumulative monomer conversion)
= V/q = 0.5 h
rT
SSl Stable SS2 Unstable SS3 Stable
Reactor 1 Tl (OC) 96.099 93.968 70.004
XF.l(%)
90.0 82.7 0.01
Reactor 2 T2 (0C) 95.167 95.526 100.40
XF,2(%)
98.6 98.1 94.6
Table 2: Parameters for the jacket and control variable constraints (same in both reactors)
po Cpo Yj T;o A U
0.997 legll
2900 J.kg-l.K 0.31 70°C 0.05 m57.4 w.m-·.f(
i
qOl q0 2 qomar qomin
tlqomaJ: ·1
tlqomin
Monomer Coolant
Coolant Inlet
Inlet
Coolant Outlet
1.26 xlO· 4 m"ls 5.595 xlO -5 m;jls 2 xl0- 4 m"ls o molls 2.5 xl0-~ m"ls -2.5 xlO -:) m;jls
1
REACTOR 2
-
Material Stream
-If- Signal Stream •
lbennocouple
Coolant Outlet
'--_ _ _ _ _Po_lymer
L -_ _ _ _ _ _ _ _ _ .. _ _ _
Figure 1: Cascade of two CSTRs for styrene polymerization
0098-13S4f94 $6.00+0.00 Pergamon Press Ltd
MODEL PREDICTIVE CONTROL OF AN OPEN-LOOP UNSTABLE TRAIN OF POLYMERIZATION REACTORS F. Gobin 1 , L. C. Zullo2 and J.-P. Calvet 2 Koninklijke/Shell-Laboratorium, Amsterdam (Shell Research B.V.) Badhuisweg 3, 1031 CM Amsterdam The Netherlands
ABSTRACT Model Predictive Control (MPC) schemes have the appealing property of combining powerful multivariate control algorithms with ease of implementation and conceptual simplicity. We have applied MPC to the model of a typical non-linear process of industrial interest, involving two Continuous Stirred-Tank Reactors (CSTR) in cascade. The reaction taking place in these reactors is the polymerization of styrene in the presence of a binary initiator mixture. Under certain conditions, such a system exhibits multiple steadystates and open-loop unstable behaviour. Because of their open-loop unstable behaviour, the reactors give rise to a difficult temperature control problem. Control is achieved by manipulating the flow rates of coolant fluid circulating inside the reactors' jackets. The present work aims at designing proper control strategies for this unstable train of reactors using a Dynamic Matrix Control (DMC) algorithm and to compare such a scheme with the use of conventional digital PID controllers. The design of the DMC controller was based on a linearization of the dynamic model around an open-loop stable operating point. The implementation of the DMC algorithm was carried out using a standard formulation.
INTRODUCTION Model-based controllers have the advantage over conventional types of controllers to be of a predictive nature. Moreover, a practical advantage in their implementation is that the algorithm computing the discrete control moves can cope not only with bounds on the manipulated variables reflecting real process constraints (e.g. bound on flow rates) but also with bounds between two consecutive moves. These represent limitations of the "speed" of control variables (e.g. rate of valve opening). The ability to handle input constraints and also (if feasible) output constraints made Model Predictive Control (MPC) very popular in industry. However, one of the drawbacks of MPC is its restriction to linear systems. Although most chemical processes have non-linear dynamics, MPC algorithms have yet been successfully used by approximating the dynamics of a process by a first or second-order linear system with delay, around the operating condition of interest. The model identification is usually carried out by perturbation of the real plant with step input changes while the output responses are recorded. This procedure is usually delicate and time-consuming. But if a reliable non-linear model is available, the identification on the plant can be avoided or reduced to a minimum. Instead, the linear approximation can be derived by linearizing the non-linear model at a given operation point. Moreover, the performance of a MPC algorithm can be tested and if necessary adapted by simulation of the plant via the rigorous non-linear model. This 'off-line' control performance analysis provides a wealth of information, such as better understanding of the process behaviour under different scenarios, e.g set point changes, disturbance rejections. These scenarios can also show if the linear approximation used for the MPC algorithms is valid and within what range. Finally, these control simulations can determine whether advanced control is really necessary or if traditional (PI, PID) type controllers are sufficient. This paper presents the proposed analysis applied to a process where nonlinearities are evident. In particular, the performance of a Dynamic Matrix Control algorighm (DMC) is compared with PIDs controllers when two polymerization reactors in series are driven from a stable to an unstable operating regime. All simulation results were obtained using the dynamic simulator SpeedUp [Pantelides, HISS]. 1 2
Ecole Nationale Su.,erieure des Industries Chimiques, 1 rue Grandville, BP 451, 54001 Nancy Cedex, France. Authors to whom correspondence should be addressed.
S525
European Symposium on Computer Aided Process Engineering-3
S526
PROCESS DESCRIPTION The process to be controlled is the homopolymerization of styrene in a cascade of two continuous stirred tank reactors (CSTRs, Figure 1). The free-radical polymerization is initiated by a mixture of two initiators having different thermal decomposition activities (tert-butyl perbenzoate and benzoyl peroxide). In previous studies of this system (Kim et al. 1990; 1991), both reactors have been found to exhibit quite complex behaviour: using the reactor residence time as a parameter, Kim found up to three steady states in the first reactor and five in the second one. In the previous studies, the reaction was carried out under adiabatic conditions. In the present work, however, we address the problem of controlling the temperature of the two-reactor system. Therefore, a detailed description of heat transfer to and from a coolant jacket has been added to the original model.
Polymerization kinetics The kinetics of the system are described in depth by Kim et al. (1991). Briefly, the kinetic assumptions made are: (i) the primary radicals generated by the decomposition of liable groups in both initiators are assumed indistinguishable, (ii) the chain transfer to solvent (ethyl benzene), the primary radical termination, and induced decomposition reactions of the initiators are assumed negligible, (iii) thermal initiation is not negligible, (iv) the gel effect is considered. The relevant kinetic scheme required for the model is given below.
k.. _ k..
+
Initiation by initiators:
fA
Thermal initiation:
3M~2PI k. Pn + M Pn + 1 (n ~ 1) Pn + M ~ Mn + PI (n ~ 1) Pn + S -CI. Mn + PI ( n ~ 1) Pn + Pm ~ Mn+m (n,m ~ 1)
Propagation : Chain transfer to monomer: Chain transfer to chain transfer agent: Chain termination:
--+
2R, fB -
2R, R
M
k.
----!. PI
where R is the primary radical, M the monomer, Pn the live polymer with n repeating units, Mn the dead polymer with n repeating units. fA and fB are the initiators having different thermal stabilities and S is the solvent.
The two-reactor model To simplify the model, we adopted a number of assumptions (see Kim et al. (1991) for details): (i) the quasi-steady-state approximation, meaning that the concentration of free radicals is constant, (ii) the longchain approximation, (iii) an empirical correlation for gel effect 3 . This last correlation implies that, for high conversion, increased viscosity leads to the termination reactions being diffusion controlled. Finally, perfect mixing and constant density are assumed. Now each reactor can be modelled as follows (both reactors have the same geometry). Mass balances for each reactant (rate constants k{.) are of Arrhenius type):
VdM/dt
= q(MJ -
M) - Vkp(T)M P
(1 )
(2) Vdf8/dt
=q(lBJ -
(3)
fB) - VkdB(T)f8
The energy balance for the reactor (heat of reaction is due to propagation only): pCpV(dT/dt)
= pCpq(TJ -
T)
+ V(-AH)kp(T)M P -
UA(T -1j)
(4)
The energy balance for the jacket : (5) Equation 5 is not present in the original model (Kim et al. (1991» and was added to design the control scheme described in this paper. Note that with the present model, polymer properties such as average chain 3 gl (X 1 ,T)
== k,jk: = exp[-2(Ax+ Bx 2 +Cx 3 )], from Hui and Hamiliec and modified by Hamer et at. (see
Kim et at. (1991».
European Symposium on Computer Aided Process Engineeriog-3
S527
length as well as polydispersity can be predicted. Also, by applying the quasi-steady-state approximation an expression for the concentration of growing polymer chain in the reactor is obtained (see Kim et al. (1991 »: (6)
Open-loop steady states This train of polymerization reactors is highly non-linear and can exhibit multiple steady states. In particular, for the residence time given in Table 1, each reactor will have three steady states, of which one is unstable. The numerical values of the feed conditions and the various process parameters (MJ' IAJ' IB/' TI and k{.},Il.H,p,Cp ) are the same as in used by Kim et al. (1991). The additional parameters used in the energy balance of the jacket are given in Table 2. From a practical point of view, the multiple steady states mean that for a given coolant flow rate (manipulated variable) the reactor can produce three different types of polymer. Although the unstable steady state can be regarded as an "exotic" phenomenon, it was noticed that this operating point yields better polymer properties. Moreover, one can see in Table 1 that, under the unstable condition, the conversion of the monomer is more evenly distributed between the reactors. Hence, rather than being "exotic", this particular unstable point can be considered the desired operating condition for the polymerization reactors.
CONTROL OPERATION AT THE UNSTABLE STEADY STATE Operating a process at an unstable operating condition is a challenging control problem. If the process is poorly controlled, even small disturbances (feed temperature and/or composition) may cause the reactor to drift away from the desired operating point. Under stable open-loop conditions, most controllers have similar performance, but if the dynamics of the process are highly non-linear or unstable, the use of advanced control techniques - in particular predictive ones - can be beneficial . It is thus desired to investigate the use of DMC when the reactors are driven from the stable to the unstable point. Since the DMC algorithm is based on a linear model, one must first linearize the non-linear model of the process. The identification of the real process can only be done at a stable operating point. The linearization of the process was derived at the steady state SS!. This task is rather tedious for complex and/or large non-linear systems. It was facilitated by means of the Control Design Interface (COl) of SpeedUp. A Fortran program of the DMC algorithm (see below) was interfaced to SpeedUp for the control simulations via the External Data Interface (EOI). A schematic DMC-SpeedUp coupling with the linearization step (via COl) is shown in Figure 2.
The DMC algorithm A complete description of the DMC algorithm (as used in this paper) can be found in Prett and Garcia (1988). For clarity, the philosophy of the predictive algorithm is sketched in Figure 3. In practice, all control algorithms are digital, i.e. the control actions are changed and held constant at every sampling time. The DMC algorithm uses the linear model to predict the trajectory of process response. Its task is to calculate, at every sampling time, the next m control moves that will minimize an objective function representing the discrepancy between the predicted trajectory and the set point over an horizon of p ~ m sampling times. The discrepancy is computed as a weighted Euclidean norm of an error vector 4 . These control moves, to be implemented, can also be included in the objective function as shown below.
(7)
Moreover, constraints can be added to this optimization problem. First, the control moves can be bounded so as to render the control action less aggressive (Il.umin(.) :5 Il.u :5 Il.u moz (.». Second, the operational constraints can be included by adding bounds on the control variables (Umin :5 u(.) :5 umoz). After some algebraic manipulations, this optimization problem leads to a classic constrained QP problem. Upon solution of the optimization, the first control move is implemented and the optimization is repeated at the next sampling time. 4\1x\l!T q = x T qT qx = x T Qx where x E R"'. In t.he DMC implementat.ion Q is usually diagonal (here it is identity).
S528
European Symposium on Computer Aided Process Engineering-3
SIMULATION RESULTS AND CONCLUSIONS Several control scenarios have been tested with the DMC algorithm. The influence offactors such as sampling time and weight on the error trajectory and control moves were also analysed. Here we only show a set point change simulation with input constraints (sampling time 10 s). Figure 4 represents the temperature profiles in both reactors with the PID and DMC implementation respectively. Comparison of performance can be seen in Figure 5, which gives a conversion - temperature plot for both reactors. It is known that for linear systems the unconstrained DMC algorithm will give the best performance. However, this does not apply to the polymerization reactors. The increase of the coolant flow rate must be done slowly (this can be achieved by imposing constraints on the control moves), otherwise reactor 1 will extinguish with no possibility to start it up again (even with the valve fully closed). This remarkable result reveals the difficulty of controlling non-linear systems away from the linearization point. It was found that the parameters of the PID controllers derived via a Cohen and Coon tuning (Stephanopoulos, 1984) were not satisfactory. It was only after repeated trials and errors that an appropriate tuning was found. As a measure of performance the time needed to reach ±O.FC and ±0.1% conv. of the desired set point was computed for the DMC and PID implementation. The results were 9000 sand 6000 s for reactors 1 and 2, respectively, with the PIDs and only 2700 sand 2900 s with the DMC implementation.
REFERENCES K. J. Kim,K. Y. Choi and J. C. Alexander, Polymer Engineering and Science, Vol.30, No.5, 1990. K. J. Kim,K. Y. Choi and J. C. Alexander, Polymer Engineering and Science, Vol. 31 , No.5, 1991.
D. M. Prett and C. E. Garcia, "Fundamental Process Control," Butterworth, 1988. G. Stephanopoulos, "Chemical Process Control: An Introduction to Theory and Practice," PrenticeHall, 1984. C. C. Pantelides, Compo and Chern. Eng., Vo1.12, 745-755, 1988.
Acknowledgement: The authors are indebted to Ms. Joanna Elder (Imperial College, London) who implemented the model in SpeedUp. Table 1: The multiple steady states in both reactors (XF,. is the cumulative monomer conversion)
= V/q = 0.5 h
rT
SSl Stable SS2 Unstable SS3 Stable
Reactor 1 Tl (OC) 96.099 93.968 70.004
XF.l(%)
90.0 82.7 0.01
Reactor 2 T2 (0C) 95.167 95.526 100.40
XF,2(%)
98.6 98.1 94.6
Table 2: Parameters for the jacket and control variable constraints (same in both reactors)
po Cpo Yj T;o A U
0.997 legll
2900 J.kg-l.K 0.31 70°C 0.05 m57.4 w.m-·.f(
i
qOl q0 2 qomar qomin
tlqomaJ: ·1
tlqomin
Monomer Coolant
Coolant Inlet
Inlet
Coolant Outlet
1.26 xlO· 4 m"ls 5.595 xlO -5 m;jls 2 xl0- 4 m"ls o molls 2.5 xl0-~ m"ls -2.5 xlO -:) m;jls
1
REACTOR 2
-
Material Stream
-If- Signal Stream •
lbennocouple
Coolant Outlet
'--_ _ _ _ _Po_lymer
L -_ _ _ _ _ _ _ _ _ .. _ _ _
Figure 1: Cascade of two CSTRs for styrene polymerization