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Modeling, simulation and nonlinear control of a gas-phase polymerization process
Computers &Chemical Engineering
Computers and Chemical Engineering 24 (2000) 945-951
ELSEVIER
www.elsevier.com/locate/compchemeng
Modeling, simulation and nonlinear control of a gas-phase polymerization process Chic Sato a,*, Tetsuya Ohtani b, Hirokazu Nishitani u a Yokogawa Electric Corporation, 2-9-32 Nakacho Musashino-shi, Tokyo 180-8750, Japan b Nara Institute of Science and Technology, 8916-5 Takayamacho Ikoma-shi, Nara 630-0101, Japan
Abstract
In an industrial gas-phase polymerization process, dynamics change globally due to grade changes and load changes, and these changes present a difficult control problem. In this paper, a nonlinear physical model was effectively used for the control system design. First, we developed a set of physical models of an industrial ethylene polymerization process with reference to McAuley's model. Parameters in the model were adjusted to simulate the actual process behavior. Second, in the control system design, the system is regarded as a two-input and two-output system. The two inputs are the feed rates of fresh hydrogen and butene, and the two outputs are cumulative melt index and density. An optimal servo controller with integral actions is designed according to the optimal regulator theory using a model linearized at a nominal operating point. Third, we examined the changes of process dynamics under typical operating points for different grade products. As a result, a nonlinear compensator was attached to the optimal servo controller to cope with remarkable changes in the process gain due to the product grade. This method resulted in good control performances during various grade changes. © 2000 Elsevier Science Ltd. All rights reserved. Keywords: Industrial polymerizationreactor; Modeling; Grade transition; Product property control; Nonlinear compensation
1. Introduction
Today's customer-oriented chemicals market demands higher quality standards and lower prices of various kinds of product grades. In this situation, it is necessary to have advanced control technologies that make grade transitions more quickly and fewer products with off-specification during grade changeovers. Recently the computation technology and software development environment of the process simulation has become very convenient. We have developed a polymerization process simulator that has more reality in the process behavior. New control systems can be connected to this simulator to evaluate control performance (Hamasaki, Shimizu, Ishikawa, Ohmura, Nakajima & Kitamura, 1994). Polymerization processes are essentially multi-input and multi-output (MIMO) systems as shown in Fig. 1 (Sato, Ohtani & Nishitani, 1999a,b). A traditional regulator control system is composed of multiple PID controllers, and these are the most widely used controllers * Corresponding author.
in the process industry. Each PID controller is designed for a single-input, single-output (SISO) process. The PID controller has the advantages of a simple structure and easily tunable parameters. However, in polymerization processes with grade transitions, the controlled variables such as temperature, pressure, composition and polymer property have strong interactions. Therefore, a multiple PID control system may not provide good control performance for all grades. In other words, a set of PID controllers designed at an operating point for a certain grade may deteriorate according to the shift of the operating point due to grade transitions. In this paper, a multivariable controller for the product property is designed on basis of a nonlinear physical model of an industrial polymerization process. In Section 2, a set of physical models of an industrial ethylene polymerization process is derived. In Section 3, process dynamics at some typical operating points for different grade products are examined. In Section 4, the nonlinear state equations are linearized at a nominal operating point. Using this linearized model, an optimal servo controller with integral actions is designed by the optimal regulator theory. Additionally, to cope with
0098-1354/00/$ - see front matter © 2000 Elsevier Science Ltd. All rights reserved. PII: S0098- 1354(00)00375-6
C. Sato et al. / Computers and Chemical Engineering 24 (2000) 945-951
946
(4)
monome-r- ~ comofloflIM
-~
~
d[M2] Vg dt - FM2 -- kpM2(T)" Y" [M2]
state
(5)
hydrogen
Mass balance of butene in the reactor
productptopelles
Vg
andproductiorate n
(4)
Mass balance of hydrogen in the reactor
d[U2] dt --FH2--RH2
(5)
(6) Instantaneous polymer property Melt index MIi[g/10min]
Fig. I. Inputs and outputs of polymerization process.
the nonlinearity of the process, a nonlinear compensator is attached to the optimal servo controller. In Section 5, the control performances of the proposed control system are evaluated in grade change simulations using the developed simulator.
[ M 2 ] . , [H2]] 3"5 MIi=exp{kT(1--~o)}'{ko+kl[---~]'K3[---~]; (6a) Density Di[g c m - 3] ( [ M 2 ] ] p4 Di = P o + P ] " l n ( M I i ) - ~P2 [ - - ~ ;
2. M o d e l i n g
of polymerization
process
(7) Cumulative polymer property Melt index MIc[g/10min]
2. I. State equations
1
The following two assumptions are made in the modeling. 1. The gases in the fluidized bed reactor and the heat exchanger are well-mixed. 2. Polymer particles in the fluidized bed reactor are well-mixed and the level of the polymer particles is regulated by the product flow. A set of nonlinear physical models of an industrial ethylene polymerization process was developed by referring to McAuley's model (McAuley & MacGregor, 1991; McAuley, MacDonald & McLellan, 1995). (1) Mass balance of ethylene in the reactor K d[Ml] = FM' _ kpM~(T) • Y" [M1] g dt (2)
(1)
Mass balance of catalyst sites in the reactor
d___Y= F v _ ka " y _ dt
(3)
(6b)
(2)
Op . Y Ws
dMI c 3.5
1 =-
dt
r
" MIi
1 1 1 3.5-• M I c 3.5
(7a)
r
Density D~[g c m - 3] 1
d-D~ dt
1 z
1 Di
1 1 r D~
(7b)
The model is composed of seven ODEs. Table 1 summarizes the operating conditions for the main product grades. Parameters in the model were adjusted to simulate the actual process behavior. For example, k o, k l , k 3 and k 7 in Eq. (6a) and Po, Pl,P2 and P4 in Eq. (6b) were respectively adjusted so that they could satisfy the corresponding equation for all grades shown in Table 1. kpM~(T) and kpM2(T) in Eqs. (1), (3) and (4) were determined in consideration of the characteristics of the catalysts. Assuming that three state variables, i.e. [M1], Y and T are tightly controlled, we have four ODEs with 1
Heat balance in the reactor
respect to [M2], [H2], MI~-~ and D~-~. We call this
dT MCp" d--}-=//feed + A H R • kpM~(T) " Y" [M1] - Qc -- npoly
(3)
set of ODEs a product property dynamic model. If this model is carefully investigated, it can be represented by the three submodels shown in Fig. 2; the dynamic
Table 1 Main product grades and their operating conditions Grade Grade Grade Grade Grade
A B C D
Op (ton h -1)
MI c (g (10 min) 1)
D c (g cm -3)
T (K)
P (Mpa)
25.0 25.0 25.0 25.0
2.00 10.00 2.00 0.50
0.93 0.93 0.92 0.94
363.15 363.15 358.15 363.15
2.06 2.06 2.06 2.06
C. Sato et al. /Computers
and Chemical
Engineering
24 (2000) 945-951
941
2.2. Development of a polymerization process simulator
Fig. 2. Product
property
dynamic
model
model from the flow rates of butene and and FH2) to the concentrations of butene ([M2] and [H2]), the static model from tions of butene and hydrogen to the
structure.
hydrogen (FM2 and hydrogen the concentrainstantaneous
product properties (MIi 315and DC ‘), and the dynamic model from the instantaneous product properties to the cumulative product properties (MI, 315and D; ‘). The first dynamic model is represented by Eqs. (4) and (5) the second static model is represented by Eqs. (6a) and (6b), and the third dynamic model is represented by Eqs. (7a) and (7b), respectively. So, the system is regarded as a two-input and two-output system. The two inputs are the feed rates of fresh butene and hydrogen, and the two outputs are cumulative melt index and density. The process nonlinearity results from the static relation from the butene and hydrogen concentrations to the instantaneous product properties.
Fig. 3. Operational
panel and a window
for grade
The polymer reactor model was implemented as a process simulator for gas-phase polyethylene polymerization process using a modeling tool of Visual Modeler provided by Omega Simulation, Ltd. Process and control equipment can be manipulated from the operating window, as shown in Fig. 3. For example, the set-point of a PID controller can be changed, or a blower can be started or stopped. The product property monitoring window is placed at the upper right comer, as shown in Fig. 3. In the two dimensional graph of the product property monitoring window, the horizontal axis represents melt index (MI) and the vertical axis represents density (D). Both the instantaneous product properties and cumulative properties are plotted so that the trajectory due to grade changes can be monitored. Using this simulator, we can perform various case studies to improve the control performances during the grade transitions.
3. Nonlinearity of the process
We examined the changes in process dynamics of the product property dynamic model shown in Fig. 2 at typical operating points for the different grade products shown in Table 1. Eqs. (4)-(7) are linearized at each operating point. Four poles and two zeros of the transfer function matrix are summarized in Table 2(a) and
transition
monitoring
(upper
right corner).
C. Sato et aL / Computers and Chemical Engineering 24 (2000) 945-951
948
Table 2 Changes in process dynamics of the product property dynamic model (a) Poles Grade A Grade B Grade C Grade D
-0.0948 -0.0984 -0.0870 -0.0990
-0.3571 -0.3571 -0.3571 -0.3571
-0.3571 -0.3571 -0.3571 -0.3571
-0.7418 -0.6350 -0.5497 - 1.0939
MIc/FM2 -0.0948 -0.3571 -0.0984 -0.3571 -0.0870 - 0.3571 -0.0990 -0.3571
MIJFm -0.7418 -0.3571 -0.6350 -0.3571 -0.5497 - 0.3571 - 1.0939 -0.3571
Dc/FM2
Dd Fm
-0.0948 -0.3571 -0.0984 -0.3571 -0.0870 - 0.3571 -0.0990 -0.3571
-0.7418 -0.3571 -0.6350 -0.3571 -0.5497 - 0.3571 - 1.0939 -0.3571
MIc/FM2 (Ratio to Grade A) -0.003012 (1.000) -0.001405 (0.466) -0.003888 (1.291) -0.004506 (1.496)
MI¢/FHz
Dc/FMz
De~Fro
(b) Zeros Grade A Grade B Grade C Grade D (c) Process gain Process gain Grade Grade Grade Grade
A B C D
-0.21413 -0.08238 -0.22315 -0.45239
(b), respectively. Four poles are the same among the four elements of the transfer function matrix, and two zeros are the same between MIdFM2 and DdFm, and between MIdFH2 and DdFH2, respectively. Table 2(c) shows the static process gain of 2 x 2 transfer function matrix. It changes remarkably due to the product grade, for example, the static process gain from FH2 to MI¢ for Grade D is about five times as high as that for Grade B. The characteristics of process dynamics are summarized as follows: 1. Two zeros are cancelled by two poles. 2. The remaining two poles move little due to the product grade. 3. The process gain changes remarkably due to the product grade.
4. Product property control
4.1. Control objectives The purpose of product property control is to drive the product property to the new target value in a short time and to produce fewer off-specification polymers. Grade transition control that causes excessive overshoots in the instantaneous product properties (MI i, Di) must be avoided.
4.2. Servo controller design If the plant's outputs are to follow a class of com-
(1.000) (0.385) (1.042) (2.113)
0.000812 0.000852 0.000878 0.000862
(1.000) (1.049) (1.082) (1.061)
-0.00360 --0.00219 --0.00383 --0.00501
(I.000) (0.609) (1.064) (1.392)
mand inputs, which are given by the outputs of a finite dimensional linear system, the controlled system is called a Servo System (Kreindler, 1969). Recently, in the presence of disturbances, the above controlled system is also called a servo system, and there have been many studies on this subject. In this paper, a servo controller is designed so that the product property could be driven to the new target value in a short time and the controlled system is stable. A method for designing an optimal servo system that minimizes the objective function comprising the quadratic forms of both the output error vector and the deviation control vector was proposed by Pak, Suzuki and Fujii (1974). In this method, a design problem of the multivariable feedback control system was reduced to the usual optimal regulator problem. However, the above optimal servo system is a controller without integral actions, so the steady state error (offset) will occur if a disturbance or model mismatch exists. Researchers proposed some methods for designing a servo system when disturbances and references were given by the outputs of the finite dimensional linear system (Davison, 1976; Ikeda, 1997). These methods are based on the lnternal Model Principle (Francis & Wonham, 1975). The servo system consists of a servocompensator and a stabilizing compensator. Fig. 4 shows the block diagram of the optimal servo system. The servo-compensator incorporates an internal model of the exogenous signal that the compensator is re-
C. Sato et al. / Computers and Chemical Engineering 24 (2000) 945-951
Servo Compensator
Plant Model for Controller Design
ks(t ) • Ks + k¢(t)" K~
(8a)
G = ks(t ) • G s + ke(t ) • G e
(8b)
K =
r
949
ks(t) =
1 + cos (t - ts) 7r ( t e - ts)
(t s < t < t~) - (t~ < t)
(8c)
(ts < t < t~)
(8d)
Fig. 4. Block diagram of optimal servo system.
quired to process. In this optimal servo system, asymptotic regulation is guaranteed independent of input disturbances and peturbations in the plant parameters. In this paper, an optimal servo system is designed using the linearized product property dynamic model with reference to a design method (Ikeda, 1997).
iI
k~(t) =
1 - cos
z~
(t~ < t)
where ts is a fixed time when a grade change begins, and te is also a fixed time when a grade change should end.
4.3. Nonlinear compensation The controller gain (K, G) can be switched to cope with the nonlinearity of the process. A gain (Ks, Gs) is obtained using the linearized model at the initial operating point, and one more gain (Ke, Ge) is calculated at the target operating point. The controller gain can be adjusted according to the following schedule (Sato & Sakawa, 1988):
5. Simulation The grade change simulations from Grade A to Grade B were performed with three kinds of controller gains. Fig. 5 shows the results. The dotted line shows the transition with the controller gain adjusted for Grade A (CASE1), the dash-dot line shows the transi-
'12
:."x
10 ~#
,=_
8 •
N 6
: . . . . . . . . . . . .
:
~. . . . . . .
-. ...........
N.,
Lg
CASE1 : CASE2 CASE3
0.93 10
15
r J,.. ~ ~r'~..; .-','.. . . . . . . . 0
5
10
O
5
10
,
, :1 Q
15
O1~
0:1 0.14
0,12
o.ti
-~0.06
o=1
0:09
0,04
C'
2~
2O
5 oo
15
10 Tinn~
4~
15
~'7~J
Fig. 5. Grade transition from Grade A to Grade B under three controllers.
!.6
950
C. Sato et al. / Computers and Chemical Engineering 24 (2000) 945-951
tion with the controller gain adjusted for Grade B (CASE2), and the solid line shows the transition with gain scheduling by Eqs. (8a)-(8d) proposed in this paper (CASE3; ts = 0, te = 7). The dashed straight line in MIc and De responses shows the target (Grade B) property. It takes shorter time for the grade transition in CASE2 than that in CASE1. During the grade change in CASE2, a smaller overshoot in MI~ than that in CASE1 but a larger deviation in D~ than that in CASE1 were observed. In another grade change from Grade A to Grade D, it also took shorter time for the transition with the controller gain adjusted to the target grade than that with the controller gain adjusted to the initial grade. Apart from this fact, it takes the shortest time for the grade transition in CASE3; during the grade change, the smallest overshoot in MI c and relatively smaller deviation in D e were observed. As a result, the proposed nonlinear compensator worked well to cope with the static process gain changes due to product grades. This method will provide good control performance during the critical grade changes.
6. Conclusions A set of physical models of an industrial ethylene polymerization process was developed. This model was composed of seven ordinary differential equations. Parameters in the model were adjusted to simulate the actual process behavior. We used this model as a virtual plant when investigating operational problems in an actual plant. Under the tight control of three state variables, we had four ODEs with respect to the butene and hydrogen concentrations, and the two cumulative polymer properties of melt index and density. This control process was regarded as a two-input, two-output system. An optimal servo controller with integral actions was designed by the optimal regulator theory using the linearized model for each product grade. The changes in process dynamics at typical operating points for different grade products were characterized by the large changes in the static process gain. As a result, a nonlinear compensator with gain scheduling was attached to the optimal servo controller to cope with the process gain changes. This gain scheduling control method provided good control performances during grade transitions. The nonlinear physical model was effectively used for the control system design.
Ken'ichi Ohmura and Goro Oguchi from Omega Simulation, Ltd. for their continuous cooperation and support.
Appendix A. Notation
DC
DI
F.2
h -1) ethylene feed rate to reactor (kgmol
h -1)
F~2
butene feed rate to reactor (kgrnol h -~) feed rate of catalyst site to reactor (kgmol h - l) enthalpy of feed to reactor (kcal h-l) Hfeed enthalpy associated with polymer leavnpoly ing reactor (kcal h -1) enthalpy of reaction (kcal kgmol -~) AHR concentration of hydrogen in reactor [H2] (kgmol m -3) k~ deactivation rate constant [l/h] kpM1 (T) temperature-dependent propagation rate constant of ethylene (m 3 kgmol -~ h-l) temperature-dependent propagation rate kl,M2 (T) constant of butene (m 3 kgrnol -l h -l) ko, kl, k3, k7 parameters in instantaneous melt index model thermal capacity of reaction vessel [kcal/ MG
F~
/q
MIc MII [M1] [M2] Op P Po,Pl,P2,P7
Qc RH2 T
To
Vg Ws Y
Acknowledgements The authors would like to thank Shintaro Miura,
cumulative density (g cm -3) instantaneous density (g cm -3) hydrogen feed rate to reactor (kgrnol
cumulative melt index (g (10 min) -~) instantaneous melt index (g (10 min) -~) concentration of ethylene in reactor (kgm o l m -a) concentration of butene in reactor (kgm o l m -3) outflow rate of polymer product (ton h -1) reactor pressure (Mpa) parameters in instantaneous density model amount of heat removed by heat exchanger (kcal h -1) rate of hydrogen consumption due to reaction (kgmol h -1) reactor temperature (K) reference temperature (K) volume of gas phase in reactor (m 3) mass of polymer in bed (ton) number of moles of catalyst sites in reactor (kgrnol) time constant of polymer phase in fluidized bed (h)
C. Sato et al./ Computers and Chemical Engineering 24 (2000) 945-951
References Davison, E. (1976). The robust control of a servo-mechanism problem for linear time invariant multivariable systems. IEEE Transactions on Automatic Control, AC-21(1), 25-34. Francis, B. A., & Wonham, W. M. (1975). The internal model principle for linear multivariable regulators. Journal of Applied Mathematics, 2(Opt), 170-194. Hamasaki, H., Shimizu, M., Ishikawa, T., Ohmura, K., Nakajima, K., & Kitamura, H. (1994). Plant operation training simulator 'PLANTUTOR'. Chemical Engineering, 39 (10), 77-83 (In Japanese). Ikeda, M. (1997). SICE Seminar Text, An Introduction to Advanced Control Theory. (In Japanese). Kreindler, E. (1969). On the linear optimal servo problem. International Journal of Control, 9(4), 465-472.
951
McAuley, K. B., & MacGregor, J. F. (1991). On-line inference of polymer properties in an industrial polyethylene reactor. American Institute of Chemical Engineering Journal, 37(6), 825-835. McAuley, K. B., MacDonald, D. A., & McLellan, P. J. (1995). Effects of operating conditions on stability of gas-phase polyethylene reactors. American Institute of Chemical Engineering Journal, 41(4), 868-879. Pak, P. S., Suzuki, Y., & Fujii, K. (1974). Synthesis of multivariable linear optimal servosystem. Transactions of SICE, 8(5), 568-575. Sato, K., & Sakawa, Y. (1988). Modeling and control of a flexible rotary crane. International Journal of Control, 48(5), 2085-2105. Sato, C., Ohtani, T., & Nishitani, H. (1999a). Optimal servo system for polymerization process. Proceedings of the 64th SCEJ Annual Conference. B206 (In Japanese). Sato, C., Ohtani, T., & Nishitani, H. (1999b). A study of nonlinearity of a polymerization process for grade change. Proceedings of the 38th SICE Annual Conference SICE'99, 725-726. (In Japanese).
Computers and Chemical Engineering 24 (2000) 945-951
ELSEVIER
www.elsevier.com/locate/compchemeng
Modeling, simulation and nonlinear control of a gas-phase polymerization process Chic Sato a,*, Tetsuya Ohtani b, Hirokazu Nishitani u a Yokogawa Electric Corporation, 2-9-32 Nakacho Musashino-shi, Tokyo 180-8750, Japan b Nara Institute of Science and Technology, 8916-5 Takayamacho Ikoma-shi, Nara 630-0101, Japan
Abstract
In an industrial gas-phase polymerization process, dynamics change globally due to grade changes and load changes, and these changes present a difficult control problem. In this paper, a nonlinear physical model was effectively used for the control system design. First, we developed a set of physical models of an industrial ethylene polymerization process with reference to McAuley's model. Parameters in the model were adjusted to simulate the actual process behavior. Second, in the control system design, the system is regarded as a two-input and two-output system. The two inputs are the feed rates of fresh hydrogen and butene, and the two outputs are cumulative melt index and density. An optimal servo controller with integral actions is designed according to the optimal regulator theory using a model linearized at a nominal operating point. Third, we examined the changes of process dynamics under typical operating points for different grade products. As a result, a nonlinear compensator was attached to the optimal servo controller to cope with remarkable changes in the process gain due to the product grade. This method resulted in good control performances during various grade changes. © 2000 Elsevier Science Ltd. All rights reserved. Keywords: Industrial polymerizationreactor; Modeling; Grade transition; Product property control; Nonlinear compensation
1. Introduction
Today's customer-oriented chemicals market demands higher quality standards and lower prices of various kinds of product grades. In this situation, it is necessary to have advanced control technologies that make grade transitions more quickly and fewer products with off-specification during grade changeovers. Recently the computation technology and software development environment of the process simulation has become very convenient. We have developed a polymerization process simulator that has more reality in the process behavior. New control systems can be connected to this simulator to evaluate control performance (Hamasaki, Shimizu, Ishikawa, Ohmura, Nakajima & Kitamura, 1994). Polymerization processes are essentially multi-input and multi-output (MIMO) systems as shown in Fig. 1 (Sato, Ohtani & Nishitani, 1999a,b). A traditional regulator control system is composed of multiple PID controllers, and these are the most widely used controllers * Corresponding author.
in the process industry. Each PID controller is designed for a single-input, single-output (SISO) process. The PID controller has the advantages of a simple structure and easily tunable parameters. However, in polymerization processes with grade transitions, the controlled variables such as temperature, pressure, composition and polymer property have strong interactions. Therefore, a multiple PID control system may not provide good control performance for all grades. In other words, a set of PID controllers designed at an operating point for a certain grade may deteriorate according to the shift of the operating point due to grade transitions. In this paper, a multivariable controller for the product property is designed on basis of a nonlinear physical model of an industrial polymerization process. In Section 2, a set of physical models of an industrial ethylene polymerization process is derived. In Section 3, process dynamics at some typical operating points for different grade products are examined. In Section 4, the nonlinear state equations are linearized at a nominal operating point. Using this linearized model, an optimal servo controller with integral actions is designed by the optimal regulator theory. Additionally, to cope with
0098-1354/00/$ - see front matter © 2000 Elsevier Science Ltd. All rights reserved. PII: S0098- 1354(00)00375-6
C. Sato et al. / Computers and Chemical Engineering 24 (2000) 945-951
946
(4)
monome-r- ~ comofloflIM
-~
~
d[M2] Vg dt - FM2 -- kpM2(T)" Y" [M2]
state
(5)
hydrogen
Mass balance of butene in the reactor
productptopelles
Vg
andproductiorate n
(4)
Mass balance of hydrogen in the reactor
d[U2] dt --FH2--RH2
(5)
(6) Instantaneous polymer property Melt index MIi[g/10min]
Fig. I. Inputs and outputs of polymerization process.
the nonlinearity of the process, a nonlinear compensator is attached to the optimal servo controller. In Section 5, the control performances of the proposed control system are evaluated in grade change simulations using the developed simulator.
[ M 2 ] . , [H2]] 3"5 MIi=exp{kT(1--~o)}'{ko+kl[---~]'K3[---~]; (6a) Density Di[g c m - 3] ( [ M 2 ] ] p4 Di = P o + P ] " l n ( M I i ) - ~P2 [ - - ~ ;
2. M o d e l i n g
of polymerization
process
(7) Cumulative polymer property Melt index MIc[g/10min]
2. I. State equations
1
The following two assumptions are made in the modeling. 1. The gases in the fluidized bed reactor and the heat exchanger are well-mixed. 2. Polymer particles in the fluidized bed reactor are well-mixed and the level of the polymer particles is regulated by the product flow. A set of nonlinear physical models of an industrial ethylene polymerization process was developed by referring to McAuley's model (McAuley & MacGregor, 1991; McAuley, MacDonald & McLellan, 1995). (1) Mass balance of ethylene in the reactor K d[Ml] = FM' _ kpM~(T) • Y" [M1] g dt (2)
(1)
Mass balance of catalyst sites in the reactor
d___Y= F v _ ka " y _ dt
(3)
(6b)
(2)
Op . Y Ws
dMI c 3.5
1 =-
dt
r
" MIi
1 1 1 3.5-• M I c 3.5
(7a)
r
Density D~[g c m - 3] 1
d-D~ dt
1 z
1 Di
1 1 r D~
(7b)
The model is composed of seven ODEs. Table 1 summarizes the operating conditions for the main product grades. Parameters in the model were adjusted to simulate the actual process behavior. For example, k o, k l , k 3 and k 7 in Eq. (6a) and Po, Pl,P2 and P4 in Eq. (6b) were respectively adjusted so that they could satisfy the corresponding equation for all grades shown in Table 1. kpM~(T) and kpM2(T) in Eqs. (1), (3) and (4) were determined in consideration of the characteristics of the catalysts. Assuming that three state variables, i.e. [M1], Y and T are tightly controlled, we have four ODEs with 1
Heat balance in the reactor
respect to [M2], [H2], MI~-~ and D~-~. We call this
dT MCp" d--}-=//feed + A H R • kpM~(T) " Y" [M1] - Qc -- npoly
(3)
set of ODEs a product property dynamic model. If this model is carefully investigated, it can be represented by the three submodels shown in Fig. 2; the dynamic
Table 1 Main product grades and their operating conditions Grade Grade Grade Grade Grade
A B C D
Op (ton h -1)
MI c (g (10 min) 1)
D c (g cm -3)
T (K)
P (Mpa)
25.0 25.0 25.0 25.0
2.00 10.00 2.00 0.50
0.93 0.93 0.92 0.94
363.15 363.15 358.15 363.15
2.06 2.06 2.06 2.06
C. Sato et al. /Computers
and Chemical
Engineering
24 (2000) 945-951
941
2.2. Development of a polymerization process simulator
Fig. 2. Product
property
dynamic
model
model from the flow rates of butene and and FH2) to the concentrations of butene ([M2] and [H2]), the static model from tions of butene and hydrogen to the
structure.
hydrogen (FM2 and hydrogen the concentrainstantaneous
product properties (MIi 315and DC ‘), and the dynamic model from the instantaneous product properties to the cumulative product properties (MI, 315and D; ‘). The first dynamic model is represented by Eqs. (4) and (5) the second static model is represented by Eqs. (6a) and (6b), and the third dynamic model is represented by Eqs. (7a) and (7b), respectively. So, the system is regarded as a two-input and two-output system. The two inputs are the feed rates of fresh butene and hydrogen, and the two outputs are cumulative melt index and density. The process nonlinearity results from the static relation from the butene and hydrogen concentrations to the instantaneous product properties.
Fig. 3. Operational
panel and a window
for grade
The polymer reactor model was implemented as a process simulator for gas-phase polyethylene polymerization process using a modeling tool of Visual Modeler provided by Omega Simulation, Ltd. Process and control equipment can be manipulated from the operating window, as shown in Fig. 3. For example, the set-point of a PID controller can be changed, or a blower can be started or stopped. The product property monitoring window is placed at the upper right comer, as shown in Fig. 3. In the two dimensional graph of the product property monitoring window, the horizontal axis represents melt index (MI) and the vertical axis represents density (D). Both the instantaneous product properties and cumulative properties are plotted so that the trajectory due to grade changes can be monitored. Using this simulator, we can perform various case studies to improve the control performances during the grade transitions.
3. Nonlinearity of the process
We examined the changes in process dynamics of the product property dynamic model shown in Fig. 2 at typical operating points for the different grade products shown in Table 1. Eqs. (4)-(7) are linearized at each operating point. Four poles and two zeros of the transfer function matrix are summarized in Table 2(a) and
transition
monitoring
(upper
right corner).
C. Sato et aL / Computers and Chemical Engineering 24 (2000) 945-951
948
Table 2 Changes in process dynamics of the product property dynamic model (a) Poles Grade A Grade B Grade C Grade D
-0.0948 -0.0984 -0.0870 -0.0990
-0.3571 -0.3571 -0.3571 -0.3571
-0.3571 -0.3571 -0.3571 -0.3571
-0.7418 -0.6350 -0.5497 - 1.0939
MIc/FM2 -0.0948 -0.3571 -0.0984 -0.3571 -0.0870 - 0.3571 -0.0990 -0.3571
MIJFm -0.7418 -0.3571 -0.6350 -0.3571 -0.5497 - 0.3571 - 1.0939 -0.3571
Dc/FM2
Dd Fm
-0.0948 -0.3571 -0.0984 -0.3571 -0.0870 - 0.3571 -0.0990 -0.3571
-0.7418 -0.3571 -0.6350 -0.3571 -0.5497 - 0.3571 - 1.0939 -0.3571
MIc/FM2 (Ratio to Grade A) -0.003012 (1.000) -0.001405 (0.466) -0.003888 (1.291) -0.004506 (1.496)
MI¢/FHz
Dc/FMz
De~Fro
(b) Zeros Grade A Grade B Grade C Grade D (c) Process gain Process gain Grade Grade Grade Grade
A B C D
-0.21413 -0.08238 -0.22315 -0.45239
(b), respectively. Four poles are the same among the four elements of the transfer function matrix, and two zeros are the same between MIdFM2 and DdFm, and between MIdFH2 and DdFH2, respectively. Table 2(c) shows the static process gain of 2 x 2 transfer function matrix. It changes remarkably due to the product grade, for example, the static process gain from FH2 to MI¢ for Grade D is about five times as high as that for Grade B. The characteristics of process dynamics are summarized as follows: 1. Two zeros are cancelled by two poles. 2. The remaining two poles move little due to the product grade. 3. The process gain changes remarkably due to the product grade.
4. Product property control
4.1. Control objectives The purpose of product property control is to drive the product property to the new target value in a short time and to produce fewer off-specification polymers. Grade transition control that causes excessive overshoots in the instantaneous product properties (MI i, Di) must be avoided.
4.2. Servo controller design If the plant's outputs are to follow a class of com-
(1.000) (0.385) (1.042) (2.113)
0.000812 0.000852 0.000878 0.000862
(1.000) (1.049) (1.082) (1.061)
-0.00360 --0.00219 --0.00383 --0.00501
(I.000) (0.609) (1.064) (1.392)
mand inputs, which are given by the outputs of a finite dimensional linear system, the controlled system is called a Servo System (Kreindler, 1969). Recently, in the presence of disturbances, the above controlled system is also called a servo system, and there have been many studies on this subject. In this paper, a servo controller is designed so that the product property could be driven to the new target value in a short time and the controlled system is stable. A method for designing an optimal servo system that minimizes the objective function comprising the quadratic forms of both the output error vector and the deviation control vector was proposed by Pak, Suzuki and Fujii (1974). In this method, a design problem of the multivariable feedback control system was reduced to the usual optimal regulator problem. However, the above optimal servo system is a controller without integral actions, so the steady state error (offset) will occur if a disturbance or model mismatch exists. Researchers proposed some methods for designing a servo system when disturbances and references were given by the outputs of the finite dimensional linear system (Davison, 1976; Ikeda, 1997). These methods are based on the lnternal Model Principle (Francis & Wonham, 1975). The servo system consists of a servocompensator and a stabilizing compensator. Fig. 4 shows the block diagram of the optimal servo system. The servo-compensator incorporates an internal model of the exogenous signal that the compensator is re-
C. Sato et al. / Computers and Chemical Engineering 24 (2000) 945-951
Servo Compensator
Plant Model for Controller Design
ks(t ) • Ks + k¢(t)" K~
(8a)
G = ks(t ) • G s + ke(t ) • G e
(8b)
K =
r
949
ks(t) =
1 + cos (t - ts) 7r ( t e - ts)
(t s < t < t~) - (t~ < t)
(8c)
(ts < t < t~)
(8d)
Fig. 4. Block diagram of optimal servo system.
quired to process. In this optimal servo system, asymptotic regulation is guaranteed independent of input disturbances and peturbations in the plant parameters. In this paper, an optimal servo system is designed using the linearized product property dynamic model with reference to a design method (Ikeda, 1997).
iI
k~(t) =
1 - cos
z~
(t~ < t)
where ts is a fixed time when a grade change begins, and te is also a fixed time when a grade change should end.
4.3. Nonlinear compensation The controller gain (K, G) can be switched to cope with the nonlinearity of the process. A gain (Ks, Gs) is obtained using the linearized model at the initial operating point, and one more gain (Ke, Ge) is calculated at the target operating point. The controller gain can be adjusted according to the following schedule (Sato & Sakawa, 1988):
5. Simulation The grade change simulations from Grade A to Grade B were performed with three kinds of controller gains. Fig. 5 shows the results. The dotted line shows the transition with the controller gain adjusted for Grade A (CASE1), the dash-dot line shows the transi-
'12
:."x
10 ~#
,=_
8 •
N 6
: . . . . . . . . . . . .
:
~. . . . . . .
-. ...........
N.,
Lg
CASE1 : CASE2 CASE3
0.93 10
15
r J,.. ~ ~r'~..; .-','.. . . . . . . . 0
5
10
O
5
10
,
, :1 Q
15
O1~
0:1 0.14
0,12
o.ti
-~0.06
o=1
0:09
0,04
C'
2~
2O
5 oo
15
10 Tinn~
4~
15
~'7~J
Fig. 5. Grade transition from Grade A to Grade B under three controllers.
!.6
950
C. Sato et al. / Computers and Chemical Engineering 24 (2000) 945-951
tion with the controller gain adjusted for Grade B (CASE2), and the solid line shows the transition with gain scheduling by Eqs. (8a)-(8d) proposed in this paper (CASE3; ts = 0, te = 7). The dashed straight line in MIc and De responses shows the target (Grade B) property. It takes shorter time for the grade transition in CASE2 than that in CASE1. During the grade change in CASE2, a smaller overshoot in MI~ than that in CASE1 but a larger deviation in D~ than that in CASE1 were observed. In another grade change from Grade A to Grade D, it also took shorter time for the transition with the controller gain adjusted to the target grade than that with the controller gain adjusted to the initial grade. Apart from this fact, it takes the shortest time for the grade transition in CASE3; during the grade change, the smallest overshoot in MI c and relatively smaller deviation in D e were observed. As a result, the proposed nonlinear compensator worked well to cope with the static process gain changes due to product grades. This method will provide good control performance during the critical grade changes.
6. Conclusions A set of physical models of an industrial ethylene polymerization process was developed. This model was composed of seven ordinary differential equations. Parameters in the model were adjusted to simulate the actual process behavior. We used this model as a virtual plant when investigating operational problems in an actual plant. Under the tight control of three state variables, we had four ODEs with respect to the butene and hydrogen concentrations, and the two cumulative polymer properties of melt index and density. This control process was regarded as a two-input, two-output system. An optimal servo controller with integral actions was designed by the optimal regulator theory using the linearized model for each product grade. The changes in process dynamics at typical operating points for different grade products were characterized by the large changes in the static process gain. As a result, a nonlinear compensator with gain scheduling was attached to the optimal servo controller to cope with the process gain changes. This gain scheduling control method provided good control performances during grade transitions. The nonlinear physical model was effectively used for the control system design.
Ken'ichi Ohmura and Goro Oguchi from Omega Simulation, Ltd. for their continuous cooperation and support.
Appendix A. Notation
DC
DI
F.2
h -1) ethylene feed rate to reactor (kgmol
h -1)
F~2
butene feed rate to reactor (kgrnol h -~) feed rate of catalyst site to reactor (kgmol h - l) enthalpy of feed to reactor (kcal h-l) Hfeed enthalpy associated with polymer leavnpoly ing reactor (kcal h -1) enthalpy of reaction (kcal kgmol -~) AHR concentration of hydrogen in reactor [H2] (kgmol m -3) k~ deactivation rate constant [l/h] kpM1 (T) temperature-dependent propagation rate constant of ethylene (m 3 kgmol -~ h-l) temperature-dependent propagation rate kl,M2 (T) constant of butene (m 3 kgrnol -l h -l) ko, kl, k3, k7 parameters in instantaneous melt index model thermal capacity of reaction vessel [kcal/ MG
F~
/q
MIc MII [M1] [M2] Op P Po,Pl,P2,P7
Qc RH2 T
To
Vg Ws Y
Acknowledgements The authors would like to thank Shintaro Miura,
cumulative density (g cm -3) instantaneous density (g cm -3) hydrogen feed rate to reactor (kgrnol
cumulative melt index (g (10 min) -~) instantaneous melt index (g (10 min) -~) concentration of ethylene in reactor (kgm o l m -a) concentration of butene in reactor (kgm o l m -3) outflow rate of polymer product (ton h -1) reactor pressure (Mpa) parameters in instantaneous density model amount of heat removed by heat exchanger (kcal h -1) rate of hydrogen consumption due to reaction (kgmol h -1) reactor temperature (K) reference temperature (K) volume of gas phase in reactor (m 3) mass of polymer in bed (ton) number of moles of catalyst sites in reactor (kgrnol) time constant of polymer phase in fluidized bed (h)
C. Sato et al./ Computers and Chemical Engineering 24 (2000) 945-951
References Davison, E. (1976). The robust control of a servo-mechanism problem for linear time invariant multivariable systems. IEEE Transactions on Automatic Control, AC-21(1), 25-34. Francis, B. A., & Wonham, W. M. (1975). The internal model principle for linear multivariable regulators. Journal of Applied Mathematics, 2(Opt), 170-194. Hamasaki, H., Shimizu, M., Ishikawa, T., Ohmura, K., Nakajima, K., & Kitamura, H. (1994). Plant operation training simulator 'PLANTUTOR'. Chemical Engineering, 39 (10), 77-83 (In Japanese). Ikeda, M. (1997). SICE Seminar Text, An Introduction to Advanced Control Theory. (In Japanese). Kreindler, E. (1969). On the linear optimal servo problem. International Journal of Control, 9(4), 465-472.
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McAuley, K. B., & MacGregor, J. F. (1991). On-line inference of polymer properties in an industrial polyethylene reactor. American Institute of Chemical Engineering Journal, 37(6), 825-835. McAuley, K. B., MacDonald, D. A., & McLellan, P. J. (1995). Effects of operating conditions on stability of gas-phase polyethylene reactors. American Institute of Chemical Engineering Journal, 41(4), 868-879. Pak, P. S., Suzuki, Y., & Fujii, K. (1974). Synthesis of multivariable linear optimal servosystem. Transactions of SICE, 8(5), 568-575. Sato, K., & Sakawa, Y. (1988). Modeling and control of a flexible rotary crane. International Journal of Control, 48(5), 2085-2105. Sato, C., Ohtani, T., & Nishitani, H. (1999a). Optimal servo system for polymerization process. Proceedings of the 64th SCEJ Annual Conference. B206 (In Japanese). Sato, C., Ohtani, T., & Nishitani, H. (1999b). A study of nonlinearity of a polymerization process for grade change. Proceedings of the 38th SICE Annual Conference SICE'99, 725-726. (In Japanese).