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Stabilization of the Dynamic Behavior of a UNIPOL Process for Polyethylene Production.

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Dev. Chem. Eng. Mineral Process. I2(I/2), pp. 199-21 6, 2004.
Stabilization of the Dynamic Behavior of a
UNIPOL Process for Polyethylene
Production
Nayef Mohamed Ghasem and Mohamed Azlan Hussain*
Department of Chemical Engineering, University of Malaya,
50603 Kuala Lumpur, Malaysia
Polyolefins have become one of the most important plastics worldwide due to
continuous improvements in catalysts and processes. Gas-phase polymerization of
olefins is one of the most important polymerization processes. Compared to other
processes such as slurry and solution polymerization, gas-phase processes have many
distinct advantages, e.g. reduced capital and operating costs. Moreover, gas-phase
polymerization offers a large variety of products, which could not be produced by
other processes. However, sheeting and agglomeration of polymer particles are two
serious problems, which can occur in modern gas-phase polymerization processes.
Overheating of particles may occur due to very high reaction rates. The temperature
of a particle can then rise above the softening temperature. In this work, recent
theories of bifurcation and chaos are used to analyze the dynamic behavior of the
UNIPOL process for the gas-phase production of polyethylene using a Ziegler-Natta
catalyst. The dynamic behavior covers a wide range of the design and operating
parameters domain for this industrially important unit. A conventional proportionalintegral-derivative (PID) controller was implemented to stabilize the desired
operating point on the unstable steady-state branch to a certain range of catalyst
injection rate. By contrast, it was found that the controlled process can go through a
period doubling sequence leading to chaotic strange attractors. The practical
implications of this analysis can be very serious, since chaos is shown to exist close to
the desired operating point where high polyethylene production rates can be
achieved.
Keywords: Polyethylene; Fluidized bed; Chaos: PID;Instabilip; UNIPOLprocess.
-~
* Authorfor correspondence (mohd-azlan@um. edu.my).
I99
N.M. Ghasenz and M.A. Hussain
Introduction
Nonlinear dynamical systems can exhibit a variety of behavior depending on the
values of the system operating parameters. For certain values of the parameters, a
nonlinear system can exhibit simple or complex (quasi-periodic, as well as strange
chaotic and nonchaotic) oscillations. In the last two decades many researchers have
studied the behavior of several non-linear dynamical systems (e.g. Jackson, 199l),
including the appearance of deterministic chaos that is characterized by the sensitive
dependence on the initial conditions of a trajectory of the system. In the case of
dissipative systems described through nonlinear ordinary differential equations and
exhibiting deterministic chaos, the system evolves in phase space over a fractal
structure called a strange or chaotic attractor that consists of a continuous stretching
and folding process. Deterministic chaos is usually associated with wild oscillations in
the variables that may be inconvenient for practical applications of the systems
exhibiting this feature. Therefore, many applications would prefer the system to have a
more regular behavior. Imagine that a vector field for the ODE x = f ( x ;p ) depends
on a set of parameters, p. Here we wish to investigate what happens to the flow of the
system when the parameters vary slightly. Do the orbits only change slightly? The
answer is sometimes, but not always. When there is a “dramatic” change in the
dynamics we say that a bifurcation has occurred. Bifurcation is a qualitative change in
dynamics occurring upon a small change in a parameter. The bifurcation diagram is a
graph that represents the qualitative behavior the system. Traditionally the abscissa of
the graph corresponds to the parameter, p, and the ordinate to the phase variable, x .
Thus, each vertical line is a picture of the vectorfield for fixed parameters, and we
stack the vectorfields with varyingp to obtain the full bifurcation picture.
Experimental evidence of sustained oscillatory behavior for a laboratory-scale
polymerization reactor was observed by Teymour and Ray (1989, 1991), and
confirmed in modeling studies. It is well known that when a highly exothermic
reaction occurs in a fluidized-bed reactor, unusual steady-state and dynamic behavior
may be observed due to the interaction between mass and heat transfer processes
(Bukur and Amundson, 1975, 1982). However, in contrast to most conventional
fluidized-bed reactor systems, it is the solid phase (polymer) which is of primary
importance in gas-phase olefin polymerization.
Chemical reactors, especially those in which rapid exothermic reactions take place,
are often difficult to control. This is the case when there are wide fluctuations in the
temperature extremes resulting from minor fluctuations in one or more operating
variables. Operation in the unstable regions may result in a poor product, runaway
temperatures, and rapid deterioration of catalyst in fluidized-bed reactors.
Recognition of the circumstances that can cause such instability, and knowledge of
how to prevent it, are important aspects of reactor design. Such reactors may exhibit
limit cycle (oscillations) at certain operating points. During such limit cycle behavior,
reactor temperatures oscillate with high amplitude, which is in many cases large
enough to pose a potential threat to thb reactor. Among those highly exothermic
reactions is the one for polyethylene production in a catalytic fluidized-bed reactor
(the UNIPOL process).
200
Stabilization of the Dynamic Behavior of a UNIPOL Process
In recent years, polyethylene has achieved its preeminent position in the
thermoplastics industry due to a remarkable catalyst and gas-solid fluidized bed
technology (Choi and Ray, 1985; McAuley et al., 1994; Lim et al., 1996; Ghasem,
1999, 2001). In addition, by employing the Ziegler-Natta catalyst, the process can
operate at relatively low pressure and temperature (20 atm, 100’C) compared to the
conventional process (2000 atm, 2OOOC).
The discovery of the process for the production of polymers in a fluidized-bed
catalytic reactor has provided a means for the production of a diverse array of
polymers. Using a gas fluidized-bed polymerization process substantially reduces the
energy requirements when compared to other processes, and most lmportantly reduces
the capital investment required to operate such a process.
Gas fluidized
polymerization plants generally operate on a continuous cycle. In one part of the
cycle, in a reactor a cycling gas stream is heated by the heat of polymerization. This
heat is removed in another part of the cycle by an external cooling system. Generally
in a gas fluidized-bed process for producing polymers from monomers, a gaseous
stream containing one or more monomers is continuously passed through a fluidized
bed under reactive conditions in the presence of a catalyst. This gaseous stream is
withdrawn from the fluidized bed and recycled back into the reactor. Simultaneously,
polymer product is withdrawn from the reactor and new monomer is added to replace
the polymerized monomer.
It is important to remove heat generated by the reaction in order to maintain the
temperature of the gaseous stream inside the reactor at a temperature below the
polymer and catalyst degradation temperatures. Further, it is important to prevent
agglomeration or formation of chunks of polymer that cannot be removed as product.
This is accomplished by controlling the temperature of the gaseous stream in the
reaction bed to a temperature below the fusion or sticlung temperature of the polymer
particles produced during the polymerization reaction. Thus, it is understood that the
amount of polymer produced in a fluidized polymerization process is directly related
to the amount of heat that can be withdrawn from a reaction zone in a fluidized bed
within the reactor.
However, improper control of the process parameters, especially the catalyst feed
rate, inlet gas feed temperature and gas superficial velocity, may lead to temperature
runaway and clusters formation in the reactor. Subsequently the plant has to be shut
down for cleaning. In addition, the situation becomes worse when the reactor bed
temperature exceeds the polyethylene softening point (approx. 125”C),where the solid
particles tends to agglomerate and may form a huge deposit in the reactor. The chaotic
system is referred to as a strange attractor. The term “strange” arises from the fact that
the attractor is a geometrical structure that will never settle down or repeat itself and
yet it is apparently bounded by a definite structure. One universal feature of chaotic
systems is their sensitive dependence on initial conditions. Other investigators of
different systems than the one investigated here have observed dynamic bifurcation
and chaotic attractors (Jaisinghani and Ray, 1977; Schmidt and Ray, 1981; Kim and
Laurence, 1998). Teymour and Ray (1985, 1991, 1992) observed chaos, intermittency
and hysteresis in the dynamic model of polymerization of vinyl acetate in CSTR
solution. Chaotic behavior of an acetyl cholinesterase enzyme system using a twocompartment model was observed (Elnashaie et al., 1995). Chaotic behavior is
201
N M . Ghasem and M.A. Hussain
observed in fluidized-bed catalytic reactors with consecutive exothermic reaction
(Elnashaie and Ajbar, 1996). Non-chaotic strange attractors are essentially complex
attractors, which look like chaotic attractors on the phase plane, Poincare map and
time traces but are not actually so (Brindley et al., 1991; Kapitanick and Elnaschie,
1991).
It is possible to control the chaotic behavior by small parametric perturbations of a
manipulated variable with suitable frequency and amplitude. By periodic forcing of
input variables such as feed temperature, the chaotic behavior can be controlled and
becomes periodic (Bandyopadhyay et al., 1993). In recent years a number of
researchers have explored ideas for the control of chaotic behavior. Shinbrot et al.
(1993) provided a review of some of these techniques
The implementation of dynamic analysis based on the theories of bibcation and
chaos is an important approach for understanding the design, operation and control of
the fluidized-bed reactor for polyethylene production. In the present work, the
bifurcation and chaotic characteristics of the UNIPOL fluidized bed catalytic reactor
are investigated under the action of a PID controller. Catalyst feed rate is chosen as
the bifurcation parameter throughout the analysis.
The UNIPOL Fluidized-Bed Reactor
A schematic diagram of the UNIPOL process is shown in Figure 1. The feed to the
reactor consists of ethylene, co-monomer, hydrogen, inert gas, and catalyst. Growing
polymer particles are fluidized by a recycle gas stream of monomers, co-monomer,
hydrogen, nitrogen and inert condensing agents such as hexane or isopentane. The
cycle gas flow provides monomer and co-monomer for polymerization, agitates the
bed, and also removes the heat of polymerization. The cycle gas exits the top of the
reactor and is then compressed and cooled before being fed into the bottom of the
fluidized bed. The heat exchanger is assumed to cool the recycle gas instantaneously
to the desired feed temperature, as in the case considered by Choi and Ray (1985).
The temperature in the reaction zone can be adjusted instantaneously by manipulating
the flow rate of the cooling water in the external heat exchanger. Alternatively,
adjusting the cooling rates in the heat exchanger by instantaneous blending of cold and
warm water streams while maintaining a constant total cooling water flow rate through
the heat exchanger (Dadebo et al., 1997). The temperature in the reaction zone can
also be adjusted by manipulating the coolant temperature of the reactor walls.
Model Development
Considering a freely bubbling fluidized bed and using the two-phase theory of
fluidization (i.e. the reactor is divided into two parts; dense phase and bubble phase)
for polyethylene polymerization (UNIPOL process), the following model equations
are obtained (Choi and Ray, 1985; Ghasem, 1999).
1. Material Balance
For the bubble phase (ethylene):
X , = X,+ (1 - X,)e-KBC
202
...( 1)
Stabilization of the Dynamic Behavior of a UNIPOL Process
Recycle gas
Blower
Heat
3
Figure 1. Schematic diagram of UNIPOL process.
The average value of
X,across the bed height:
j?, = jI X , d { = X ,
+(-)
0
1-X,
KB
(l-e-KB)
...(2)
For the emulsion phase (ethylene):
-a,- a ( -~X,) + pr(i - X,)(I - e - K B )
dr
(-81 X , )
-<XI X 2 e
6 X , 2 X 2 e(-6'x3)
-
I.
.(3)
e+x,
For the emulsion phase (catalyst):
a 2 --
dr
4c
M(1-6') (1- smr)ps
to
< XI X2'
e(-6/x1)
e+x,
. . .(4)
203
N.M. Ghasem and M A . Hussain
2. Energy Balance
For the bubble phase (ethylene):
X, = X,
+ (YF - X 3 )
e- KH
... ( 5 )
clx4
The average value of X , across the bed height:
. . .(6)
For the dense phase (ethylene):
...(7)
3. Proportional-Integral-Derivative (PID) Controller
The manipulated variable of the dimensionless reactor wall cooling temperature
(assumed equal the reactor coolant temperature), Y , , is related to the error
( X s p- X,) by the classical PID control law:
...(8 )
By defining the following variable:
r
X, = K~
J(xsP-x3)dt
. .(9)
*
0
Substituting Equation (9) into Equation (8):
Y,= Yo+ K,(X,-X,)+X,+Kd-
204
a
3
dr
. . .(10)
Stabilization of the Dynamic Behavior of a UNIPOL Process
By substituting Equation (10) into Equation (7) and rearranging, the system is readily
transformed to a 4* order system consisting of Equations (3) and (4) together with
Equations (1 1) and (1 2) given below:
.. .(11)
.. .( 12)
Numerical values of the physical parameters and reactor dimensions are given in
Table 1.
Table 1. Physical constants and parameters.
0.44 callg K
0.456 callg K
250 cm
DG
Ea
H
KPO
6.0~10" cm2 I s
9000 cal I mol
600 cm
4 . 1 6 7 ~ 1 0cm3
~ lgcat s
0.95 glcm3
2.37 glcm3
0.029 g I cm
1 . 1 6 ~ 1 g1cm.s
0~
916 callg
205
N.M. Ghasem and .MA. Hussain
Simulation parameters are estimated by the empirical correlations (Choi and Ray,
1985; McAuley et al., 1994; Kunii an Levenspiel, 1991), values for the present case
are shown in Table 2.
Table 2. Simulation parameters estimated by the empirical correlation (Choi and
Ray, 1985; McAuley et al., 1990; Kunii and Levenspiel, 1991).
56.0 cm
0.380
1 . 3 2 ~ 1 0 -callcm3
~
sK
3 . 8 1 ~ 1 0 -callcm2
~
s K
1.04x10-* s-l
192.96 cmls
15.75 cmls
5.2 c m l s
31.08 cmls
Equations (3, 4, 7, 12) represent a 4* order system consisting of four ordinary
differential equations and the others are algebraic equations. The model equations
were solved to construct static and dynamic bifurcation characteristics of industrial
fluidized-bed polyethylene reactors, using the software AUTO86 by Doedel and
Kerneves (1986). The AUTO86 code uses a continuation method in which the
computation begins at a hown solution point and continues to points along a specific
branch of solutions. AUTO86 can be used to trace the steady-state solution branches,
to detect bihcation points (multiple steady-state solutions), and to compute the
bifurcation branches. It also locates Hopf bifurcation points and traces periodic
solution branches (PD). Afier the period doubling (PD), various methods of
integration to characterize the complex dynamics is used to improve the calculations
such as subroutine IVPAG in the IMSL library (IMSL,1985).
Results and Discussion
The kinetic behavior of the catalyst and the effect of reactor temperature on the
product properties require, in some cases, operation of the reactor at a temperature
just below the softening point of the polymer. By contrast, operating the system closer
to the softening point will cause temperature runaway, since the system is not stable
near this point. For this reason, a suitable controller is required to stabilize the
unstable steady states to avoid polymer melting and to achieve higher polyethylene
production rate. The conventional PID controller implemented for stabilization of a
206
Stabilization of the Dynamic Behavior of a UNIPOL Process
polyethylene fluidized-bed reactor stabilized the desired operating point and revealed
interesting dynamics appearing in the instability region which leads to period adding,
period doubling and chaotic behavior. A suitable operating point (OP) is selected on
the unstable steady-state branch (Ghasem, 2000) corresponding to ( X 3 = 1.307,
qc =0.35 k g l h , Yo = 1.15, YF =1.2, u, /u,f = 6 ) . This point is just below the
softening point of the polymer, whch is approximately 125°C (the type of polymer
considered by Choi and Ray 1985; McAuley et al., 1994; Ghasem, 1999, 2001).
1. Static and Dynamic Bifurcation Diagram
Figure 2 shows the bikcation diagram for the effect of catalyst feed rate (qc)on
dimensionless reactor dense-phase temperature (X,)under the action of PID control.
There is only one HB point. The diagram shows a stable steady-state up to HB = 0.36
kgh. At the HB point, it loses its stability and becomes unstable. A stable periodic
branch emanates from the HB point, loses its stability, and becomes an unstable
periodic branch at qc = 0.384 kgih. The unstable periodic branch regains its stability
and becomes stable after qc = 0.4541 kgh. The entire region of thls diagram can be
divided into four different sub-regions as follows:
qc < HB = 0.36 kgh.
This region is a unique stable steady-state. This region is suitable for operation
without the possibility of polymer melting, since the temperature is stable and
below the polyethylene melting point (X,< 1.333; i.e. T, < 125OC).
.
2.w
.**..***..*.*..**
ZPD = 0.4541 1
0
0
0
0
0
0
unstnble 0
'."
HE = 0.36
e0PD = 0.3843
.**
*.
o%
8.m
0.31
9.41
0
.. . .
.*
11.51
0.611
8.70
a.m
4, [kglhl
Figure 2. Bifurcation diagram for the effect of catalyst feed rate versus dimensionless
reactor temperature)3'(
under PID control.
( Y F =1.2, u,Iu,,,,
= 6 , Yo =1.15,C0 =0.02glcm3,
K d = l . o ~ l O s- -~' , K c =O.l,Ki =0.05)
207
N.M. Ghasem and M.A. Hussain
0.36 < qc< PDI = 0.3843 kgh.
In this region there is a unique unstable steady-state surrounded by stable periodic
attractor. This region is not suitable for operation since temperature oscillation
exceeds the melting temperature of the polymer.
PD1= 0.3843 qc
< PD2= 0.4541 kgh.
In this region, an unstable periodic attractor surrounds the unstable steady-state.
This region is not suitable for operation for the above-mentioned reasons.
Detailed analysis of this region is presented in the following sections.
For qc = PD2> 0.4541 kg/h.
In this region, the unstable steady-state is surrounded by a stable periodic branch.
This region is not suitable for operation since the temperature oscillates above the
polymer melting temperature.
Two Parameter Continuation Diagram
A two-parameter continuation diagram can give a clear picture of the bifurcation
behavior of the system in a condensed form. The graph can be used to predict the
main features of the one-parameter bifurcation diagram. Two parameter continuation
diagrams can be used to locate the loci of the static limit point (in the present
bifurcation diagram no static limit points exist) and Hopf bifurcation points for two
manipulated variables at fixed values of the other parameters.
Figure 3 is a two parameter continuation diagram (a) (K,vs 4,) ; (b) ( K i vsq,) ;
(c) ( K dvs 4,) ; showing the loci of the HB point of Figure 2. Figure 3a shows that the
increase in K, increases the range of catalyst feed rate at which stable steady states
exist by pushing the HB point to higher values of catalyst feed rate. A higher value of
K i decreases this ranges (see Figure 3b). The increase in the derivative action ( K d )
decreases the suitable operating stable steady-state region by pulling the HB point to
smaller values of qc (see Figure 3c). Large values of K, may be used to improve the
speed of the closed loop response and increase the stable steady-state region.
3. Chaotic Attractor and the Connection between Chaotic and Periodic Attractors
A stable periodic branch means there is limit cycle behavior. Whereas an unstable
periodic branch needs to be further analyzed using dynamic simulation in order to
visualize the shape of the time trace and phase plane, which can be a guide to the
behavior of the system in h s region.
Various methods of characterizing the complex dynamics of this system (e.g.
Poincare bifurcation diagrams, return point histogram, phase plane and time trace)
could be used. Catalyst feed rate ( qc ) is used as the bifurcation parameter throughout
this analysis. AUTO86 detects the period doubling point. Figures 4 to 8 were
generated using FORTRAN and Subroutine DGEAR recently hown as IVPAG in the
new IMSL library.
208
._ 1
Stabilization of the Dynamic Behavior of a UNIPOL Process
.
.
I
..". .
.
.
.
I
.
.
I
.,
'. ..
.
, ,
~
,
,
%.
.
.
I
..m
.....
.."
.
.
I
Catalyst feed rate (I@)
Catalyst feed rate (kern)
@)
(a)
.oj
*,-
..-......--'\,
..O
Catalyst feed rate (ko)
(c)
Figure 3. Twoparameters continuation diagramfor the loci of HB point;
(a!Kc vs q c ; (6) K, vs q c ; (c! Kd vs q c
Figure 4 is a Poincare bifurcation diagram obtained by integrating the system of
the four ordinary differential equation using a DGEAR subroutine. This figure is
constructed for the intersection of the trajectories with fEed hyper plane (X,=O.815).
The diagram covers the range of catalyst feed rate between qc= 0.382 and qc = 0.456
kgh. On this scale, it can be observed that only the periodic branch emanating from
HB point (period one) bifurcates through period adding, and period doubling leading
to strips of chaos. The behavior of the system then looks like an alternation of period
doubling and intermittent chaos and emerges as a period one attractor after qc = 0.456
kg/h, i.e. the stable periodic branch. It is clear that the behavior is quite complex and
more enlargement of specific regions is required in order to be able to identify and
analyze the fine structures of the graph. Further enlargement for finer structures is
shown in Figure 5 .
Figure 5a shows that as the value of qc increases, the PI (period one) periodic
branch starting at the HB point loses its stability and the PI attractor is interrupted by
the appearance of a P5 (period five) attractor that goes through a sequence of period
doubling bifurcation leading to a very thin strip of chaos. The graph is thus
characterized by the occurrence of the phenomena of bi-stability over different
regions, where two attractors coexist for the same bifurcation parameter.
209
N.M. Ghasem and M.A. Hussain
1.30 .~
0.382
0.307
0.412
0.426
0.441
0.466
qc IWl
Figure 4. One-dimensional Poincare bifircation diagram X3vs catalyst feed rate
(intersectionwith the Poincareplane at XI= 0.815).
0.384
0.W
0.388
0.W
0.300
0.393 0.m 0.3M
qc FJal
1.42
0.400
Pc I y h l
(a)
1.40
1.38
x3 1.38
. .,
1.30
,
.,
. .. .
1.34
1.32
1.30
0.400 0.40s 0.410
qc FJhI
(C)
0.41s
0.420
0.420
0.430 0.440
qc 1-
0.450
0.400
(a
Figure 5 Poincare bifivcation diagramsfor diifserent sections of Figure 4
for the efect of catalystfeed rate on reactor temperature 0.
210
Stabifization of the Dynamic Behavior of a UNIPOL Process
Globally, it can be seen that the system is alternating between PI, Ps, period
doubling (such as at qc = 0.388 kgh) and very thm strips of chaos. Figure 5b shows
that at qc = 0.39kgh, P5attractor cascade through period doubling bifurcation leading
to chaos, chaos persists up to critical catalyst feed rate, at which point order emerges
out of chaos in the form of a periodic orbit, which quickly undergoes a series of period
halving ending in the recovery of the P7 orbit (i.e. at qc = 0.3915 kgh). The latter
gives way to chaos through the same intermittency mechanism. Figure 5c starts with a
strip of chaos, by increasing qe,the system regains its stability and Ps attractor appears
(i.e. at qc = 0.40125 kgh) which goes through a sequence of period doubling leading
to chaos. Figure 5d starts with P3 attractor, which bifurcates through period doubling
to strips of chaos interrupted by periodic windows and ends with PI. In general, it can
be seen that the system starts with period one at qc = 0.382k g h (the range of period
one starts with HB and ends with PD,= 0.3843)and goes through repeated alternation
between period adding, period-doubling bifurcation cascading into strips of chaos, and
ends with period one corresponding to the stable periodic branch after PDI = 0.4541.
:: 1
:: 1
.........
...............
q
1.38
1.30
1;.
100200300400600(100
.
.....
...._. . . . . ...; ... .( . . .
"
'
..
.+rq-qlT
1 w 2 0 0 3 0 0 4 o D 5 0 0 ~
Tima Ial
M
100200300400500800
l W 2 0 0 3 W 4 0 0 ~ ~
Time [h]
(4
T h e [a1
(d)
Figure 6. Return point histogram ( X , vs. time) at diferent values of catalyst feed
rate (a) qc= 0.396 kgh; (b) qc = 0.3967; (c) qc= 0.3972 kdh; (4 qc= 0.3998 kdh.
N.M. Ghasem and M.A. Hussain
5
0
M
8
0
6
6
7
0
5
Time [hl
0
5
6
~
Time P]
8
6
7
0
(a
(el
Figure 7. Time trace (X, vs. time) at dyerent values of catahst feed rate
(a) qc= 0.396 kdh; (b) qc= 0.3967; (c) qc= 0.3972 kdh; (4 qc = 0.3998 kdh.
2.0
1.2
x3;qi-Fi
0.2
0.4
0.8
0.8
1.0
0.2
0.4
Xl
0.6
0.6
1.0
0.8
1.0
X1
1.2
1.0
1.0
0.2
0.4
X1
0.6
0.6
X1
(GI
(d)
0.8
1.0
0.2
0.4
Figure 8. Phase plane (X,vs. X , ) at dyerent values of catalyst feed rate (a) qc =
0.396 kdh; (b) qc= 0.3967; (c) qc= 0.3972 kdh; (4 qc = 0.3998 kgdh.
212
Stabilization of the Dynamic Behavior of a UNIPOL Process
4. Return Points, Time Trace and Phase Plane
Chaotic behavior can be seen if a return point hstogram is studied, as in Figure 6,
where a plot of the dimensionless reactor temperature (X3)at the return map-Poincare
map- vs time provides more insight into the time evolution of the dynamic behavior.
Figures 6(a), (b) and (d) show the return points hstograms for X,vs. time, at different
values of catalyst feed rate (0.396, 0.3967, and 0.3998) respectively. Chaotic
behavior exists as shown from the accumulation points. On the contrary, at catalyst
feed rate of qc = 0.3972 kgh, Figure 6(c) shows three lines representing period three.
Time trace and phase plane for selected catalyst feed rates (0.396, 0.3967, 0.3972,
0.3998) are shown in Figures 7 and 8, respectively. The time traces and phase planes
confirms the presence of period doubling and chaotic behavior.
Conclusions
T h s paper examined some dynamical characteristics induced by the PID control of a
fluidized-bed reactor for polyethylene production using the two-phase theory of
fluidization. Chaotic regime was found to evolve throughout the period doubling
cascade. The structure of the strange attractor was analyzed using various qualitative
techniques, which demonstrated the complexity of the dynamics. Through the use of
the Poincare bifurcation diagram, various types of bibcation including period
doubling and period adding bifurcation were observed. The chaotic regime was also
shown to be interspersed with various periodic windows. The inclusion of the
derivative action does not change these mechanisms of transition to chaos. Larger
proportional gains are required to lmprove the closed loop performance of the system.
Small parametric perturbation of an input variable can be used to control the chaotic
behavior to become periodic.
Nomenclature
A
Cross-sectional area of the bed (cm’)
Cnrb Monomer concentration in the bubble phase (g/cm3)
C
,,
C,
Monomer concentration in the dense phase (g/cm3)
Reference gas concentration (g/cm3)
C,
Heat capacity of gas (caVg K)
C
,
D,
Heat capacity of polymer cal/g K)
Bed diameter (cm)
Bubble diameter (cm)
Gas diffusion coeficient (m2/s)
Particle diameter (cm)
E,
Activation energy for propagation (caymole)
D
d,
DG
213
N.M. Ghasem and M.A. Hussain
H
*be
Bed height (cm)
Overall coefficient of heat exchange between bubble and emulsion phase
(cal/cm2s K)
Heat of reaction (caVg)
Wall heat transfer coefficient (cal/cm2 s K)
Coefficient of gas interchange between bubble and emulsion phase, s-'
Proportional gain
Integral gain
Derivative gain
Reaction rate constant at reference temperature (cm3 /gcat s)
Catalyst injection rate (g/s)
Gas constant (cal/mole K )
Bubble phase temperature (K)
Emulsion phase temperature ( K )
Temperature of the feed gas ( K )
Controller set point temperature (K)
Reactor wall temperature (K)
Reference temperature (300K)
Reference time (1.08 x lo5 sec)
Velocity of bubble rising through the bed (cm 1s)
Upward velocity of gas through the emulsion phase ( c d s )
Inlet gas velocity ( c d s )
Minimum fluidization velocity ( c d s )
Catalyst concentration (g catalysvg polymer)
Dimensionless monomer concentration in the dense phase (C,,,JC,)
Dimensionless catalyst concentration (XaJ
Dimensionless reactor temperature (T3ITrcf)
Dimensionless monomer concentration in the bubble phase ( C d C , )
Dimensionless bubble phase temperature (TdTref)
Reactor temperature set point (Tsp/Tref)
Dimensionless gas feed temperature (TplTref)
Dimensionless temperature of the coolant when the controller is off (To/Tref)
Dimensionless reactor wall coolant temperature (TJTrcf)
Variable fluidized bed height (cm)
214
Stabilization of the Dynamic Behavior of a UNIPOL Process
Abbreviations
HB
Hopf bihcation
PD
Period doubling
Greek Symbols
Void fraction of the bed at minimum fluidization
E,,,~
E,,,,~
Polymer void fraction
pg
Density of catalyst (g/cm3)
Density of gas (g/cm3)
ppr
Density of polymer (g/cm3)
pcnr
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216
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