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

код для вставкиСкачатьDev. 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 References Bandyopadhyay, J.K., Kumar, V.R, and Kulkami, B.D.1993. 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