Simulação e Controle de Processo do Polietileno

Effects of Operating Conditions on Stability of GasPhase Pol yet hylene Reactors K 3 McAuley D A Macdonald and P J McLellan Dept of Chemical Engineering Queens University Kingston Ontario Canada K7L 3N6 Fluidizedbed polyetliylene reactors are prone to unstable behavior and temperature oscillations Choi and Ray 1YNSb Their work is atended to show the effects of ethy lene feed system operation reactor cooling system design catalyst properties and gas composition on reactor stability and dynamics The analysis is peformed using a well mixed model because heat and masstransfer resistances between mirltiple phases are small and ure not required to account for the observed bifurcution phenomena The addition of a gas recycle and heat exchanger system to the model significantly ajjects dynamic performance including the formation of limit cycles The size and dynamics of the heat exchanger however huve little effect on the overall stability In contrast aid tomation of the ethylene feed system to replace the monomer in the reactor as it is consumed leads to substantially different dynamic behavior than if the ethylene feed is maintained nt a constant rate Catalyst properties multiple sites actiuation energy mid deactivation significantly affect dynamics and stability whereas comonomer and other gases uffect them only mildly The results confirm that without proper temperature con trol gasphase polyethylene reactors are prone to instability limit cycles and excursions toward unacceptable hightemperature steady states Introduction Polyethylene is the most popular of all synthetic comniod ity polymers with current worldwide production of more than 30 billion tonnes per year A large proportion of this polyeth ylene is produced in gasphase reactors using ZieglerNatta catalysts One benefit of gasphase polyethylene production is that there is no solvent present in the reactor system to bc recovered and processcd Another advantage is that gasphase processes usually operate at more moderate temperatures and pressures than highpressure or liquidphase polymerization systems A disadvantage of gasphase systems is that the tem perature in the rcaetion zone must be maintained above the dew point of the reactants to avoid condensation and below the melting point of the polymer to prevent particle melting and agglomeration Excellent reviews of industrial gasphase polyethylene production technology have been provided by Choi and Ray 198Sa and Xie et al 1994 To ensure sufficient polymerization rates and maintain the temperature below the polymer melting point commercial Corrcywndence conccming hi article hould hc addrrssed to K B McAuley gasphase fluidizedbed polyethylene reactors are operated in a relativcly narrow temperature range between 75 and 110C Xie et al 1994 Even within this range significant tempera ture excursions must he avoided because they can lead to low catalyst productivity and changes in product properties Mc Auley and MacGregor 1991 1992 1993 Past modeling studies Choi and Ray 198Sb McAulcy 1992 havc indicated that gasphase polyethylene rcactors are prone to unstablc behavior and temperature oscillations The goal of this arti cle is to extend this earlier work to show the effects of ethy lene fccd system operation and reactor cooling system pa rameters on thc thermal behavior of the reactor We also investigate the importance of catalyst deactivation and multi ple catalyst sites and show the effects of hydrogen inert and comonomcr concentrations on reactor stability and dynamics Dynamics and Stability of Polymerization Reactors Van Heerden 1953 was one of the first rcscarchers to study the problems of multiple steady states and instability in chemical reactors Later Uppal et al 1974 1976 used bifur 868 April 1995 Vol 41 No 4 AIChE Journal cation theory to study multiplicity and stability in their analy sis of the autothermal CSTR problem This seminal work prompted a rapid expansion in chemical reactor stability and multiplicity research in the late 1970s and early 1980s Mor bidelli et al 1987 and Razon and Schmitz 1987 have pro vided excellent reviews of this research Important new re sults on the application of singularity theory to determine the steadystate multiplicity patterns of differcnt reactor configu rations are still being developed Lovo and Balakotaiah 1992 W H Ray and coworkers at the University of Wisconsin have performed a large portion of the multiplicity and stabil ity research specific to polymerization reactors In polymer ization systems multiplicity and oscillations can arise for a number of reasons including particle nucleation and growth phenomena in emulsion polymerizations Rawlings 1985 gel effect Jaisinghani and Ray 1977 Schmidt and Ray 1981 Schmidt et al 1984 Adebekun et al 1989 and autothermal effects Hamcr et al 1981 Teymour and Ray 1989 1992a The multiplicities and oscillations predicted for polyethylene reactors in the current article arise due to autothermal ef fects Teymour and Ray 198Y 1992a have studied autothermal limit cycles and multiple steady states in the solution poly merization of vinyl acetate in a CSTR A laboratoryscale polymerization reactor was used to validate their model and to confirm the existence of limit cycles and multiple steady states Teymour and Ray 199 extended their model to demonstrate that the dynamics of fullscale industrial reac tors can be even more complex than their labscale counter parts For example their fullscale model predicts period doubling bifurcations for the limit cycles and possibly even chaos whereas these phenomena were not predicted at the laboratory scale The difference in dynamics between small and large reactors results from the difference in volume to surface area In labscale reactors the heat capacity of the reactor walls and the rate of heat loss to the surroundings are relatively large and help to damp out temperature oscilla tions Feedback temperature controllers can often be used to sta bilize reactors that arc openloop unstable In their study of a fullscale polypropylene reactor Choi and Ray 1988 have shown that the steadystate operating point of industrial in terest is almost always openloop unstable Without tempera ture control the only stable steady states occur when the level in the reactor is controlled but the reactor pressure is not Pressure can only be controlled if a stabilizing feedback tem perature controller is added to the system Even though tem perature control can be used to stabilize openloop unstable systems a good understanding of the underlying steadystate and transient behavior of the system is essential This infor mation is useful for designing control schemes and for pre dicting and avoiding undesirable oscillations or runaway phe nomena when the temperature controller fails or saturates Steadystate multiplicity in polyethylene reactors has been examined on two levels at the smallscale of individual poly mer particles and at the macroscale of the whole reactor Hutchinson and Ray 1987 modeled heat generation in and dissipation from particles in the fluidized bed and predicted multiple steady states for individual polymer particles They showed that particle ignition and overheating leading to an undesirable high temperature steady state can occur early in the lifetime of a polymer particle when the ratio of the mass of catalyst to the particle surface area is high As the parti cles grow their surface area increases and the catalyst de cays reducing the risk of particle melting Hutchinson and Ray 1987 showed that particle overheating is largely unre lated to the efficiency of the macroscale heat removal system of the reactor Even injection of liquid condensate onto the particles which might occur when fluidized bed reactors are operated in condensed mode has little or no effect on parti cle overheating Hutchinson and Ray 1991 Hence the solu tion to the polymer particle ignition problem is one of proper catalyst design At the larger scale of the entire fluidized bed reactor in stability multiple steady states and limit cycles have been predicted using mechanistic models In the very first article on the dynamics of polyethylene reactors Choi and Ray 1985b used a dynamic model accounting for separate bub ble and emulsion phases in the bed to predict up to three steady states in a polyethylene reactor Their analysis which focused on the reactor alone without the associated heat ex changer predicts that a Hopf bifurcation leads to oscillatory behavior Talbot 1990 showed that a simplified backmixed model also predicted multiple steady states In simulations of ethylenecuolefin copolymerizations McAuley 1992 used a wellmixed model to predict that gasphase polyethylene re actors are prone to openloop unstable behavior in operating regions of industrial interest and showed that the system can be stabilized using a PID feedback controller In their dynamic analysis Choi and Ray 198 used a comprehensive twophase model that accounts for heat and masstransfer resistances between the emulsion and bubble phases in the fluidized bed Recently McAuley et al 1994 compared steadystate predictions of a twophase model and a simplified wellmixed model They showed that tempera ture and concentration predictions of the two models differ by less than 2 or 3C and 2 respectively in the operating range of industrial interest Hence at steady state industrial fluidizedbed polyethylene reactors can be accurately mod eled using a simplified wellmixed model The current article demonstrates that dynamic behavior and stability predictions from a simplified wellmixed model of the reactor are similar to those of Choi and Raystwophase model As such we used the simplified model for all subse quent analysis First the behavior of the reactor is analyzed on its own and it is shown that either one or three steady states is possible depending on the catalyst feed rate The local stability of the multiple steady states is analyzed to show that the low temperature steady state may be either stable or unstablc The middle steady state however is always unsta ble and the high temperature steady state is always stable Next a recycle stream and external cooler are added to the model reflecting the usual industrial situation We show that the behavior of the combined reactor and recycle system is very different from that of the reactor alone There is a range of catalyst feed rates where no stable steady state exists and within this range limit cycle behavior is obtained The effects of adding an ethylene concentration controller to the combined reactor and recycle system are examined to demonstrate that the reactor feed system strategy can have a significant effect on stability and multiplicity The effects of AIChE Journal April 1995 Vol 41 No 4 869 l BLEED cooling water PRODUCT FRESH FEED ethylene L nitrogen hydrogen L butenel Figure 1 Gasphase polyethylene reactor system heat exchanger dynamics and size are investigated and are shown to have very little influence on the overall dynamics and stability of the system In the original work of Choi and Ray 1985b only ethylene homopolymerization was modeled using a nondeactivating singlesite ZieglerNatta catalyst In the current article we demonstrate the importance of cata lyst deactivation and multiple catalyst sites and investigate the effects of inerts hydrogen and comonomer to the reac tor We show that while catalyst properties and the ethylene feed system have a significant influence addition of comonomers and other gases to the reactor system have only a mild effect on multiplicity and stability SinglePass CSTR Model for the Fluidized Bed A dynamic model of the polyethylene reactor shown within the dashed system boundary in Figure 1 can be derived by assuming that both the gas and the solid phases are well mixed that the temperature within the reactor is uniform that the mass of polymer in the reactor is constant due to perfect bedlevel control and that there is only one type of catalyst site d Y Of F k d Y Y dt B In the mass balance on the ethylene Eq 11 M is the con ccntration of ethylene in the reactor V is the volume of the gas phase in the reactor F is the volumetric feed rate oi the monomer to the reactor and R is the rate of ethylene consumption due to reaction where Y is the number of moles of catalyst sites in the reac tor and kT is a temperaturedependent propagation rate constant where R is the ideal gas constant E is the activation energy for propagation and Tref is a reference temperature at which the value of the rate constant is known In the catalyst mass balance Eq 2 F is the molar feed rate of catalyst sites to the reactor that can be determined from the mass flow rate of catalyst to the reactor F and the active site concentration on the catalyst a F Fca 6 In Eq 2 k is a deactivation rate constant 0 Rhmwl is the outflow rate of the polymer product from the reactor and B is the mass of polymer in the bed In the reactor energy balance Eq 3 T is the reactor temperature MC is the thermal capacitance of the reaction vessel and Cpp0 is the heat capacity of the polymer The contribution of the gas to the heat capacity of thc gasphase reactor and its contents has been neglected H is the enthalpy of the feed to the reactor and Htop is the enthalpy of gas leaving the reactor H the rate of heat generation by reaction is related to the enthalpy of reaction A HR by where m is the molecular weight of ethylene H is the enthalpy associated with the polymer leaving the reactor In the development of the dynamic reactor model Eqs 131 it has also been assumed that the quantity of unreacted monomer leaving the reactor with the polymer is negligible The rate of heat loss through the reactor wall is also ne glected because it is much smaller than the other terms in Eq 3 Choi and Rays 1985b polyethylene reactor model con sisting of five differential equations is more complicated than the current model because separate monomer and energy balances are required for the bubble and emulsion phases The assumption of uniform monomer concentration and tem perature in the gas McAuley et al 1994 leads to the cur rent wellmixed model requiring only three differential equa tions 870 April 1995 Vol 41 No 4 AIChE Journal 1440 1080 Y 720 I 360 0 0 5 Fc kghr 10 15 Figure 2 Multiple steady states and stability for a sin glepass gasphase polyethylene reactor k 0 E 9000 calmol Stable steady state unstable steady state Hopf bifurcation point The steadystate behavior predicted by the wellmixed model is shown in Figure 2 indicating that three possible steady states can occur for catalyst feed rates between 082 and 66 kgh For feed rates below 082 kgh only a single low temperature steady state exists and above 66 kgh there is only a high temperature steady state Since the melting point of polyethylene is approximately 400 K the high tem perature steady state predicted by the model would not actu ally be obtained Instead the fluidized polymer particles would melt and agglomerate and the wellmixed model would no longer be valid The solid portions of the curve in Figure 2 indicate stable steady states whereas the dashed portion is unstable For catalyst feed rates greater than 47 kgh the lower portion of the steadystate curve is unstable The insta bility arises at a Hopf bifurcation point which was deter mined using the AUTO software of Doedel 1986 Attempt ing to operate the reactor with a catalyst feed rate of 5 kgh results in oscillation away from the low temperature unstable steadystate operating point and runaway toward the high temperature steady state Therefore feedback temperature control is required for safe operation of the reactor at cata lyst feed rates beyond thc Hopf bifurcation point The multiplicity and stability behavior shown in Figure 2 is very similar to that determined by Choi and Ray 198 us ing their twophase model It appears that a complicated model accounting for interactions between separate bubble and emulsion phases is not required to predict overall reactor dynamics and stability in the operating region of industrial interest because resistances to heat and mass transfer be tween the two phases are small McAuley et al 1994 have Table 1 Reactor Operating Parameters Used in the Single Pass Model 0548 molkg B w 70 tonne 110 d m o l K CPI 085 calg K CP POI 9000 calmol E kd 0s1 F V 1 28000 Ls A HR 894 calg 85 Lmol s 2805 gmol 14000 kcalK k 360 K m MI I 03 molL KC To 298 K T f 360 K vg 500 m3 shown that significant deviations between wellmixed and twophase polyethylene reactor models can occur for high temperature reactor operation Therefore we expect our model predictions of steadystate operating temperatures will not be accurate for middle and high temperature steadystate portions of the curves This is not a problem however be cause we are only interested in the portion of the steadystate operating curve that is below 400 K The model parameters used to obtain Figure 2 are given in Table 1 Behavior of the Combined Reactor and Cooling System Mass balances Most fluidized bed polyethylene reactors are cooled using a heat exchanger on the recycle gas line as shown in Figure 1 Since the singlepass conversion in these reactors is usually quite low 25 the recycle stream is much larger than the fresh feed stream Inert components and impurities are pre vented from building up in the system by removing gas from the reactor either along with the product or in a separate bleed stream In the current model we assume that a bleed stream is present and that any gas exiting with the polymer is immediately captured and recycled to the reactor We also neglect the short timedelay associated with the recycle gas flow through the heat exchanger and recycle lines as well as any unreacted monomer or comonomer that leaves the reac tor dissolved in the polymer particles Using these assump tions a dynamic ethylene mass balance can be written for the combined reactor and cooling system 11 where F M L j is the molar feed rate of fresh ethylene to the reactor and bM is the outflow rate of ethylene in the bleed stream In some industrial ethylene polymerization systems the time delay associated with transport of unreacted monomer through the recycle system and heat exchanger can be large and can dominate the dynamics This is especially true if the heat exchanger is used as a condenser or separator as in the BASF or AMOCO gasphase processes Xie et al 1994 In the current article only systems with negligible AIChE Journal April 1995 Vol 41 No 4 871 Table 2 Operating Parameters Used in the Combined Re AU 114X 10 calsK actor and Heat Exchanger Model 77 calrnol K CZ 69 calmol K c C I 110 calmol K CMZ 240 calmol K c 75 atm5 rnoIs 8500 mols 5 134 mols 5 6 10 calsK K 5737 molatrnsf 0088 Lrnol s 037 Lmol s 1 0 Lrnol s 85 Lrnol s 3 Lmol s 172 I 2805 gmol inw2 562 gmol Pu 1 5 767 atm P 17 atm 293 K 293 K 05 360 s FMlO k f l k 2 khl k k 360 K kp12 360 K k 360 K kp2 360 K F c h k k2 40X T T P TI 1500 s monomer recycle time lags are considered To control the ethylene pressure in response to changes in the ethylene con sumption rate the following PI control law has been imple mentcd in most of the simulations in this article where F is the ethylene feed rate to the reactor when the controller is turned off and et is the error between the ethylene partialpressure setpoint plsp and the actual ethy lene partial pressure which can be determined from the ideal gas law Values for the tuning parameters K and T are given in Table 2 Simulations indicate that multiplicity and stability results are not very sensitive to the tuning of the ethylene partial pressure controller Thc flow of ethylene in the bleed stream b is equal to the mole fraction of ethylene in the gasphase multiplied by b the total bleed stream flow rate that depends on the reac tor pressure P b V p C u d m where 5 and C are the bleed stream valve position and valve coefficient respectively and P is the pressure down stream of the bleed valve Values for Vp C P and all other parameters used to simulate the reactor behavior are given in Table 2 Inert components such as nitrogen are added to gasphase polyethylene reactors either as a carrier for the catalyst or to help remove the heat of reaction from the polymer parti cles Hydrogen is added to the reactor to control the molecu lar weight of the polymer Butenc or other comonomers can be added to control the density of the polymer McAulcy et al 1990 The following mass balances can be written for these additional gasphase components 17 wherc I H 2 and M are the concentrations of inerts hydrogen and eomonomer respectively in the gas phase The bleed flow rate of each component can be determined by multiplying the respective mole fraction by the total bleed rate The rate of consumption of hydrogen by reaction R is much smaller than the other terms in Eq 16 and can be ncglected ZieglerNatta catalysts used for gavphase ethylcnc poly merization can have multiple types of active sites each with different rate constants and activation energies for propaga tion and deactivation reactions The current model considers a twosite catalyst with temporary site deactivation due to chain transfer reactions with hydrogen McAulcy et al 1990 The following mass balances can be written for each type of site where N is the number of moles of sites of type i deactivated by hydrogen N can be obtained using the stationary state hypothesis Y KfH 2 I N i 12 19 k M k R B where k is the rate constant for site deactivation by hydro gen and k and khl are rate constants for subsequent site reactivation reactions with cocatalyst and ethylene respec tively and R is the concentration of cocatalyst in the reac tor Using this twosite catalyst the rate of ethylene con sumption is and the rate of butene consumption is 21 where kPIJ is a pseudopropagation rate constant for con sumption of monomer i at catalyst sitcs of type j 872 April 1995 Vol 41 No 4 AIChE Journal Energy balances Separate energy balances on the reactor and external heat exchanger are required to model the combined system The reactor energy balance is similar to Eq 3 except that the enthalpy of the gas stream entering the reactor is divided into two parts the enthalpy of the fresh feed stream H and that of the recycle stream H Thc enthalpy of the gas entering in the fresh feed stream Hr is Hj FMlcq Tre 23 and the enthalpy of the gas entering the reactor with the re cycle stream is The recycle flow rate F is controlled at a constant value by compressors on the recycle line to achieve good fluidization and mixing in the bed The cnthalpy of the gas leaving the reactor is where 7 is the firstorder time constant for the exchanger This approach gives only an approximate heatremoval rate from the system However the firstorder approximation pro vides a more realistic method for predicting interactions be tween the reactor and external exchanger than that used by Choi and Ray 1985b who assumed that the heat exchanger could instantaneously cool the recycle stream to a desired reactor inlet temperature If Eq 28 is used to predict the heatremoval rate in the exchanger then the enthalpy of the recycle stream entering the reactor can be determined from where H which is the enthalpy associated with the bleed stream is very small and can be neglected Stability and Multiplicity Results for the Combined Reactor and Cooling System Constant ethylene feed policy The steadystate reactor temperature obtained for differ ent catalyst feed rates using a constant fresh ethylene feed rate of 134 mols is shown in Figure 3 A singlesitc catalyst and H the enthalpy associated with the polymer product stream is given by Eq 10 The external heat exchanger has been modeled as a coun tercurrent singlepass exchanger with the recycle gas on the tube side and cooling water on the shell side Steadystate analysis of such a system reveals that the gasside outlet tem perature is where I where T is the cooling water inlet temperature T is the temperature of the gas entering the heat exchanger F and F are gas and cooling water flow rates Cp and C are the respective heat capacities A is the area available for heat transfer and U is a heattransfer coefficient To avoid using a complicated partial differential equation model for the dynamics of the shell and tube heat exchanger the approach taken in the current study is to determine the steadystate heatremoval rate using Eq 26 and then deter mine the dynamic heatremoval rate Qd using a firstorder approximation h I s I 0 0 1 2 3 4 5 Fc kglhr Figure 3 Steady states and stability for the combined reactor and cooling system for a constant eth ylene feed rate of 134 mols k 00001 s E 9000 cavmoi Stable steady state unstable steady stdte Hopf bifurcdtion point AIChE Journal April 1995 Vol 41 No 4 873 300 J 0 1 2 3 4 4 time x10 sec 3590 h 3585 t 4 3580 0 1 2 3 4 time x10 sec Figure 4 Dynamic response of the reactor system to a cooling water temperature disturbance a F 15 kgh b F 25 kgh E 9000 calmol kd 00001s is used in the analysis ethylene is the only gasphase reactant and nitrogen is fed to the reactor at a rate of 2 mols Notice that multiple steady states are not observed and that unstable reactor behavior is predicted for catalyst feed rates between 003 and 21 kgh Figure 4a and b show the openloop reactor behavior in re sponse to a small cooling water disturbance for catalyst feed rates of 15 and 25 kgh respectively In both simulations the reactor operates at the unstable steady state for the first 1000 s At this time the cooling water temperature increases from 293 to 295 K and remains at the new level for 100 s before returning to 293 K As predicted in Figure 3 reactor operation without a feedback temperature controller at a cat alyst feed rate of 15 kgh leads to unstable oscillatory behav ior Limit cycle behavior like that shown in Figure 4a al though predicted by the dynamic model would not be ob served in a gasphase polyethylene reactor Once the reactor temperature reached the melting point of the polymer parti cles 400 K agglomeration and loss of fluidization would occur rapidly leading to reactor shutdown The cooling water temperature disturbance has only a minor effect on the reac tor temperature when the catalyst feed rate is 25 kgh Fig ure 4b The reactor temperature increases by approximately 03 K and then oscillates back to the original steady state The unstable steady states in Figure 3 and the limit cycles in Figure 4a can be explained mathematically by the presence of Hopf bifurcation points where complex conjugate pairs of eigenvalues of the Jacobian matrix for the dynamic model cross the imaginary axis Physically the oscillatory behavior can be explained by positive feedback between the reactor temperature and the reaction rate If the reactor tempera ture is above the unstable steadystate temperature then the heatrcmoval rate in the heat exchanger is larger than the steadystate heatgeneration rate McAuley 1992 As a re sult thc reactor temperaturc begins to decrease decreasing the rate of reaction The product outflow rate from the reac tor is reduced thereby reducing the rate at which catalyst flows from the reactor with the polymer product Thus cata lyst and monomer begin to accumulate in the reactor When enough catalyst and monomer have accumulated the rate of reaction begins to increase increasing the reactor tempera ture and the product outflow rate When the monomer con centration is very low and most of the catalyst has left the reactor in the product outflow stream the reaction rate and reactor temperature begin to fall and the cycle begins again This type of unstable oscillatory behavior does not occur for higher catalyst feed rates as in Figure 4b where the reactor operates in a monomerstarved mode Sufficient catalyst is present in the reactor at all times so that the concentration of ethylene in the system remains very low Such low ethylene concentrations are not observed in industrial polyethylene re actors because the ethylene pressure in the reactor is main tained using a feedback controller that manipulates the fresh ethylene feed rate As demonstrated in the ncxt section the presence of an ethylene pressurc controller can lead to multi ple steady states and higher operating temperatures because the controller ensures that there is always a ready supply of ethylcne in the reactor Reactor operation using an ethylene partialpressure con troller Addition of an ethylene partialpressure controller Eq 12 to the system has a significant effect on multiplicity and sta bility As ethylene is consumed by the reaction the controller increases its feed rate to maintain thc ethylene partial pres sure at the desired setpoint The stability and multiplicity be havior for the reactor system is shown in Figure 5 for cthy lene partialpressure setpoints of 3 767 and 15 atm Either one or three steady states is possible depending on the ethy lene partialpressure setpoint and on the catalyst feed rate Figure 5 illustrates that high ethylene concentrations in the reactor make the system more prone to runaway to unaccept able high temperature steady states At an ethylene pressure of 15 atm the lower steadystate branch exists only for cata lyst feed rates up to 48 kgh whereas for an ethylene pres sure of 3 atm the lower steadystate branch extends to a cat alyst feed rate of 146 kgh Figure 6 shows the effect of the activation energy of the catalyst on stability and multiplicity for reactor operation at an ethylene pressure setpoint of 767 atm All of the catalysts shown in Figure 6 have the same propagation rate constant at 360 K but exhibit different activation energies For a cata lyst with a low activation energy 5000 calmol single stable steady states are observed for a given catalyst feed rate In creasing the catalyst feed rate results in a monotonic increase in the steadystate temperature and can result in operating 874 April 1995 Vol 41 No 4 AIChE Journal 1440 1080 720 c 360 0 0 5 10 15 20 Fc kghr Figure 5 Steady states and stability for the combined reactor and cooling system using an ethylene partial pressure controller k d 00001 s E 9000 calmol Stable steady state unstable steady state Hopf bifurcation point temperatures above the melting point of the polymer if cata lyst feed rates are too large Catalysts with larger activation energies are more sensitive to changes in reactor tempera ture and exhibit multiple steady states As the activation en ergy increases the range of catalyst feed rates for which a high temperature steady state exists becomes larger Al though the low temperature branch of the steadystate curve is stable disturbances in the catalyst feed rate catalyst prop erties or cooling system could lead to reactor runaway to ward the high temperature steady state In Figure 7 multiplicity and stability information for the system is presented in a slightly different way Activation en ergy is plotted as the bifurcation parameter along the ab scissa and steadystate operating temperatures are shown for several fixedcatalyst feed rates For catalyst feed rates of 50 and 80 kgh either a single low temperature steady state occurs or there are three steady states with the lower and upper steady states being stable and the middle steady state unstable For intermediate catalyst feed rates of 58 and 70 kgh there are certain catalyst feed rates for which a single unstable steady state is observed Limit cycle behavior is ob served at these catalyst feed rates Figure 8 summarizes the multiplicity and stability behavior from Figures 6 and 7 In region A there is a unique unstable steady state Region B 144 108 h s 720 360 0 I I I I 0 4 a 12 Fc kghr Figure 6 Effect of activation energy on stability and multiplicity k d 00001 s pMMlSp 767 atm Stable steady state unstable steady state Hopf bifurcation point has three steady states with the upper and lower stable and the middle unstable Region C also has three steady states with both the middle and lower unstable and the upper sta ble Regions D and E correspond to unique high temperature and low temperature steady states respectively Effect of heat exchanger size and operating conditions The area available for heat transfer AU and the firstorder time constant T have very little effect on either the location or the stability of the steady states AU was varied over two orders of magnitude from 114X lo5 to 114X lo6 and T was varied from 100 s to 3600 s In all cases the steadystate curves and stability predictions were nearly coincident with the corresponding curves in Figures 5 and 6 A secondorder overdamped model of the heat exchanger was also tested and had very little effect One explanation for the insensitivity of the reactor behavior to the heat exchanger size and dynamics is that instability and multiple steady states arise due to au tothermal feedback within the reactor itself The cooling water temperature T is often used as a ma nipulated variable for feedback control of the reactor tem perature McAuley 1992 As expected Figure 9 shows that the cooling water temperature has a significant effect on the steadystate reactor temperature For the conditions simu lated in Figure 9 cooling water temperatures greater than AIChE Journal April 1995 Vol 41 No 4 875 1440r I I 0 II Fc80 0 0 5 10 15 0 5 10 15 Ea kcallrnol Ea kcalhnol Figure 7 Effect of catalyst feed rate on stability and multiplicity kd 00001 s pMrsp 767 atm Stable eidy state unstable s t u d y Ftdte Hopf hilurcation point 298 K result in an undesirable hightemperature steady state Even in the limit when the cooling water freezes a hightem perature steady state exists From Figure 9 it appears that if the reactor experiences an excursion toward a hightempera ture steady state then manipulation of thc cooling water temperature alone may not be sufficient to bring the temper ature back to the desired level Importance of multiple catalyst sites The results in Figures 2 to 8 have been obtained using a catalyst with a single type of active sitc In Figure 10 multi plicity and stability behavior are shown for a twosite ZieglerNatta catalyst in which half of the sites have activa tion energy E and the other half have activation energy Ea2 Both sites have the same deactivation rate constant k 00001 s and the same propagation rate constant at 360 K The ethylene partial pressure controller setpoint is 767 atm and the catalyst feed rate is 48 kgh The results in Figure 10 demonstrate what happens when the active sites are al lowed to have different activation energies for the propaga tion reaction When both activation energies are low a unique stable low temperature steady state is observed When both activation energies are high a unique stable high tempera ture steady state is observed In the region between the dashed and solid lines there is a unique unstable steady state and the reactor experiences limit cycle behavior If the cata 876 April 1995 20 16 12 L f m L L 8 Y 0 v 4 0 T D B i Ea kcalhol Figure 8 Stability and multiplicity regions for the com bined reactor and recycle system lyst feed rate were allowed to change along a third axis a threedimensional bifurcation diagram could be constructed wherein Figure 10 would correspond to the plane where F 48 kgh and Figure 8 would correspond to the diagonal plane where E EU2 Eflect of catalyst deactivation In Figures 2 to 10 the rate of catalyst deactivation was assumed to be independent of temperature Figure 10 demonstrates the effect of catalyst deactivation on stability and multiplicity behavior for a singlesite catalyst with an ac tivation energy for propagation of 9000 calmol As shown in Figure 11 a catalyst with a constant deactivation rate con stant of 00001 s leads to an improvement in reactor stabil ity over a nondeactivating catalyst since the lower portion of the steadystate curve extends over a larger range of catalyst feed rates When temperaturedependent deactivation is added to the reactor model with k00001 s at 360 K and an activation energy for deactivation of 13000 calmol multiple steady states are not observed However the single steady state is openloop unstable for catalyst feed rates be tween 21 and 188 kgh Thus even with a catalyst that de activates at high temperatures the reactor would be prone to limit cycles which would lead to dynamic operating tempera tures that are well beyond the melting temperature of the polymer particles Vol 41 No 4 AIChE Journal I 4 O 0 c 1200 1000 800 9 v I Tw K Figure 9 Effect of cooling water temperature on steadystate operating temperature k 00001 s pMlsp 767 atm E 9000 call mol F 58 kgh Stable steady state unstable steady state Hopf bifurcation point Effect of hydrogen inerts and comonomer Gasphase polyethylene reactors are operated with other components in the reactor in addition to ethylene Nitrogen or other inerts may be added as a carrier for the catalyst or to assist in heat removal from the polymer particles Hydro gen is added to control molecular weight Comonomers such as butene and hexene are added to produce linear low den sity polyethylene copolymers Figure 12 shows that adding ni trogen and hydrogen to the reactor at a rate of 2 mols has very little effect on the multiplicity and stability behavior of the reactor In Figure 12 the ethylene partial pressure is con upper stable 1 Eal kcalmol Figure 10 Effect of activation energies of active sites on reactor stability k 00001 s pMls 767 atm F 48 kgh 400 I kd fT x I I 2 o o l L r T r T T r r 4 0 4 a 12 16 20 Fc kgihr Figure 11 Effect of catalyst deactivation on multiplicity and stability pMls 767 atm E 9000 cal mol F58 kgh Stable steady state unstable cteady stdte Hopf bifurcation point trolled at 767 atm and kd 0 Compared with the k 0 case in Figure 10 there is very little change in the steadystate behavior Slight differences are caused by the different heat capacities of the gases and by the deactivating influence of hydrogen on the catalyst Eq 18 Figure 12 shows the effect of butene addition to the reac tor using a catalyst with k 00001 s and E 9000 calmol The comonomer has a very mild effect on the reac tor operation for the propagation rate constants given in Table 2 but does not significantly alter the multiplicity and stability behavior of the reactor system Butene is consumed to a much smaller extent than ethylene so the reactor behav ior is far more sensitive to the ethylene concentration than the butene concentration Conclusions A gasphase ethylene polymerization model has been ana lyzed to determine the effects of reactor operating conditions on dynamics and stability The reactor model employed as sumed that both the gas and polymer phases in the reactor are well mixed Our analysis of ethylene homopolymerization in the reactor alone gives similar results to those of Choi and Ray 1985b confirming that mass and heattransfer limita tions between multiple phases are not responsible for the ob served bifurcation behavior AIChE Journal April 1995 Vol 41 No 4 877 1440 1080 720 360 0 I 360 I Fm2 0 Fm2 1 0 molls d 0 4 8 12 Fc kglhr Figure 12 Effect of nitrogen and hydrogen on stability and multiplicity Stable steady state unstable steady state Ilopf bifurcation point The reactor model was extended to account for the effects of the external heat exchanger and the recycle stream Ac counting for the recycle stream is important because the re cycle to fresh feed ratio in gasphase polyethylene reactors is on the order of 50 to 1 Xie et al 1994 The combined sys tem was shown to exhibit unstable behavior and limit cycles Because the unstable behavior arises due to positive feed back between the reactor temperature and polymerization rate the size and dynamics of the heat exchanger have very little effect on either the stability or multiplicity of the sys tem It was shown that the presence of an ethylene partialpres sure controller had a significant effect on multiplicity and stability Higher partialpressure setpoints made the reactor more prone to run away toward undesirable high tempera ture steady states by ensuring that ample monomer is always available for the reaction The catalyst fced rate activation energy and deactivation properties were also shown to have a significant influence on reactor behavior If a multiplesite catalyst is used for the polymerization then both the distri bution and kinetic behavior of the individual sites on the cat alyst can affect the dynamic behavior and stability of the sys tem Hydrogen inert components and butene comonomer were shown to have only a very minor effect on system behav ior The results in this study demonstrate that without feed 878 April 1995 0 L p L 0 4 8 12 Fc kglhr Figure 13 Effect of butene comonomer on stability and multiplicity Stable steady 5tatC unztahle steady state v Hopf bifurcation point back temperature control gasphase polyethylene reactors are prone to unstable steady states limit cycles and excursions toward unacceptably high temperature steady states Minor design changes to the external heat exchanger cannot be used to ensure openloop stability so feedback temperature con trol is very important in gasphase polyethylene reactors Without feedback temperature control disturbances in the catalyst feed rate catalyst properties or ethylene concentra tion in the reactor could cause the system to move away from a desirable steady state into an unacceptable operating regime Such information about the underlying properties of the system is very important to the design of reactor operat ing policies and to the implementation of stabilizing feedback temperature controllers Acknowledgment The authors thank the Principals Development Fund of Queens University and the Natural Science and Engineering Research Coun cil of Canada for support of this research Notation K ethylene partialpressure controller gain p partial pressure of component i R rate of consumption of component i by reaction t time Vol 41 No 4 AIChE Journai Y Y2 quantity of active catalyst sites of type 1 and type 2 respec tively moll Greek letters y dimensionless group defined in Eq 17 T integral time for ethylene partialpressure controller Subscripts 1 M1 ethylene 2 M 2 butene comonomer 0 reactor inlet condition f fresh feed stream I inert component r due to reaction ss steady state w cooling water g0 gas recycle stream entering the reactor top condition of stream leaving the top of the reactor catalyst site of type i Literature Cited Adebekun A K K M Kwalik and F J Schork SteadyState Multiplicity During Solution Polymerization of Methyl Methacry late in a CSTRChem Eng Sci 4410 2269 1989 Choi K Y and W H Ray Recent Developments in Transition Metal Catalyzed Olefin PolymerizationA Survey I Ethylene Polymerization J Macromol Sci Rev Macromol Chern Phys C25 I 1985a Choi K Y and W H Ray The Dynamic Behavior of Fluidized Bed Reactors for Solid Catalyzed Gas Phase Olefin Polymeriza tion Chem Eng Sci 4112 2261 1985b Choi K Y and W H Ray The Dynamic Behavior of Continuous StirredBed Reactors for the Solid Catalyzed Gas Phase Polymer ization of Propylene Chem Eng Sci 4310 2587 1988 Doedel E J AUTO Software for Continuation and Bifurcation Problems in Ordinary Differential Equations California Institute of Technology Pasadena CA 1986 Hamer J W T A Akramov and W H Ray The Dynamic Behav ior of Continuous Polymerization Reactors11 Nonisothermal Solution Homopolymerization and Copolymerization in a CSTR Chem Eng Sci 36 1897 1981 Hutchinson R A and W H Ray Polymerization of Olefins through Heterogeneous Catalysis VII Particle Ignition and Ex tinction Phenomena J Appl Poly Sci 34 657 1987 Hutchinson R A and W H Ray Polymerization of Olefins through Heterogeneous CatalysisThe Effect of Condensation Cooling on Particle Ignition J Appl Po Sci 43 1387 1991 Jaisinghdni R and W H Ray On the Dynamic Behavior of a Class of Homogeneous Continuous Stirred Tank Polymerization Reactors Chem Eng Sci 32 811 1977 Lmo M and V Balakotaiah Multiplicity Features of Adiabatic Autothermal Reactors AZChE J 381 101 1992 McAuley K B Modelling Estimation and Control of Product Properties in a Gas Phase Polyethylene ReactorPhD Thesis MC Master Univ Hamilton Ont Canada 1992 McAuley K B and J F MacGregor OnLine Inference of Poly mer Properties in an Industrial Polyethylene Reactor AIChE J 376 825 1991 McAuley K B and J F MacGregor Optimal Grade Transitions in a Gas Phase Polyethylene Reactor AIChE J 38 1564 1992 McAuley K B and J F MacGregor Nonlinear Product Property Control in Industrial Gas Phase Polyethylene Reactors AIChE J 395 855 1993 McAuley K B J F MacGregor and A E Hamielec A hnetic Model for Industrial Gas Phase Ethylene Copolymerization AIChE J 366 837 1990 McAuley K B J P Talbot and T J Harris A Comparison of TwoPhase and WellMixed Models for Fluidized Bed Polyethy lene Reactors Chem Eng Sci 4913 2035 1994 Morbidelli M A Varma and R Ark Reactor SteadyState Multi plicity and Stability Chemical and Reaction Engineering Chap 15 J J Carberry and A Varma eds Marcel Dekker New York 1987 Rawlings J B Simulation and Stability of Continuous Emulsion Polymerization Reactors PhD Thesis Univ of Wisconsin Madi son 1985 Razon L F and R A Schmitz Multiplicities and Instabilities in Chemically Reacting SystemsA Review Chem Eng Sci 425 1005 1987 Schmidt A D and W H Ray The Dynamic Behavior of Continu ous Polymerization Reactors I Isothermal Solution Polymeriza tion in a CSTRChem Eng Sci 36 1401 1981 Schmidt A D A B Clinch and W H Ray The Dynamic Behav ior of Continuous Polymerization Reactors 111 An Experimental Study of Multiple Steady States in Solution Polymerization Chem Eng Sci 393 419 1984 Talbot J P The Dynamic Modelling and Particle Effects on a Fluidised Bed Polyethylene Reactor PhD Thesis Queens Univ Kingston Ont Canada 1990 Teymour F and W H Ray The Dynamic Behavior of Continuous Solution Polymerization Reactors IV Dynamic Stability and Bi furcation Analysis of an Experimental Reactor Chem Eng Sci 449 1967 1989 Teymour F and W H Ray The Dynamic Behavior of Continuous Polymerization Reactors V Experimental Investigation of Limit Cycle Behavior for Vinyl Acetate PolymerizationChem Eng Sci 471516 4121 1992a Teymour F and W H Ray The Dynamic Behavior of Continuous Polymerization Reactors VI Complex Dynamics in FullScale Re actors Chem Eng Sci 471516 4133 1992b Uppal A W H Ray and A B Poore On the Dynamic Behavior of Continuous Stirred Tank Reactors Chem Eng Sci 29 967 1974 Uppal A W H Ray and A B Poore The Classification of the Dynamic Behavior of Continuous Stirred Tank ReactorsIn fluence of Reactor Residence Time Chem En Sci 31 205 1976 Van Heerden C Auto thermic Processes Properties and Reactor Design Ind Eng Chem 45 1242 1953 Xie T K B McAuley C C Hsu and D W Bacon Gas Phase Ethylene Polymerization Production Processes Polymer Proper ties and Reactor Modelling Ind Eng Chem Res 333 449 1994 Manuscript received Nov 29 1993 and revision received Apr 22 1994 AIChE Journal April 1995 Vol 41 No 4 879 UNIVERSIDADE FEDERAL DA PARAÍBA CENTRO DE TECNOLOGIA DEPARTAMENTO DE ENGENHARIA QUÍMICA INSTRUMENTAÇÃO E CONTROLE DE PROCESSOS PROJETO BASE DA DISCIPLINA DE INSTRUMENTAÇÃO E CONTROLE DE PROCESSOS QUÍMICOS POLIETILENO Docente Prof Dr Arioston Araújo de Morais Junior Discentes Jéssica Beatriz Torres Apolinário Suwelane Gomes Vieira Sistema de reator de polimerização Setembro de 2024 João pessoa PB 1 Introdução O presente trabalho toma como base o estudo por K B McAuley D A Macdonald e P J McLellan apresentado no artigo Effects of operating conditions on stability of gasphase polyethylene reactors onde aborda um reator de polimerização de polietileno em fase gasosa operado como um leito fluidizado que é amplamente utilizado na indústria devido à sua eficiência e menor custo operacional Nesse processo convertese o monômero de etileno em polímero na presença de um catalisador ZieglerNatta com o sistema funcionando em fase gasosa não necessitando o uso de solventes Uma corrente de gás reciclado contendo etileno inertes e às vezes hidrogênio é continuamente alimentada ao reator onde a polimerização ocorre na superfície das partículas de catalisador suspensas no leito fluidizado Neste tipo de reator a temperatura é um dos fatores mais críticos uma vez que a polimerização de etileno se trata de uma reação altamente exotérmica O processo deve ser mantido em uma faixa de temperatura estreita entre 75C e 110C abaixo do ponto de fusão do polímero de forma a evitar problemas como a fusão das partículas e a aglomeração do polímero que podem comprometer o funcionamento do reator Para isso o sistema conta com trocadores de calor na linha de reciclagem onde o calor gerado é removido durante a polimerização mantendose a temperatura dentro dos limites adequados Este estudo foca nos desafios de controle e estabilidade do reator tendo em vista que sem um controle adequado o sistema pode apresentar oscilações térmicas e até mesmo mudanças bruscas para estados de alta temperatura o que pode afetar a qualidade do produto e a operação segura A desativação do catalisador ao longo do tempo também representa um desafio adicional pois influencia diretamente a produtividade do reator O trabalho também explora os efeitos de parâmetros operacionais a exemplo da alimentação de etileno e a eficiência do sistema de resfriamento assim como investiga o impacto de componentes como hidrogênio inertes e comonômeros na dinâmica e estabilidade do reator 2 Desenvolvimento Variáveis Controladas Temperatura do reator concentração de etileno peso molecular do polímero Variáveis Manipuladas Taxa de alimentação de etileno taxa de alimentação de catalisador taxa de reciclagem taxa de hidrogênio temperatura da água de resfriamento Distúrbios Variações na composição do gás temperatura ambiente variações de pressão desativação do catalisador 21 Balanço de massa Assumindo que Tanto o gás quanto às fases sólidas está bem misturado que a temperatura dentro do reator é uniforme A massa de polímero no reator é constante devido ao controle perfeito do nível do leito Modelo CSTR monofásico para leito fluidizado 𝑑𝑀1 𝑑𝑡 𝐹𝑀1 𝑀10 𝑀1 𝑅𝑀1 𝑉𝑔 𝐸𝑞 1 Na equação 1 temos o balanço de massa do etileno onde M1 Concentração do etileno no reato FM1 é a vazão volumétrica de alimentação do monômero para o reator Vg Volume da fase gasosa no reator RM1 Taxa de consumo de etileno devido à reação Eq 2 𝑅𝑀1 𝑀1 𝑘𝑝0 𝑒 𝐸𝑎 𝑅 1 𝑇 1 𝑇𝑓 𝑌 𝐸𝑞 2 Temos que R Constante dos gases ideais Ea Energia de ativação para propagação Tref Temperatura de referência na qual o valor da constante é conhecido 𝑑𝑌 𝑑𝑡 𝐹𝑐 𝑎𝑐 𝑘𝑑 𝑌 𝐸𝑞 3 𝐹𝑌 𝐹𝑐 𝑎𝑐 𝐸𝑞4 No balanço de massa do catalisador Eq 3 temos FY Taxa de alimentação molar dos sítios do catalisador para o reator que pode ser determinada pela Eq 4 kd Constante de taxa de desativação Para os inertes gases que não participam da reação como o nitrogênio o balanço de massa pode ser escrito da seguinte forma 𝑑𝐼 𝑑𝑡 𝐹𝐼 𝐼0 𝐼 𝑏𝑡 𝑉𝑔 𝐸𝑞 5 Onde I é a concentração de inertes no reator I0 é a concentração de inertes na alimentação FI é a taxa volumétrica de alimentação dos inertes bt é a taxa de saída do reator Para o polímero produto o balanço de massa é dado por 𝑑𝑃 𝑑𝑡 𝑅𝑀1 𝑉𝑔 𝐹𝑃 𝐸𝑞 6 Onde P é a quantidade de polímero no reator RM1 é a taxa de consumo de etileno pela reação de polimerização que gera o polímero FP é a taxa de saída do polímero O hidrogênio que é frequentemente utilizado para controlar o peso molecular também pode ter um balanço de massa 𝑑𝐻2 𝑑𝑡 𝐹𝐼 𝐻20 𝐻2 𝑅𝐻2 𝑉𝑔 𝐸𝑞 7 Onde H2 é a concentração de hidrogênio no reator H20 é a concentração de hidrogênio na alimentação FH2 é a taxa de alimentação de hidrogênio RH2 é a taxa de consumo de hidrogênio por reações de transferência de cadeia 22 Balanço de Energia dT dt H0 Htop Hr Hp Mr Cpr BW Cp pol Eq8 No balanço de energia do reator Eq 8 temse T Temperatura do reator MrCpr Capacitância térmica do recipiente de reação Cppol Capacidade térmica do polímero H0 Entalpia de da alimentação do reator Eq 9 Htop Entalpia do gás que sai do reator Eq 10 Hr Taxa de geração de calor pela reação Eq 11 que está relacionada à entalpia de reação ΔHR Hp Entalpia associada ao polímero que sai do reator Eq 12 𝐻0 𝐹𝑣1 𝑀10 𝐶𝑝𝑔 𝑇0 𝑇𝑟𝑒𝑓 𝐸𝑞9 𝐻𝑡𝑜𝑝 𝐹𝑉1 𝑀1 𝐶𝑝𝑔 𝑇 𝑇𝑟𝑒𝑓 𝐸𝑞 10 𝐻𝑟 𝐻𝑟 𝐾𝑝 𝑌 𝑀1 𝑚𝑤1 𝐸𝑞11 𝐻𝑝 𝑂𝑝 𝐶𝑝𝑝𝑜𝑙 𝑇 𝑇𝑟𝑒𝑓 𝐸𝑞 12 O balanço de energia para o fluido de resfriamento Tw1 considera dois mecanismos de transferência de calor Calor trocado entre o fluido de resfriamento que entra e o fluido já presente no sistema Calor trocado entre o fluido de resfriamento e o gás no reator através de uma superfície de troca térmica 𝑑𝐸𝑤 𝑑𝑡 𝐹𝑤𝑡 𝐶𝑝𝑤 𝑇𝑤𝑖 𝑇𝑤1 UA 𝑇𝑤1 𝑇𝑔1 Eq 13 Onde dEwdt é a taxa de variação de energia no fluido de resfriamento Fw é a taxa de fluxo de massa do fluido de resfriamento kgs Cpw é a capacidade calorífica específica do fluido de resfriamento JkgK Twi é a temperatura de entrada do fluido de resfriamento K Tw1 é a temperatura do fluido de resfriamento no sistema K UA é o coeficiente global de transferência de calor Tg1 é a temperatura do gás no reator K Temos que a energia interna do fluido de resfriamento é dada por 𝐸𝑤 𝑚𝑤 𝐶𝑝𝑤 𝑇𝑤1 Eq 14 onde mw é a massa do fluido de resfriamento e que mwFwMw Eq 15 fluxo de massa dividido pela massa molar substituindo a Eq 14 e Eq 15 na Eq 13 o balanço de energia pode ser expressado por 𝑑𝑇𝑤1 𝑑𝑡 𝐹𝑤𝑡 𝑀𝑤 𝑇𝑤𝑖 𝑇𝑤1 𝑈𝐴 𝑀𝑤𝐶𝑝𝑤 𝑇𝑤1 𝑇𝑔1 Eq 16 Agora para o gás no reator o balanço de energia considera dois termos principais Calor trocado entre o gás que entra e o gás no reator Calor transferido do fluido de resfriamento para o gás no reator 𝑑𝐸𝑔 𝑑𝑡 𝐹𝑔𝑡 𝐶𝑝𝑔 𝑇 𝑇𝑔1 UA 𝑇𝑤1 𝑇𝑔1 Eq 17 Onde dEgdt é a taxa de variação de energia no gás do reator Fg é a taxa de fluxo de massa do gás mols Cpg é a capacidade calorífica específica do gás JmolK T é a temperatura do reator principal Tg1 é a temperatura do gás no reator UA é o coeficiente global de transferência de calor Tw1 é a temperatura do fluido de resfriamento Temos que a energia interna do gás no reator é dada por 𝐸𝑔 𝑚𝑔 𝐶𝑝𝑔 𝑇𝑔1 Eq 18 onde mg é a massa do gás e que mgFgMg Eq 19 substituindo a Eq 18 e a Eq 19 na Eq 17 o balanço de energia pode ser expressado por 𝑑𝑇𝑔1 𝑑𝑡 𝐹𝑔𝑡 𝑀𝑔 𝑇 𝑇𝑔1 𝑈𝐴 𝑀𝑔𝐶𝑝𝑔 𝑇𝑤1 𝑇𝑔1 Eq 20 23 Variáveis constantes Essas variáveis são parâmetros físicos operacionais ou propriedades fixas que não mudam com o tempo no modelo Vg Volume do gás no reator 500 m³ Vp Volume de polímero 05 m³ Pv Pressão do reator 17 atm Bw Massa de polímero 70 toneladas Cpm1 Capacidade calorífica específica do monômero 11 JmolK Cv Coeficiente de fluxo 75 atm05 Cpw Capacidade calorífica da água 4186 JkgK CpIn Capacidade calorífica de inertes 69 JkgK Cppol Capacidade calorífica do polímero 085 JgK Mw1 Massa molar do monômero 2805 gmol MrCpr Capacidade calorífica total 1410⁷ JK Hreac Entalpia de reação 894 kJkg UA Coeficiente global de transferência de calor 11410⁶ JsK FIn Taxa de alimentação de inertes 5 mols FM1 Taxa de alimentação de monômero 190 mols Fg Fluxo de gás no reator 8500 mols Fw Fluxo de água de resfriamento 31110⁵ kgs RR Constante dos gases 82057510⁵ m³atmmolK R Constante universal dos gases 8314 JmolK Tf Temperatura de referência 360 K Twi Temperatura de entrada da água de resfriamento 28956 K 24 Variáveis dependentes do tempo Estas são as variáveis de estado do reator In Concentração de inertes no reator M1 Concentração de monômero no reator Y1 e Y2 Concentrações dos dois tipos de sítios catalíticos ativos T Temperatura do reator Tw1 Temperatura da água de resfriamento dentro do reator Tg1 Temperatura do gás no reator Equações Algébricas 𝑏𝑡 𝑉𝑝 𝐶𝑣 𝑀1 𝐼𝑛 𝑅𝑅 𝑇 𝑃𝑉 2 𝑅𝑀1 𝑀1 𝑘𝑝0 𝑒 𝐸𝑎 𝑅 1 𝑇 1 𝑇𝑓 𝑌 𝐶𝑝𝑔 𝑀1 𝑀1 𝐼𝑛 𝐶𝑝𝑀1 𝐼𝑛 𝑀1 𝐼𝑛 𝐶𝑝𝐼𝑛 𝐻𝑓 𝐹𝑀1 𝐶𝑝𝑀1 𝑇0 𝑇𝑓 𝐹𝐼𝑛 𝐶𝑝𝐼𝑛 𝑇0 𝑇𝑓 𝐻𝑔1 𝐹𝑔 𝐶𝑝𝑔 𝑇0𝑔1 𝑇𝑓 𝐻𝑔0 𝐹𝑔 𝑏𝑡 𝐶𝑝𝑔 𝑇 𝑇𝑓 𝐻𝑟 𝐻𝑅 𝑀𝑊1 𝑅𝑀1 𝐻𝑝𝑜𝑙 𝐶𝑝𝑝𝑜𝑙 𝑇 𝑇𝑓 𝑀𝑊1 𝑅𝑀1 25 Linearização e TLP 251 Balanço de Massa Geral 𝑑𝑀𝑡𝑜𝑡𝑎𝑙 𝑑𝑡 𝐹𝑒 𝐹𝑠 Onde Mtotal é a massa total no reator gás e sólido Fe são os fluxos de entrada de todos os componentes monômero inertes hidrogênio catalisador Fs são os fluxos de saída produto gases de sangria resíduos Ao somar todos os componentes monômero catalisador inertes hidrogênio e polímero o balanço de massa global para o sistema é 𝑑𝑀𝑡𝑜𝑡𝑎𝑙 𝑑𝑡 𝐹𝑀1 𝐹𝑐 𝐹𝐼𝑛 𝐹𝐻2𝑒 𝑅𝑀1𝑉𝑔 𝑏𝑡𝑉𝑔 𝑅𝐻2𝑉𝑔 𝐹𝑃𝑠 𝑑𝑀𝑡𝑜𝑡𝑎𝑙 𝑑𝑡 𝐹𝑀1 𝑀10 𝑀1 𝐹𝑐 𝑎𝑐 𝑘𝑑𝑌 𝐹𝐼𝑛𝐼0 𝐼 𝑏1𝑉𝑔 𝐹𝐻2 𝐻20 𝐻2 𝑅𝑀1𝑉𝑔 𝐹𝑃 𝑀𝑡𝑜𝑡𝑎𝑙 𝑉𝑔 𝑀1 𝑀𝑤1 𝑌 𝑀𝑤𝑐 𝐼 𝑉𝑔 𝑀𝑤𝐼 𝑃 𝑀𝑤𝑝 𝑑𝑉𝑔 𝑀1 𝑀𝑤1 𝑌 𝑀𝑤𝑐 𝐼 𝑉𝑔 𝑀𝑤𝐼 𝑃 𝑀𝑤𝑝 𝑑𝑡 𝐹𝑀1 𝑀10 𝑀1 𝐹𝑐 𝑎𝑐 𝑘𝑑𝑌 𝐹𝐼𝑛𝐼0 𝐼 𝑏1𝑉𝑔 𝐹𝐻2 𝐻20 𝐻2 𝑅𝑀1𝑉𝑔 𝐹𝑃 𝑑𝑉𝑔 𝑀1 𝑀𝑤1 𝑑𝑡 𝑑𝑌 𝑀𝑤𝑐 𝑑𝑡 𝑑𝐼 𝑉𝑔 𝑀𝑤𝐼 𝑑𝑡 𝑑𝑃 𝑀𝑤𝑝 𝑑𝑡 𝐹𝑀1 𝑀10 𝑀1 𝐹𝑐 𝑎𝑐 𝑘𝑑𝑌 𝐹𝐼𝑛𝐼0 𝐼 𝑏1𝑉𝑔 𝐹𝐻2 𝐻20 𝐻2 𝑅𝑀1𝑉𝑔 𝐹𝑃 𝑉𝑔 𝑀𝑤1 𝑑𝑀1 𝑑𝑡 𝑀𝑤𝑐 𝑑𝑌 𝑑𝑡 𝑉𝑔 𝑀𝑤𝐼 𝑑𝐼 𝑑𝑡 𝑀𝑤𝑝 𝑑𝑃 𝑑𝑡 𝐹𝑀1 𝑀10 𝑀1 𝐹𝑐 𝑎𝑐 𝑘𝑑𝑌 𝐹𝐼𝑛𝐼0 𝐼 𝑏1𝑉𝑔 𝐹𝐻2 𝐻20 𝐻2 𝑅𝑀1𝑉𝑔 𝐹𝑃 𝑉𝑔 𝑀𝑤1 𝑑𝑀1 𝑑𝑡 𝐹𝑀1 𝑀10 𝑀1 𝐹𝑐 𝑎𝑐 𝑘𝑑𝑌 𝐹𝐼𝑛𝐼0 𝐼 𝑏1𝑉𝑔 𝐹𝐻2 𝐻20 𝐻2 𝑅𝑀1 𝑉𝑔 𝐹𝑃 𝑀𝑤𝑐 𝐹𝑐 𝑘𝑑𝑌 𝑉𝑔 𝑀𝑤𝐼 𝐹𝐼𝑛𝐼0 𝐼 𝑏1𝑉𝑔 𝑀𝑤𝑝 𝑅𝑀1 𝑉𝑔 𝐹𝑃 𝑉𝑔 𝑀𝑤1 𝑑𝑀1 𝑑𝑡 𝐹𝑀1 𝑀10 𝑀1 𝐹𝐻2 𝐻20 𝐻2 𝑅𝑀1 𝑉𝑔 𝐹𝑃 1 𝑀𝑤𝑐 𝐹𝑐 𝑎𝑐 𝑘𝑑𝑌 1 𝑉𝑔 𝑀𝑤𝐼 𝐹𝐼𝑛𝐼0 𝐼 𝑏1𝑉𝑔 𝑀𝑤𝑝 𝑅𝑀1 𝑉𝑔 𝑀𝑤𝑝 𝐹𝑃 𝑉𝑔 𝑀𝑤1 𝑑𝑀1 𝑑𝑡 𝐹𝑀1 𝑀10 𝑀1 𝐹𝐻2 𝐻20 𝐻2 1 𝑀𝑤𝑐 𝐹𝑐 𝑎𝑐 𝑘𝑑𝑌 1 𝑉𝑔 𝑀𝑤𝐼 𝐹𝐼𝑛𝐼0 𝐼 𝑏1𝑉𝑔 𝑀𝑤𝑝 1 𝑅𝑀1 𝑉𝑔 𝑀𝑤𝑝 1 𝐹𝑃 𝑑𝑀1𝑡 𝑑𝑡 𝐹𝑀1𝑡 𝑀10 𝑀1𝑡 𝑉𝑔 𝑀𝑤1 𝐹𝐻2 𝐻20 𝐻2 𝑉𝑔 𝑀𝑤1 1 𝑀𝑤𝑐 𝐹𝑐 𝑎𝑐 𝑘𝑑𝑌𝑡 𝑉𝑔 𝑀𝑤1 𝐹𝐼𝑛𝐼0 𝐼𝑡 1 𝑉𝑔 𝑀𝑤𝐼 𝑉𝑔 𝑀𝑤1 𝑉𝑝 2 𝐶𝑣 𝑀1𝑡 𝐼𝑡 𝑅𝑅 𝑇 𝑃𝑉 2 𝑉𝑝 𝑀𝑤𝐼 1 𝑉𝑔 𝑀𝑤1 𝑀𝑤𝑝 1 𝑀1𝑡 𝑘𝑝0 𝑒 𝐸𝑎 𝑅 1 𝑇 1 𝑇𝑓 𝑌𝑡 𝑉𝑔 𝑉𝑔 𝑀𝑤1 𝑀𝑤𝑝 1 𝐹𝑃 𝑉𝑔 𝑀𝑤1 𝑑𝑀1𝑡 𝑑𝑡 𝐺 𝑓𝑡𝐹𝑀1𝑡 𝑀1𝑡 𝑌𝑡 𝐼𝑡 𝑑𝑀1𝑡 𝑑𝑡 𝐺 𝑔𝐹𝑀1𝑡𝑀1𝑡𝑌𝑡𝐼𝑡 𝑔𝐹𝑀10𝑀10 𝑌0𝐼0 𝑑𝑀1𝑡 𝑑𝑡 𝑑𝐺 𝑑𝐹𝑀1 𝐹𝑀1𝑡 𝐹𝑀10 𝑑𝐺 𝑑𝑀1 𝑀1𝑡 𝑀10 𝑑𝐺 𝑑𝑌 𝑌𝑡 𝑌0 𝑑𝐺 𝑑𝐼 𝐼𝑡 𝐼0 𝑑𝐺 𝑑𝐹𝑀1 𝐶1 𝑑𝐺 𝑑𝑀1 𝐶2 𝑑𝐺 𝑑𝑌 𝐶3 𝑑𝐺 𝑑𝐼 𝐶4 𝐶1 𝑀10 𝑀1𝑡 𝑉𝑔 𝑀𝑤1 𝐶2 𝐹𝑀1𝑡 𝑉𝑔 𝑀𝑤1 𝑉𝑝 2 𝐶𝑣 𝑅𝑅 𝑇 𝑉𝑝 𝑀𝑤𝐼 1 2 𝑉𝑔 𝑀𝑤1 𝑀1𝑡 𝐼𝑡 𝑅𝑅 𝑇 𝑃𝑉 2 𝑀𝑤𝑝 1 𝑘𝑝0 𝑒 𝐸𝑎 𝑅 1 𝑇 1 𝑇𝑓 𝑌𝑡 𝑉𝑔 𝑉𝑔 𝑀𝑤1 𝐶3 1 𝑀𝑤𝑐 𝑘𝑑 𝑉𝑔 𝑀𝑤1 𝑀𝑤𝑝 1 𝑀1𝑡 𝑘𝑝0 𝑒 𝐸𝑎 𝑅 1 𝑇 1 𝑇𝑓 𝑉𝑔 𝑉𝑔 𝑀𝑤1 𝐶4 𝐹𝐼𝑛 1 𝑉𝑔 𝑀𝑤𝐼 𝑉𝑔 𝑀𝑤1 𝑉𝑝 2 𝐶𝑣 𝑅𝑅 𝑇 𝑉𝑝 𝑀𝑤𝐼 1 2 𝑉𝑔 𝑀𝑤1 𝑀1𝑡 𝐼𝑡 𝑅𝑅 𝑇 𝑃𝑉 2 E introduzindo as variáveis de desvios temos 𝑑𝑀1𝑡 𝑑𝑡 𝐶1𝐹𝑀1𝑡 𝐹𝑀10 𝐶2𝑀1𝑡 𝑀10 𝐶3𝑌𝑡 𝑌0 𝐶4𝐼𝑡 𝐼0 𝑑𝛤𝑀1𝑡 𝑑𝑡 𝐶1 𝐹𝑀1𝑡 𝐶2 𝛤𝑀1 𝐶3𝛤𝑌 𝐶4𝛤𝐼 Dividindo tudo por C2 e isolando 𝛤𝑀1 temos 1 𝐶2 𝑑𝛤𝑀1𝑡 𝑑𝑡 𝛤𝑀1𝑡 𝐶1 𝐶2 𝐹𝑀1𝑡 𝐶3 𝐶2 𝛤𝑌𝑡 𝐶4 𝐶2 𝛤𝐼𝑡 1 𝐶2 𝜏1 𝐶1 𝐶2 𝐾1 𝐶3 𝐶2 𝐾2 𝐶4 𝐶2 𝐾3 Aplicar a TLP na função 𝛤𝑀1𝑠𝜏1𝑠 1 𝐾1 𝐹𝑀1𝑠 𝐾2 𝛤𝑌𝑠 𝐾3 𝛤𝐼𝑠 𝛤𝑀1𝑠 𝐾1 𝐹𝑀1𝑠 𝜏1𝑠 1 𝐾2 𝛤𝑌𝑠 𝜏1𝑠 1 𝐾3 𝛤𝐼𝑠 𝜏1𝑠 1 252 Balanço de Energia Balanço de energia interno do gás 𝑑𝑇𝑔1 𝑑𝑡 𝐹𝑔𝑡 𝑀𝑔 𝑇 𝑇𝑔1 𝑈𝐴 𝑀𝑔 𝐶𝑝𝑔 𝑇𝑤1 𝑇𝑔1 Fg taxa de fluxo de massa do gás mols Cpg capacidade calorífica específica do gás JmolK T temperatura do reator principal Tg1 temperatura do gás no reator Tw1 temperatura do fluido de resfriamento no sistema K UA coeficiente global de transferência de calor 𝑑𝑥1 𝑑𝑡 𝑥1 𝑑𝑇𝑔1𝑡 𝑑𝑡 𝑓1 𝑡𝐹𝑔𝑡 𝑇𝑤1𝑡𝑇𝑔1𝑡 𝑓1 𝑡0 𝐹𝑔0𝑇𝑤10𝑇𝑔10 𝜗𝑓1 𝜗𝐹𝑔 𝐹𝑔𝑡 𝐹𝑔0 𝜗𝑓1 𝜗𝑇𝑤1 𝑇𝑤1𝑡 𝑇𝑤10 𝜗𝑓1 𝜗𝑇𝑔1 𝑇𝑔1𝑡 𝑇𝑔10 𝐶5 𝐹𝑔𝑡 𝐶6 Γ𝑤1 𝐶7Γ𝑔1 𝐶5 𝑇 𝑇𝑔1 𝑀𝑔 𝐶6 𝑈𝐴 𝑀𝑔 𝐶𝑝𝑔 𝐶7 𝐹𝑔 𝑀𝑔 𝑈𝐴 𝑀𝑔 𝐶𝑝𝑔 𝑑 Γ𝑔1𝑡 𝑑𝑡 𝐶5 𝐹𝑔𝑡 𝐶6 Γ𝑤1𝑡 𝐶7 Γ𝑔1𝑡 𝑑 Γ𝑔1𝑡 𝑑𝑡 𝐶6 Γ𝑤1𝑡 𝐶5 𝐹𝑔𝑡 𝐶7 Γ𝑔1𝑡 dividindo por C6 1 𝐶6 𝑑 Γ𝑔1𝑡 𝑑𝑡 Γ𝑔1𝑡 𝐶5 𝐶6 𝐹𝑔𝑡 𝐶7 𝐶6 Γ𝑔1𝑡 Aplicar a TLP na função 1 𝐶6 𝜏2 𝐶5 𝐶6 𝐾4 𝐶7 𝐶6 𝐾5 𝛤𝑤1𝑠𝜏2𝑠 1 𝐾4 𝐹𝑔𝑠 𝐾5 𝛤𝑔1𝑠 𝛤𝑤1𝑠 𝐾4 𝐹𝑔𝑠 𝜏2𝑠 1 𝐾5 𝛤𝑔1𝑠 𝜏2𝑠 1 Balanço de energia interno do fluído de resfriamento 𝑑𝑇𝑤1 𝑑𝑡 𝐹𝑤 𝑀𝑤 𝑇𝑤𝑖 𝑇𝑤1 𝑈𝐴 𝑀𝑤𝐶𝑝𝑤 𝑇𝑤1 𝑇𝑔1 𝑑𝑥2 𝑑𝑡 𝑥2 𝑑𝑇𝑤1𝑡 𝑑𝑡 𝑓2 𝑡 𝐹𝑤𝑡𝑇𝑤1𝑡 𝑇𝑔1𝑡 𝑡 𝐹𝑤0𝑇𝑤10𝑇𝑔10 𝜗𝑓2 𝜗𝐹𝑤 𝐹𝑤𝑡 𝐹𝑤0 𝜗𝑓2 𝜗𝑇𝑤1 𝑇𝑤1𝑡 𝑇𝑤10 𝜗𝑓2 𝜗𝑇𝑔1 𝑇𝑔1𝑡 𝑇𝑔10 𝐶8 𝐹𝑤𝑡 𝐶9 Γ𝑤1 𝐶10Γ𝑔1 𝐶8 𝑇 𝑇𝑤10 𝑀𝑤 𝐶9 𝑈𝐴 𝑀𝑤 𝐶𝑝𝑤 𝐹𝑤 𝑀𝑤 𝐶10 𝑈𝐴 𝑀𝑤 𝐶𝑝𝑤 𝑑 Γ𝑤1𝑡 𝑑𝑡 𝐶8 𝐹𝑤𝑡 𝐶9 Γ𝑤1𝑡 𝐶10 Γ𝑔1𝑡 𝑑 Γ𝑤1𝑡 𝑑𝑡 𝐶9 Γ𝑤1𝑡 𝐶8 𝐹𝑤𝑡 𝐶10 Γ𝑔1𝑡 dividindo por C9 1 𝐶9 𝑑 Γ𝑤1𝑡 𝑑𝑡 Γ𝑤1𝑡 𝐶8 𝐶9 𝐹𝑤𝑡 𝐶10 𝐶9 Γ𝑔1𝑡 𝜏 𝑑 Γ𝑤1𝑡 𝑑𝑡 Γ𝑤1𝑡 𝐾6 𝐹𝑤𝑡 𝐾7 Γ𝑔1𝑡 Aplicar a TLP na função 1 𝐶9 𝜏3 𝐶8 𝐶9 𝐾6 𝐶10 𝐶9 𝐾7 𝛤𝑤1𝑠𝜏3𝑠 1 𝐾6 𝐹𝑔𝑠 𝐾7 𝛤𝑔1𝑠 Balanço de energia do reator dT dt H0 Htop Hr Hp Mr Cpr BW Cp pol 𝐻0 𝐹𝑣1 𝑀10 𝐶𝑝𝑔 𝑇0 𝑇𝑟𝑒𝑓 𝐻𝑡𝑜𝑝 𝐹𝑉1 𝑀1 𝐶𝑝𝑔 𝑇 𝑇𝑟𝑒𝑓 𝐻𝑟 𝐻𝑟 𝐾𝑝 𝑌 𝑀1 𝑚𝑤1 𝐻𝑝 𝑂𝑝 𝐶𝑝𝑝𝑜𝑙 𝑇 𝑇𝑟𝑒𝑓 𝑑𝑇 𝑑𝑡 𝐹𝑣1𝑀10𝐶𝑝𝑔 𝑇0 𝑇𝑟𝑒𝑓 𝐹𝑣1 𝑀1 𝐶𝑝𝑔 𝑇 𝑇𝑟𝑒𝑓 𝑀𝑟 𝐶𝑝𝑟 𝐵𝑊 𝐶𝑝𝑝𝑜𝑙 𝐻𝑟 𝐾𝑝𝑇 𝑌 𝑀1𝑚𝑤1 𝑂𝑝 𝐶𝑝𝑝𝑜𝑙 𝑇 𝑇𝑟𝑒𝑓 𝑀𝑟 𝐶𝑝𝑟 𝐵𝑊 𝐶𝑝𝑝𝑜𝑙 𝑑𝑥3 𝑑𝑡 𝑥3 𝑑𝑇𝑡 𝑑𝑡 𝑓3 𝑡 𝐹𝑣1𝑡 𝑇𝑡𝑇𝑟𝑒𝑓𝑡 𝑡𝐹𝑣10 𝑇0𝑇𝑟𝑒𝑓0 𝜗𝑓3 𝜗𝐹𝑣1 𝐹𝑣1𝑡 𝐹𝑣10 𝜗𝑓3 𝜗𝑇 𝑇𝑡 𝑇0 𝜗𝑓3 𝜗𝑇𝑟𝑒𝑓 𝑇𝑟𝑒𝑓𝑡 𝑇𝑟𝑒𝑓0 𝐶11 𝐹𝑣1𝑡 𝐶12 Γ𝐶13Γ𝑟𝑒𝑓 𝐶11 𝑇0 𝑇𝑟𝑒𝑓0 𝑇 𝑇𝑟𝑒𝑓 𝑀𝑟 𝐶𝑝𝑟 𝐵𝑊 𝐶𝑝𝑝𝑜𝑙 𝐶12 𝐹𝑣1 𝑀10 𝐶𝑝𝑔 𝑀𝑟 𝐶𝑝𝑟 𝐵𝑊 𝐶𝑝𝑝𝑜𝑙 𝑂𝑝𝐶𝑝𝑝𝑜𝑙 𝑀𝑟 𝐶𝑝𝑟 𝐵𝑊 𝐶𝑝𝑝𝑜𝑙 𝐶8 𝐹𝑣1 𝑀10 𝐶𝑝𝑔 𝑂𝑝𝐶𝑝𝑝𝑜𝑙 𝑀𝑟 𝐶𝑝𝑟 𝐵𝑊 𝐶𝑝𝑝𝑜𝑙 2 𝐶12 𝐹𝑣1 𝑀10 𝐶𝑝𝑔 𝐹𝑣1 𝑀1 𝐶𝑝𝑔 𝑀𝑟 𝐶𝑝𝑟 𝐵𝑊 𝐶𝑝𝑝𝑜𝑙 𝑂𝑝𝐶𝑝𝑝𝑜𝑙 𝑀𝑟 𝐶𝑝𝑟 𝐵𝑊 𝐶𝑝𝑝𝑜𝑙 𝐶14 𝐹𝑣1 𝑀10 𝐶𝑝𝑔 𝐹𝑣1 𝑀1 𝐶𝑝𝑔 𝑂𝑝𝐶𝑝𝑝𝑜𝑙 𝑀𝑟 𝐶𝑝𝑟 𝐵𝑊 𝐶𝑝𝑝𝑜𝑙 2 𝑑 Γ𝑡 𝑑𝑡 𝐶11 𝐹𝑣1𝑡 𝐶12 Γ𝑡 𝐶13 Γ𝑟𝑒𝑓𝑡 𝑑 Γ𝑡 𝑑𝑡 𝐶12 Γ𝑡 𝐶11 𝐹𝑣1𝑡 𝐶13 Γ𝑟𝑒𝑓𝑡 dividindo por C12 1 𝐶12 𝑑 Γ𝑡 𝑑𝑡 Γ𝑡 𝐶11 𝐶12 𝐹𝑣1𝑡 𝐶13 𝐶12 Γ𝑟𝑒𝑓𝑡 𝛤𝑡𝑠𝜏4𝑠 1 𝐾8 𝐹𝑣1𝑡 𝐾9 Γ𝑟𝑒𝑓𝑡 𝛤𝑡𝑠 𝐾8 𝐹𝑣1𝑡 𝜏4𝑠 1 𝐾9 Γ𝑟𝑒𝑓𝑡 𝜏4𝑠 1 3 Resultado e Discussão Foram encontradas as constantes 𝐶1 0 𝐶2 00693 𝐶3 0 𝐶4 49439 𝐶5 1 𝐶6 330434783 𝐶7 347434783 𝐶8 05556 𝐶9 172929076 𝐶10 151298 𝐶11 0 𝐶12 0 𝐶13 0 𝐶14 0 K Ganho do sistema 105000 Tau Constante de tempo 40000 segundos Considerando Fw a variável manipulada e Tg a variável controlada A escolha da temperatura do gás de reciclo Tg como a variável controlada está ligada à sua influência direta na eficiência do processo de polimerização na estabilidade térmica do reator e na segurança operacional Controlar Tg permite ajustar indiretamente outras variáveis críticas do sistema como a temperatura interna do reator e a taxa de polimerização Gráfico 1 Comparação RungeKutta e Laplace Podese observar que pelo método RungeKutta o sistema rapidamente dissipa o calor e estabiliza a uma temperatura mais baixa 292 K No método laplace o sistema modelado analiticamente pode ter uma resposta menos dinâmica o que leva a uma subida a 251 K e estabilização posterior Os dois métodos podem estar lidando de forma diferente com alguns aspectos do modelo como a capacidade térmica do gás e do fluido refrigerante que afetam o equilíbrio térmico e a Troca de calor entre o gás reciclado e o sistema de refrigeração que pode ter uma dinâmica complexa que o método de Laplace não capta completamente Em resumo o método de RungeKutta pode ser mais sensível às condições iniciais resultando em uma queda abrupta para um valor próximo ao ponto de equilíbrio 292 K enquanto o método de Laplace devido à sua natureza analítica não necessariamente lida com as condições iniciais da mesma forma o que pode resultar na variação mais suave observada no gráfico Os efeitos dos distúrbios no sistema ou mudanças rápidas na entrada como a vazão de fluido refrigerante RungeKutta Captura uma queda brusca na temperatura para 292 K possivelmente refletindo uma rápida dissipação de calor ou ajuste do sistema ao novo equilíbrio O comportamento abrupto pode ser devido à precisão numérica em capturar pequenos transientes ou distúrbios no sistema Laplace Mostra um aumento inicial para 251 K seguido de estabilização Isso pode ser resultado da linearização que não reflete totalmente a dinâmica real do sistema suavizando variações bruscas ou comportamentos nãolineares Logo o método RungeKutta é mais adequado para capturar comportamentos rápidos enquanto o Laplace fornece uma solução mais simplificada e aproximada Gráfico 2 Conversão do reagente me função da temperatura Conforme o esperado da equação de Arrhenius a taxa de reação aumenta exponencialmente com a temperatura À medida que a temperatura aumenta a conversão de reagentes também cresce Válvula de controle 𝑎1 𝜗𝑉𝑡 𝜗𝑡 𝑎0𝑉𝑡 𝑏0𝐶𝑡 𝑎1 𝑎0 𝜗𝑉𝑡 𝜗𝑡 𝑉𝑡 𝑏0 𝑎0 𝑐𝑡 𝑎1 𝑎0 𝑆𝑉𝑆 𝑉𝑆 𝑏0 𝑎0 𝑐𝑆 𝑉𝑆 𝑎1 𝑎0 𝑆 1 𝑏0 𝑎0 𝑐𝑆 𝑉𝑆 𝑏0 𝑎0 𝑐𝑆 𝑎1 𝑎0𝑆 1 𝑡𝑣 𝑎1 𝑎0 𝐾𝑣 𝑏0 𝑎0 𝑉𝑆 𝐾𝑣 𝑐𝑆 𝑡𝑣 𝑆 1 Para calcular 𝑡𝑣 temos 𝑉𝑆 𝐾𝑣 𝑌𝑆 𝑡𝑣 𝑆 1 𝐶𝑆 𝑀 𝑆 𝑀 1 𝑉𝑆 𝐾𝑣 𝑡𝑣 𝑆 1𝑆 𝑉𝑆 𝐾𝑡 𝑡𝑣 𝑆 1 𝑡𝑣 Em frações parciais 𝑉𝑆 𝐾𝑡 𝑡𝑣 𝑆 1 𝑡𝑣 𝑆 𝐶1 𝑆 1 𝑡𝑣 𝐶2 𝑆 𝑆 0 𝐾𝑡 𝑡𝑣 𝑆 𝑆 1 𝑡𝑣 𝑆 𝑆𝐶1 𝑆 1 𝑡𝑣 𝑆𝐶2 𝑆 𝐾𝑡 𝑡𝑣 1 𝑡𝑣 0 𝐶2 𝐾𝑣 𝐶2 𝑆 1 𝑡𝑣 𝐾𝑣 𝑡𝑣 𝑆 1 𝑡𝑣 𝑆 1 𝑡𝑣 𝑆 𝐶1 𝑆 1 𝑡𝑣 𝑆 1 𝑡𝑣 𝐶2 𝑆 1 𝑡𝑣 𝑆 𝐾𝑣 𝑡𝑣 𝑆 𝐶1 𝐾𝑣 𝑡𝑣 1 𝑡𝑣 𝐶1 𝐾𝑣 𝐶1 𝑉𝑆 𝐾𝑣 𝑡𝑣 𝑆 1 𝑡𝑣 𝑆 𝐾𝑣 𝑆 1 𝑡𝑣 𝐾𝑣 𝑆 Aplicando a inversa TLP 𝑉𝑡 𝐾𝑣 1 𝑒 𝑡 𝑡𝑣 𝑉𝑡 𝐾𝑣 1 𝑒 𝑡 𝑡𝑣 Encontrando Kc e Kcu pelo método e substituição direta 1 𝐺𝑣 𝐺𝑐 𝐺𝑇 𝐺𝑝 0 1 𝐾𝑐 1216639 04𝑠 1 3 205𝑠 1 360 28956𝑠 1 0 𝐾𝑐 1216639 3 360 131397012 𝐾𝑐 04 𝑠 1 205 𝑠 1 28956 𝑠 1 23755𝑠3 71027𝑠2 29201𝑠 1 1 131397012 𝐾𝑐 23755𝑠3 71027𝑠2 29201𝑠 0 Substituindo 𝑠 𝑖𝑤𝑢 1 131397012 𝐾𝑐 23755𝑖𝑤𝑢3 71027𝑖𝑤𝑢2 29201𝑖𝑤𝑢 0 𝑖3 𝑖 𝑖2 1 1 131397012 𝐾𝑐 23755𝑤𝑢3𝑖 71027𝑤𝑢2 29201𝑖𝑤𝑢 0 Parte real 1 131397012 𝐾𝑐 71027𝑤𝑢2 0 𝐾𝑐 1 71027𝑤𝑢2 131397012 Parte imaginária 23755𝑤𝑢3𝑖 29201𝑖𝑤𝑢 0 23755𝑤𝑢2 29201𝑖𝑤𝑢 0 Para 𝑤𝑢 0 𝐾𝑐 1 131397012 000000008 Para 𝑤𝑢2 23755𝑤𝑢2 29201 0 𝑤𝑢 11097 parte real positiva Logo temos que 𝐾𝑐 1 71027811092 131397012 00000664 𝐾𝑐𝑢 000000008 𝐾𝑐 00000664 ANÁLISE DO GRÁFICO Arranjo de Routh 1 131397012 𝐾𝑐 23755𝑠3 71027𝑠2 29201𝑠 0 𝑎3 23755 𝑎2 71027 𝑎1 29201 𝑎0 1 131397012 𝐾𝑐 Linha Arranjo 1 𝑎3 𝑎1 2 𝑎2 𝑎0 3 𝑏3 3 𝑐3 𝑏3 𝑎2𝑎1 𝑎3𝑎0 𝑎2 71027 29201 23755 1 131397012 𝐾𝑐 71027 𝑏3 𝑎2𝑎1 𝑎3𝑎0 𝑎2 71027 29201 23755 1 131397012 𝐾𝑐 71027 20724 439 𝐾𝑐 𝑐3 𝑏3𝑎0 𝑎2𝑏2 𝑏3 𝑏3𝑎0 𝑎2 0 𝑏3 𝑎0 1 131397012 𝐾𝑐 Linha Arranjo 1 23755 29201 2 71027 1 131397012 𝐾𝑐 3 20727 439 𝐾𝑐 3 1 131397012 𝐾𝑐 Para ser estável 𝑏𝑛 0 20727 439 𝐾𝑐 0 𝐾𝑐 1 71027811092 131397012 00000664 𝐾𝑐𝑢 𝑐𝑛 0 1 1313970112 𝐾𝑐 0 𝐾𝑐 1 131397012 000000008 O intervalo pelo método de Routh foi igual ao da substituição direta logo a análise é a mesma 000000008 𝐾𝑐 00000664 𝑤𝑢 11097 𝐾𝑐𝑢 00000664 𝜏𝑢 2𝜋 𝑤𝑢 5703 ANÁLISE DO GRÁFICO 𝑓𝑚á𝑥 190 𝑚𝑜𝑙 𝑠 taxa de alimentação máxima de etileno 𝐺𝑙 𝐹𝑚á𝑥 𝐶𝑣 𝑚á𝑥 ganho do sistema Como 𝐶𝑣 𝑚á𝑥 75 𝑎𝑡𝑚05 logo 𝐶𝑣 𝑚á𝑥 𝑄 Δ𝑃 𝐺𝑓 Coeficiente de vazão máxima da válvula Q é a vazão volumétrica neste caso a vazão de etileno convertida para galões por minuto ΔP é a queda de pressão em psi 𝐺𝑓 é o fator de correção de densidade definido como 𝐺𝑓 𝜌𝑓 𝜌𝐻2𝑂 𝜌𝑓 7374 𝑙𝑏𝑚 𝑓𝑡3 densidade do etileno 𝜌𝐻2𝑂 624 𝑙𝑏𝑚 𝑓𝑡3 densidade da água Logo 𝐺𝑓 7374 624 118 Substituindo na equação de 𝐶𝑣 𝑚á𝑥 Como não existe uma válvula com essa capacidade máxima de 3226 𝑔𝑎𝑙 min 𝑝𝑠𝑖05 escolhese então uma válvula disponível de capacidade imediatamente superior Pela tabela é possível encontrar uma de capacidade de 46 𝑔𝑝𝑚 𝑝𝑠𝑖05 com isso é possível encontrar o ganho da válvula Válvula de igual porcentagem e ar para abrir As válvulas de igual porcentagem apresentam algumas vantagens específicas para o processo de resfriamento em um reator de polietileno Segurança em Caso de Falha A válvula com atuação ar para abrir utiliza a pressão de ar para se manter aberta o que significa que em caso de perda de sinal de ar ou energia ela se fechará automaticamente Esse recurso de segurança é importante em processos de resfriamento de reatores pois o fechamento automático evita o excesso de resfriamento que poderia resultar em uma queda rápida de temperatura afetando a estabilidade da reação e a qualidade do polietileno produzido Estabilidade no Controle de Vazão A válvula de igual porcentagem ajusta o fluxo em função da abertura de forma proporcional o que facilita o controle em processos com grande variação de demanda como o resfriamento em um reator exotérmico Esse comportamento é importante para evitar flutuações de temperatura no reator garantindo um resfriamento eficiente Versatilidade e Adaptação A válvula de igual porcentagem é adequada para lidar com a faixa de operação do reator respondendo de maneira adaptável a diferentes condições como variações na taxa de reação e geração de calor contribuindo para a estabilidade do processo e a segurança operacional Estabilidade Térmica e Redução de Sobrecargas Com a atuação ar para abrir a válvula abre gradualmente conforme a demanda de resfriamento aumen proporcionando uma resposta proporcional que reduz o risco de sobrecargas térmicas no sistema de resfriamento e assegura uma transição mais suave na vazão de fluido refrigerante Considerando rangeabilidade 50 e MV0 o valor da variável manipulada vazão no estado estacionário temos que 𝐺𝑣𝑠 𝐾𝑣 𝜏𝑣𝑆 1 𝐾𝑣 ln𝛼 100 𝑀𝑉0 ln50 100 311105 𝐾𝑔 𝑠 1216639 𝐾𝑔 𝑠 Encontrando 𝜏𝑣 Por tentativa e erro adicionamos 4 valores e de acordo com o gráfico é possível observar que a curva laranja respondeu mais rápido onde foi utilizado o valor de 𝜏𝑣 04 com o perfil mais próximo da válvula escolhida Logo 𝐺𝑣𝑆 1216639 04𝑆 1 O sistema GX da Fisher é uma válvula de controle com atuador integrado de última geração projetada para controlar eficientemente uma ampla variedade de líquidos gases e vapores Fabricado em aço carbono aço inoxidável duplex ou ligas especiais o GX oferece desempenho confiável e durável em serviços críticos como redução de ruído controle de cavitação ambientes erosivos baixa vazão e condições com opções de características de vazão em porcentagem igual linear ou espacial o GX se adapta a diferentes demandas de controle Sua construção otimizada reduz a complexidade e o número de peças resultando em menor custo de manutenção Características Fácil dimensionamento e seleção Não é necessário fazer o dimensionamento do atuador a seleção é automática O atuador aperfeiçoado permite a utilização de uma ampla gama de suprimentos de ar Projetado para fácil manutenção Baixo custo durante toda a vida útil Design simples e robusto Atuador pneumático compacto com várias molas Disponível com Controlador de válvula digital DVC2000 ou DVC6200 de fácil calibração Design de alta capacidade Passagem de fluxo da válvula aperfeiçoada para estabilidade do fluxo Ampla gama de materiais incluindo ligas especiais Capacidades de vedação Classes II IV V e VI Rangeabilidade de 501 igual porcentagem Vedação de metal dos foles opcional Montagem ISO 5210 F7 disponível para uso com atuadores elétricos SensoresTransmissores 𝑎1 𝜗𝑦𝑚é𝑑𝑡 𝜗𝑡 𝑎0𝑦𝑚é𝑑 𝑡 𝑏0𝑦𝑡 𝑎1 𝑎0 𝜗 𝑦𝑚é𝑑 𝑡 𝜗𝑡 𝑦𝑚é𝑑𝑡 𝑏0 𝑎0 𝑦𝑡 𝑎1 𝑎0 𝑆𝑌𝑚𝑒𝑑𝑆 𝑌𝑚𝑒𝑑𝑆 𝑏0 𝑎0 𝑌𝑆 transformada de laplace 𝑌𝑚𝑒𝑑𝑆 𝑎1 𝑎0 𝑆 1 𝑏0 𝑎0 𝑌𝑆 𝑌𝑚𝑒𝑑𝑆 𝑏0 𝑎0 𝑌𝑆 𝑎1 𝑎0 𝑆 1 𝑡𝑡 𝑎1 𝑎0 𝐾𝑡 𝑏0 𝑎0 𝐾𝑡 100 0𝑆𝑇 17 0𝑏𝑎𝑟 6 𝑃𝑎𝑟𝑎 𝑀 1 𝑌𝑆 𝑀 𝑆 𝑌𝑚é𝑑𝑆 𝐾𝑇 𝑡𝑡 𝑆 1𝑆 𝑌𝑚é𝑑 𝐾𝑡 𝑡𝑡 𝑆 1 𝑡𝑡 𝑆 Em frações parciais 𝑌𝑚é𝑑𝑆 𝐾𝑇 𝑡𝑡 𝑆 1 𝑡𝑡 𝑆 𝐶1 𝑆 1 𝑡𝑡 𝐶2 𝑆 S 0 𝐾𝑇 𝑡𝑡 𝑆 𝑆 1 𝑡𝑡 𝑆 𝑆𝐶1 𝑆 1 𝑡𝑡 𝑆 𝑆𝐶2 𝑆 𝐾𝑇 𝑡𝑡 1 𝑡𝑡 0 𝐶2 𝐾𝑇 𝐶2 S 1 𝑡𝑡 𝐾𝑇 𝑡𝑡 𝑆 1 𝑡𝑡 𝑆 1 𝑡𝑡 𝑆 𝐶1 𝑆 1 𝑡𝑡 𝑆 1 𝑡𝑡 𝐶2 𝑆 1 𝑡𝑡 𝑆 𝐾𝑇 𝑡𝑡 𝑆 𝐶1 𝐾𝑇 𝑡𝑡 1 𝑡𝑡 𝐶1 𝐾𝑇 𝐶1 Aplicando a inversa TLP 𝑌𝑚𝑒𝑑𝑡 𝐾𝑇 1 𝑒 𝑡 𝑡𝑡 𝑌𝑚𝑒𝑑𝑡 𝐾𝑇 1 𝑒 𝑡 𝑡𝑡 Considerando o tempo morto temos que Desprezando o valor do atraso podemos estabelecer a seguinte função de transferência 𝑌𝑚𝑒𝑑𝑆 𝐾𝑇 𝑌𝑆 𝑒𝜃𝑆 𝑡𝑡 𝑆 1 𝐺𝑇 𝑌𝑚𝑒𝑑𝑆 𝑌𝑠 𝐾𝑇 𝑌𝑆𝑒𝜃𝑆 𝑡𝑡 𝑆 1 𝑌𝑆 𝑀 𝑆 𝑀 1 𝑌𝑚𝑒𝑑𝑆 𝐾𝑇 𝑒𝜃𝑆 𝑡𝑡 𝑆 1𝑆 𝑌𝑚𝑒𝑑𝑆 𝐾𝑇 𝑡𝑡 𝑆 1 𝑡𝑡 𝑆 𝑒𝜃𝑆 Em frações parciais 𝑌𝑚𝑒𝑑𝑆 𝐾𝑇 𝑡𝑡 𝑆 1 𝑡𝑡 𝑆 𝑒𝜃𝑆 𝐶1 𝑆 1 𝑡𝑡 𝐶2 𝑆 S 0 𝐾𝑇 𝑡𝑡 𝑆 𝑆 1 𝑡𝑡 𝑆 𝑒𝜃𝑆 𝑆𝐶1 𝑆 1 𝑡𝑡 𝑆𝐶2 𝑆 𝐾𝑇 𝑡𝑡 𝑆 𝑒𝜃𝑆 𝐶1 𝐾𝑇 𝑡𝑡 1 𝑡𝑡 𝑒 𝜃 1 𝑡𝑡 𝐶1 𝐾𝑇 𝑒 𝜃1 𝑡𝑡 𝐶1 𝑌𝑚𝑒𝑑𝑆 𝐾𝑇 𝑡𝑡 𝑆 1 𝑡𝑡 𝑆 𝐾𝑇 𝑒 𝜃1 𝑡𝑡 𝑆 1 𝑡𝑡 𝐾𝑇 𝑆 Aplicando a inversa TLP 𝑌𝑚𝑒𝑑𝑡 𝐾𝑇 1 𝑒 𝜃1 𝑡𝑡𝑒 𝜃𝑡 𝑡𝑡 𝑌𝑚𝑒𝑑𝑡 𝐾𝑇 1 𝑒 𝜃𝑡 𝑡𝑡 𝑡 𝑡𝑡 𝑌𝑚𝑒𝑑𝑡 𝐾𝑇 06 06 1 𝑒 𝜃𝑡𝑡 𝑡𝑡 ln06 ln 1 𝑒 𝜃𝑡𝑡 𝑡𝑡 𝑡𝑡 ln06 𝜃 𝑡𝑡 𝑡𝑡1 ln06 0 𝑡𝑡 𝜃 1 ln06 𝜃 3𝑠 𝑡𝑡 3 1 ln06 204 Logo a função de transferência do sensor é 𝐺𝑇 6 𝑒2𝑆 205 𝑆 1 Simulação sensor com e sem delay Sensor de Pressão G14 12 MPa 5V USPG41 para Gás e Líquidos Descrição O Sensor de Pressão G14 12 MPa USPG41 é uma excelente escolha para projetos de automação e robótica sendo capaz de medir a pressão de gases não corrosivos água óleo e outros líquidos não corrosivos Este modelo destacase pela sua facilidade de integração com microcontroladores como Arduino Raspberry Pi ESP32 entre outros Ele possui apenas três fios para conexão VCC vermelho GND preto e OUT amarelo ou azul O sensor opera dentro da faixa de pressão de 0 a 12 MPa e deve ser utilizado em temperaturas entre 0 e 85 o que o torna inadequado para aplicações automotivas Com uma construção totalmente metálica em liga de aço carbono o sensor apresenta rosca G14 com diâmetro de 13 mm permitindo uma fácil integração em diversas estruturas Especificações Modelo USPG41 Tensão de operação 5V DC Tensão de saída 05 a 45V DC Corrente de trabalho 10 mA Faixa de Pressão 0 a 12 MPa Pressão máxima 24 MPa Temperatura de trabalho 0 a 85 Faixa de erro 15 Conexões VCC vermelho GND preto OUT amarelo ou azul Diâmetro da rosca 13mm Base hexagonal 24mm Comprimento do fio 18cm Dimensões CxD 51x24mm Peso 42g Faixa de Medição O sensor USPG41 possui uma faixa de medição de 0 a 12 MPa megapascal o que permite medições precisas e confiáveis dentro desse intervalo Esta capacidade é especialmente útil em aplicações industriais que operam com pressões controladas como no processo de produção de cumeno Span Variação da Faixa de Medição O span do sensor USPG41 é de 12 MPa indicando a totalidade da pressão que ele pode detectar e converter em um sinal elétrico Essa variação abrange desde 0 MPa até 12 MPa cobrindo integralmente a faixa necessária para o monitoramento em processos de produção Fundo de Escala O fundo de escala do sensor USPG41 é de 0 MPa permitindo que ele detecte pressões a partir do zero absoluto Essa característica é essencial para aplicações que requerem alta precisão na medição de baixas pressões sendo ideal para monitorar variações sutis em diferentes estágios do processo de produção de cumeno Aplicações no Processo de Produção de Polietileno Controle de Pressão em Reatores Monitorar a pressão dentro de reatores químicos onde a polimerização do etileno ocorre garantindo que as condições de pressão estejam dentro dos limites desejados para otimizar a reação Sistema de Transporte Medir a pressão em tubulações que transportam gás ou líquido assegurando que não haja vazamentos ou bloqueios Monitoramento de Válvulas Integrar o sensor ao sistema de controle de válvulas para garantir que a pressão esteja ajustada corretamente durante o processo de produção Sistemas de Armazenamento Monitorar a pressão em tanques de armazenamento para evitar sobrepressão e garantir a segurança do sistema Controle de Qualidade Usar o sensor para monitorar a pressão em diferentes estágios do processo para garantir a consistência na qualidade do polietileno produzido Sintonia dos controladores Os controladores foram avaliados segundo o overshoot percentual tempo de estabilização e os critérios de integrais do erro mostradas abaixo 𝐼𝐴𝐸 0 𝑒𝑡𝑑𝑡 𝐼𝑆𝐸 0 𝑒𝑡2𝑑𝑡 𝐼𝑇𝐴𝐸 0 𝑒𝑡𝑡𝑑𝑡 Método de sintonia Zn malha fechada Kc 𝜏𝑢 𝜏𝑑 Zn 𝐾𝑐𝑢 00000664 P 00000332 𝑤𝑢 11097 PI 000003018 47525 𝜏𝑢 5703 PID 0000039066 28515 07128 Comparação entre os controladores sem distúrbios Gráfico comparação set point P PI e PID IAE ISE ITAE ov Te P A partir das equações acima A partir das equações acima A partir das equações acima De acordo com o gráfico De acordo com o gráfico PI A partir das equações acima A partir das equações acima A partir das equações acima De acordo com o gráfico De acordo com o gráfico PID A partir das equações acima A partir das equações acima A partir das equações acima De acordo com o gráfico De acordo com o gráfico O controlador PID e P se mostraram ter bom desempenho tendo o PID um menor tempo de estabilização e o P menor overshoot Já o PI teve um desempenho abaixo com uma grande oscilação e um tempo de estabilização muito longo em comparação com outros controladores ajustar de acordo com nossos resultados Inserção de um distúrbio de 20 em q Gráfico comparação set point P PI e PID IAE ISE ITAE ov Te P A partir das equações acima A partir das equações acima A partir das equações acima De acordo com o gráfico De acordo com o gráfico PI A partir das equações acima A partir das equações acima A partir das equações acima De acordo com o gráfico De acordo com o gráfico PID A partir das equações acima A partir das equações acima A partir das equações acima De acordo com o gráfico De acordo com o gráfico O controlador PID e P se mostraram ter bom desempenho tendo o PID um menor tempo de estabilização e o P menor overshoot Já o PI teve um desempenho abaixo com uma grande oscilação e um tempo de estabilização muito longo em comparação com outros controladores ajustar de acordo com nossos resultados Método RELE Malha RELE com o gráfico desse tipo abaixo RELE Kcu De acordo com a simulação P Calculado de acordo com os valores encontrados d De acordo com a simulação PI Calculado de acordo com os valores encontrados Calculado de acordo com os valores encontrados Pu De acordo com a simulação PID Calculado de acordo com os valores encontrados Calculado de acordo com os valores encontrados Calculado de acordo com os valores encontrados Análise gráfico sem inserção de distúrbio IAE ISE ITAE ov Te P A partir das equações acima A partir das equações acima A partir das equações acima De acordo com o gráfico De acordo com o gráfico PI A partir das equações acima A partir das equações acima A partir das equações acima De acordo com o gráfico De acordo com o gráfico PID A partir das equações acima A partir das equações acima A partir das equações acima De acordo com o gráfico De acordo com o gráfico O controlador PID e P se mostraram ter bom desempenho tendo o PID um menor tempo de estabilização e o P menor overshoot Já o PI teve um desempenho abaixo com uma grande oscilação e um tempo de estabilização muito longo em comparação com outros controladores ajustar de acordo com nossos resultados Inserção de um distúrbio de 20 em q Análise gráfico sem inserção de distúrbio IAE ISE ITAE ov Te P A partir das equações acima A partir das equações acima A partir das equações acima De acordo com o gráfico De acordo com o gráfico PI A partir das equações acima A partir das equações acima A partir das equações acima De acordo com o gráfico De acordo com o gráfico PID A partir das equações acima A partir das equações acima A partir das equações acima De acordo com o gráfico De acordo com o gráfico O controlador PID e P se mostraram ter bom desempenho tendo o PID um menor tempo de estabilização e o P menor overshoot Já o PI teve um desempenho abaixo com uma grande oscilação e um tempo de estabilização muito longo em comparação com outros controladores ajustar de acordo com nossos resultados Método curva de reação Adicionar simulação e gráfico Determinação dos parametros 𝜃 𝜏 e K