Process Design for Operability: Louis P. Russo and B. Wayne Bequette

Transcription

1 Process Design for Operability: A Styrene Polymerization Application Louis P. Russo an B. Wayne Bequette Department of Chemical Engineering, Rensselaer Polytechnic nstitute, Troy, NY USA Abstract: Steay-state multiplicity analysis is use to stuy the operability of a jackete exothermic styrene polymerization CSTR. Low an high temperature infeasibility regions are shown to occur uner certain esign conitions. The approach evelope is base on using a combination of analytical an numerical techniques to guie the analysis, rather than using a purely numerical approach. A reactor esign use by previous researchers to show poor closeloop performance from a single input-single output control strategy is stuie in this paper. t is shown how this poor performance coul have been preicte, an eliminate, at the process esign stage using our bifurcation-base approach. 1 NTRODUCTON The integration of process esign an control (or operability) has been recognize as an important objective for many years. A number of relevant techniques have been propose, incluing exibility analysis, resiliency, an steay-state an ynamic \back-o"analysis. n this paper we present a bifurcation-base approach to unerstan how esign parameters inuence the characteristic input/output behavior of process systems. We stuy the steay-state multiplicity behavior of a jackete exothermic styrene polymerization CSTR. We are motivate in our stuies to consier the attributes of \process controllability" an \process operability" in orer to unerstan how moeling ecisions (an ultimately, esign an operation ecisions) eect the operating characteristics of the chemical reactor. Since the structure an properties of polymers are etermine primarily by the operation of the chemical reactor in which they are mae, there is signicant motivation to optimize the process esign an operation to improve controllability. Previous researchers showe poor single input-single output control performance for a given set of polymerization CSTR esign parameters. n this work we show how this poor performance coul have been preicte, an eliminate, at the process esign stage using our bifurcation-base approach. 2 NDUSTRALLY RELEANT APPLCATON Although a etaile analysis of the bifurcation behavior of a jackete exothermic CSTR was presente in Russo an Bequette (1995, 1996), the kinetic mechanism stuie up to now is quite simple in nature an hence is not generally sucient to capture the behavior of some inustrially relevant chemistries. Therefore, it is important to examine a chemical \system" in which these bifurcation results can be of practical usage. The prouction of synthetic polymers is an example of a chemical system that is important from an inustrial viewpoint. Synthetic polymer prouction excees one hunre million metric tons per year worlwie (Ray, 199). Hence, there is sucient incentive to obtain a eeper unerstaning of the parameters which inuence the steay-state an ynamic behavior of polymerization chemical reactors. The structure an properties of the polymer are etermine primarily by the operation of the reactor in which they are mae. Therefore, it is important to have tight control of the operating conitions insie the process. The emphasis in this paper is on an unerstaning of the bifurcation behavior of exothermic polymerization reactors. Ray (199) serves as an excellent review of polymerization reaction moeling an control, while Billmeyer (198) presents a more general overview of polymer science. Russo (1996) reviews the bifurcation behavior of polymerization reactors. MOTATON Kim et al. (1991) note that poor heat transfer from the reactor to the cooling jacket in polymerization systems is one of the major causes for strong process nonlinearity. A major problem is that inustrial scale reactors have heat transfer area per volume ratios which are much smaller than lab-scale units. Perhaps the easiest an most common approach to regulate reactor temperature is to utilize a cooling jacket to remove the energy liberate by the reaction mixture. A \col" liqui (water, Dowtherm, glycol, etc.) is pumpe through the jacket to remove excess heat from the reactor. The cooling jacket owrate is the primary manipulate variable use to control the reactor temperature, which in turn species the conversion an thus the polymer properties. Congaliis et al. (1989) assume that a \properly tune" P controller coul be use to manipulate the coolant ow rate in orer to obtain any value of the reactor jacket temperature. However, it was emonstrate in Russo an Bequette (1995) that for even simple 1st orer kinetics (A! B) the presence of infeasible reactor operating regions is possible. Pinto an Ray (1995) stresse that the role of physical constraints on the moel has been largely

2 overlooke in the past. These constraints, taken with the mathematical moel, create a new moel which may or may not present completely ierent bifurcation structures that can improve the esign an operation of real processes (Pinto an Ray, 1995). STYRENE POLYMERZATON CSTR MODEL n this section the features of the styrene polymerization moel of Hialgo an Brosilow (199) are escribe. The free raical polymerization kinetic mechanism is moele as: M + R P n + M P n + P m P n + P m k?! 2 R (initiator ecomp.) k i?! P1 (chain initiation) k p?! Pn+1 (propagation) k t?! Tn + T m (termination) k tc?! T n+m (termination) where is the initiator, R is the raical prouce by initiator ecomposition, M is the monomer species, P is the growing polymer chain, an T is the terminate polymer chain (cf. Billmeyer (198) for a iscussion on free raical polymerization kinetics). A number of assumptions were mae by Hialgo an Brosilow (199) in obtaining the styrene polymerization CSTR moel. The reaer is referre to Russo (1996) an Hialgo an Brosilow (199) for etails concerning these moeling assumptions an a erivation of the moeling equations. The moel for polystyrene polymerization is: C m T C i (QiCif? (Qi + Qm + Qs)Ci)? k C i (1) (QmCmf? (Qi + Qm + Qs)Cm)? k pc mc gp (2) (Qi + Qm + Qs)(Tf? T ) T c Qc(Tcf? Tc)? UA C p (T? Tc) + (?Hr) C p k pc mc gp () + UA (T? T c) () c cc pc c C gp r h 2fkC i i where C i, C m, T, T c, C gp, an Q c represent the concentration of initiator, concentration of monomer, reactor temperature, cooling jacket temperature, concentration of growing polymer, an cooling jacket owrate, respectively. The initiator is azobisisobutyronitrile (ABN) issolve in benzene, while the monomer is styrene an the solvent is benzene. Five ierent sets of numerical values of the process parameters are given in Hialgo an Brosilow (199). k t (5) The four-state polymerization CSTR moel can be extene in orer to calculate the number-average molecular weight (NAMW) M n. Following Schmi an Ray (1981) an Russo (1996), two polymer chain moment equations are appene to the current moel: ktcc gp 2? Qt (6) k pc mc gp? Qt 1 (7) where is the total molar concentration of ea polymer an 1 is the total molar concentration of monomer units present as polymer (Schmi an Ray, 1981). The number-average molecular weight (NAMW) M n is ene as: M n P1 i1 MiNi P1 i1 Ni (8) where M i is the molecular weight of the ith polymer an N i is the number of molecules of type i (Billmeyer, 198). For most polymers the number-average molecular weight (NAMW) lies close to the peak of the molecular weight istribution curve (Billmeyer, 198). The number-average molecular weight (NAMW) can be obtaine from the moment equations by noting that Mn is simply: M n Total mass of polymer 1 MWmono Total moles of polymer (9) where MW mono is the molecular weight of the repeating monomer unit, which in our case is styrene (?CH 2CH(C 6H 5)?) with MW styrene 1:15g/mol. Equations 1 { 5 can be written in the following imensionless form: x x 1 x 2 qix1f? (qi + qm + qs)x1? (x)x1 (1) qmx2f? (qi + qm + qs)x2? pp(x)x2x5 (11) (qi+qm+qs)(xf?x)+pp(x)x2x5?(x?x) x (12) 1[qc(xf? x) + 2(x? x)] (1) x 5 r 2f(x ) x1 (1) t t(x ) where x 1, x 2, x, x, x 5, an are the imensionless initiator concentration, monomer concentration, reactor temperature, cooling jacket temperature, growing polymer concentration, an cooling jacket owrate, respectively. The imensionless variables an parameters are given in Russo (1996). The imensionless moment equations are: x tt(x)x2 5? x 6 (15) x 7 pp(x)x2x5? x7 (16) The number-average molecular weight (NAMW) Mn is: x 7 M n MW mono x 7 x 6 (17) The ratio of x 6 is the imensionless number-average molecular weight (NAMW).

3 5 STEADY-STATE MULTPLCTY ANALYSS 5.1 ntrouction The eect of a number of ierent parameter variations on the multiplicity behavior of the polymerization CSTR is consiere in the following sections. These results are similar in nature to those obtaine previously with the lower orer CSTR moel. Fortunately, the steay-state moeling equations for the four-state polymerization system can be \collapse" into a single nonlinear algebraic equation. The result is: h(x ; ; p) q mx 2f p p(x )(x ) q i + q m + q s + p p(x )(x ) +(x f? x ) + (qi + qm + qs)(xf? x) 2 + qc (18) where (x ) is r 2f (x )q ix 1f (x ) t t(x )(q i + q m + q s + (x )) x5 (19) where x (imensionless reactor temperature) is the controlle output, (imensionless cooling jacket owrate) is the manipulate input, an p is a vector of parameters. 5.2 Determination of Global Multiplicity Diagrams Elementary catastrophe theory allows one to characterize the number of multiple steay-state solutions to equations 1 { 1. The appearance or isappearance of output multiplicities occurs when higher orer erivatives of h with respect to x are @x 2 : : k+1 h 6 k+1 A multiplicity of k + 1 solutions exists aroun the coimension k singular point satisfying equation 2. The more complicate structure of the polymerization CSTR moeling equations make an analytical treatment icult. The highest orer singularity foun numerically correspons to k 2, hence up to three ierent steay-state imensionless reactor temperatures may exist for a given imensionless coolant owrate. The etermination of regions of feasible an infeasible reactor operation are important for feeback control purposes, since one can see irectly if the current operating conitions are \feasible" in a steay-state sense. Therefore, the global parameter space is ivie into regions with ierent types of multiplicity behavior. This current stuy portrays the singularity loci on the - 2-D cross section of the parameter space. is the initiator issociation Damkohler number, while (imensionless heat of reaction) is a irect measure of the exothermicity of the reaction. Since we are intereste in how changes in process parameters inuence the location an existence of these singularities on the steay-state input-output (x - ) curves, the singularity locus is constructe over the bouns on the imensionless coolant owrate @2 2 (21) 2 [; 1) We enote,, an x that satisfy the hysteresis conition (equation 21) as?hyst, hyst, an x?hyst. The singularity locus escribe by equation 21 bouns the region of output multiplicity behavior in the parameter space. A coimension 1 fol bifurcates from the bouns on the cooling jacket owrate. This lower orer singularity separates the global multiplicity behavior in the - parameter @x (22)! 1 (2) Equations 22 an 2 satisfy the \-isjoint" an \1- isjoint" loci, respectively. n general, equations 22 an 2 can be satise for \low" an \high" temperature limit points, as approaches zero or innity, respectively. Hence, there are a total of four curves in the - space that bifurcate from the singularity locus given in equation Global Multiplicity Behavior The coimension 2 (cusp) singularity locus (which speci- es the set of hysteresis points) an coimension 1 (fol) \-isjoint" an \1-isjoint" loci (which specify the set of limit points) ivie the - parameter space into ve ierent regions as shown in Figure 1. These regions correspon to ve ierent steay-state input-output curves, shown in Figures 2-6. Region in the - parameter space correspons to a monotonic relationship between x an, while in region output multiplicity exists between x an (inverse S shape input-output curve). Regions an exhibit low temperature an high temperature infeasibilities, respectively, corresponing to \-isjoint" an \1-isjoint" bifurcations. The presence of \-isjoint" behavior is associate with overcooling of the reactor by the cooling jacket, whereas \1-isjoint" behavior occurs when there is not sucient cooling by the cooling jacket (the heat transfer characteristics of the reactor are poor, as seen in large scale inustrial reactors). Region is an overlap of regions an, where low an high temperature infeasibilities exist when the \-isjoint" an \1-isjoint" loci intersect. The steay-state input-output (T - Q c) curve is shown in Figure 7, where the reactor temperature is shown as a function of the cooling jacket owrate. Figure 8 is a plot of the steay-state number-average molecular weight (NAMW) Mn as a function of the coolant owrate for the low an high temperature branches. Notice that the \high" temperature infeasibility region shown in Figure 7 has resulte in a corresponing infeasibility region of number-average molecular weights. alues of the NAMW from 8 to 12 o not exist uner these operating conitions.

4 Hialgo an Brosilow (199) claime that this reactor coul not be operate safely using only the coolant owrate as the manipulate variable for feeback control, but i not present an explanation. t is clear that the problem is that near the operating range of interest (corresponing to x :75 or T 8 o C) the gain between the reactor temperature an the cooling jacket owrate is approaching zero as one moves into the infeasibility region (cf. Figure 7). Hialgo an Brosilow (199) also place a maximum constraint on the cooling jacket owrate of Q c 6 liters min. Notice from Figure 7 that this constraint virtually eliminates any ability to operate in the reactor range of interest with only the cooling jacket owrate. 6 NFLUENCE OF PROCESS PARAMETERS ON THE MULTPLCTY BEHAOR 6.1 Eect of Reactor Scaleup Let us consier the eect of reactor sizing on the hysteresis locus behavior. n Russo an Bequette (1995) we showe that when the cooling jacket fee temperature was equal to the reactor fee temperature, the Damkohler number along the hysteresis locus ( hyst) is a constant. The imensionless heat of reaction along the hysteresis locus ( hyst) changes by a factor of q+ q (recall that is the imensionless heat transfer coecient an q is the imensionless owrate). hyst(! 1) q + hyst( ) (2) q Hence, in this case we can clearly see how the multiplicity behavior changes as a function of the imensionless heat transfer coecient (). The hysteresis locus length is once again relate to the value of the imensionless heat transfer coecient () an the process owrates. Russo (1996) calculate values of hyst( ) an hyst(! 1) for a number of process owrates an imensionless heat transfer coecients an foun that hyst(! 1) hyst( ) qi + qm + qs + q i + q m + q s (25) which is the same relationship that was seen in the simpler three-state CSTR moel presente in Russo an Bequette (1995). f the resience time is hel constant upon reactor scaleup, then 1 D r, where D r is the iameter of the reactor. Therefore, a \larger" reactor will have a lower value of, resulting in a shorter hysteresis locus. This larger reactor has less ecient cooling characteristics, which can shift the reactor operation into an unesirable region. A reactor volume of liters correspons to a :77 when the resience time is hel constant on scaleup, whereas a reactor volume of 15 liters correspons to :928. As one can see, the larger reactor has a shorter hysteresis locus. t is clear that if we scale own the reactor size \enough" the reactor operation will move from region through region into region, resulting in multiple steay-state behavior. The multiplicity behavior in this case is a result of the \low" temperature operating region becoming feasible as the heat transfer characteristics improve when we \scaleown" the reactor size. 6.2 Eect of Process Flowrates on the Multiplicity Behavior n Russo an Bequette (1995) we saw that when the cooling jacket fee temperature is equal to the reactor fee temperature, then the value of the Damkohler number along the hysteresis locus ( hyst) was linearly proportional to the process owrate (q). Therefore, if the reactor operation is in a region of multiple steay-states, one coul simply ecrease the process owrate such that hyst <, which will move the reactor operation region into region, where the steay-state input/output curve is monotonic. For the purposes of this case stuy, we choose the cooling jacket fee temperature equal to the reactor fee temperature. We want to unerstan how the various process owrates (initiator an monomer) inuence the multiplicity behavior. The Damkohler number along the hysteresis locus (?hyst) is not simply a linear function of the owrate as it was in the three-state CSTR moel. For example, notice the behavior of?hyst in Figure 9. We see that as one varies the initiator owrate (q i) that?hyst initially ecreases an then it levels out an starts increasing (for a xe value of the solvent owrate q s). This behavior is unique to the four-state polymerization CSTR moel, in that there are multiple values of the initiator owrate for the same hysteresis Damkohler number. There exist certain operating regimes in which multiple steay-states can be avoie by either increasing the initiator owrate \enough" (thus ecreasing the resience time of the reactor for a lower conversion), or by ecreasing the initiator owrate \enough" (thus effectively \quenching" the reaction ue to an absence of raicals). n aition, notice on Figure 9 that ecreasing the monomer owrate (q m) tens to ecrease?hyst. Therefore, one can remove multiple steay-state behavior by ecreasing the monomer owrate. This gure allows one to n process operating conitions to guarantee a monotonic steay-state input-output curve over a range of operating conitions. For example, consier a Damkohler number of :15. For monomer owrates of q m : an below (with a xe solvent owrate of q s :8571) we see that there exists a wie range of initiator owrates which guarantee operation with a monotonic steay-state input-output curve. t is well-known that the amount of initiator present in the system generally controls the polymer properties (Hialgo an Brosilow, 199), hence the initiator owrate may only be varie over a narrow range to obtain the esire polymer properties. 6. Eect of Process Operation Changes We want to explore the eect of process operation changes on the bifurcation behavior of the polymerization CSTR moel. The reactor operation falls in region (\high" temperature infeasibility region); the corresponing input-output (T - Q c) curve is shown in Figure 7. Consier a change in the cooling jacket fee temperature for a liter reactor. f we ecrease the cooling jacket fee temperature from x f?1:12 to x f?1:5, the reactor operation shifts from region to region, as shown in Figure 1. Once again, we have multiple steay-

5 states, shown in Figure 11, which necessitates the use of feeback control to stabilize the system. However, we have remove the \high" temperature infeasibility region. Finally, we re-examine Figure 7. Russo (1996) has shown that increasing the cooling jacket fee temperature (from x f?1:12 to x f :) shifts the reactor operation from region to region (monotonic steay-state inputoutput curve). However, we lose the ability to operate the reactor in a \low" temperature operating region. 7 CONCLUSONS n this work the role of physical constraints on the steaystate input-output behavior of a styrene polymerization moel was stuie. n particular, an analysis of the steay-state multiplicity behavior was conucte, an the eect of imensionless values of the heat of reaction, overall heat transfer coecient, cooling jacket fee temperature, an physical limits on the cooling jacket owrate were consiere. The hysteresis Damkohler number (?hyst) was not a linear function of the process owrates, unlike the A! B reaction network case. We saw uner certain conitions that scaling up CSTR volume resulte in a region of high temperature infeasibility which was not present in a CSTR with a smaller volume. Decreasing the cooling jacket fee temperature was also foun to move the steay-state input-output behavior into a globally feasible operating region. ACKNOWLEDGMENT Russo, L.P. (1996). Ph.D. Thesis. Rensselaer Polytechnic nstitute, Troy NY. Russo, L.P. an B.W. Bequette (1995). mpact of Process Design on the Multiplicity Behavior of a Jackete Exothermic CSTR, AChE J., 1, Russo, L.P. an B.W. Bequette (1996). Eect of Process Design on the Open-Loop Behavior of a Jackete Exothermic CSTR, Comp. Chem. Eng., 2, Schmi, A.D. an W.H. Ray (1981). The Dynamic Behavior of Continuous Polymerization Reactors -. sothermal Solution Polymerization in a CSTR. Chem. Eng. Sci., 6, φ 2.x β 1 Hysteresis locus "-isjoint" loci " -isjoint" loci Figure 1: Global multiplicity behavior for the styrene polymerization CSTR moel, x f :, : This work was partially supporte by a grant from the Petroleum Research Founation, aministere by the American Chemical Society. A portion of this work was performe while Wayne Bequette was on sabbatical at Northwestern University. REFERENCES x Billmeyer, F.W. (198). Wiley an Sons. Textbook of Polymer Science., Congaliis, J.P., J.R. Richars, an W.H. Ray (1989). Feeforwar an Feeback Control of a Solution Copolymerization Reactor. AChE J., 5, Hialgo, P.M. an C.B. Brosilow (199). Nonlinear Moel Preictive Control of Styrene Polymerization at Unstable Operating Points. Comp. Chem. Eng., 1, Figure 2: :18, 8. x Kim, K.J., K.Y. Choi, an J.C. Alexaner (1991). Dynamics of a Cascae of Two Continuous Stirre Tank Polymerization Reactors with a Binary nitiator Mixture. Polym. Eng. Sci., 1, -52. Pinto, J.C. an W.H. Ray (1995). The Dynamic Behavior of Continuous Solution Polymerization Reactors -. A Full Bifurcation Analysis of a Lab-scale Copolymerization Reactor. Chem. Eng. Sci., 5, Figure : :1, Ray, W.H. (199). Moeling an Control of Polymerization Reactors. n Dynamics an Control of Chemical Reactors, Distillation Columns, an Batch Processes (DYCORD+ 1992), J.G. Balchen (E.), Pergamon Press,

6 x Figure : :, x1 NAMW (g/mol) Q c (liters/min) Figure 8: Number-average molecular weight (NAMW) versus cooling jacket owrate, :1688, 1:1796, x f?1: x 2 5x1 - φ -hyst q m.2 q m. q m. q m.5 q m Figure 5: :, Figure 9: Damkohler number hysteresis locus as a function of the initiator owrate for various values of the monomer owrate, x f :, q s :8571. q i x 5 2 φ Hysteresis locus "-isjoint" loci " -isjoint" loci Figure 6: :1, Figure 1: Global multiplicity behavior for the styrene polymerization CSTR moel, x f?1:5. 12 β T ( o C) Q c (liters/min) Figure 7: Steay-state reactor temperature versus cooling jacket owrate, :1688, 1:1796, x f?1: T ( o C) Q c (liters/min) Figure 11: Steay-state reactor temperature versus cooling jacket owrate, :1688, 1:1796, x f?1:5. 6 8