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Ind. Eng. Chem. Res. 1993,32, 882-893
Optimal Design and Operation of Batch Reactors. 2. A Case Study Masoud Soroush and Costas Kravaris' Department of Chemical Engineering, The University of Michigan, Ann Arbor, Michigan 48109-2136
In this work, the optimal design and operation methodology developed in part 1of this study will be illustrated in a batch polymerization reactor system (polymerization of methyl methacrylate (MMA) initiated by azobis(isobutyronitri1e) (AIBN) in toluene). Following the steps of the methodology, first a mathematical model of the process (which accounts for gel effect and volume shrinkage) is developed and then the performance indices and mode of operation are selected. The optimal loading conditions and reactor temperature profile (which minimize the polydispersity of the final polymer product and maximize the monomer conversion constrained to a desired value of the weight-average molecular weight at the end of the batch time) are computed numerically. After designing the heating/cooling equipment, its flexibility, controllability, and safety, as well as the feasibility of the computed optimal operating conditions, are investigated. A nonlinear temperature controller is finally synthesized to enforce the reactor startup conditions and the optimal temperature profile.
Introduction The use of polymers in our society has always had a growing trend. The production rate of polymeric materials is well over 100 billion pounds per year, and this figure is expected to increase significantly in the future as higher strength plastics and composite materials replace metals in automobiles, aircraft, and other products. At this scale of production, even relatively minor improvements in the design and operation of polymerizationprocesses are very significant. Many polymeric products are low-volume speciality materials, frequently copolymers, designed to perform a specified function. Because of these features, the most prevalent mode of polymerization is in batch reactors, which allow for great flexibility. Capital and/or operating costs are not the principal design objective for these reactors. Instead the need is for production of a complex product with desirable properties. Therefore, apart from a safe and economic operation, the main objective of polymerization reactor design and operation is to obtain acceptable product quality. Quality is normally related to molecular and/or macroscopic architecture of the product and is mainly determined at the synthesis stage. Unlike other simpler products, such architecture cannot normally be modified in postpolymerization finishing. The importance of the efficient design of polymerization reactors is well-recognized in polymer engineering (e.g., MacGregor, 1986;Malone and McKenna, 1990,Tirrell, et al., 1987). Two key factors that should be considered in the selection of a design candidate are (1) the product quality as reflected in the polymer molecular weight and composition distribution (Malone and McKenna, 1990) and (2)the stability and control of the polymerization reactor (MacGregor, 1986;Pollock et al., 1982). Pollock et al. (1982)have reported a case in which control could not eliminate the sustained oscillation in continuous emulsion polymerization reactor trains. However, a small change in the design eliminated the oscillation. Free-radical polymerization reactors are recognized as highly complex processes (Ray, 1986;Bailagou and Soong, 1985a). Specifically, the continuous stirred tank polymerization reactors may exhibit multiple steady states, parametric sensitivity, and limit cycles; as a result the models representing these processes are highly nonlinear
* To whom correspondence should be addressed.
0
1
2
3
rm,hr
4
5
Figure 1. Profiles of the rate of heat generation by the polymerization reactions for different initial monomer volume fractions.
(Schmidt and Ray, 1981;Hamer et al., 1981;Schmidt et al., 1981). A major source of the nonlinearity is the autocatalytic nature of the polymerization reactions (the gel effect). The gel effect frequently causes uncontrollable reactions, resulting in excessive temperature rise, rapid conversion, and plugging of equipment. Moreover, in contrast to classical reactions (as in reactions of the type A B), the maximum rate of heat production by the polymerization reactions does not occur at the beginning of a batch cycle (when the concentrations of the reactants are at the highest level). Indeed, the rate of heat production by the free-radical polymerization reactions becomes maximum some time during the period of autoacceleration. The profiles of the rate of heat production by methyl methacrylate solution polymerization reactions (under isothermal condition, T = 50 "C, and Ci(0) = 0.05 kgmol m-3) for different initial monomer volume fractions are depicted in Figure 1. In this case study, the proposed design and operation methodology developed in part 1 is used to obtain optimal operating conditions of a pilot size polymerization reactor and to design the reactor including its heating/cooling (HI C) and/or feeding systems. The designed reactor was built at the Chemical Process Control Laboratory of The University of Michigan and was used for experimental control studies (see Soroush and Kravaris, 1992).
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0888-5885/93/2632-0882$04.00/00 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, No. 5, 1993 883 Table I. Free Radical Reaction Mechanism a. initiation reactions ki
1-21 ""
1+M-P, b. propagation reactions hi1
Pn + M
P n tI c. termination by coupling reactions -+
kc
0 to find the value of ~j such that Uj(t) A"ui(t) minimizes 31,which is easier to use and converges much faster. Here the parameter ~j is chosen based on Kirk's approach. V. Compute a new optimizing varible history from
+
subject to i = 31(Z,%)
303 I '2f I 373
(optimizing variable constraint)
-zi(t) I 0, i = 1,...,4
g,(z(O),O) =
(inner system)
(state inequality constraints)
1:; 1 z4(o)
=0
(initial constraints)
df(z(tf),tf)= [t - tfl = 0 (terminal constraint) The following numerical algorithm was used for computing the solution of the optimization problem of eq A.l. Numerical Algorithm. It involvesthe followingsteps: (a) Make an initial guess wl for the concentration of monomer at t = 0. (b)Set ~ ~ (=0wl, ) ~ ~ (=0 (e,) - emol)/ei,Z 3 ( 0 ) = Zq(0) = 0.
I. Make an initial guess W ( t )for the optimizing varible history (e.g., a constant W). 11. Integrate the system of equations i = '&(z(t),"Uj(t)) forward with the specified initial conditions z(0) and the optimizing varible history from step I. Record the solution as zJ(t). 111. Set Aj(tf), using the zj(tf) obtained in step 11, according to
zc'+'(t) = 'UJ(t)+ AWj(t) VI. Repeat the computations starting at step 11. Continue until IIA"UJ(t)llmC 1.0 X 1k2. (c) Calculate + ( w ~A - Ag(O)B,/Bi. (d)Make another initial guess wl+1 for the concentration of monomer at t = 0. (e) Set zl(0) = w+1, z d 0 ) = (0, - & w + i ) / B i , zd0) = zd0) = 0. (0 Perform steps I-VI. (g) Calculate +(w+d h ( 0 )- A2(O)B,JJBi. +(wl) and (w1+1) should not have the same signs, otherwise repeat steps d-g. (h) Set w1+2 = (wl + w1+1)/2, zl(0) = ~ + 2 z, d 0 ) = (8, Bmw1+2)/6i,z3(0) = z d 0 ) = 0, perform steps I-VI, and then calculate $ ( w ~ + ~9) Al(0)- ~~(o)e,/e~. (i) If I+(w1+2)15 1.0 X 10-5then stop, otherwise go to step
+-
j.
6)If +(wd and +(01+2) have same signs then set 01= W+Z, otherwise wl+1 = w1+2. Go to step h. Literature Cited Baillagou, P. E.; Soong, D. S. Major Factors Contributing to the Nonlinear Kinetics of Free-Radical Polymerization. Chem. Eng. Sci. 1985a, 40, 75-87.
Ind. Eng. Chem. Res., Vol. 32, No. 5,1993 893 Baillagou, P. E.; Soong, D. S. Molecular Weight Distribution of Products of Free Radical Nonisothermal Polymerization with Gel Effect. Simulation for Polymerization of Poly(methy1methacrylate). Chem. Eng. Sci. 198Sb, 40, 87-104. Brandrup, J.; Immergut, E. H. Polymer Handbook, 3rd ed.; Wiley: New York, 1989; pp 77-86. Bryson, A. E.; Ho, Yu-Chi. Applied Optimal Control; Hemisphere: New York, 1975; pp 216-232. Chiu, W. Y.; Carratt, G. M.; Soong, D. S. A Computer Model for the Gel Effect in Free-RadicalPolymerization. MacromolecuZes 1983, 16, 348-357. Cuthrell,J. E.; Biegler, L. T. SimultaneousOptimizationand Solution Methods for Batch Reactor Control Profies. Comput. Chem.Eng. 1989, 13, 49-62. Hamer, J. W.; Akramov, T. A.; Ray, W. H. The Dynamic Behavior of Continuous PolvmerizationReactors-11. Chem.E n-s Sci. 1981, 36,1879-1914. Hicks, J.; Mohan, A,; Ray, W. H. The Optimal Control of Polymerization Reactors. Can. J. Chem. E m . 1969,47, 590-597. Jutan, A.; Uppal, A. Combined Feedforward-FeedbackServoControl Scheme for an Exothermic Batch Reactor. Ind. Eng. Chem. Process Des. Res. 1984,23, 597-609. Keller, H. B. Numerical Methods for Two-point Boundary Value Problems; Blaisdelk Waltham, 1968; pp 54-73. Kirk, D. E. Optimum Control Theory: An Introduction, 2nd ed.; Prentice-Hall: Englewood Cliffs, NJ, 1970; pp 329-342. Liptak, B. G. Controllingand OptimizingChemical Reactors. Chem. Eng. 1986, May 26,6941. MacGregor, J. F. Control of Polymerization Reactors. h o c . IFAC PRP-6 Automation 1986,31-34. McGearvy, C. On-Line Computer Control System for Chemical Reaction Processes. Comput. Chem. Eng. 1983, 7,529-566. Malone, M. F.; McKenna, T. F. Process design for Polymer Production. In Foundations of Computer-Aided Process Design; Siirola, J. J., Grossmann, I. E., Stephanopoulos, G., Eds.; CACHE Elsevier: New York, 1990; pp 469-483. Marten, T. J.; Hamielec,A. E. High Conversion Diffusion-Controlled Polymerization. ACS Symp. Ser. 1979, 104, 43-70.
Morari, M.; Stephanopoulos, G. Studies in the Synthesis of Control Structures for Chemical Processes. Part I1 Structural Aspect and the Synthesisof AltemativeFeasibleControlSchemes. AZChE J. 1980,26,232-246. Nunes, R.W.; Martin, J. R.;Johnson, J. F. Influences of Molecular Weight and Molecular Weight Distribution on Mechanical P r o p erties of Polymers. Polym. Eng. Sci. 1982,4, 206-228. Pollock, M. J.; MacGregor,J. F.; Hamielec, A. E. ContinuousPolyvinyl Acetate Emulsion Polymerization Reactors: Dynamic Modeling of MW and Particle SizeDevelopmentand Application to Reactor System Design. ACS Symp. Ser. 1982, 197-209. Ray, W. H. On the Mathematical Modeling of Polymerization Reactors. J. Macromol. Sci.-Rev. Macromol. Chem. 1972, C8 (11, 1-56. Ray, W. H. Polymerization Reactor Control. IEEE Control Syst. Mag. 1986, 6 (4), 3-8. Sage, P. S.; White 111, C. C. Optimum Systems Control, 2nd ed.; Prentice-Hall: Englewood Cliffs, NJ, 1977; pp 300-315. Schmidt, A. D.; Clinch, A. B.; Ray, W. H. The Dynamic Behavior of Continuous Polymerization Reactors-111. Chem. Eng. Sci. 1981, 39,419-432. Schmidt, A. D.; Ray, W. H. The Dynamic Behavior of Continous Polymerization Reactors-I. Chem. Eng. Sci. 1981, 36, 14011410. Soroush,M.; Kravarie, C. Nonlinear Controlof a Batch Polymerization Reactor: An Experimental Study. AIChE J. 1992,38,1429-1448. Thomas, I. M.; Kiparissides, C. Computation of the Near-Optimal Temperature and Initiator Policies for a Batch Polymerization Reactor. Can. J. Chem. Eng. 1984,62,284-291. Tirrell, M.; Galvan, R.;Laurence, R. L. Polymerization Reactors. In Chemical Reaction and Reactor Engineering; Carberry, J. J., Varma, A., Eds.; Marcel Dekker: Basel, New York, 1987;pp 735770. Received for review June 5, 1992 Revised manuscript received January 12, 1993 Accepted January 25, 1993
