Macroscopic Mechanistic Modeling and Optimization of a Self-Initiated High-Temperature Polymerization Reactor

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1 211 American Conol Conference on O'Farrell Seet, San Francisco, CA, USA June 29 - July 1, 211 Macroscopic Mechanistic Moeling an Optimization of a Self-Initiate High-Temperature Polymerization Reactor Thomas Rier, Sriraj Srinivasan, Masou Soroush, George A. Kalfas, Michael C. Gray, an Anrew M. Rappe Absact This paper presents a macroscopic mechanistic mathematical moeling an optimization stuy of a batch polymerization reactor in which self-initiate free-raical polymerization of n-butyl acrylate at 14 an 16 C takes place. The moel is obtaine using a comprehensive free-raical polymerization reaction mechanism. The rate constant of the monomer self-initiation is estimate from monomer conversion measurements. The moel is valiate using a ifferent set of conversion measurements. The valiation results show that the macroscopic mechanistic moel is accurate enough for optimization of the self-initiate polymerization reactor to prouce high quality acrylic resins. The moel is then use to calculate an optimal batch-reactor temperature profile that yiels an en-batch polymer prouct with esire properties (conversion an number-average molecular weight). I. INTRODUCTION Acrylic resins are wiely use in automobile paint an coating formulations as weather-proofing agents an constitute a multi-billion ollar global market that is continuously growing [1], [2]. In the last several ecaes, because of singent environmental regulations in North America, the volatile organic content in paints an coatings has been reuce to less than 3 g/l [3]. Acrylic resins with high molecular weight an high solvent content were replace with low molecular weight, low solvent content resins. Hightemperature (greater than 1 C) free-raical polymerization processes were evelope to prouce these resins in place of conventional low-temperature polymerization processes. Initiators such as organic peroxies an azoniiles have been commonly use to initiate high-temperature polymerization of alkyl acrylates, but their high costs an tenency to prouce coloration when they are left unreacte in the en-prouct have mae them less atactive to use [4], [5]. Spontaneous thermal polymerization of alkyl acrylates was foun to take place in the absence of any known ae external initiators [6]. Such a process coul provie an atactive alternative for the prouction of low molecular weight polymers of alkyl acrylates. Initial stuies [7], [8] T. Rier, S. Srinivasan an M. Soroush are with the Department of Chemical an Biological Engineering, Drexel University, Philaelphia, PA 1914, U.S.A. G.A. Kalfas an M.C. Gray are with DuPont Experimental Station, Wilmington, DE 1988, USA A.M. Rappe is with Makineni Theoretical Laboratories, Department of Chemisy, University of Pennsylvania, Philaelphia, PA, USA Corresponing author: M. Soroush; (phone); (fax); ms1@rexel.eu This work was supporte by the National Science Founation Grant CBET using eleconspray ionization-fourier ansform mass specomey (ESI-FTMS), nuclear magnetic resonance (NMR) specoscopy an etaile analytic macroscopic moeling were unable to reveal the ue nature of the initiating species an the mechanism of initiation. More recently, using quantum chemical calculations [9], [1], it has been shown that monomer self-initiation in alkyl acrylate polymerization is the ominant initiating mechanism. Further evience to the occurrence of the self-initiation mechanism was shown by accurately fitting the compute suctures to maix-assiste laser esorption ionization-time of flight mass specomey [11] an ESI-FTMS specum [12] of poly-alkyl acrylate samples that were generate using the spontaneous thermal polymerization process. Macroscopic kinetic moels are important tools to unerstan complex polymerization processes [12],. The influence of various parameters on the overall kinetics an thermoynamics of the process can be ientifie with the evelopment of reasonably accurate mechanistic moels. The avantage of mechanistic over semi-empirical moels is in their ability to account for the rate constants of iniviual reactions an provie information about the impact of each of these reactions on the overall polymerization kinetics [14]. Also, the kinetics compute using quantum chemical calculations can be use in the mechanistic moels to improve the preiction of variables such as monomer conversion, average molecular weights an chain branches per 1 monomer units [15]. A macroscopic moel can be use to calculate optimal reactor operating conitions, esign moel-base conollers, conuct experimental esign, an esign proucts with improve quality [16]. A macroscopic mechanistic moel can be evelope using several approaches. The black box approach is base upon empirical correlations that are obtaine via well-esigne experiments, but have limite preictive power. On the other han, the etaile analytic approach is capable of escribing the composition of polymer mass at any time instant by using the information about elementary kinetic mechanisms. However, the lack of aequate quantitative knowlege on many iniviual reactions in a polymerization process has limite the preiction capabilities of the etaile analytical moels. A thir moeling metho is that of tenency moeling, in which number of monomer units in polymer chains is not accounte for, an the Schulz- Flory isibution is use to obtain the moments of live an ea polymer chains. While this simplification reuces the complexity involve in solving the moel equations, /11/$ AACC 371

2 the tenency moeling approach provies little information on the iniviual raicals an their chain length uring polymerization [17]. Propagation rate constants in free-raical polymerization of styrene, methyl methacrylate an alkyl acrylates have been measure through pulse laser polymerization-size exclusion chromatography (PLP-SEC) technique [18], [19], [2]. Because this technique is unable to accurately etermine rate constants at high temperatures, alternative methos shoul be use for the estimation of the rate constants in high-temperature polymerization. One of these alternative methos is macroscopic kinetic moeling combine with estimation of the reaction rate constants from measurements obtaine from polymer sample analyses [8]. Previous macroscopic mechanistic moeling stuies of spontaneous thermal polymerization of styrene an methyl methacrylate have provie qualitative evience to the occurrence of selfinitiation [21], [22]. Styrene an methyl methacrylate selfinitiation rate constants have been reporte [22], [8] to have an orer greater than two. Extensive stuies of macroscopic kinetic moeling of high-temperature polymerization of alkyl acrylates, initiate by ae thermal initiators, have reporte rate constants of various reactions [8], [23]. Rantow et al. [8] explore spontaneous polymerization of n-butyl acrylate using a etaile analytic moeling approach an were able to preict the activation energy an rate constants of spontaneous initiation, backbiting an β-scission reactions. However, no funamental unerstaning of the initiation mechanism was obtaine from this stuy. In this paper, a macroscopic mechanistic kinetic moel is evelope for self-initiate polymerization of n-butyl acrylate (nba) in a batch reactor. The rate constant of the selfinitiation reaction is estimate from monomer conversion measurements, an the moel is valiate using a ifferent set of measurements. The moel is then use to calculate an optimal temperature profile for a batch polymerization reactor. The organization of the rest of this paper is as follows. Section II escribes the evelopment of the mathematical moel. Section III iscusses the parameter estimation. Section IV presents the optimization stuy. Finally, concluing remarks are given in Section V. A. Reaction Mechanisms II. MATHEMATICAL MODELING The reaction scheme consiere here for thermally selfinitiate free-raical polymerization of nba involves the following reactions (Table 1): thermal self-initiation, chain propagation, ina-molecular chain ansfer to polymer (backbiting), inter-molecular chain ansfer to polymer, chain ansfer to solvent, an chain ansfer to monomer, β- scission, termination by isproportionation an termination by combination. Here, M an S represent the monomer an solvent, respectively. D n enotes a ea polymer chain with n monomer units but without a terminal ouble bon. U n is a ea polymer chain with n monomer units an a terminal TABLE I REACTION MECHANISMS a. Monomer self-initiation reactions k 3M i R 1 + R2 b. Propagation reactions R n + M k p R n+1 R n + M kt p R n+1 (+LCB) + M kt p +1 (+SCB) + U m k mac c. Backbiting reactions (n > 2) R n+m k bb R n. β scission reactions (n > 3) k β U k β U n 2 + R2 k β U n m + Rm k β U m + m e. Intermolecular chain ansfer to polymer reactions + D k P m Dn + Rm k P + U m Dn + Rm f. Chain ansfer to monomer reactions + M k M Dn + R1 + M k t M Dn + R1 + M k t M Dn + R1 g. Chain ansfer to solvent reactions + S k S Dn + R + S k t S Dn + R + S k t S Dn + R h. Termination by coupling reactions R n + R m R n + R m R n + R m + Rm + Rm k t c k t tc k t tc tc tc + Rm ktc tt i. Termination by isproportionation reactions + Rm k t + k R t t m + Rm kt t + Rm + Rm + Rm t t kt tt 372

3 ouble bon. is a seconary raical with n monomer units. is a tertiary raical with n monomer units forme by the intermolecular chain ansfer to polymer reactions. represents a tertiary raical with n monomer units generate by the backbiting reactions. SCB enotes a short branching point, an LCB a long chain branching point. The self-initiation reaction is consiere to occur via the iraical mechanism of self-initiation reporte in [1]. The overall self-initiation reaction is inclue in the mechanism given above. It follows a secon orer reaction. The seconary raicals forme by the self-initiation reaction react with a monomer, leaing to chain propagation. Tertiary raicals forme via chain ansfer to polymer an backbiting either propagates to form short chain branches or unergo β-scission. A β-scission reaction generates a ea polymer with a terminal ouble bon an a seconary raical. The β- scission reactions atibute to lowering of the polymer chain length an the average molecular weights. B. Rate Equations On the basis of the reaction mechanisms liste in Table 1, assuming elementary reactions one can erive rate expressions for the various species involve. The prouction rates of M, S, R, R 1, R 2, R 3, R, R, R, D an U, respectively, are: r M = 3k i [M] 2 k p [M][R ] k t p[m]([r ] + [R ]) k M [M][R ] k t M [M]([R ] + [R ]) r S = k S [S][R ] k t S [S]([R ] + [R ]) r R = k S [S][R ] + k t S [S]([R ] + [R ]) k p [M][R ] k mac [R ][U].5k t [R ][R ].5k t t([r ] + [R ])[R ] r R 1 = k i [M] 2 + k p [M]([R ] [R 1]) + k M [M]([R ] R 1]) + k t M [M]([R ] + [R ]) k S [S][R 1] k mac [U][R 1].5k t [R ][R 1].5k t t([r ] +[R ])[R 1] k P [R 1]([U] + [D]) r R 2 = k i [M] 2 + k p [M]([R 1] [R 2]) + k β [R ] k mac [U][R 2] k P ([D] + [U])[R 2] (k M [M] + k S [S])[R 2].5k t [R ][R 2].5k t t([r ] + [R ])[R 2] r R 3 = k p [M]([R 2] [R 3]) k mac [U][R 3] k P ([D] +[U])[R 3] (k M [M] + k S [S])[R 3].5k t [R ][R 3].5k t t([r ] + [R ])[R 3] r R = 2k i [M] 2 + k t p[m]([r ] + [R ]) k mac [U][R ] k bb [R ] + 2k β ([R ] + [R ]) k P ([D] +[U])[R ] + (k t M [M] + k t S [S])([R ] +[R ]) k t [R ] 2.5k t t([r ] + [R ])[R ] r R = kp[m][r t ] 2k β [R ] + k P ([D] + [U])[R ] (k t M [M] + k t S [S])[R ].5kt[R t ][R ].5kt tt (2[R ] + [R ])[R ] + k mac [U][R ] r R = kp[m][r t ] + k bb [R ] 2k β [R ] (k t M [M] + k t S [S])[R ].5kt[R t ] [R ].5kt tt ([R ] + 2[R ])[R ] r D = k P [R ][U] + (k M [M] + k S [S])[R ] +(k t M [M] + k t S [S])([R ] + [R ]) +.5k t [R ] 2 +.5kt[R t ]([R ] + [R ]) +.5kt tt ([R ][R ] + [R ] 2 + [R ] 2 ) r U = 2k β ([R ] + [R ]) k P [R ][U] k mac [R ][U] +.5k t [R ] 2 +.5kt t [R ] ([R ] + [R ]) +.5kt tt ([R ][R ] +[R ] 2 + [R ] 2 ) where [z] enotes the molar concenation of species z, R is a seconary raical, R is a tertiary raical forme by the intermolecular chain ansfer to polymer reactions, an R represents a tertiary raical generate by the backbiting reactions. C. Batch Reactor Moel Mole balances for all species involve yiel a mathematical moel of the batch reactor: [M] = r M, [M]() = [M] [S] = r S, [S]() = [S] [R] = r R, [R]() = [R1] = r R 1, [R1]() = [R2] = r R 2, [R2]() = [R3] = r R 3, [R3]() = [R ] = r R, [R ]() = [R ] = r R, [R ]() = [R ] = r R, [R ]() = [D] = r D, [D]() = [U] = r U, [U]() = where [M] an [S] are the initial concenations of M an S, respectively. The number-average molecular weight of the ea polymer chains, M n, an the monomer conversion,, are then obtaine from: M n = [M] [M] M m [D] + [U] = 1 [M] [M] 373

4 TABLE II REACTION RATE CONSTANT VALUES. Parameter Frequency Factor Activation Ref. Energy kj.mol 1 k p L.mol 1.s [28] kp t L.mol 1.s [29] k bb s [3] k M L.mol 1.s [12] k t L.mol 1.s [31] kt tt L.mol 1.s k β s k P L.mol 1.s [32] C S Measurements Moel Preiction Time (min) Fig. 1. Moel preiction fitte to the measurement set 1 at 14 C TABLE III OTHER KINETIC PARAMETER VALUES. Parameter Dimensionless Value δ s.1 δ st.7 δ t.9 γ III. PARAMETER ESTIMATION Parameter estimation was carrie out using least-squareerror minimization an measurements of monomer conversion from isothermal spontaneous thermal polymerization of nba (4 wt% nba an 6 wt% xylene) at 14 an 16 C [8]. The minimization was conucte using the optimization routine fminsearch from the Optimization Toolbox of MATLAB R21b. Except for the frequency factor an activation energy of the self-initiation reaction that were estimate in this stuy, the other reaction kinetic parameter values were taken from the literature (Table 2). Two ifference sets of conversion measurements (set 1 an set 2) were use in this stuy. Each set inclue gravimeic measurements of monomer conversion from isothermal batch operations at 14 an 16 C. First, the self-initiation reaction rate constant at 14 C was estimate from set-1 conversion measurements obtaine at 14 C. Secon, the self-initiation reaction rate constant at 16 C was estimate from set-1 conversion measurements from batch operations at 16 C. Thir, using the Arrhenius plot, the frequency factor an activation energy of the selfinitiation reaction were calculate: Activation Energy = kj.mol 1 Frequency Factor = L.mol 1.s 1 The quality of the moel fitting the conversion ata set 1 can be seen in Figures 1 an 2. The secon set of conversion ata (set 2) at 14 an 16 C was then use for the moel valiation. The valiation results are shown in Figures 3 an 4; as can be seen, the moel preicts the conversion reasonably well. IV. BATCH REACTOR OPTIMIZATION Using the reactor moel presente in Section IIIB, we calculate an optimal temperature profile, T (t), for a batch.2 Measurements Moel Preiction Fig. 2. Moel preiction fitte to the measurement set 1 at 16 C Measurements Moel Preiction Fig. 3. Moel valiation: moel preiction an measurement set 2 at 14 C Measurements Moel Preiction Fig. 4. Moel valiation: moel preiction an measurement set 2 at 16 C 374

5 TABLE IV REACTION RATE CONSTANT DEFINITIONS/RELATIONS. k t = k tc + k t kt t = kt t c + kt t kt tt = kt tt c + kt tt k t = δ s k t kt tt = δ t kt tt kt t = δ st k t kt tt k tc = (1 δ s )k t kt tt c = (1 δ t )k t tt kt t c = (1 δ st ) k t kt tt k S = C S k p k t M = kt p k p k M k t S = kt p k S k p k mac = γk p T (K) Time(min) Fig. 5. Initial guess for the reactor temperature profile 49 nba polymerization reactor with a batch cycle time of 7 hours (incluing the heating-up an cooling-own perios). The batch reactor temperature shoul be K at the beginning an the en of each batch cycle; T () = K an T (7 36) = K. The reactor is initially loae with monomer (4 wt.%) an solvent (6 wt.%). The optimal temperature profile shoul minimize the performance inex: [ [1 (t f )] M ] 2 n(t f ) 5 where t f = 7 36 s. In other wors, the optimal temperature profile shoul yiel a polymer prouct with a monomer conversion an a number-average molecular weight as close as possible to one an 5, kg.kmol 1, respectively. To simplify the optimization problem, we parameterize the reactor temperature profile using a fourth-orer polynomial in time; that is, where T (t) = at 4 + bt 3 + ct e e = K, = (at 3 f + bt 2 f + ct f ) K.s 1 This parameterization of the optimal temperature profile reuce the search for an optimal profile to a search for three constants, a, b an c. The optimization routine fminsearch from the Optimization Toolbox of MATLAB R21b with the initial guesses: a = b = c = was use to calculate the optimal values of a, b an c. These initial guesses correspon to the initial-guess temperature profile shown in Figure T (K) Fig. 6. Time(min) Optimal reactor temperature profile The optimization routine calculate the optimal values: a = b = c = which correspon to the optimal temperature profile shown in Figure 6. Enforcing the optimal temperature profile to the reactor leas to the prouction of a polymer with (t f ) =.814 an M n (t f ) = 4918 kg.kmol 1. The optimization results shown herein were for a batch reactor; they were presente to just show an application of the moel. A realistic case is that of a semi-batch reactor in which the monomer is ae to the reactor over time such that the mass fraction of free (unreacte) monomer in the reactor never excees 1%. This upper limit on the monomer mass fraction is to ensure the reactor safety. In this case, the entire solvent is loae initially, an the optimization variables are the reactor temperature an the monomer fee flow rate. V. CONCLUSIONS A macroscopic mechanistic mathematical moeling an optimization stuy of a batch polymerization reactor in which self-initiate free-raical polymerization of n-butyl acrylate at 14 an 16 C takes place, was presente. 375

6 The moel was obtaine using a comprehensive free-raical polymerization reaction mechanism. The rate constant of the monomer self-initiation was estimate from monomer conversion measurements. The moel valiation results showe that the macroscopic mechanistic moel is accurate enough for optimizing the self-initiate polymerization process to prouce high quality acrylic resins. On the basis of the moel, optimization was conucte, an an optimal batchreactor temperature profile that yiels an en-batch polymer prouct with esire properties (conversion an numberaverage molecular weight) was calculate. Other applications of the moel inclue moel-base conoller esign, reactor scaleup an safety analysis. VI. ACKNOWLEDGEMENT This material is base upon work supporte by the National Science Founation uner Grant No. CBET The authors thank Mona Bavarian for her invaluable help with the simulations, an gratefully acknowlege reviewers comments. 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