HUNGAR~JOURNAL OF INDUS1RIAL CHEMIS1RY VESZPREM Vol. 30. pp. 53 - 59 (2002) PARAMETRIC SENSITIVITY OF FREE RADICAL POLYMERIZATION ASSOCIATED WITH GEL AND GLASS EFFECTS C. PE1RILA and S. CURTBANU (Department of Chemical Engineering, Technical University "Gh. Aso.chi", IASI B-dul D. Mangeron 71A, 6600 IASI, ROMANIA) Received: August 17,2001 The parametric sensitivity is studied of an isothermal homopolymerization system exhibiting the gel and glass effects. As example, the initiated radical polymerization of methyl methacrylate to be achieved in a batch bulk process is considered. For the diffusion controlled reactions, a simple model was proposed containing the dependencies of termination and propagation rate constants with monomer conversion, initiator concentration and temperature. The easiness of handling it and the good agreement between simulation and experimental results are the main features of these models, recommending them for complex engineering studies. The sensitivities of the state variables (initiator concentration, monomer conversion and distribution moments of the chain length) and the model outputs (numerical and gravimetrical average polymerization degrees) with respect to various parameters are computed. It is found that the temperature has the strongest influence in the polymerization process, so it is necessary the stabilization of this parameter with a control loop. The analysis shows that the propagation and initiation activation energies are two of the most important parameters governing the system performance. Almost all the system parameters have the greatest influence at the gel effect moment. Using a calculus based on sensitivity functions, the quantitative estimation of the parameter influence on the system state and output variables are made. Keywords: free radical polymerization, sensitivity analysis, metyl methacrylate Introduction Several constraints are faced in the design of chain polymerization reactors, particularly those exhibiting the gel effect: the large heat of reaction, the low thermal diffusivity and the high viscosity of the reaction mixture. Hence, in the design or operation of systems of this type it becomes important to know, a priori, the limits on values of the various parameters associated with the reaction system, arising because of these constraints. These are referred to as parametric sensitivities boundaries and represent conditions at which different state variables, like conversion or molecular weight become extremely sensitive to small changes in the input parameters. These sensitivity studies also help in identifying parameters which are most critical~ and which must, therefore, be controlled or estimated precisely. Considerable work has been reported on the parametric sensitivity of chemical systems, but less concerning polymerization processes because of the complexity of these systems and the difficulties involved by their models. Studies on parametric sensitivity of polymerization systems started with those of Biesenberger et al. [1]. These prior works on thermal runaway in chain polymerizations and copolymerizations were limited to systems which do not exhibit the gel effect. Baillagou and Soong [2, 3 J realize a parametric sensitivity study for a system exhibiting the gel effect, but the study is somewhat restrictive because it follows an intuitive criteria. Tjahjadi et al. [4] have studied temperature and molecular weight sensitivities ·in plug-flow (or well-mixed batch) homopolymerization reactors in the absence of the gel effect. Their work is mathematically more precise and less intuitive in nature and uses as example the low density JX>lyethylene system. Kapoor, Gupta and Varma (5] developed .a mathematical technique for studying the parametnc sensitivity of reactors including chain polymerization systems exhibiting the gel and glass effects. They used the model of Chiu et al. [6] for the gel and glass effects. This model considers diffusional effects as an integral part of termination and propagation reactions just from the beginning of the polymerization process. The sensitivities of the two temperature maxima with respect Contactinformation: E-mail: scurtean@ch.tuiasi.ro; Silvia"Curteanu, Etemitate str., nr.69a. 6600 IASI. ROMANIA 54 to various parameters are computed. It is found that all the sensitivities of the gel effect induced temperature peak attain their maximum at the same conditions. This sensitivity boundary is associated with high conversions and high molecular weights. The present work is a frrst part of a complex study starting with parametric sensitivity of the isothermal bulk free radical polymerization of methyl methacrylate in well-mixed batch reactors. It must be emphasized that new correlations for diffusion controlled reactions (gel and glass effects) were used. Also, a simple algorithm for sensitivity calculus was used, different of that of the above attempts. To provide practical constraints on reactor design and operation, the qualitative and quantitative estimations of the various parameters influences on the system state and output are made. The quantitative predictions related to polymerization process are very important. Kinetic Model The initiated radical polymerization of methyl methacrylate (MMA) is considered to be achieved in a batch bulk process. For this reaction, the following kinetic diagram is used: Initiation Propagation {l~2R* R'"+M~If * k * pn +M~~+l Chain transfer to monomer * . k * ~ +M~Da+f; Termination by disproportionation • .. k P,. + pm ~Dlt + Dm where I, M and R* represent the initiator. monomer~ and primary radical~ respectively; /!* and Dn are the macroradical and the dead polymer with n monomer units; kcs, ~. kp, kt.. k. are rate constants for initiator decomposition .. initiation. propagation, chain transfer to monomer. and termination by disproportionationy respectively. Based on the kinetic diagram~ one can write the material balance equations for monomer conversion (x), concentration of the initiator (I). and moments of radicals (At) and dead polymer (IJJ (k = 0~1.2). which give the distribution of the chain length: dl . 1-x dt = -ktll-lE I+ £X Ao(kp + k,..) (l) dx - = (A: + k )(1- x) 1 dt , ,. '"0 (2) dp, i.-x 1-x (8) - 2 =k1Ao~-J-i.2AoE--(kP +k,,.)+ktmM0--~ dt l+ex l+ex It is assumed that no monomer is consumed in the initiation process and that the quasi-steady-state approximation for the initiator fragment balance is also valid. The E is a parameter accounting for the volume variation during polymerization and t represents time. · To quantify the gel and glass effects, the following dependencies are proposed: k1 = k10 exp(Ao + ~ · x + ~ · x 2 + A3 • x 3 ) (9) kP = kpo exp(B0 + B1 • x + B2 • x 2 + B3 • x 3 ) (10) where k{o and kpo are the rate constants for termination and propagation reaction~ in ·the absence of gel and glass effects and Ao. Ah A2, A3, B0, Bh B2 and B3 are empirical constants. For the rate constant of the chain transfer to monomer, a similar decrease to that of propagation rate constant was proposed [7] because both reactions involve the same diffusion mechanism - the monomer molecules migrating toward the growing macroradicals. k k =k _L tm tmO k pO (11) The empirical parameters depend on initial concentration of the initiator. In. and temperature (T) and can be determined by minimizing the least square errors between experimental conversion data and model predictions [8]. By automatic processing of numerous experimental data {9~ 12]. the best correlations between empirical coefficients of the models (9) and (10} and parameters T and ~ are obtained. A B b. d 2 10 g , =a+-+c·l0 +-+e·l0 + !·-+-+ T T 2 T T 3 h 13 • 1; . 10 + . o+l·-+J·- T T 2 (12) 55 Table 1 Numerical values of constants in gel and glass. effect models Ao A1 Az A3 Bo B1 B2 B~ a 6671.494 -62659.663 143947.312 -83550.066 5216.517 -5143.078 115683.384 -65355.732 b -109.433 -0.674 2549.132 -3015.263 c -59.575 574.171 -1355.835 809.742 d 0.983 12.599 -87.026 86.865 e 0.177 -1.749 4.226 -2.583 f 0.658 -0.535 -12.439 15.375 g -0.0688 0.588 -1.276 0.754 h -0.000175 0.00177 -0.00436 0.00271 -0.001009 0.0018 0.0140 -0.0188 j -0.000937 -0.0551 0.295 -0.278 corr 0.976 0.981 0.974 0.974 Table 1 contains numerical values for the constants appearing in relations (12). The parameter carr shows the agreement with experimental data. The kinetic model associated with gel and glass effects models were rigorously verified by comparing simulation results with experimental data [ 13]. For the model of MMA polymerization one considers: • the state variable vector z = [I X A.O A.l A.2 J..lO J.!.l J.A2]T = -85.750 -47.281 2399.339 -2798.386 -46.544 471.286 -1092.004 637.698 0.954 13.722 -91.356 91.571 0.138 -1.434 3.409 -2.045 0.514 -0.307 -11.475 14.038 -0.0257 -0.151 1.264 -1.268 -0.000137 0.00145 -0.00353 0.00216 -0.000774 0.00126 0.0134 -0.0174 -0.0022 -0.0338 0.225 -0.226 0.961 0.969 0.974 0.977 i = 1,8 j = 1,11 (17) The matrix S; can be calculated using the parametric sensitivity equation [14]: asz (t) = - - = = _P_= f .. (z(t),p,t)·Spz(t)+ JP dt .. (18) =[zl,z2,z3,z4,z5,z6,z7,z8]T (13) with the initial condition s;(o) = 0. • the model output vector containing numerical and gravimetrical average polymerization degrees y = [DPn, DPw]T = [yl, y2]T (14) • the vector of parameters p = fr,f,£,k~,kt00 ,k;:,k!0 ,Ed,EI'EP,Etmf = [ph pz, P3• P4• Ps• P6• P1• pg, pg, Pto. Pn]T (15) It should be noted that the reaction temperature, T, is an important operating parameter. Even the model describes satisfactory the evolution of MMA polymerization, establishing' of numerical values of kinetic parameters by experimental techniques may be influenced by uncertainties. Therefore, it is necessary to determine the influence of these uncertainties on the model results. The parametric sensitivity calculus The parametric sensitivity matrix of the state, S; , has the following components: szl J!ll szl Pz szl Pu sz = ~z1 j = !! Jlj szz PI sz: Pz szz Pn (16) sz.. Pt sz. 112 sz. Pu whose elements are depicted by: In the Eq.(18), f z and f P are Jacobean matrices of -- - - the vector function f(z(t),p,t) with respect to z(t) and p. The components f1o f2, ... ,f8 of the vectorial function /(z(t),p,t) are the right members of Eqs.( 1)- (8). = The parametric sensitivity matrix of the output, S; ~ has the form: with elements defined by: s Yl (t) A OJ.t (t) Pj = opj {19) k = 1,2 j = l,ll (20) The parametric sensitivity of the ~output is simply obtained by: s;(t) = gz(~(t),p,t))·S;(t) (21) with ;z jacobian matrix of .the output vector. y. with respect to z(t). To compare and order the influence of parameters on system state and output, the dimensionless parametric sensitivity matrices are defined: S:: = L·:. ]a(s:. ..!L] <22) P {;)pi = P1 z,(t) 56 Table 2 Parameters used in MMA polymerization k~ = 1.053 xl015 s"1 (for initiation with AIBN) k!o = 4.917 x1£ .• and :l£;· prove that the increase of F, produces the decrease of the polymerization degrees on the entire time domain. with the greatest decrease at the gel effect moment. 58 SDP. T ' sfPw 5000 4000 3000 1 2000 1000 0 20 ~1000 -2000 Time [min] Fig.S The variation in time of sensitivity functions s~P,. (1). s~P .. (2) Arx 0.16 ...-----------------..., 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0~~~~~~--~~~~~ 0 20 40 60 80 100 Time [min] Fig.6 The conversion variation caused by temperature . increase; solid line - calculus based on sensitivity functions; dotted line - calculus based on model simulations The quantitative estimation of the parameter influence on the system states and outputs With the addition of sensitivity functions there were calculated the deviation of the states z when each "p" parameter changes: (24) Similar for the system outputs ( A.Py = s; · Ap ). The first example of using sensitivity functions involves a small variation of temperature, 0.2% (about 0.7°C. respectively). Fig.6 shows the conversion variation caused by temperature change ( Arx) calculated with sensitivity . functions (solid line): A1 x = s; ·AT and by model simulation (dotted line). The good agreement between the two curves in Fig.6 illustrates the correctness of the sensitivity function Conversion 1~------------------~---------. 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 -l------.-------r-----t-----1 40 45 59 55 60 Time [min] Fig.7Th.e influence of temperature increase on monomer conversion; 1 - initial conversion; 2 - disturbed conversion caused by temperature change values. Consequently, these functions can be used to provide quantitative information about the polymerization system. The maximum value of the curves in Fig.6 (0.15) means that a temperature increase of 0.2% at t = 55 min (gel effect) determines a conversion increase of 0.15. This is clearly emphasized in Fig. 7 where curve noted 1 represents the initial conversion, and curve 2 is the conversion resulted with temperature change. For the simultaneous variation of the three parameters with greatest influence on the states - T, Ed, Ep- with variations of 0.2%, -0.1 %, 0.6%, a conversion increase of 0.4 at the gel effect and a decrease of this state of 0.6 at the end of the reaction take place. For the most sensitive parameters- T, Ed, Ep- very small changes have been considered. Because of strong nonlinear model of the polymerization process in connection with T, Ep Ed, we cannot appreciate quantitatively the influence of thyse parameters if their changes are greater. But, the sensitivity functions offer important qualitative information. The temperature is one of the parameters with significant influence on polymerization degrees - the outputs of the polymerization system under study. An increase of temperatpre with 0.2% determines an initial decrease of polymerization degrees, followed by an important increase at the gel effect moment (approximately with 450 units for DP n and 2700 units for DP w). Greater values of polymerization degrees mean. in fact, earlier appearance of the gel effect due to the temperature increase. After the gel effect, the increase of temperature results the decrease of polymerization degrees with 300 units for DP n and approximately 700 units for DP w· The fo~lowing example involves a variation of 2% for the parameter c. This change of e parameter determines an increase of IXJlymerization degrees of llOO units'" approximately7 for DPn (Figs.8 and 9) and 5800forDP,.. !l.£DPn 1200 ! 1000 800 600 400 200 0 0 20 40 60 80 100 Time [min] Fig~8 The variation of numerical average polymerization degree caused by £ parameter change; solid line - calculus based on sensitivity functions; dotted line - calculus based on model simulations Conclusions This paper is dealing with parametric sensitivity of isothermal batch bulk polymerization of methyl methacrylate. The kinetic model including the mathematical relations for diffusion controlled phenomena describes quite well the polymerization process, continuously on the whole conversion domain and for different reaction conditions. The termination and propagation rate constants dependencies of monomer conversion, initial initiator concentration and temperature are the suitaple form for the sensitivity analysis. The sensitivities of the state variables (initiator concentration, monomer conversion and distribution moments of the chain length) and the model outputs (numerical and gravimetrical polymerization degrees) with respect to various parameters (temperature, initiator efficiency, volume variation parameter, frequency factors, activation energies) are computed. It is found that the temperature has the strongest influence on the state and output of the polymerization system. Therefore, the thermal regime of the reactor have to be carefully controlled. Other two parameters influencing significantly the polymerization process are the propagation and initiation activation energies. Consequently, it is important to have precise estimation of these two parameters for good design and operation of the reactor. Generally, diagrams of parametric sensitivities show a positive maximum or a negative minimum representing an increase or a decrease of the system variables when the considered parameter increases. These values represent the generalized sensitivity boundaries for reactor design or operation and correspond to the gel effect. A calculus based on sensitivity functions are used to made quantitative estimations of the parameter 59 DPn ,5200 2 4700 ------------------------- 4200 3700 3200 2700 2200 40 45 50 55 60 Time [min] Fig.9 The influence of£ parameter change on numerical average polymerization degree; 1 - initial DP n; 2 - disturbed DPn influence on the system state and output variables. 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