Free radical polymerizations associated with the trommsdorff effect under semibatch reactor conditions. I: Modeling

Free Radical Polymerizations Associated With the Trommsdorff Effect Under Semibatch Reactor Conditions. I: Modeling ASIT B. RAY, D. N. SARAF, and SANTOSH K. GUFTA* Department of Chemical Engineering Indian Institute of Technology Kanpur-208 01 6, India Several important polymerizations [e.g., poly(methy1 methacrylate) (PMMA), polystyrene] exhibit gel and glass effects during polymerization. These are associ- ated with the decrease of the diffusivities of the macroradicals and monomer molecules with increasing viscosities of the reaction mass. A new model has been developed to account for the gel and glass effects, both under batch and semibatch reactor operations. The free volume theory of Vrentas and Duda has been used to account for the changes in the diffusion coefficients. The model parameters are tuned using experimental data on the isothermal bulk and solution polymeriza- tions of MMA in batch reactors. Conversion histories and molecular weights have been obtained for cases where solvent, initiator or monomer are added to (or removed from) the reaction mass after some reaction has taken place. In addition, studies involving step changes in temperature have been carried out. It is found that the conversion us. time behavior after such step changes depend upon a complex interplay of three factors: polymer concentration, molecular weight, and the temperature at the point these changes are effected. INTRODUCTION temperature, T , and the reaction mixture becomes n recent years, polymer reaction engineers have I started focusing their attention on the mathemati- cal modeling of various physical phenomena in in- dustrial polymerization reactors. One such phe- nomenon, diffusion-controlled reactions under semi- batch reactor conditions, plays an important role in free radical chain growth polymerizations and is the focus of the present study. The three most important reaction rate constants, k,, k,, and k, (= k,,+ k td) (see Table I), in free radical polymerization are associated with diffusional limitations (1-7). The gel or Trommsdorff effect repre- sents the effect of decreasing diffusivity of macroradi- cals, P, (due to increasing viscosity of the reaction mass) on the termination rate constants, k,, and ktd and was first observed by Norrish and Smith (6), and Tromsdorff et aL (7). It is exhibited (8, 9) as a sudden increase in the rate of polymerization and the weight average chain length, pw. In the later stages of reac- tion, the glass transition temperature, Tg. of the reac- tion mass becomes higher than the polymerization ‘To whom correspondence should be addressed, glassy in nature. Then the propagation rate constant, k,, also becomes diffusion controlled. As a conse- quence, the reaction stops short of complete monomer conversion. This effect is referred to as the glass effect. In addition to these, the initiator efficiency, f, decreases with time at high monomer conversions. This is referred to as the cage effect. We have as- sumed f to be constant, following several workers ( 1-51. to keep the model simple. Several workers have attempted to model the gel and glass effects using semi-empirical, time varying rate constants for the termination and propagation reactions. These have been reviewed by O’Driscoll (10) and Hamielec (1 1). Most of these models involve discontinuities in either the monomer conversion or the chain length, or both, at which the diffusional restrictions become operative. This artificiality causes problems in the use of these models for control and optimization work. Chiu et aL (21, in a major departure from the past, considered diffusional limitation as an integral part of the termination and propagation reactions from the very beginning of polymerization. They used the free volume theory of Fujita and Doolittle [as discussed by 1290 POLYMER ENGlNEERlNG AND SCIENCE, AUGUST 1995, Vol. 35, No. 16 Free Radical Polymerizations Associated with the Trornmsdorff Efect. I Table 1. Kinetic Scheme for Polymerization of MMA. Initiation Propagation kfC Termination by combination Pn + Pm + Dn + rn Termination by disproportionation Pn+Pm --f Dn+Dm kf Chain transfer to monomer Pn+M + P, +On k s Chain transfer to monomer Pn+S-+S‘+D, S .+ M ‘ZtS + PI or ks P,+M + D,+P, via solvent Bueche (1211 for the diffusivity. Recently, Achilias and Kiparissides (13, 14). developed the model of Chiu et aL further. In their theory, there was only one curve-fit parameter, the remaining being obtained using independent measurements on nonreacting systems. Their theory used the diffusion theory of Vrentas and Duda (15-17), and the theory of excess chain-end mobility (18). This class of theories has been used in several studies having an engineering bias. Even though Louie and Soong (19, 20) used the model of Chiu et aL (2) for the simulation and opti- mization of semibatch reactors, such an extension to semibatch operation is questionable. This arises from the use of the initial initiator concentration, [I],, in the model to account for the molecular weight depen- dence of the diffusivity of polymer radicals. I t is un- clear as to what should be used in place of [ I], in the model after the addition of initiator, monomer or solvent to the reaction mass. This is an important question, since such additions are made routinely in industrial reactor for optimal performance. A similar controversy arises in the application of the theory of Achilias and Kiparissides (13, 14) to semibatch reac- tors. Here, an empirical curve-fit parameter, jC,,, is correlated to the initial number average chain length, pn,o. This study attempts to resolve this important question using a new model. The model parameters are then obtained (“tuned’) using curve-fit of experi- mental data on MMA polymerization in isothermal batch reactors (in the absence of monomer, solvent or initiator additions). The model is then used to predict how the monomer conversion and molecular weights would change after intermediate addition of pure components to the reaction mass. The behavior of the system under step changes in temperature is also predicted. These predictions need to be con- firmed experimentally. However, even without this, one can use this model for model-based optimal con- trol with adaptation, provided we “tune” the parame ters periodically on-line. Adaptive optimal control (using intuitively meaningful models) is being prac- ticed in the chemical and petrochemical industry, and can be used in the polymerization industxy as well. FORMULATION The kinetic scheme of free radical polymerization of methyl methacrylate is summarized in Table I (see Nomenclature for definitions). This mechanism de- scribes several other free radical chain polymeriza- tion systems as well. The mass balance equations for a well-mixed semibatch reactor in which components can be added or vaporization can take place, can easily be written. These can be summed up to give the equations for the moments of the radical (A,, A, , A, 1 and dead ( p,, p p2 macromolecular species. The complete set of equations [ordinary dif- ferential equations (ODES)] for a fairly general set of reactor operating conditions are presented in Table 2. The equations are written in terms of total moles of the various components, since it is easier to work with these for semibatch reactors. The volume occu- pied by the initiator, I, is neglected. The volume, V,, of the liquid reaction mixture is expressed as a sum of the volumes of the three pure components: monomer, polymer, and solvent. It is assumed that there is no polymer at the beginning. The equations in Table 2 account for the addition of (liquid) initia- tor, monomer, and solvent, as well as for the vapor- ization of monomer and solvent through the terms Rli(t), RLm( t ) , R J t), and Rum( t) , Rus( t ) respectively. The pressure in the vapor space (of volume V’) above the liquid reaction mass can change with time due to vaporization, as well as due to release of vapor at some desired rate, V,C t). This necessitates the intro- duction of equations for three additional variables, M U , S u , and NU. The monomer conversion, x, in such reactors can be written in terms of the total monomer ( JmI) added till time t as (1 ) The most important part of the model are the equa- tions for the gel and glass effects. The model of Achil- ias and Kiparissides (13) was modified so as to apply for semibatch reactors. The rate constant, k, (= k,,, since k,, = 0 for PMMA) has been written (13) using the free volume theory of Vrentas and Duda ( 15- 17) as where i,b is given in Table 3 ( E q c). The same notation has been used as in Ref. 13. We now use the proce- dure of Chiu et aL (2) to simplify Eq 2. We define a reference (subscript: ref) state as Cp, = 4, = 0, = 1. POLYMER ENGINEERING AND SCIENCE, AUGUST 1995, Vol. 35, No. 16 1291 Asit B. Ray, D. N . Saraf. and Santosh K. Gupta Table 3. Gel and Glass Effect Equations. Table 2. Model Equations for MMA Polymerization in Semibatch Reactors. dM AoM RM A 0 dt VI ' VI Vl 2. - = - ( k , + k,) __ - k - - k,S - dS dt 4. - = Rl,(t) - R,,(t) dA, RM A A 0 4 6. -=k,-+k M - - - k - dt Vl V, V, 7. dS " S V 15. - = R - dt "' V J ( M v + S v + N Y ) dN N" 16. -= - V J dt ( M v + S " + N Y ) X 17. T = T r e f + ___ 18. F l = S(MWs)+.$,(MW,) FiCFmrx S ( M W ) M(MWm) ( C m - M ) ( M W m ) 19. v , = L + - - - - + Ps Pm PD dP M v + S Y + N Y dT RT dt Vg 20. -= R- + - (Rvm + R,, - V,) dt vg Pm Ps PP 24. 6,= 1 - 6, - 6s Equation 2 can be rewritten in terms of this as where, I J J ~ ~ ~ = y/VJp. The parameter, Bt. defined in Eq 3 is somewhat similar to 8,,c,s [ = r,$'3Dref] used by Chiu, Carratt, and Soong (CCS) (2). The term, r:, in Eq 3 is expected to be insensitive (2) to the monomer conversion, x, temperature, T, and the number aver- age chain length, pn, of the polymer present. The term, Dref, is a function of T and pn, r e f , and, accord- ing to the free volume theory of Vrentas and Duda ( 15- 17). is inversely proportional to p:. r e f . Thus, the term D,eJp?,reJ, is expected to be a function of T only. 8, in Eq 3 is, therefore, expected to be a func- tion of T alone, as indicated in Eq 3. It may be empha- sized that in the earlier theory of Chiu et al, the term Ot,ccs was a function of T and was proportional to [ I]; while in the present model 8, is a function of T alone-the earlier dependence on[ I]; having been replaced by the term pz, the square of the current value of the number average chain length. Use of p,, in the expression for 8, enables modeling of semi- batch reactor operation. In a similar manner (13). we can write an expres sion for k,. This is given as Eq b in Table 3, with OP(T) = f 2 / 3 D $ . fm and DreJ are associated with the diffusion-cum-reaction of macroradicals with monomer. The complete set of equations for the gel and glass effects is given in Table 3. I t may be added that a similar expression could also be written for the diffusional effects on the initia- tor efficiency, f. This would introduce some addi- 1292 POLYMER ENGINEERING AND SCIENCE, AUGUST 1995, Vol. 35, No. 16 tional tuning parameter in our model, which, we believe, are not necessary at this stage. What could possibly be achieved by incorporating such an equa- tion accounting for the decrease in f at high conver- sions, can be alternatively accomplished by the equa- tions for k, and k, alone with slightly different values of the parameters Q t and 0,. Table 4 compiles (4, 14, 21-29) the values of the various parameters and the several correlations used. Numerical integration of the equations in Tables 2 and 3 is done using the double-precision version of Gear’s algorithm (NAG library routine D02EJF). A tolerance of is used. Decrease of tolerance did not affect the results much. The initial conditions used are given by ( ,y, M ”, S’, N U not used): at t = O : R,,(t) and R,,(t) were taken to be zero in the pre- sent simulations. Also, the energy equation was not used. The continuity conditions accounting for sud- den additions of (liquid) monomer, solvent, and initia- tor (at the temperature of operation) in a semibatch reactor can easily be written and are of the form y(tkf) = y(t;) + y,, where y, moles of pure y are added at time t,. The initiator efficiency, gel, and glass effect parame ters, f, Q , , and 0,. can be obtained by curve-fitting (tuning) available experimental data (8, 9) on MMA polymerization under isothermal conditions in a batch reactor. The Box complex technique (30) is used. We minimize the normalized sum of square errors Free Radical Polymerizations Associated with the TFornmsdorff Effect. I POLYMER ENGlNEERlNG AND SCIENCE, AUGUST 1995, Vol. 35, No. 76 1293 where “exp” and “th” indicate experimental and theo- retical values, respectively, and N, and NMn are the number of data points available for x and M,. Alter- natively, we could have used conversion and the weight average molecular weight, M,, in our error function, E, to obtain good fits with M, data while overpredicting M,. In fact, no simultaneous good fit for x, M,,, and M , histories can be obtained using a single k,. The computer code incorporating the Box complex procedure took a CPU time of 74 s on a supermini HP 9000/850S computer to converge to Table 4. Parameters Used for Polymerization of MMA (4, 14,21-29). prn = 966.5 - 1.1 (T - 273.1) kg/m3 pp = 1200 kg/rn3 pQ = 844.1 8 - 1.07165(T - 323.1 ) kg/rn3 (Ref. 23) (Ref. 2) (Ref. 22) (Benzene) C ,=CP,,=1676J/kg-K 8’ =2017.17 (benzene at 65°C) J/kg-K kg0,014.917 x 10’ m3/mol-s k,,,,-9.8 x l o 4 m3/mol-s %i = 1.69 x 1014 s- ’; for BPO k: = 1.053 x 1 015 s ~ ’ ; for AlBN ktc = 0.0 k, = 0.0 ki = kp k, = 0.0 E d = 125.40 kJ/rnol; for BPO Ed = 128.45 kJ/rnol; for AlBN E, = 18.22 kJ/rnol E,, =2.937 kJ/mol (MW,) =0.10013 kg/rnol (MW,) =0.07811 kg/mol (Ref. 23) (Ref. 24) (Ref. 25) (Ref. 21) (Ref. 21) (Ref. 21) (Ref. 25) (Ref. 21) (Ref. 21) (Ref. 21) Constitutive Parameters for the Gel and Glass Effects qz =8.22 x m3/kg (Ref. 26) v: =7.70 x m3/kg (Ref. 27) v: =9.01 x m3/kg (Benzene) (Ref. 28) M,p =0.18781 kg/rnol (Ref. 28) y= 1 (Ref. 14) V,, = 0.149 + 2.9 X 10 - 4[T(K) - 273.1 1 (Ref. 29) V,,=O.O194+1.3x 10-4[T(K) -273.1 - 1051; (Ref. 29) for T < (105+273.1) K V,,=0.025+1.0 x 10-3[T(K) - 171.11 (Benzene) (Ref. 4) Best-Fit Correlations (BFCs) From This Study: f = 0.52997; for AlBN (bulk polymerization) f = - 0.1 78974ff + 0.5414; for BPO-Benzene log,,[O,(T), S] = 1.2365506976057 X 10’ - 1.02779282951 74 X 1 O 5 (1 / T ) +2.2665992726301 X l o 6 (1 / T ’ ) log,,[O,(T), s] =1.58323 x 10’- 1.24953 x l o 5 (1/T) +2.5596 X l o 7 (1 / T ‘ ) + Ps Prn optimal values of the three parameters for one of the experimental runs, in 30 iterations. RESULTS AND DISCUSSION The values of Q t . 0,. and f , which minimize E are computed for each of the different experimental runs of Balke and Hamielec (8) and of Schulz and Har- borth (91, to give what we term as “individually opti- mized’ parameters (IOP). These are listed in Table 5 along with values of the error, E,,,, and the values of N, and NMn used for curve fitting. The conversion histories computed using these parameters (solid curves) are compared with the experimental results in Fig. 1 for one set of experimental conditions (8). The model results for the number and weight average molecular weights for bulk polymerization of MMA with AIBN using the IOP values of Table 5, are shown in Fig. 2 for the same set of conditions. The theoreti- cal results compare favorably with experimental re- Asit B. Ray, D. N . SaraJ and Santosh K. Gupta Table 5. Optimal Values of the Parameters for the Individual Experimental Runs (8,9) for MMA Polymerization. - - D - AIBN 90°C - f," = o Ill, IOP BFC T f EBFC t%) (mol m-3 ) ff N, NMn 0, ts) 0, ts) f ElOP 0, (s) 0, ts) 50* 25.8 20.18 15.48 15.48 15.48 70* 25.8 90* 25.8 50@ 41.3 70@ 41.3 0.0 36 0 0.0 16 0 0.0 17 0 0.0 19 19 0.0 12 12 0.0 18 18 0.0 1 1 1 1 0.0 13 0 0.2 14 0 0.4 20 0 0.6 18 0 0.8 18 0 0.9 18 0 0.0 12 0 0.2 16 0 0.4 18 0 0.6 18 0 0.8 18 0 0.9 18 0 5.45941 X 6.89852 X 10l6 0.4207 0.639 4.50790 X 5.79066 X 10l6 0.52997 1.681 4.98751 x 4.69629 x 10l6 0.4386 0.389 0.229 4.69177 X lo2' 6.65225 X 10l6 0.3928 0.795 0.559 5.38182 X 10l6 5.66049 X 10" 0.6335 0.849 4.23997 X 10l6 3.63103 X 10" 0.52997 1.457 4.48209 X 10l6 2.54906 X lo1' 0.6493 0.447 1.188 3.74311 x 10" 1.70677 x 10' 0.5700 0,909 3.19535 x 10l2 2.14411 x 10' 0.52997 1.259 3.24727 X 10l2 2.74496 X 10' 0.6049 0.266 0.479 4.30814 X 4.93027 X 10l6 0.5026 0.092 4.50790 X 5.79066 X 10l6 0.54150 0.542 5.53051 X 10" 4.31431 X 10l6 0.3646 0.185 0.50570 0.646 7.35777 X 0.71221 X 10l6 0.3865 0.323 0.46991 0.219 4.00000 X lo2' 1 .OOOOO X 1017 0.5101 0.029 0.43412 0.027 9.14076 X 1 .OOOOO X lot5 0.4426 0.031 0.39832 0.035 8.34569 X 3.28803 X loi5 0.3902 0.055 0.38042 0.048 3.95185 X 10l6 2.63082 X 10" 0.6542 0.079 4.23997 X 10l6 3.63103 X 10" 0.54150 0.261 7.35327 X lot6 2.13873 X 10" 0.5451 0.417 0.50570 0.975 1.32050 X 1017 5.07391 X 10'' 0.5116 0.134 0.46991 0.297 4.1 9775 X 1017 3.72875 X 10" 0.4229 0.038 0.43412 0.071 3.77101 X 1017 1.00000 X 10'' 0.4141 0.030 0.39832 0.031 3.79900 X 1 Oi7 4.78629 X 10" 0.31 50 0.043 0.38042 0.168 'AIBN (8). @ BPO-Benzene (9). 'Y 1 t AlBN 50°C n "-- 0 80 160 240 320 LOO Time,min Fig. 1 . Conversion histories for AIBN polymerized MMA at 90°C for two initiator loadings. f: = 0. Data points of Balke and Hamielec (8) shown. Solid curves represent model r e sulk using individually optimized parameters (fop) given in Table 5. Dotted curves correspond to model results using the bestJt correlations ( B E ) in Table 4. sults (8) on dt) as well as M,,(t). Results on M, show larger discrepancies, specially in the gel-effect region. This is similar to what was observed by Chiu et aL (2) and Achilias and Kiparissides (13, 14). Simi- lar agreement between theory and experiment is also observed at other conditions (8, 9) but plots are not provided for the sake of brevity (they can be supplied on request). Better agreement of x, M,, as well as M, cannot be achieved using a single k,. It must be mentioned that the curve-fit achieved with our model is as good as observed with earlier theories for isothermal batch reactors. Our model, however, has the advantage that O t is independent of the initial conditions and so can be used for describing the operation of semibatch reactors as well as batch reactors. It may be added that the value of EIop/(Nw + NM,) gives an idea of the 95% confidence limit for the mol/m3 o a,b : 25.8 a c , d : 15.48 I 10' 3 =. f e 10) 10'; I I I I I I " ' 0.2 0.4 0.6 0.0 1.0 Conversion Fig. 2. M,, and M , for AIBN polymerized MMA at 90°C for three initiator loadings. fz = 0. Notation as in Q. 1 . ExperC mental data of Ealke and Hamielec (8) also shown. parameters, or of the variance, but minimization of ElOp is preferred since the error terms are normal- ized. This model of estimating errors is particularly suited to our case where the values of M,, are far larger than those of x, and normalization avoids the errors in M,, to dominate EIop. The same final values for the parameters are obtained when we start with different initial guesses for them, confirming that the final values correspond to a global minima of EIop. Table 5 shows the values of Elop to be fairly low for most cases. It is necessary to develop correlations for O t . ep, and f in terms of appropriate variables (like temperature, solvent fraction, etc.) so that we can generate results 1294 POLYMER ENGINEERING AND SCIENCE, AUGUST 1995, Vol. 35, No. 76 Free Radical Polymerizations Associated with the TrornmsdorffEffect. I for polymerizations under conditions for which exper- imental data are not available, even if this worsens the curve-fit. It has already been mentioned that O t and OP are expected to be functions of temperature alone for a given polymer. Similarly, we would expect f for any initiator to be a function off:, solvent used and possibly, of temperature, since these influence the viscosity of the medium which, in turn, decides the strength of the cage effect. It has been reported (2, 13, 21, 22) that f is not too sensitive a function of temperature for T between 50 and 90°C, and so we have Jf,") for a given initiator and solvent. We at- tempted to develop some correlations along these lines for MMA polymerization for $,(T), OJT), and Jinitiator, solvent, f:). A regression analysis gives the equation for f for this system as given in Table 4. I t is to be mentioned that use of the two correlations for f gives fairly good agreement with experimental data in the pre gel-effect region. Since f accounts for several physical steps (e.g., diffusion of radicals away from each other, further reactions of radicals, etc.), it is not surprising that f differs for the two initiators, and is a function off,". Table 4 also gives the best-fit correlations (BFCs) obtained from plots of the individ- ually optimized values of 8( and OP us. 1/T. I t is observed (see Table 5) that for values off," > 0.4, the individually optimized values of 8, and OP differ s u b stantially from the values predicted by the BFCs. In fact, these points were given much less weightage while obtaining the BFCs since negligible amounts of gel and glass-effects are exhibited for these condi- tions. The values of the error, E,,,, associated with the use of the correlations are given in Table 5. Fig ures 1 and 2 (dotted curves) show the model results using these correlations. For this as well as the other cases studied (8.9). the agreement with experimental results is still fairly good. It is found that the curve-fit (or value of E ) is somewhat sensitive to the value of O t . This is also true of the theories of Chiu et aL (2) and Achilias and Kiparissides (13). and is a common undesirable feature of this set of models. There is one problem that remains before one can simulate semibatch reactors, viz, the value of f," to be used after the addition of pure components to the reaction mass. One plausible model is described here. In a semibatch reactor, SMWJ kg of solvent, M(MW,) kg of monomer, and (5, - M ) ( M W , ) kg of polymer are present at any time, t (these values get updated with the addition of pure components). The mass of initiator is neglected. We can envisage an equivalent initial mixture corresponding to this "state" as one comprising of S(MW,) kg of solvent (since solvent is not consumed by reaction) and [ M ( M W , ) + ( c M - M)(MW,)I = 5 , (MW,) kg of monomer. Thus, we could use an equivalent value, fZequivalent given by Eq a in Table 4 to estimate f for semibatch reactor operation. This equation predicts f," to be the actual initial value (since [,= M,) for a batch reactor. Similarly, for a semibatch reactor in which R,,, R,,, R,,, and R,, are zero, f," remains at its initial value till some pure components are (in- stantaneously) added, after which an updated value of f," is used. An alternative would be to model f using an adaptation of the newer theory of Achilias and Kiparissides (14). and relate it to current values of appropriate variables as has been done in this paper for k , and k,. We now present model results for semibatch reac- tor operation, using the parameters tuned on batch reactor data. Only a single instantaneous intermedi- ate addition (IA) or removal (IR) of a pure component (at the same temperature as the reaction mass) is considered. The effect of adding (or removing by flash- ing) pure monomer to the reaction mass at different times is shown in Figs. 3 and 4. These reflect the interesting interplay of the effects of polymer concen- tration and molecular weight (Mu) on the viscosity, ""1 AIBN-MMA 7OoC 0.80 c I 1 1 1 1 : I Tirne,min Fig. 3. Effect of intermediate addition (IA) and removal (IR) of pure monomer on the conversion histories for AIBN initt ated bulk polymerization of MMA. Solid curves denote non M / I R results a t three values of [I]o, Dotted curves give results with lA and IR of monomer as described in the text. 3 I: I C I" 0 0.2 0.4 0.6 0.8 1 .o Conversion Fg. 4. Effect of lA and IR of pure monomer for the case of Fg. 3 on M,, and M , . POLYMER ENGINEERING AND SCIENCE, AUGUST 1995, Vol. 35, No. 16 1295 Asit B. Ray, D. N. Saraf; and Santosh K. Gupta and so on the gel effect. 1 m3 of an initial mixture is polymerized at 70°C with [ I ] , = 20.18 mol/m3. At t = 25 min (conversion = 16.43%. point a) 2.6972 mol of pure monomer (at 70°C) are added instantaneously to this mixture. This amount is chosen so if it had been added at t = 0, the initial value of [I], would have been 15.48 mol/m3. The instantaneous addition of monomer leads to a sudden drop in the value of the monomer conversion ( Eq 1) to point d, since the conversion is based on the total amount of liquid monomer added till time t. The gel effect is found to get delayed with respect to polymerization with [I], = 15.48 mol/m3 and without any LA. This is because the polymer concentration as well as M , (see Fig. 4 ) are lower at point a“ than at point a‘. The viscosity of the reaction mass at point a“ is, thus, lower than at point a’, leading to lower diffusional resistances. If, on the other hand, we add the same amount of monomer later on during the gel effect (at point b: t = 68.5 min, monomer conversion = 58.03%). we find that the gel effect occurs earlier than in the case with [I], = 15.48 mol/m3 but without IA. This is because the polymer concentration as well as M , are higher at point b than at point b’. The viscosity corre- sponding to point b is higher than at point b‘, leading to diffusional limitations being higher at b . A similar explanation can be given for the effect of monomer flashing. We, thus, observe that instanta- neous addition of monomer influences the gel effect through the viscosity and we must look at both the polymer concentration as well as the value of M , to predict the future course of polymerization. Figures 5 and 6 show the effect of adding 10.32 mol of AIBN at t = 25 min (point a in the pre-gel effect region where the monomer conversion is 14.63%) to the reaction mass which starts with 15.48 mol of AIBN in 1 m3 of AIBN-Mh4A mixture. It is observed that the gel effect occurs even earlier than when the initial mixture has 25.8 mol m-3 of AIBN. This is because the viscosity of the reaction mass and so the diffusional resistance corresponding to point a is higher than at point a’ ( Fg. 6) because of the higher value of M , (the effect of the slightly higher polymer concentration at point a’ is dominated over by the much higher value of M J It may be noted that the final value of M , for the IA case is different. Thus, intermediate addition of AIBN may have important influences on the product properties. If 10.32 mol of AIBN are added later [e.g., at t = 74.25 min (point b where the conversion is 58.14%)], negligible changes in the conversion or M,, histories are observed. How- ever, M,, is again found to change slightly. Negligible changes in the conversion, M , as well as M , histo- ries are observed when 10.32 mol of AIBN are added still later at t = 85 min (point c: conversion = 90.56%). This is because point c lies in the glassy region, and addition of initiator at this stage does not serve much purpose since diffusional limitations are very high. The effect of adding or removing pure solvent to a partially reacted BPO-Benzene-Mh4A system is shown in Figs. 7 and 8. A 1 m3 starting mixture having I, = 41.3 mol, with f,” = 0.1 is used. Isothermal poly- merization at 50°C is carried out till time t = 120 min (point a). At this time, 1.351 mol of pure solvent (at 50°C) is added instantaneously (this choice effectively makes f,” = 0.2 after the addition), or 1.081 mol of pure solvent is removed (to make f,” = 0 after the removal). It is to be noted that the gel effect corre- sponding to -f: = 0 or 0.2 with no IA or intermediate removal (IR) is not duplicated since the conditions of viscosity at t = 120 min for the various cases differ. In fact, similar explanations can be made for the course of polymerization following IR or IA as in the earlier cases. Figures 7 and 8 also show the sensitivity of the M , history when IA or IR is done at a later time (e.g., at point b: t = 325 min, conversion = 56.01%. or at point c: t = 400 min, conversion = 94.53%). It is to 1 o7 AIBN 7OoC 0 20 LO 60 80 100 T i m e p i n Fig. 5. A l B N initiated M M A polymerization with f,” = 0 at 70°C. Solid curves represent cases with [ I ] , = 15.48 and 25.8 mol/rn3 with no intermediate addition (IA) of pure cornpo nents. Dotted curue shows results when 10.32 mol ofinitiator is added at t= 25 min (point a) to a system which started with 15.48 mol AIBN in 1 m3 AIBN-MMA mixture at t = 0. 0 0.2 0.4 0.6 0.8 1.0 Conversion Fig. 6. M, and M , us. conversion for the case of Fg. 5. Curves for IA of 10.32 rnol of A l B N at point b ( t = 74.25 mid also shown. 1296 POLYMER ENGINEERING AND SCIENCE, AUGUST 1995, Vol. 35, No. 16 Free Radical Polymerizations Associated with the TrommsdorflEffect. 1 C U 0, E u .- 1.0- 0.8 - 0.6- 0.4 - BPO-BZ-MMA S O T with 1A _ _ _ _ 0 120 240 360 480 600 Time,min Fg. 7. BPO initiated MMA polymerization in benzene at 50"C, with I(t = 0) = 41.3 mol, V,(t = 0) = 1 m3. Solid curves repre sent cases without IA of solvent, with f: = 0, 0.1 and 0.2. Dotted curves show results when 1.351 mol pure solvent is added instantaneously (IA,J or 1.081 mol is removed ( I@ at points a, b, or c. 3 f H r c Mn I b 0.2 0.C 0.6 0.8 1.0 105' ' I I I I I 0 Conversion Fg. 8. Variation of M, and M, for the system described in Fig. 7. be observed that addition of pure solvent even at a late stage (point c) leads to higher (= 100%) final monomer conversions, as well as different values M, and M,. An important inference from these results is that the continuous vaporization of solvent inevitably present in any industrial semibatch reactor may have important consequences on M,, and on product properties, and needs to be accounted for properly in any model for such reactors (through the term R,, in Table 2). using appropriate correlations for mass transfer coefficients. The effect of step changes in temperature at some point during polymerization is shown in Figs. 9 and 10 (results on M, are not being presented). Isother- mal polymerization of an AIBN-MMA mixture with [ I ] , = 25.8 mol/m3 is carried out at 70°C. The tem- perature is suddenly increased (SI, step increase) to 90°C at point a ( t = 25 min, conversion = 18.29%). It is found that the gel effect occurs earlier compared to the isothermal 70°C case because of the higher rates associated with higher temperatures. It is also found 1.0 a. C ' f I - - 1 1 1 ~ : x - a m o ~ / r n ' 1 - Isothermal .- s I I 3 1 0.60- I b" I I / / I , I I 0 70 140 210 2 80 350 m. 9. Effect of step increase (SI) and decrease (SO) of tern peraturefrom 70 to 90°C or 50°C at different locations, on the conversion. Solid curves indicate isothermal results. SI and SD results shown by dotted curves. Time.min ? I 1.0 10 5~ 0 0.2 0.4 0.6 0.8 Convers ion Fg. 10. Effect of step increase (SI) and decrease (SO) of temperaturefrom 70 to 90°C or 50°C at different locations on M, . Notation same as in Fig. 9. that the gel effect occurs later than for the 90°C isothermal case since M, and polymer concentration are both lower at point a than at a". When the temperature of the reaction mass is decreased sud- denly (SD, step decrease) from 70 to 50°C at point a, the reaction slows down (Fig. 9) because of lower temperatures. However, for this SD case, the gel ef- fect occurs later than for the 50°C isothermal case. This is because the effect of lower M, at point a (compared to point a') dominates over the effect of higher polymer concentration at that point. The vis- cosity is, thus, lower at point a. This leads to the gel effect occurring later than for the isothermal 50°C case. Step increases in temperature at point b lead to similar results. However, a step decrease in tempera- ture from 70 to 50°C at point b ( t = 63.5 min, conver- sion = 60.4%) leads to the gel effect occurring earlier POLYMER ENGINEERING AND SCIENCE, AUGUST 1995, Vol. 35, No. 16 1297 Asit B. Ray, D. N . Saraf, and Santosh K. Gupta than for the 50°C isothermal case. This is because the value of M, at point b is only slightly lower than at point b', but the polymer concentration is much higher. The latter effect predominates and, therefore, the viscosity at point b is higher. Decreasing the temperature near the glassy region (point c, t = 75 min, conversion = 90.68%) leads to an almost instan- taneous attainment of the glassy region correspond- ing to 50°C. while a step increase in temperature at point c from 70 to 90°C leads to a lowering of the viscosity of the reaction mass primarily due to higher temperatures (at point c, M , is higher and polymer concentration is very slightly lower than at point c', so the increase in conversion for the SI case from point c in Fig. 9 can only be because of the lowering of viscosity due to higher temperatures thereafter). Thus, we observe that the polymer concentration, temperature, and the M, at the point of step change are important in deciding the further course of reac- tion. We would like to add that some recent (31) experi- mental results taken in our laboratory on a well- stirred, PC-interfaced, 1-liter SS Parr reactor do in- deed confirm the theoretically predicted SD curve from point a inFig. 9. More detailed experimental tests are in progress and will form the subject of discussion of part I1 of this series. CONCLUSIONS A new model is described in this work which can be used to simulate the operation of batch as well as semibatch reactors. The model-parameters have been tuned using isothermal experimental data on batch reactors. The variation of conversion and molecular weights are then predicted for simple operations in the semibatch mode, e.g., instantaneous addition and flashing of solvent, monomer, and initiator, step in- crease and decrease in temperature, etc. It is found that conversion histories depend upon an interesting and complex interplay of three factors: polymer con- centration, molecular weight, and temperature at the time these changes are effected. Detailed explana- tions are offered to explain model results in terms of the viscosity of the reaction mass. The model, after experimental confirmation, is well suited for use for on-line adaptive control and optimization. ACKNOWLEDGMENT This work was supported through financial support received from the Council of Scientific and Industrial Research, New Delhi, India through research scheme NO. 22(0232)/93/EMR-I1. NOMENCLATURE A =Jacket area, m2. CFm, Ci,qp, C;,gs =Specific heats of pure (liquid) monomer, polymer or solvent, J kg-' K-'. mixture at any time t, J kg-' K-' . =Specific heat of liquid reaction liq Cp. mix Dn f f," AHr, p r o p 1 kt. kt. 0 ks, kp, kjl o , etc. M M U Mn N U P pn R R S" S' t T Tref =Specific heats of pure (vapor) monomer or solvent, J kg- ' K- '. =Dead polymer molecule having n repeating units. =Activation energies for the reactions in Table 1 , kJ mol-'. =Mass of liquid in reactor at time t, =Initiator efficiency. =Initial volume fraction of solvent =Enthalpy change of propagation =Moles of initiator at any time t , =Initial molar concentration of =Rate constants for the reactions kg. in reaction mixture. reaction, J mol-'. mol. initiator, mol m-3. in Table 1 at any time t , s-' or m3 mo1-ls-l. glass effects) rate constants, m3 mol-'s- ' . =Intrinsic (in absence of gel and =ktc + ktd: 'tc, o + 0. =frequency factors for intrinsic rate constants, s-' or m3 mo1-l s-l. =Moles of monomer in liquid phase, mol. =Moles of monomer in vapor phase, mol. =Number average molecular weight = (MWm)(hl + p l ) / ( h o + po), kg mol-'. =Weight average molecular weight = ( M W m ) ( A 2 + p2)/(A1 + pl). kg mol-'. =Molecular weights of pure monomer and solvent, kg mol- I . =Total moles of inert in vapor phase, mol. =Total pressure, atm. =Growing polymer radical having n =Primary radical. =Universal gas constant, atm-m3 mol--' K-'. =Rate of continuous addition of (liquid) initiator, monomer, or solvent to reactor, mol s - ' . solvent, rnol s - ' . mol. mol. repeat units. =Rate of evaporation of monomer or =Moles of solvent in liquid phase, =Moles of solvent in vapor phase, =Solvent radical. =Time, s. =Temperature, K. =Reference temperature, K. 1298 POLYMER ENGINEERING AND SCIENCE, AUGUST 1995, Vol. 35, No. 16 Free Radical Polymerizations Associated with the Trommsdor-Effect. 1 =Temperature of continuous liquid feed, K. T/ TJ =Jacket fluid temperature, K. U Vl vg v7- X =Monomer conversion (molar). =Overall heat transfer coefficient, J =Volume of liquid at time t, m3. =Volume of vapor space at time t, =Rate of vapor release from reactor, m-2 s - l K-1 m3. mol s-'. Greek Letters e p . ot =Adjustable parameters in the = kth ( k = 0.1.2,. . .) moment of live model, s. (P,) polymer radicals = Ez= nkpn, mol. =Latent heat of vaporization for monomer or solvent at Tref, J mol-'. = kth (k = 0, 1.2,. . .) moment of dead ( 0,) polymer chains = E:= , nkDn, mol. =Number average chain length at time t= (A , + p l ) / ( A o + k0). =Density of pure (liquid) monomer, polymer or solvent at temperature T (at time t ) , kg m-3. polymer or solvent in liquid at time t. AZf, AFf P k Pn Prnq P p . Ps 4,. 4p* 4 s =Volume fractions of monomer, Subscript / Superscript 0 l iq =Liquid phase. VaP =Vapor phase. =Initial value or value without gel and glass effects. REFERENCES 1. J. N. Cardenas and K. F. ODriscoll. J. 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