SIMULATION OF EMULSION COPOLYMERIZATION REACTIONS IN A CONTINUOUS PULSED SIEVE-PLATE COLUMN REACTOR

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1 Brazilian Journal of Chemical Engineering ISSN Printed in Brazil Vol., No. 3, , July - Setember 4 SIMULATION OF EMULSION COPOLYMERIZATION REACTIONS IN A CONTINUOUS PULSED SIEVE-PLATE COLUMN REACTOR C. Sayer and R. Giudici * Deartamento de Engenharia Química, Escola Politécnica, Universidade de São Paulo, Caixa Postal 6548, CEP , São Paulo - SP, Brazil. Phone () , Fax () , csayer@lsc.qi.e.us.br rgiudici@us.br (Received: February, 3 ; Acceted: Aril 3, 4) Abstract - This work addressed the viability of using a ulsed sieve-late column reactor to carry out continuous vinyl acetate/butyl acrylate emulsion coolymerization reactions. A rigorous mathematical model of emulsion coolymerization reactions in a tubular reactor with axial disersion was used for this urose. Oerational conditions were defined to attain high monomer conversions at the reactor outlet in a relatively short residence time and, at the same time, roduce a coolymer with a more homogeneous comosition. Keywords: continuous ulsed sieve-late column (PSPC), emuls ion coolymerization, modeling. INTRODUCTION In revious work (Palma et al., a, b; Sayer et al., a, b) a new tye of reactor, the ulsed sieve-late column (PSPC), had been studied for emulsion vinyl acetate homoolymerization reactions. It had been observed that transforming batch emulsion olymerization reactions into continuous reactions in a PSPC may be an attractive way to intensify this rocess since: ) high conversions can be achieved in short residence times; ) the rocess is safer in the PSPC than in a stirred tank batch or continuous reactor, due to the high heat exchange caacity of the PSPC and 3) due to the continuous oeration, variations in roduct quality, which are commonly found from one batch to the next, are avoided. In batch coolymerizations carried out with comonomers that have very different reactivity ratios, a coolymer comosition drift is observed during the reaction (Araújo, 997). This same behavior is also observed in continuous reactions in tubular reactors that are not erfectly mixed. Van den Boomen et al. (999) and Scholtens et al. () conducted exerimental studies of styrene/methyl acrylate emulsion coolymerization reactions (reactivity ratios r Sty/MA =.73 and r MA/Sty =.9) in a continuous ulsed column reactor and verified that for this system, which does not show differences in order of magnitude between the reactivity ratios, the coolymer comosition can be controlled using lateral monomer feed streams. This work addresses the viability of using the PSPC for vinyl acetate/butyl acrylate emulsion coolymerization reactions. This system was chosen due to its imortance in the aint and aer coatings industries and for having very different reactivity ratios (r VA/BA =.37 and r BA/VA = 6.35). A rigorous *To whom corresondence should be addressed

2 46 C. Sayer and R. Giudici dynamic mathematical model of emulsion coolymerization reactions was used in this study. The effect of various rocess variables axial disersion, maniulated through the amlitude of the ulses; initiator and emulsifier concentrations; and lateral monomer feed streams on conversion and coolymer comosition was verified. DESCRIPTION OF THE PSPC REACTOR The PSPC (Figure ) is comosed of five stainless steel sections, each one with a length of mm and an internal diameter of 4. mm. The lates are stainless steel disks, each with 39 3 mm holes in a triangular arrangement, resulting in a free area of.3% on each late. Plate sacing is 5 mm, resulting in a free volume of 95.7% of the reactor. In order to revent demulsif ication and the consequent lugging of the reactor, a ulsation is suerosed on the nominal axial flow. The ulsator is located at the bottom of the column and the stroke length and ulsation frequency may be adjusted within the ranges of 5 to 5 mm and. and 3.5 Hz, resectively. Reactants are fed in the bottom and at oints throughout the PSPC. A. PSPC, B.C. feeding tanks, D. imeller, E. re-mixer, F. reheater, G. ulsator, H. samling oints, I. jacket inlets, J. jacket outlets, K. balances, L. metering ums, M. comuter. Figure : Diagram of the exerimental setu: Brazilian Journal of Chemical Engineering

3 Simulation of Emulsion Coolymerization Reactions 46 MATHEMATICAL MODELING The mathematical model is based on a revious model of Sayer et al. (a) of emulsion homoolymerization reactions in the PSPC. In the elaboration/comlementation of this model the following assumtions are made: Monomer concentrations in olymer articles, monomer drolets and aqueous hase are at thermodynamic equilibrium and are comuted using the iterative rocedure roosed by Omi et al. (985). Total mass of olymer roduced in the aqueous hase is negligible. Particle nucleation occurs through micelar and homogeneous mechanisms. Particle coagulation deends on initiator concentration. The average number of radicals er olymer article is comuted using the method roosed by Ugelstad et al. (967). Polymer articles are sherical and monodiserse. The critical micelar concentration and the surface covered by one mol of emulsifier are constant. The seudo-steady-state assumtion is valid for olymer radicals. Kinetic constants do not deend on chain length. Kinetic constants in the aqueous and olymer hases are the same. Radicals generated by initiation or chain transfer to monomer and olymer have similar reactivities. Reactions are conducted at constant temerature. Radial mixing is erfect. Average net forward velocity is constant. Axial mixing is reresented by an effective axial disersion coefficient. The boundary conditions of Danckwerts (953) were alied to the axial disersion model. The method of lines was used to solve the second-order artial differential equations. The balance equations of continuous homoolymerization reactions in the PSPC were resented in Sayer et al. (a) and are shown in the Aendix. In the resent work the following average coefficients were used to describe the coolymerization reactions: Proagation: [ ] = ( Ai A + Bi B ) [ ] () i= A,B k M k P k P i Chain transfer to monomer: f[ ] = ( fai A + fbi B ) [ ] () i= A,B k M k P k P i Termination: k = k P + k P P + k P (3) t taa A tab A B tbb B where P A and P B are the robabilities that the radicals contain monomeric units of tye A or B at the active extremity. All other variables are described in the Aendix. The model develoed was imlemented in FORTRAN and differential and algebraic equations were solved by the DASSL solver (Petzold, 98). The model had been validated reviously with exerimental data on vinyl acetate emulsion homoolymerization reactions conducted at 55 o C in the PSPC (Sayer et al., a). All simulations were carried out using the method of lines with 5 internal oints evenly distributed in the axial direction. RESULTS AND DISCUSSION In order to evaluate how the butyl acrylate comonomer affects the kinetics of vinyl acetate emulsion olymerization reactions, the results of a vinyl acetate/butyl acrylate coolymerization reaction were comared with those of vinyl acetate (VA) and butyl acrylate (BA) homoolymerization reactions (see formulations in Table ). In the coolymerization 5 wt.% of monomer is comosed of butyl acrylate. Tables and 3 show the oerational conditions and the kinetic constants and arameters used in the simulations. Table : Formulation of VA/BA emulsion homo- and coolymerization reactions. Reactant VA BA VA/BA Mass fraction VA BA Water SLS Na S O Brazilian Journal of Chemical Engineering Vol., No. 3, July - Setember 4

4 46 C. Sayer and R. Giudici Table : Oerational conditions. Temerature ( o C) 55 Residence time (min.) 3 Stroke length (mm) 5 Peclet number 9 Table 3: Parameters used in the simulations (A - vinyl acetate/b - butyl acrylate). Parameter Value Reference k d (/s).8. 7 ex[-755/t] Rawlings and Ray, 988 k AA (cm 3 /mol s).445. ex[-49/t] Hutchinson et al., 995 k BB (cm 3 /mol s) ex[-37/t] McKenna et al., 995 k AB (cm 3 /mol s) k AA /r A k BA (cm 3 /mol s) k BB /r B k faa (cm 3 /mol s) k AA Chatterjee et al., 977 k fbb (cm 3 /mol s) k BB Brandru and Immergut, 989 k fab (cm 3 /mol s) k faa /r A k fba (cm 3 /mol s) k fbb /r B k taa (cm 3 /mol s) ex[-884/t] Baad et al., 98 k tbb (cm 3 /mol s) ex[-49/t] McKenna et al., 995 k tab (cm 3 /mol s) k taa a s (cm /mol) N A Min and Ray, 978 r m (cm) Min and Ray, 978 [E] CMC (mol/cm 3 ) Unzueta and Forcada, 997 D WA (cm /s).. -5 Min and Ray, 978 D WB (cm /s).. -5 Gardon, 968 D A (cm /s).. -6 Min and Ray, 978 D B (cm /s).. -7 Gardon, 968 f absa f absb f absma f absmb c ca (/s) c cb (/s). - c homa (/s) c homb (/s) j crita 6 Gilbert, 995 j critb 8 Gilbert, 995 j za 8 Gilbert, 995 j zb Gilbert, 995 PM A (g/mol) 86.9 Brandru and Immergut, 989 PM B (g/mol) 8.7 Brandru and Immergut, 989 PM E (g/mol) Brandru and Immergut, 989 PM w (g/mol) 8. Perry and Chilton, 98 ρ ma (g/cm 3 ).933 Brandru and Immergut, 989 ρ mb (g/cm 3 ).8 Brandru and Immergut, 989 ρ A (g/cm 3 ).3 Araújo et al., ρ B (g/cm 3 ).933 Brandru and Immergut, 989 ρ w (g/cm 3 ).8 Perry and Chilton, 98 d k A 34.7 Gardon, 968 d k B 75. Unzueta and Forcada, 997 k A 9.5 Gardon, 968 k B 46. Unzueta and Forcada, 997 r A.37 Gugliotta et al., 995 r B 6.35 Gugliotta et al., 995 Brazilian Journal of Chemical Engineering

5 Simulation of Emulsion Coolymerization Reactions 463 A Comarison of VA/BA Homo - and Co- Polymerization Reactions Figure a comares the evolution of the overall conversions in the PSPC during emulsion VA/BA (85/5) coolymerization and VA and BA homoolymerization reactions. It can be observed that while in the VA and BA homoolymerizations conversions above 95% are reached at the reactor outlet, in the VA/BA (85/5) coolymerization the conversion at the reactor outlet is below %. This occurs because the more reactive monomer in this coolymerization (BA) acts as a retardant of the roagation of the other monomer (VA). This can be verified in Figure b, which shows the evolution of the VA and BA fractional conversions in the PSPC during VA/BA (85/5) emulsion coolymerization reactions. The significant reduction in the reaction rate of the coolymerization reaction comared to those of the coolymerizations may jeoardize this tye of reactions in the PSPC, since very long residence times would be required in order to achieve comlete monomer conversion. Therefore, in order to enhance the reaction rate, simulations resented below were carried out using a new reaction formulation with higher emulsifier and initiator concentrations, as shown in Table 4. VA (ex.) Conversion BA VA Conversion BA. VA/BA (85/5) Fractional reactor length. VA Fractional reactor length (a) (b) Figure : Conversions in the PSPC during VA and BA emulsion homo- and coolymerization reactions. (a) Overall conversions; (b) Fractional conversions. Table 4: New formulation of VA/BA emulsion coolymerization reactions. Reactant Mass fraction VA.576 BA.79 Water.7969 SLS.9 Na S O 8.57 Figure 3a shows the evolution of overall and fractional conversions in the reactor of a reaction carried out in the PSPC using the new formulation. It might be observed that by increasing emulsifier and initiator concentrations, it is ossible to reach high conversions in a relatively low residence time. On the other hand, coolymer comosition (Figure 3b) shows a significant drift along the reactor length, resulting in the roduction of a quite heterogeneous coolymer. The effect of Axial Disersion Figure 4a shows the effect of axial disersion, which is varied through the Peclet number (Pe), on the comosition of the coolymer formed in the Brazilian Journal of Chemical Engineering Vol., No. 3, July - Setember 4

6 464 C. Sayer and R. Giudici PSPC. It might be observed that a coolymer with a constant comosition is only formed when the reactor is oerated under conditions that are close to those of erfect mixing (Pe =.4). Nevertheless, those conditions result in very low conversions, as shown in Figure 4b, which shows the effect of Pe on conversion in the PSPC. The effect of Lateral Feed Streams A way of minimizing the coolymer comosition drift along the reactor length without significantly affecting the conversion is to use lateral feed streams of the more reactive monomer. Figures 5a and b show the evolution of overall and fractional conversions and coolymer comosition in the PSPC of a reaction erformed with two lateral BA feed streams. It might be observed that with this rocedure the coolymer comosition shows a much less accentuated drift in the reactor, resulting in the roduction of a more homogeneous coolymer. Overall (x) and fractional (x VA, x BA ) conversions x BA x x VA Fractional reactor length (a) (b) Figure 3: Conversions and coolymer comosition in the PSPC during VA and BA emulsion coolymerization reactions new formulation (Table 4). (a) Overall and fractional conversions; (b) Coolymer comosition (mass fraction of VA). Brazilian Journal of Chemical Engineering

7 x V A Simulation of Emulsion Coolymerization Reactions Comosition (y VA ) Pe =.4 Pe = 6..4 Pe = 38. Pe = Fractional reactor length (a).8 Conversion.6.4 Pe =.4 Pe = 6.. Pe = 38. Pe = Fractional reactor length (b) Figure 4: Effect of Peclet number on coolymer comosition and overall conversion in the PSPC during VA and BA emulsion coolymerization reactions. (a) Coolymer comosition (mass fraction of VA); (b) Overall and fractional conversions. Overall (x) and fractional (x VA, x BA ) conversions x BA x x VA Fractional reactor length (a) Brazilian Journal of Chemical Engineering Vol., No. 3, July - Setember 4

8 466 C. Sayer and R. Giudici (b) Figure 5: Conversions and coolymer comosition in the PSPC during VA and BA emulsion coolymerization reactions with two lateral BA feed streams. (a) Overall and fractional conversions; (b) Coolymer comosition (mass fraction of VA). CONCLUSIONS This work involved a comutational study of vinyl acetate/butyl acrylate emulsion coolymerization reactions in the ulsed sieve-late column reactor (PSPC). It was observed that due to the significant difference between the reactivity ratios of these two monomers, vinyl acetate/butyl acrylate emulsion coolymerization reactions are significantly slower than the vinyl acetate or butyl acrylate coolymerization reactions carried out under the same oerational conditions. Therefore, a new formulation with higher initiator and emulsifier concentrations was tested. With this new formulation it was ossible to achieve high conversions (~%) in a relatively low residence time (3 minutes). Nevertheless, also due to the significant difference between the reactivity ratios of these two monomers, the coolymer comosition showed a considerable drift along the reactor length, thereby resulting in a quite heterogeneous coolymer. It was verified that this comosition drift can be minimized using lateral feed streams of the more reactive monomer in the coolymerization, in this case the butyl acrylate. ACKNOWLEDGEMENTS The financial suort of FAPESP Fundação de Amarao à Pesquisa do Estado de São Paulo and CNPq Conselho Nacional de Desenvolvimento Científico e Tecnológico are gratefully acknowledged. REFERENCES Araújo, O., Coolimerização e Terolimerização em Emulsão: Trabalho Exerimental e Estudo de Asectos Relacionados à Modelagem Matemática do Processo, Ph.D. Diss., Universidade De São Paulo, Brazil, 997. Araújo, P.H.H., De La Cal, J.C., Asua, J.M. and Pinto, J.C., Modeling Particle Size Distribution (PSD) in Emulsion Coolymerization Reactions in a Continuous Loo Reactor, Comuter-Aided Chem. Eng., vol. 8, Elsevier Science, ,. Baad, W., Moritz, H.U. and Reichert, K.H., Kinetics of High Conversion Polymerization of Vinyl Acetate Effects of Mixing and Reactor Tye on Polymer Proerties, J. Al. Polym. Sci., vol. 7, , 98. Van Den Boomen, F.H.A.M., Meuldijk, J. and Thoenes, D., Emulsion Coolymerization in a Flexible Continuously Oerated Reactor, Chem. Eng. Sci., vol. 54, , 999. Brandru, J. and Immergut, E.H., Polymer Handbook, 3rd ed., Wiley, New York, 989. Chatterjee, A., Park, W.S. and Graessley, W.W., Brazilian Journal of Chemical Engineering

9 Simulation of Emulsion Coolymerization Reactions 467 Free Radical Polymerization with Long Chain Branching: Continuous Polymerization of Vinyl Acetate in t-butanol, Chem. Eng. Sci., vol. 3, , 977. Danckwerts, P.V., Continuous Flow Systems. Distribution of Residence Times, Chem. Eng. Sci., vol.,. -3, 953. Gardon, J.L., Emulsion Polymerization. II. Review of Exerimental Data in the Context of the Revised Smith-Ewart Theory, J. Pol. Sci. A, vol. 6, , 968. Gilbert, R.G., Emulsion Polymerization. st ed. London, Academic Press, 995. Gugliotta, L.M., Arzamendi, G. and Asua, J.M., Choice of Monomer Partition Model in Mathematical Modeling of Emulsion Coolymerization Systems, J. Al. Polym. Sci., vol. 55,. 7-39, 995. Hutchinson, R.A., Paquet, D.A., Mcminn, J.H., Beuermann, S., Fuller, R.E. and Jackson, C., Dechema Monogr. vol. 3,. 467, 995. McKenna, T.F., Graillat, C. and Guillot, J., Contributions to Defining the Rate Constants for the Coolymerization of Butyl Acrylate and Vinyl Acetate, Pol. Bul. vol. 34, , 995. Melis, S., Kemmere, M., Meuldijk, J., Storti, G. and Morbidelli, M., A Model for the Coagulation of the Polyvinyl Acetate Particles in Emulsion. Chem. Eng. Sci. vol. 55,. 3-3,. Min, K.W. and Ray, W.H., The Comuter Simulation of Batch Emulsion Polymerization Reactors Through a Detailed Mathematical Model, J.Al.Polym. Sci., vol.,. 89-, 978. Omi, S., Kushibiki, K., Negishi, M. and Iso, M., Generalized Comuter Modeling of Semi-Batch, n-comonent Emulsion Coolymerization Systems and Its Alications. Zairyo Gijutsu. vol.3,. 46, 985. Palma, M., Sayer, C. and Giudici, R., Dechema Monogr., vol. 37,. 65, a. Palma, M., Miranda, S. C. F., Sayer, C. and Giudici, R., Comaração Entre Reações Contínuas de Polimerização em Emulsão em uma Coluna Pulsada com Pratos Perfurados com Reações em Batelada, 6 th CBPOL. Gramado, Brazil, -5th November, b. Perry, R.H. and Chilton, C.H., Manual de Engenharia Química, 5 th Ed., Guanabara Dois, Rio de Janeiro, 98. Petzold, L.R., A Descrition of Dassl: A Differential Algebraic System Solver, Sandia National Laboratories, Reort # SAND8-8637, 98. Rawlings, J.B. and Ray, W.R., The Modelling of Batch and Continuous Emulsion Polymerization Reactors: II. Comarison with Exerimental Data from Continuous Stirred Tank Reactors, Polym. Eng. Sci., vol. 8, , 988. Sayer, C., Palma, M. and Giudici, R., Modeling Continuous Vinyl Acetate Emulsion Polymerization Reactions in a Pulsed Sieve Plate Column, Ind. Eng. Chem. Res., vol. 4, , a. Sayer, C., Palma, M. and Giudici, R., Kinetics of Vinyl Acetate Emulsion Polymerizations in a Pulsed Tubular Reactor. Comarison between Exerimental and Simulation Results. Braz. J. Chem. Eng., vol. 9, , b (3 rd ENPROMER). Scholtens, C.A., Meuldijk, J. and Drinkenburg, A.A.H., Production of Coolymers with a Predefined Intermolecular Chemical Comosition Distribution by Emulsion Polymerization in a Continuously Oerated Reactor, Chem. Eng. Sci., vol. 56, ,. Unzueta, E. and Forcada, J., Modeling the Effect of Mixed Emulsifier Systems in Emulsion Coolymerization, J. Al. Polym. Sci., vol. 66, , 997. Ugelstad, J., Moek, P.C. and Aasen, J.O., Kinetics of Emulsion Polymerization, J. Polym. Sci, vol. 5,. 8-87, 967. Brazilian Journal of Chemical Engineering Vol., No. 3, July - Setember 4

10 468 C. Sayer and R. Giudici APPENDIX EQUATIONS OF THE EMULSION POLYMERIZATION MODEL In order to model continuous reactions carried out in the PSPC, a dynamic axially disersed lug-flow model was imlemented. Given the assumtions shown in Section 3, the oulation and mass balance equations for a continuous emulsion olymerization system are resented below: a) Polymer Particle Concentration - N vz De N N N + = t z z ( ) = R + R c N mic hom c (A) where v z is the axial velocity and De is the effective disersion coefficient in cm /s and v q e z = ( πrcol fvoid ) col De vz Pe (A) a = (A3) where q e is the inlet flow rate of the reactor in cm 3 /s; r col and a col are, resectively, the radius and the length of the PSPC in cm and f void is the void fraction of the PSPC. Pe is the Peclet number, R mic and R hom reresent the micelar and homogeneous nucleation rates and c c (N ) reresents the rate of article coalescence. Micelar nucleation: m mic abs ent mic R = k R N (A4) where R ent is the concentration of radicals in the aqueous hase that may enter olymer articles and k m abs is the absortion rate coefficient of radicals from the aqueous hase by the micelles given by the Smoluchowski equation: k m abs w m m abs = 4π D r Naf (A5) where D w, r m and f m abs reresent, resectively, the diffusion coefficient in the aqueous hase, the radius of one micelle and the absortion efficiency of radicals from the aqueous hase by the micelles. Variable N mic is the concentration of micelles formed when the emulsifier concentration in the aqueous hase [E] aq is above the critical micelar concentration (CMC): Nmic = Mmicvaq (A6) where M mic is the number of micelles er volume of the aqueous hase and v R,, v and v aq are, resectively, the volumes of the reactor and of the olymer and aqueous hases. Homogeneous nucleation: hom hom aq jcrit aq R = c k [M] R Nav (A7) where c hom is the coefficient of homogeneous nucleation and k [M] aq is the roagation rate in the aqueous hase. Variable R jcrit is the concentration of radicals with the critical length j crit in the aqueous hase. Particle coalescence: In order to avoid using rather comlex mathematical exressions like the extensions of the DLVO model (Derjaguin-Landau-Verwey- Overbeek) that involve a significant number of often unknown arameters (Gilbert, 995; Araújo et al., ), a simle equation for the rate constant of article coalescence was used. This equation is an extremely simlified version of the DLVO model and the only effect that is accounted for in this equation is the initiator concentration in the aqueous hase, since increasing electrolyte concentrations enhances the article coagulation rate (Melis et al., ): [ ] c c R c = c I v Na (A8) where c c is the only unknown arameter that must be adjusted in order to reresent exerimental data. b) Initiator Concentration in the Reactor - [I] [ I] [ I] [ I] + vz De = [ Ik ] t z z d (A9) Brazilian Journal of Chemical Engineering

11 Simulation of Emulsion Coolymerization Reactions 469 c) Emulsifier Concentration in the Reactor - [E] [ E] [ E] [ E] + vz De = t z z (A) In emulsion olymerization systems the emulsifier might be in three different forms: a) free in the aqueous hase ([E] aq ); b) in the form of micelles in the aqueous hase (M mic ); and c) adsorbed on olymer articles ([E] ads ) and monomer drolets. As monomer drolets are significantly larger than olymer articles, they have a much smaller total surface area and, therefore, the amount of emulsifier adsorbed on monomer drolets might be neglected and [ E] aq aq +γ mic aq + ads [E] v M v a N [E] = (A) v R where [E] aq is the concentration of emulsifier in the aqueous hase, γ is the number of moles of emulsifier er micelle and [E] ads is the emulsifier adsorbed on olymer articles er unit surface. Variable γ is comuted by s m 4πr γ= (A) a where a s is the area covered by one mol of emulsifier and a is the surface area of one olymer article swollen with monomer: a = 4π r (A3) r is the radius of one olymer article swollen with monomer, comuted by r = 3 3v ( N v ) R 4π (A4) Variables [E] aq, [E] ads and M mic must be comuted for three different situations: a) the emulsifier concentration is above the critical micelar concentration ([E] CMC ), so the emulsifier concentration in the aqueous hase ([E] aq ) equals [E] CMC, the emulsifier concentration adsorbed on olymer articles ([E] ad ) equals the saturation concentration ([E] sat ads=/a s ) and M mic is comuted with Eq. (A); b) the emulsifier concentration is below [E] CMC and the emulsifier concentration is large enough to saturate olymer articles, so M mic equals zero, [E] ad equals [E] sat ads (assuming that the emulsifier is absorbed on olymer articles than in the aqueous hase) and [E] aq is comuted with Eq. (A); or c) the emulsifier concentration is below [E] CMC and the emulsifier concentration is not high enough to saturate olymer articles, so M mic and [E] aq equal zero and [E] ad is comuted with Eq. (A). d) Monomer Concentration in the Reactor - [M] [ M] [ M] [ M] + vz De = t z z ( [ ] [ ] f ) nn k M k M + Na (A5) [M] is the monomer concentration in olymer articles. The monomer concentrations in the different hases in emulsion olymerization reactions were comuted by the iterative rocedure roosed by Omi et al. (985). e) Polymer mass in the reactor - P: P P P nn + v De = k M PM t z z Na z [ ] m f) Water Fraction in the Reactor - f w : φ w t + v φ z φ De z w w z = g) Comutation of the Average Number of Radicals Per Polymer Particle (A6) (A7) The method roosed by Ugelstad et al. (967) was used for the iterative comutation of the average number of radicals er olymer article n : h 4 n = h 4 m+ + h 4 m+ + m (A8) where h and m are the relative absortion/termination and desortion/termination coefficients, defined as Brazilian Journal of Chemical Engineering Vol., No. 3, July - Setember 4

12 47 C. Sayer and R. Giudici 8kabsRentNav h = kxgeln t v R t R (A9) knav m = (A) kxgeln v k abs is the coefficient of the rate of absortion of radicals from the aqueous hase by the olymer articles, given by the Smoluchowski equation: k = 4π D r Naf (A) abs w abs where R ent is the concentration of radicals in the aqueous hase that may enter olymer articles. R ent is comuted by the iterative rocedure develoed by Araújo et al. () based on two different critical radical lengths roosed by Gilbert (995): NvR Ψ= k [M] aq + kabs + v Na N v + k + aq m mic R abs kr t aq vaqna (A) aq Rent = R k [M] aq + kr t aq jcrit j= jz k [M] Ψ aq k[m] ( j jz + ) ( jz ) (A5) j ( j) z k[m] aq Raq = R + + R ent (A6) k[m] j aq + kr = t aq where R aq is the concentration of radicals in the aqueous hase; j z and j crit are the critical lengths for radical entry into micelles and olymer articles and for homogeneous nucleation (Gilbert, 995), which deend on the solubility of the monomer in the aqueous hase; k t R aq and k [M] aq are termination and roagation rates in the aqueous hase er unit radical; f is the efficiency in forming radicals by decomosition of the initiator and k is the coefficient of the rate of desortion of radicals from olymer articles: k = k[m] f k m ( kmβ m + k [M] ) (A7) R Iv Rfk k nn d v R + vaq vaqna = k [M] aq+ kr t aq aq R jcrit = R k[m] aq + kr t aq k [M] Ψ aq ( jcrit jz) k [M] ( jz ) (A3) (A4) where k m and β m are the exit rate of a monomeric radical from a olymer article and the robability that a radical in the aqueous hase will react by roagation or by termination: k m β = m Dw 3 r k m = D + Dk w m ( k [M] w + kr t aq) ( ψ) (A8) (A9) where D is the diffusion coefficient of a radical in olymer articles in cm /s. Brazilian Journal of Chemical Engineering