1238
Ind. Eng. Chem. Res. 1997, 36, 1238-1246
Emulsion Copolymerization of Acrylonitrile and Butadiene. Mathematical Model of an Industrial Reactor Jorge R. Vega, Luis M. Gugliotta, Raquel O. Bielsa, Marcelo C. Brandolini, and Gregorio R. Meira* INTEC (Universidad Nacional del Litoral and CONICET), Gu¨ emes 3450, 3000 Santa Fe, Argentina
The batch emulsion copolymerization of acrylonitrile and butadiene is modeled, with the aim of simulating an industrial process and of improving the final polymer quality. The mathematical model is an extended version of that developed by Gugliotta et al. for a continuous emulsion polymerization of styrene and butadiene. Due to the relatively high solubility of acrylonitrile in water, the following effects are included: (a) the acrylonitrile homopolymerization in the water phase; (b) the desorption of acrylonitrile radicals from the polymer particles; and (c) homogeneous particle nucleation. Apart from simulating two typical batch operations, the model allowed us to develop and simulate a semibatch operation aimed at reducing the copolymer compositional drift. A (suboptimal) semibatch operation was implemented that involved a series of impulsive acrylonitrile additions. Introduction In this work, the industrial emulsion polymerization of acrylonitrile (A) and butadiene (B) was investigated, with the aim of predicting the evolution of the main process variables and of improving the product quality. To this effect, the mathematical model of a “cold” acrylonitrile-butadiene rubber (NBR) emulsion process was developed. The model is an extension of that presented in our recent work (Gugliotta et al., 1995), which in turn was adapted from Broadhead (1984), Broadhead et al. (1985), Kozub (1989), Huo et al. (1988), and Ugelstad et al. (1983). In Gugliotta et al. (1995), a review of these previous publications is presented, and for this reason, they will not be reconsidered here. The model in Gugliotta et al. (1995) simulates the emulsion polymerization of styrene and B carried out in a train of continuous-stirred-tank reactors. Its main characteristics are (a) the particle size distribution (PSD) of the polymer latex is calculated through a population balance that involves a set of partial differential equations with a generation term that only considers the classical mechanism of micellar nucleation, (b) impurities in the water and in the polymer phases deactivate free radicals and oxidize the initiator catalyst, (c) a polymerizing impurity in the polymer phase reacts with the accumulated copolymer, affecting molecular weights and branching frequencies, and (d) the low water solubility of the comonomers and of the modifier determine a negligible polymerization in the water phase, a negligible radical desorption, and a negligible homogeneous nucleation. The model in Gugliotta et al. (1995) was subdivided into three sections: (1) a basic module that calculates the evolution of conversion, the average copolymer composition, the number of polymer particles, the average number of radicals per particle, and the reagent concentrations in all three phases; (2) a PSD module that determines the distribution of the particle volumes; and (3) a molecular weight module that estimates not only the average molecular weights, M h n and M h w, but also the average number of tri- and tetrafunctional branches per molecule (B h N3 and B h N4, respectively). * Corresponding author. Telephone: 54-42-559175. Fax: 5442-550944. E-mail:
[email protected]. S0888-5885(96)00534-9 CCC: $14.00
For more water-soluble monomers, other physicochemical processes such as the monomer polymerization in the aqueous phase, the homogeneous particle nucleation, and the desorption of free radicals, can be included in the model. Some publications that have investigated such processes are Guyot (1983), Fitch and Tsai (1971), Kiparissides et al. (1979), Pollock (1983), Nomura et al. (1983), and Asua et al. (1989). The particular case of emulsion polymerizations involving the A monomer have been considered in Lin et al. (1981), Guillot (1981), Guyot (1983), Guyot et al. (1984), Shvetsov (1986), Shvetsov et al. (1987), Mead and Poehlein (1988), Nomura et al. (1994), Vindevoghel and Guyot (1995), and Nishida et al. (1995). As far as we are aware, this is the first publication where a mathematical model for an industrial NBR emulsion process is presented. Mathematical Model The model was developed to simulate an emulsion semibatch process, with possible addition of reagents along the reaction. Since the model is an extension of that in Gugliotta et al. (1995), only the differences with respect to that publication will be stressed. Like in Gugliotta et al. (1995), we shall also subdivide the model into three modules. The PSD module and the molecular weight module are almost identical as in our previous work, and for this reason, they will not be further discussed. All of the model extensions were incorporated into the basic module. The extensions include the following: (1) the polymerization of A in the aqueous phase; (2) the deactivation of growing free radicals by O2 both in the water and in the polymer phases; (3) the desorption of primary A radicals from the polymer particles; and (4) the homogeneous nucleation mechanism. In Appendix A, the kinetic scheme adopted in the basic module is presented. (A different kinetics is assumed for the molecular weight module, but it shall not be discussed here.) In the aqueous phase, the deactivation of free radicals of any chain length and the propagation and termination of A macroradicals are permitted. The initiator and its associated redox couple are assumed to react only in the aqueous phase; A and © 1997 American Chemical Society
Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1239 Table 1. Initial Loads for the Investigated Plant Experiments expt 1 batch
expt 2 batch
expt 3 semibatch
reagent
kg
mol
kg
mol
kg
mol
acrylonitrile butadiene emulsifier initiatord oxidizing agentg reducing agenth modifieri water
1 953 4 347b 234.7 0.25e 0.20 4.22 23.62 11 340
36 849 80 366b 813.8 1.289e 0.719 27.40 116.70 630 000
2 016 4 284c 235.7 0.25f 0.22 4.45 22.18 11 064
38 038 79 201c 817.3 1.289f 0.791 28.90 109.60 614 667
1 386a 4 284b 220.5 0.25 0.22 4.45 22.18 11 064
26 151a 79 201b 765 1.289 0.791 28.90 109.60 614 666
a Three intermediate A additions were implemented: 250 kg (4717 mol) at x ) 16%, 250 kg (4717 mol) at x ) 28%, and 200 kg (3774 mol) at x ) 49%. b Containing vinylacetylene impurity. c Without vinylacetylene impurity. d Diisobutyl hydroperoxide. e Two intermediate initiator additions were applied: 0.050 kg (0.258 mol) at t ) 450 min and 0.050 kg (0.258 mol) at t ) 630 min. f One intermediate initiator addition was applied: 0.075 kg (0.387 mol) at t ) 600 min. g SO4Fe‚7H2O. h Sodium formaldehydesulfoxylate. i tert-Dodecyl mercaptan.
O2 are assumed to react both in the water and polymer phases, and B is assumed to react only in the polymer phase. The equations for the instantaneous partitioning of reagents among the three phases are as in Gugliotta et al. (1995) and will not be repeated here. All other basic module equations are presented in Appendix B. The employed symbols are indicated in the Nomenclature section. In eqs B.1-B.14, the main mass balances are presented. In spite of its relatively low influence on the polymerization rates, the modifier balance is also included in the basic module (eq B.8), due to its indirect effect on the average desorption rate coefficient, kde, which in turn affects the average number of radicals per particle, n j. To evaluate the polymerization rates in the polymer phase, n j and the total number of polymer particles, Np, must be calculated. For obtaining n j , eqs B.21-B.28 (which are based on Huo et al. (1988)), together with eqs B.29-B.33 (that allow us to obtain kde), are required. For calculating Np, the particle generation rate, Ngen, has to be integrated (eq B.14). In the expressions for Ngen (eqs B.38-B.42), both the micellar and the homogeneous nucleation mechanisms are considered. Finally, eqs B.44-B.46 allow us to compute the total conversion, x, the average mass fraction of polymerized A in the accumulated copolymer, p j A, and the mass fraction of polymer produced in the aqueous phase, wAb,w. It should be emphasized that the proposed basic module extensions are not considered in the molecular weight module, and for this reason are expected to have only a slight indirect effect on the predicted molecular weights and branching frequencies. Not including homopolymerization in the water phase in the molecular weight calculations may be justified by the fact that the contribution of poly(acrylonitrile) oligomers toward molecular weights and branching frequencies is expected to be relatively important only at the very beginning of the polymerization but negligible at moderate or high conversions. Not including A radical desorption in the molecular weight calculations is justified by the fact that most of the accumulated polymer is generated by transfer to the modifier. Simulated Process Experimental Work. Polymerizations were carried out with the temperature controlled at 10 °C, in a 21 000-dm3 stirred-tank reactor. Two batch and one semibatch experiments were investigated. In all three cases, a BJLT grade NBR was produced. In Table 1,
the applied recipes are shown. The batch experiments 1 and 2 correspond to two typical industrial operations. The semibatch experiment 3 was especially designed and implemented to reduce the copolymer compositional drift. For nearly identical recipes and conditions, relatively large batch-to-batch differences in the polymerization rate were normally observed. This can be explained by the presence of varying amounts of O2 and other unmeasured impurities that inevitably contaminated the initial load. To compensate for the reduced polymerization rates, intermediate initiator injections were applied. In experiments 1 and 3, B monomer was known to contain a polymerizing and branching impurity: vinylacetylene. Latex samples were taken along the reaction to measure the conversion, average molecular weights, copolymer composition, and average particle size. For the conversion measurements, samples were collected into specially-designed 20-cm3 stainless steel bottles, fitted with a pressure-resistant rubber septa (to avoid B loss) and small quantities of “short-stop” (or deactivating agent). After an initial pipe purge, fresh reactor latex was directly delivered into such bottles through an adapted syringe needle. For the polymer-quality measurements, about 0.5 dm3 of latex was collected into open glass bottles containing short-stop. Water vapor was then bubbled into the rubber latex to strip off the residual monomers. In the plant, the conversion (x) was gravimetrically determined according to ASTM B 1417-80, and the copolymer composition (p j A) was estimated through the Kjeldahl method (Kolthoff et al., 1969). In our laboratories, the average molecular weights were measured through a Waters ALC220 size-exclusion chromatograph, and the (unswollen) number-average particle diameter (d h p,unsw) was obtained by scanning electron microscopy and UV-vis turbidimetry (Llosent et al., 1996). The experimental results are indicated in Figures 1 and 4. The higher M h w values in experiment 1 can be attributed to the (known) presence of vinylacetylene in that case. Even though not shown, the following measurements were also carried out in our laboratories: (a) the percent of macrogel by gravimetry according to ASTM D 3616-77 and (b) glass transition temperature through a Mettler TA3000 DSC. Model Adjustment and Batch Operations. The model parameters were adjusted on the basis of experiments 1 and 2. Except for the impurities concentration, a single set of physicochemical parameters (not presented here for proprietary reasons) was required to
1240 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997
Figure 1. Main reaction variables for the batch experiments 1 and 2. Measurements are indicated by dots, while the model predictions are shown in the continuous trace. Dashed trace: model predictions assuming absence of O2. Dotted lines: model predictions assuming the absence of vinylacetylene. (Experiment 2 was not contaminated with vinylacetylene, and therefore, the complete model predictions coincide with the predictions assuming the absence of such an impurity.)
adequately reproduce the experimental data. Most of the parameters were directly adopted from Broadhead (1984) and Brandrup and Immergut (1989). The unknown and/or highly-sensitive parameters were adjusted as indicated below, in a similar fashion as in Gugliotta et al. (1995). (1) The conversions, x, were fit via a simultaneous adjustment of the following: the emulsifier surface coverage capacity, AS; the diffusion coefficient of A radicals in the polymer phase, DpA; and the initial moles 0 of O2 in the reactor charge, NO . 2 (2) The copolymer compositions, p j A, were fit by adjustment of the B reactivity ratio, rB. (3) The unswollen number-average particle diameters, d h p,unsw, were fit by adjustment of the B homopropagation rate, kpBB. (4) The number-average molecular weights, M h n, were fit by adjustment of the rate constant corresponding to the chain-transfer reaction from B-ended radicals to the modifier, kfBX. (5) The weight-average molecular weights, M h w, were fit by adjustment of the initial vinylacetylene concentration (see the molecular weight module described in Gugliotta et al. (1995)). All of the adjusted parameters resulted within reported literature ranges. In parts a-d and e-h of Figure 1, the results of experiments 1 and 2 are presented, respectively. The model predictions (in continuous trace) are compared with the measurements (in dots). To illustrate the contribution of O2 toward the model predictions, computer simulations assuming the absence of O2 are also included (in dashed lines) in Figure 1. Similarly, the model predictions assuming the absence of vinylacetyl-
Figure 2. More theoretical predictions for the batch experiments 1 and 2. Continuous trace: complete model. Dashed trace: assuming the absence of O2. Dotted lines: assuming the absence of vinylacetylene.
Figure 3. Feed profiles of A for the semibatch experiment 3. (a) Theoretical (optimal) profile. (b) Applied (suboptimal) profile.
ene in experiment 1 are shown (in dotted lines) in Figures 1a-d. The model estimates for other predicted variables in experiments 1 and 2 are presented in parts a-e and f-j of Figure 2, respectively. In the case of Figure 2a,f, the asterisks do not represent independent measurements of the number of particles, Np, since they were indirectly calculated from the measurements of x and d h p,unsw. The following comments can be made: (1) Monomeric A radicals are easily desorbed from the polymer particles, while monomeric B radicals practically do not desorb. Also, since kpAB[B]p . kpBA[A]p, the fraction of B-ended radicals in the polymer particles (with respect to the total amount of free radicals in the particles) is high during most of the polymerization. This determines that the global system behavior re-
Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1241
Figure 4. Main reaction variables for the semibatch experiment 3. Measurements are indicated by dots, while the model predictions are shown in the continuous trace.
sembles a case II kinetics with n j = 0.5 more than a case I kinetics with n j , 1. (2) The model predicts that all the intermediate initiator feeds were applied after the nucleation stage had ended. For this reason, the number of polymer particles remained unchanged by the initiator additions, and therefore, the total effect was relatively reduced (only small step changes in n j that correspond with the initiator injections are visible in Figure 2b,g). (3) The effects of O2 are more evident in experiment 2, where parameter adjustment suggests a higher initial concentration of this impurity (9.4 × 10-4 mol/dm3 in experiment 2 vs 9.4 × 10-5 mol/dm3 in experiment 1). O2 delays the start of the polymerization and distorts the whole shape of x(t), with the final conversion being slightly higher than in the O2-free case. This last effect may be explained by the higher Np values when the O2 concentration is increased. To understand this, consider first the O2-free case. In the absence of O2, the change in the recipe has a negligible effect both on n j and Np, as indicated by the dashed curves of parts a, b, f, and g of Figure 2. Since O2 preferentially dissolves in the organic phase, a higher free-radical deactivation in the polymer phase than in the water phase is expected to occur. For this reason, as the global O2 concentration is increased, n j decreases, the particle growth becomes slower, the consumption of the emulsifier and micelles to cover the slowly-increasing particle area is reduced, more micelles can be transformed into polymer particles, and Np increases. The decreased polymerization rate per particle with a presence of O2 also reduces the average particle diameters. (4) PAN homopolymer is only produced in the aqueous phase at a very low rate along the polymerization. During the initial inhibition period, the copolymerization in the particles is very slow due to the low values of n j and Np. For this reason, both the overall polymer
composition, p j A, and the fraction of PAN with respect to the total polymer, wAb,w, are initially high. However, soon after the inhibition period is over, the A homopolymerization rate is rapidly overpassed by the main copolymerization rate. After an initial transient, a relatively slow compositional drift is observed in p j A, and this is a consequence of the reactivity ratios (rA ) 0.03 and rB ) 0.25) and of the monomer partitions. For the investigated system, the composition of the instantaneously-produced copolymer is always close to that of the azeotrope (approximately 42% of bound A). Quality requirements on the compositional drift are quite strict, however, and even variations of a few percent may affect the polymer quality. The total amount of PAN in the final product is low (below 0.2%). (5) M h n changes very moderately along the polymerization and even exhibits a slight maximum. The mathematical model provides an explanation for this (rather unusual) behavior: M h n values are mainly determined by the rates of propagation and of transfer to the modifier and to the monomers; and it so happens that their variations tend to compensate with each other along the polymerization. Since the homopolymerization of A in the water phase is not considered in the molecular weight module, errors in excess in M h n are expected at the very beginning of the polymerization. (6) As expected, M h w and the branching frequencies, h N4, all increase along the polymerization as a B h N3 and B result of the reactions with the accumulated polymer. These changes are more pronounced in experiment 1, due to the presence of vinylacetylene. Semibatch Operation for Composition Control. Experiment 3 was designed to reduce the (already quite moderate) compositional drift observed in the batch experiments. Minimum-time strategies for composition control require that the reactor must be initially charged with all of the less reactive monomer (B) and with the necessary amount of the more reactive monomer (A) so as to initially produce the desired copolymer composition. Then, the remaining amount of the more reactive monomer may be added to maintain constant the comonomer concentrations ratio in the polymer phase (Hamielec and MacGregor, 1983). To calculate the optimal feed profile of A, a closedloop numerical procedure was applied on the mathematical model, with the aim of maintaining a constant [A]p/[B]p ratio such that the required instantaneous copolymer composition was produced. In Figure 3a, the continuous feed profile of A, FA,in|theor, that is required for producing a constant global polymer composition, p jA ) 33.5%, is presented. Due to the difficulties with the practical application of such a profile, the continuous A addition was replaced by the train of the three impulsive additions that are represented in Figure 3b. Such impulsive (open-loop) additions were applied on a conversion basis, to avoid the batch-to-batch changes in reaction time due to impurities. In Figure 4, the results of experiment 3 are presented. For the computer simulations, the impurities concentrations were a posteriori adjusted to fit the experimental data. The polymer compositional drift was reduced from 39.0% to 34.2% in experiment 1 and 35.5% to 33.3% in experiment 3. The predicted values of p j A in Figure 4b are not perfectly constant due to the model errors and because a rather crude suboptimal feed profile had in practice been applied.
1242
Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997
Concluding Remarks The proposed mathematical model adequately predicts the evolution of the main variables in an industrial NBR process. A single set of kinetic and physicochemical parameters proved to be adequate for simulating the three investigated experiments, and only the individual adjustment of the impurities concentration was required. In particular, O2 was seen to distort the complete conversion profile by affecting the number of particles produced during the nucleation period. Also, a polymerizing and branching impurity (vinylacetylene) was seen to seriously affect the molecular weights. In spite of the relatively high water solubility of acrylonitrile, the mass fraction of PAN homopolymer in the end product is, in practice, negligible. Also, the copolymer compositional drift was moderate in the batch cases since the experiments were carried out near to the azeotropic concentration. This should not be the case of other NBR grades, however, where semibatch (rather than batch) operations should be more fully justified. More generally, the present use of process computers for the sequencing and control of discontinuous NBR processes greatly facilitates the feasibility of on-line calculation and implementation of semibatch operations aimed at improving the productivity and product quality. Acknowledgment We are grateful to PASA S.A., in particular to J. L. Azum, E. O. Iturralde, and R. A. Castillo of that company, for providing us with the experimental data and the polymer samples, to CONICET, Universidad Nacional del Litoral, and PASA S.A. for their financial support, and to J. L. Castan˜eda for his help with the analytical measurements. Nomenclature Reagents and Products A, B ) acrylonitrile and butadiene A•, B• ) monomeric radicals A•n, B•n ) A-ended and B-ended propagating radicals of chain length n A•n,w ) PAN macroradical produced in the aqueous phase E ) emulsifier I ) initiator K•O ) stable free radical Pn ) copolymer produced in the polymer phase Pn,w ) PAN produced in the aqueous phase Ra ) reducing agent R•c ) primary initiator radical produced in the water phase R•n ) propagating copolymer radical in polymer phase, representing either A•n or B•n X ) modifier X• ) modifier radical Variables Ap ) total surface area of the polymer particles (dm2) Am ) total surface area of the emulsifier micelles (dm2) B h N3 ) trifunctional branching frequency, in moles of branches per mole of polymer (dimensionless) B h N4 ) tetrafunctional branching frequency, in moles of branches per mole of polymer (dimensionless) d h p ) monomer-swollen number-average particle diameter (dm) d h p,unsw ) unswollen number-average particle diameter (dm) Fi,in ) inlet molar flow rate of reagent i (mol/min) H ) variable defined by eq B.40 (dm-1) [i]j ) concentration of species i in phase j (mol/dm3)
m ) defined by eq B.25 (dimensionless) h w ) number- and weight-average molecular weights M h n, M (g/mol) n j ) average number of radicals per particle (dimensionless) n j A, n j B ) average number of A-ended or B-ended radicals per particle, respectively (dimensionless) Ngen ) particle generation rate (min-1) Ni ) moles of reagent i (mol) NAb,j, NBb,j ) moles of A and B polymerized in phase j (mol) Np ) total number of polymer particles in the reactor (dimensionless) p ) defined by eq B.28 (dimensionless) p j A ) average mass fraction of polymerized A in the accumulated polymer (dimensionless) qH2O,in ) inlet flow of pure water (dm3/min) RpA,w, RpA,p ) polymerization rates of A in the water and polymer phases (mol/(dm3 min)) RpB,p ) polymerization rate of B in the polymer phase (mol/ (dm3 min)) RI ) initiation rate (mol/min) t ) time (min) VH2O ) pure water volume (dm3) vj p ) swollen number-average volume of the latex particles (dm3) Vp ) polymer-phase volume (dm3) VT ) total reaction volume (dm3) Vw ) aqueous-phase volume (dm3) wAb,w ) mass fraction of the polymer produced in the aqueous phase (dimensionless) W ) defined by eq B.27 (dimensionless) x ) conversion (dimensionless) Y ) defined by eq B.26 (dimensionless) Kinetic and Physicochemical Parameters AS ) emulsifier surface coverage capacity (dm2) DpA, DwA ) diffusion coefficients of A radicals in the polymer and water phases (dm2/min) [E]CMC ) emulsifier critical micellar concentration (mol/ dm3) k1 ) rate constant for generation of primary free radicals (dm3/(mol min)) k2 ) rate constant of the reduction reaction in the initiation mechanism (dm3/(mol min)) ka ) rate constant of radical absorption into particles (dm/ min) kde ) average rate constant of radical desorption from the particles (min-1) kdeA•, kdeB• ) desorption rate constants of the A and B monomer radicals (min-1) kfAA, kfAB, kfBA, kfBB ) rate constants of the transfer reactions from the A- or B-ended radicals to the A or B monomers (dm3/(mol min)) kfAX, kfBX ) rate constants of chain transfer from the Aand B-ended radicals to the modifier (dm3/(mol min)) km ) rate constant of radical absorption into the emulsifier micelles (dm/min) kh ) rate coefficient for homogeneous nucleation, as defined in eq B.37 (min-1) kh0 ) homogeneous nucleation rate constant (min-1) kpAA, kpAB, kpBA, kpBB ) propagation rate constants in the polymer phase (dm3/(mol min)) kpAA,w ) A homopropagation rate constant in the water phase (dm3/(mol min)) kpc ) rate constant for generation of primary monomeric free radicals (dm3/(mol min)) ktp ) rate constant for radical termination in the polymer phase (dm3/(mol min)) ktpOR ) rate constant for radical deactivation by O2 in the polymer phase (dm3/2/(mol1/2 min)) ktw ) rate constant for radical termination in the water phase (dm3/(mol min))
Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1243 ktwOFe ) rate constant for the oxidation of Fe2+ in the water phase (dm3/4/(mol1/4 min)) ktwOR ) rate constant for radical termination with O2 in the water phase (dm3/2/(mol1/2 min)) KAwp ) partition coefficient of A between the water and the polymer phases (dimensionless) L ) average diffusion path length of the free radicals in the aqueous phase (dm) MA, MB ) molecular weights of A and B (g/mol) NAv ) Avogadro’s constant (mol-1) rA, rB ) reactivity ratios of A and B (dimensionless)
ktw
A•n,w + A•m,w 98 Pn,w + Pm,w or Pn+m,w
Polymer-Phase Reactions. These include the following:
propagation kpAA
A•n + A 98 A•n+1 kpBA
B•n + A 98 A•n+1
Greek Symbols R, R′ ) variables defined by eqs B.23 and B.24, respectively (dimensionless) β ) variable defined by eq B.33 (dimensionless) ) relationship defined by eq B.41 (dimensionless) µ ) relationship defined by eq B.42 (dm-1) F ) global absorption rate of free radicals (mol/(dm3 min)) FA, FB ) densities of A and B (g/dm3) Fp ) polymer density (g/dm3) φP,p ) polymer volume fraction in the polymer phase (dimensionless) ω0A ) variable defined by eq B.32 (dimensionless) Subscripts
(A.8)
kpAB
A•n + B 98 B•n+1 kpBB
B•n + B 98 B•n+1
(A.9) (A.10) (A.11) (A.12)
termination ktp
R•n + R•m 98 Pn + Pm or Pn+m
(A.13)
where R•n indistinctly represents A•n or B•n in the polymer phase and Pn is the inactive polymer.
m ) monomer phase p ) polymer phase w ) water phase
transfer to the monomers kfAA
Appendix A: Kinetic Mechanism Adopted for the Basic Module
A•n + A 98 Pn + A•
Aqueous-Phase Reactions. Initiation. Equations A.1 and A.2 represent the production of primary radicals via a redox couple, while eqs A.3 and A.4 represent the production of the first adducts:
B•n + A 98 Pn + A•
k1
I + Fe2+ 98 R•c + Fe3+ + OHk2
Fe3+ + Ra 98 Fe2+ + Ra+ kpc
R•c + A 98 A•1,w kpc
R•c + B 98 B•1,w
(A.1) (A.2) (A.3) (A.4) Fe2+
where I is the initiator (diisobutyl hydroperoxide), is the ferrous ion produced from SO4Fe‚7H2O, R•c is a primary radical, Ra is the reducing agent (sodium formaldehydesulfoxylate), A and B are the comonomers, and A•1,w and B•1,w are the monomeric primary radicals. Deactivation by Oxygen. This occurs via ktw
OR
A•n,w + 1/2O2 98 Pn,w + K•O
(A.5)
ktw
OFe
Fe2+ + 1/4O2 + 1/2H2O 98 Fe3+ + OH- (A.6) where A•n,w is a growing poly(acrylonitrile) (PAN) macroradical of chain length n ()1, 2, ...) in the aqueous phase, Pn,w is a PAN molecule of chain length n produced in the aqueous phase, and K•O is a stable free radical. Homopropagation and Homotermination in the Aqueous Phase. These occur via kpAA,w
A•n,w + A 98 A•n+1,w
(A.7)
kfBA
kfAB
A•n + B 98 Pn + B• kfBB
B•n + B 98 Pn + B•
(A.14) (A.15) (A.16) (A.17)
transfer to the modifier (X) kfAX
A•n + X 98 Pn + X• kfBX
B•n + X 98 Pn + X•
(A.18) (A.19)
deactivation by oxygen ktp
OR
R•n + 1/2O2 98 K•O + Pn
(A.20)
Appendix B: The Basic Module Differential Equations. These include the following:
dNA ) FA,in - RpA,pVp - RpA,wVw dt
(B.1)
dNB ) FB,in - RpB,pVp dt
(B.2)
dNI ) FI,in - k1[I]wNFe2+ dt dNFe ) FFe,in dt
(B.3) (B.4)
1244 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997
dNFe2+ ) FFe2+,in - k1[I]wNFe2+ - ktwOFe[O2]w1/4NFe2+ + dt NFe - NFe2+ k2 NRa (B.5) Vw dNRa NFe - NFe2+ ) FRa,in - k2 NRa dt Vw
(B.6)
Morton et al. (1954), originally proposed for homopolymer systems (see Ugelstad et al. (1983)). Average Number of Radicals per Particle. In the aqueous phase, the following balance of radicals may be written (Huo et al., 1988):
n j Np Ap ) ka [R•]w + ktw[R•]w2 + RI + kde NAvVw VT ktwOR[O2]w1/2[R•]w (B.20)
dNE ) FE,in dt
(B.7) where the initiation rate RI is defined by
n j Np dNX kfAXkpBA[A]p + kfBXkpAB[B]p ) FX,in [X]p dt NAv kpBA[A]p + kpAB[B]p (B.8) dVH2O dt dNO2 dt
) qH2O,in
(B.9)
NFe2+ RI ) k1 [I]w Vw
The average number of radicals per particle is calculated through
n j ) 0.5
1/2
) FO2,in - ktwOR[R ]w[O2]w Vw •
ktwOFe[O2]w1/4NFe2+ - ktpOR[O2]p1/2
n j Np (B.10) NAv
dNAb,p ) RpA,pVp dt dNAb,w ) RpA,wVw dt
R ) R′ + mn j - R2Y - RW
R′ ) (B.12) (B.13)
dNp ) Ngen dt
(B.14)
RpA,pVp )
j Np kpAAkpBB(rA[A]p + [A]p[B]p) n kpBBrA[A]p + kpAArB[B]p NAv
(B.15)
RpB,pVp )
j Np kpAAkpBB(rB[B]p2 + [A]p[B]p) n kpBBrA[A]p + kpAArB[B]p NAv
(B.16)
RpA,wVw ) kpAA,w[A]w[R•]wVw
(B.17)
The volumes of the polymer and aqueous phases are calculated by
V H 2O MA[A]w MB[B]w 1FA FB
Y)
The partitioning of A, B, I, X, and O2 into the three phases was calculated as in Gugliotta et al. (1995). In particular, the monomer concentrations were obtained from an extension to the copolymer systems developed by Guillot (1981) of a thermodynamic treatment by
(B.24)
kdevj p N ktp Av
(B.25)
ktpktwNp kaAp 2 vj pVwNAv2 VT
( )
W)
p)m+
ktwOR[O2]w1/2
(B.27)
kaAp VT ktpOR[O2]p1/2vj p ktp
(B.26)
NAv
(B.28)
Desorption Rate Coefficient. Only monomeric radicals are allowed to desorb. Following Nomura et al. (1983), the average desorption rate coefficient for monomeric radicals, kde, is calculated through
(B.18)
(B.19)
RIVwvj p N 2 Npktp Av
m)
2
Vw )
(B.23)
(B.11)
dNBb,p ) RpB,pVp dt
MA(NAb,p + NAb,w) + MBNBb,p FpφP,p
2R (B.22) 2R p+ p + 1 + [2R/(p + 2 + ...)]
with
The rates of polymerization in the polymer and aqueous phases can be calculated through
Vp )
(B.21)
kde )
n jA n jB kdeA• + k n jA + n jB n jA + n j B deB•
(B.29)
with
n j B kpAArB[B]p ) n j A kpBBrA[A]p
(B.30)
For the NBR system, kdeB• ≈ 0. The parameter kdeA• is obtained through an extension to a copolymerization system of the model proposed by Asua et al. (1989), resulting in
kdeA• )
(
Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1245
)
n jB [A]p kfAA + kfBA ω0A n jA
µ)
βω0A + kpAA[A]p + kpAB[B]p + kfAX[X]p
(B.31)
ω0A ) 2
d hp
β)
(
)
w kpAA [A]w
Ap Am w kpAA [A]w + ka + km + kh VT VT
(B.42)
From eqs B.34 and B.35, the total concentration of free radicals in the aqueous phase results:
with
12DwAKAwp 2DwAKAwp 1+ DpA
kh0 km
(B.32)
(B.33)
(
)
Ap Am + kh + ktwOR[O2]w1/2 - ka + k m V VT T • [R ]w ) + 2ktw
(x(
Ap Am ka + km + kh + ktwOR[O2]w1/2 VT VT
)
2
(
))
n j Np + 4ktw RI + kde / NAvVw (2ktw)
(B.43)
Particle Generation Rate and Free-Radical Concentration in the Aqueous Phase. During the nucleation period, a radical balance in the aqueous phase yields
Conversion, Copolymer Composition, and Mass Fraction of Copolymer Produced in the Aqueous Phase. The total conversion, the average mass fraction of A in the accumulated copolymer, and the mass fraction of copolymer produced in the aqueous phase can be computed from
n j Np ) F + ktw[R•]w2 + ktwOR[O2]w1/2[R•]w RI + kde NAvVw (B.34)
x)
MA(NAb,p + NAb,w) + MBNBb,p MA(NAb,p + NAb,w + NA) + MB(NBb,p + NB)
with
Ap Am F ) ka [R•]w + km [R•]w + kh[R•]w (B.35) VT VT where F is the global absorption rate of free radicals. Admitting that particles are formed via the micellar and the homogeneous mechanisms, the particle generation rate, Ngen, is calculated from (Pollock, 1983)
(
)
Am Ngen ) km + kh [R•]wVwNAv VT
(B.36)
To estimate the kh coefficient in eqs B.35 and B.36, Fitch and Tsai (1971) have proposed
[
kh ) kh0 1 -
]
LAp 4VT
(B.37)
with the homogeneous nucleation stopping when kh e 0. Replacing eqs B.35 and B.37 in eq B.36, the following is obtained:
Ngen )
FVwNAv Ap 1+ Am + HVT
(B.38)
with
Am ) AS(NE - [E]CMCVw) - Ap
H)
{(
µ 10
)
LAp 4VT
)
if LAp e 4VT
(B.39)
(B.40)
if LAp > 4VT ka km
(B.41)
p jA )
MA(NAb,p + NAb,w) MA(NAb,p + NAb,w) + MBNBb,p
wAb,w )
(B.44) (B.45)
MANAb,w (B.46) MA(NAb,p + NAb,w) + MBNBb,p
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Received for review August 28, 1996 Revised manuscript received November 12, 1996 Accepted November 13, 1996X IE9605342 X Abstract published in Advance ACS Abstracts, February 15, 1997.
