1822
Znd. Eng. Chem. Res. 1993,32,1822-1830
Modeling and Experimental Studies of Aqueous Suspension Polymerization Processes. 1. Modeling and Simulations George K a l f a s and W.H a r m o n Ray' Department of Chemical Engineering, University of Wisconsin-Madison, Madison, Wisconsin 53706
The modeling and simulation of free radical polymerizations in different processes (bulk, solution, suspension, emulsion, dispersion) should not require altered sets of kinetic rate constants for the same monomer or monomer mixture system. Appropriate handling of the different physical picture in each process and a n accurate description of all the elemental steps (physical and chemical phenomena) can result in universal models built on the common kinetic mechanism of free-radical polymerizations. In this work we have extended our homogeneous free radical polymerization model to consider the suspension polymerization of partially water-soluble monomers. Simulation results from this model are compared to experimental results from the literature and from our own suspension polymerization experiments. Introduction In aqueous suspension polymerization of the bead type, one or more liquid monomers are polymerized while dispersed into droplets of 0.1-5 mm in diameter by a combination of strong stirring and the use of small amounts of suspending agents (stabilizers). If the monomer is a solvent for the polymer the particles pass througha viscous syrupy state and finally form small and clear spheres (beads). Initiators used in suspension polymerization are oilsoluble, and polymerization takes place within the monomer droplets. Typical initiator compositions are 0.1-0.5 w t 7% based on monomer. A simple calculation shows that the monomer droplets are large enough to contain a very large number of free radicals (-lo*). Therefore the kinetic mechanism is the same as that of bulk polymerization (Munzer and Trommsdorff, 1977), and the same kind of dependence of polymerization rate on initiator concentration and temperature is observed. Bead BUSpension processes are considered to be water-cooled microbulk polymerizations in the monomer droplets. A more detailed description of all the aspects of suspension polymerization processes may be found in a recent review article (Yuan et al., 1991). When the solubility of the monomer in the aqueous phase is negligible (as it is for styrene), quantitative predictions are possible for batch suspension polymerizations using the same models and rate constants used for the corresponding homogeneous (bulk or solution) polymerizations. However, in the case of monomers which are partially dissolved in the aqueous phase, quantitative agreement between the experimental results and the model simulation requires modified rate constants. Values reported for the propagation rate constant, k,, in the polymerization of vinyl acetate vary significantly between bulk and suspension processes (Taylor and Reichert, 1985). Monomer solubility also affects the polymerization rates, the composition, and the molecular weights of suspension copolymers. In the suspension copolymerization of styrene-acrylonitrile, the apparent reactivity ratios differ from those reported from bulk or solution processes (Mino, 19561, and beads of different sizes have been found to contain different molecular weight polymer (Bahargava et al., 1979). In this paper we first present a simpler version of our homogeneous free-radical polymerization models from the *To whom correspondence should be addressed.
CAD software package POLYRED (Christiansen et al., 1990). Then we extend this model to account for the partition of the monomer(s) into two phases and the monomer transport between them in order to model suspension homopolymerizations and copolymerizations of partially water-soluble monomers. The monomer partition is calculated based on an extension of the FloryHuggins theory of mixing. Simulation results from these models are given here. The extended model provides a more precise representation of the physical picture of the bead suspension polymerizationprocess. Using this model, one may obtain good quantitative agreement with experimental results, without modifications of the kinetic rate constants. Our two-phase model treats the liquid-liquid equilibrium in a similar way to the monomer partition model proposed by Delgado et al. (1988) for miniemulsion polymerizations. However, the polymerization kinetics modeling equations are different from ours. In miniemulsion polymerization, with monomer droplets extremelly small in volume, the reaction terms are calculated based on a discrete number of radicals per particle in the same way as in emulsion polymerization models. In our model for suspension polymerization, the concentration of radicals may be used in the calculation of the reaction terms just as in bulk polymerization, because the droplets in suspension polymerization are large enough to contain a large number of radicals.
Free-Radical Polymerization Kinetics
A kinetic model for free-radical polymerizations adapted from a comprehensive model for addition polymerizations (Arriola, 1989) is the basis of our polymerization model. Kinetic Scheme. The kinetic mechanism for freeradical copolymerization is listed in Table I, where D, refers to dead polymer with chain length n and Pi refers to live polymer radicals of length n with monomer type i at the growing end. This scheme is a generalization of the homopolymerization case. The followingassumptions are incorporated into the model: (i) The long-chain hypothesis (LCH) is assumed valid. (ii) The reactor steady-state approximation (RSSA) for all radical species is used. (iii) Non-polymer radicals react very rapidly with monomer to form live polymer chains. The copolymerization kinetics model contains an additional assumption besides those for the homopolymerization case.
0888-588519312632-1822$04.00/0 0 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, No. 9,1993 1823 The live and total polymer number chain length distribution (NCLD) moments are defined as follows:
Table I. Reaction Mechanism of Free-Radical CoDolymerization initiation:
OD
live moments:
fkd
peroxide or azo compound decomposition
I-2R' R' + Mi
-
font
propagation:
+ Mi
-
kPY
Pl
(D
total moments:
,
.
-
P;
by inhibition
k+i P;+X-D,
spontaneous
.% Pk D,
-
chain transfer:
-
k-4
to monomer
P;+Mj
to solvent
PI + S --c D,
to agent
P;
k..,
reinitiation:
C~'([P,I + [D~I)
where [P,] and [D,] represent the concentrations of live and dead polymer chains with length n, respectively. The relationships between the average polymer properties and the moments are given by
Pi+l
+ P,. %D, + D,
with disproportionation
xi =
n=l
termination: with coupling
C~'[P,I n=l
C + Mj- Pi P;
=
,
Zi
special initiation
pi
D,+Mj'
+ S'
+ A kq. -,D, + A'
Mj', s',A', or H'+ Mi
-
fat
,
P',
(iv) Quasi-stationarity of the instantaneous copolymer composition and radical concentrations is assumed. A t any instant all live chains are treated as having identical composition and sequence lengths. When this assumption is satisfied, it is possible to model the kinetics with the use of apparent rate constants. This method yields a set of apparent rate constants by averaging the reaction rate constants based on the monomer mixture and instantaneous live copolymer compositions. The apparent rate constants are then used to model the copolymerizationas a pseudo-homopolymerizationsystem. Detailed examples of this approach are available in the literature (Hamielec and MacGregor, 1983;Kuo and Chen, 1981; Kuo et al., 1984). While peroxide or azo compound decomposition (Rd = 2fkdc1) is the most common type of initiation reaction, a "special" initiation reaction has been included in the mechanism to allow for other types of initiations. The special initiation rate (Ri= gkictc$j hut) may be configured for a second initiator decomposition (Ri = gk,c$), thermal initiation (Ri = kit$), or photoinitiation (Ri = gkic$hvc). The modeling equations are derived from mass balances for the non-polymer species and population balances for the polymer species. The large number of polymer species balances (one for each chain length) are lumped to a few ode's describing the rates of change of the lower moments of the chain length distribution using the method of moments (Ray, 1972) (see Appendix). Finally the rates of change of the polymer properties are expressed in terms of the rates of change of the polymer moments.
DP, = A,& Zp = X,A& (3) where DPn is the number chain length, and Z, is the polydispersity index of the chain length distribution. The rates of change of the average polymer properties can be expressed in terms of the moment rates by a simple differentiation of the equations (3). Copolymerization Composition and Sequence Lengths. The prediction of the copolymer composition molar fractions, Fpi, and sequence length averages, SLi, i = 1,2,...,N ( N is the number of monomers), is based on the assumption that the cross-propagation reactions determine the system state at any instant. Radicals and growing polymer species of any type are assumed at their steady-state concentration (RSSA). Under these assumptions the polymer radical concentrations are calculated by solving a system of N linear homogeneous equations (see Appendix). The instant copolymer composition, InstFpi, results also from the solution of another set of Nlinear homogeneous equations. The average instant sequence lengths are easily calculated based on probability arguments. The average cumulative copolymer composition and sequence lengths are calculated by integration of the corresponding instantaneous quantities. Copolymerization Kinetic Parameters. For copolymer systems, each set of homopolymer rate constants is determined separately, with the cross reaction rate constants specified by the reactivity ratio matrices. A set of apparent rate constants is calculated in order to be used in the calculations of the chain length distribution moment rates. The homopolymer rate constants and the cross-reaction rate constants are averaged , the over the comonomer mixture composition, F M ~and instant radical composition, F h , both expressed in mole fractions. The apparent rate constants are defined as follows:
Gel Effect Correlations. Diffusion limitations at high conversions lead to dramatic decrease in the termination rate resulting in an autoacceleration of the polymerization, a phenomenon known as Tromsdorff, Norrish-Smith, or gel effect. At even higher conversions, propagation becomes diffusion controlled and eventually polymerization stops (glass effect). The presence of the gel effect introduces severe nonlinearities to the models of free-radical polymerizations.
1824 Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993
Several gel effect correlations, some empirical and others semiempirical, have appeared in the literature. In our modeling work we have employed three of them: a freevolume-based correlation introduced for the bulk polymerization of methyl methacrylate (MMA) by Ross and Laurence (1976) and extended to the solution polymerization by Schmidt and Ray (1981),an empirical correlation for styrene polymerization by Hamer and Ray (1983),and a diffusion-based correlation for MMA polymerization by Chiu et al. (1983),extended to copolymerizationby Sharma and Soane (Soong) (1988). Gel and glass effect factors may be optionally applied to any of the free-radical mechanism reactions. For copolymersystems, a compositionweighted gel effect factor is applied to the apparent rate constants.
Homogeneous Well-Mixed Tank Reactor Model The two-phase model is an extension of a homogeneous well-mixed tank reactor model. Thus a description of the homogeneous reactor model features is needed first. The polymerization reactor is modeled as a well-mixed tank with one inlet and one outlet stream. The model can be used to describe different reaction media, outflow types, and thermal conditions. Some of the user configurable options available in the POLYRED package include the following: (i) Optional presence of a solvent allows the reactor model to describe both bulk and solution reactors. (ii) Batch, semibatch, or continuous flow operation, including startup. (iii) Underflow or overflow under constant volume or pressure. (iv) Isothermal, driven, adiabatic, or nonisothermal operation. (v) One or more reactor units may be incorporated into a flowsheet. The use of multistream units like stream mixers and splitter1 separators allow for flowsheets with intermediate feeds and recycle streams. The reacting mixture is modeled as a single homogeneous phase, consistingmainly of monomer, solvent and polymer. In copolymerizations, a mixture of up to five monomers is allowed. Only monomer(s), solvent, and polymer are assumed to have a significant volume, with other species (initiator, transfer agent, inhibitor) having negligible volume contributions. Reactant volume in the well-mixed tank reactor is calculated from the total mass balance: dmldt = mh - mout(=) d(Vp)ldt = Qinph- Qp ( 5 ) where p is the density of the reacting mixture calculated at the reactor temperature, pressure, and mixture composition. The full form of the energy balance around the stirred tank reactor is d(Vpe)
dt
+ c-dT = Qhpheh - Qpe + Einput dt
where c is the external heat capacitance of the reactor walls, the stirrer etc. Heat transfer is considered both to/from a cooling/ heating jacket and to/from the environment. Miscellaneous energy input to the system, such as that from stirring or radiation, is considered through the term Einput. To have a consistent definition of the fractional monomer conversion to polymer, xp, in all inflow-outflow conditions, xp is defined at any instance as the weight fraction of the monomer units incorporated into polymer
chains over the total monomer units in the reactor: xp
=
WP wM+
(7) wP
where V is the volume of the reacting mixture, Q is the volumetric flow rate, p is the density, ws is the weight fraction of the solvent, MM is the monomer molecular weight, and RX1 is the rate of change of the first moment of the polymer chain length distribution (i.e,, the polymerization rate in mol/L). In the copolymerization case individual monomer balances for each monomer in the mixture are required:
where FM~ is the mole fraction of monomer i in the monomer mixture, InstFp, is the mole fraction of the monomer i in the instantly formed polymer, and MM is the average monomer mixture molecular weight. To complete the model, it is necessary to specify the rates of evolution of the nonpolymer species weight fractions (solvent, initiator, transfer agent, inhibitor, and activator).
Suspension Polymerization Model For partially water-soluble monomers, the single-phase tank reactor model presented in the previous section predicts faster polymerization rates than those observed experimentally. The experimental calculation of the monomer conversion is based on the total monomer charge of the reactor. However, since oil-soluble initiator@)are used and radicals are formed only in the organic phase, the amount of monomer dissolved in the inert aqueous phase does not react until it is transported back to the polymerizing droplets. The reason for the monomer transport is a shift in the thermodynamic equilibrium between the two phases as the fraction of polymer in the droplets increases. Modeling of Interphase Monomer Transport. In the model presented here it is assumed that thermodynamic equilibrium between the aqueous and the organic phase is established before the polymerization starts. During the dispersion of the monomer droplets the aqueous phase becomes saturated with dissolved monomer. As the polymerization proceeds in the monomer droplets, the monomer concentration decreases from its equilibrium value (with respect to the dissolved monomer in the aqueous phase) forcing mass transfer from the aqueous phase to the droplets. For the rate of the monomer transport the following expression is used:
where is the overall mass transfer coefficient of monomer i [cmlsl from the aqueous phase "a" to the droplets "d", Ad is the interfacial surface area of the
Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993 1825 droplets [cmz], c$ is the concentration of monomer i in the droplets a t equilibrium with the amount of monomer i dissolved in the aqueous phase [mol/Ll, and Ci,d is the actual concentration of monomer i in the droplets [mol/Ll. The equilibrium concentrations are calculated based on the Flory-Huggins lattice theory of polymer solutions and the modifications by Ugelstad et al. (1980). The free energy of mixing in the two phases is given below: (i) Organic phase-droplets (monomer(s)+ polymer + swelling agent):
d[Vp(l- ws)(l- xp)]/dt = Qinp"(l - W;)(I -$) Qp(1- Ws)(l- xp) - VMMRX,
+
N ill
Additionalbalances are needed for each monomer which is partially dissolved in the aqueous phase: d[w,~WaI/dt
W&QF
) --
WM
where 4i,d represents the volume fraction of species i in phase d, xij is the Flory-Huggins interaction parameter of species i and species j , and mij (=vi/vj)is the ratio of the molar volumes of species i and species j (equivalent segment fraction). The last term of eq 11 accounting for the interfacial tension energy may be neglected for droplets of the size encountered in suspension polymerizations (rd > 50 pm). Also, the presence of water in the organic phase is neglected in this relation. (ii) Aqueous phase (monomer(s) + water):
- Wa,MQa -
i = 1, ...,N (15) ki,a4AdM%(C$l- Ci,d), where wa,M is the weight fraction of monomer in the aqueousphase, Wa is the totalweight of the aqueous phase, and Qa is the exit mass flow rate of the aqueous phase. In the copolymerization model a correction is also required to the individual monomer balances (eq 9), in order to account for the transfer of each monomer between the two phases: d( VpO- wsx; - np)Fy dt Q"ph(l - w;)(l-
N
N-1
- ci,d) (14)
xki,a+AdMM,(c::
-
N
-
z$)%- Qp(1- w,)(l-
~p)Fq
-
@M
i = 1,...,N (16) The energy balance should include the aqueous phase which is assumed to be at the same temperature as the organic phase:
+
+
d(Vpe) d(Waea) c-d T = pp"e" - Qpe QFeF dt dt dt Qaea+ Einput - [UjAj(T- Ti) + UaAa(T- Tall + Veprd (17) The assumption that the monomer droplets have the same temperature as the surrounding aqueous phase may be justified from order of magnitude arguments. The Ranz-Marshall correlation for the Nusselt number (Ranz and Marshall, 1952) provides an estimate for the heat transfer coefficient, h, a t the boundary layer of a droplet:
+-
+
N
N-1
N
Modificationsof the Tank Reactor Model Balances. A number of modifications are required to the reactor model presented in the previous section in order to simulate the two-phase system. First the monomer conversion must be calculated by including the monomer dissolved in the aqueous phase. The total conversion is then defined as follows:
.P"
=
WP
WM+ Wp+
WM
(13)
where WMis the weight of the monomer dissolved in the aqueous phase. The total monomer balance (eq 8) is modified by taking into account the monomer transport from the inert (continuous) phase through the interface area, Ad:
Nu = 2.0 0.6Pr'/3Re1/2 = hd/k, (18) To be conservative, assume a droplet at rest with Nu = 2. Using the value of the heat conductivity of water and with a droplet size d = 1mm, we estimate the heat-transfer coefficient to be 0.03 cal s-l cm-2 K-I. An energy balance around the monomer droplet gives a temperature rise across the boundary layer as A T = (-AH)dRX,/Gh (19) The maximum polymerization rate, RX1, observed in MMA polymerizations with benzoyl peroxide initiator has been mol/(L/s). Substituting the values given above for h and d yields a temperature rise at the boundary layer of less than 0.1K. Thus the polymerization rate has to increase a t least by 1order of magnitude before significant difference in the temperature of the two phases can be expected. Results and Discussion Although there are more extensive comparisons to our experimental results in part 2 of this work, here we will illustrate the features of the model with only two monomer systems.
1826 Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993 1.0 :
.
,
Mm = 0.2
.... .. .. ....... . .... . .... . . .. ........
0.8
-
___----___
/
M/w
/
0
0.1
0'
VaCBatChsuspensim PolymerizatimEaperiments
' / . ' ' I ' . " ' ' . 0.0
0.2
0.6
0.4
.
I " .
MonomerComdonint~c3ea
Figure 1. Effect of the monomer-to-waterratioon the totalmonomer conversion. Here the monomer partition is always assumed at equilibrium. Vinyl acetate suspensionpolymerizationat 60 4 and 0.5 wt 5% BPO (initiator).
ii
O.OQ)!'
0.0
\ I "
I 0.2
"
'
I 0.4
"
'
I 0.6
"
'
; 0.8
.
.
.
.
.
.
.
.
"
\I' Y
1.0
Monomer C h d o n in the particles
Figure 2. Effect of the mass-transfer coefficient on the amount of monomer remaining in the aqueousphase. Vinyl acetate suspension polymerization at 60 4 and 0.5 w t 5% BPO (initiator).
First, the homopolymerization of vinyl acetate in batch suspension reactors is investigated through a series of simulations. Vinyl acetate is partially soluble in water (2.5-4 w t % at polymerization temperatures) and water is also dissolved in the organic phase (-2 wt '% ) (Luskin, 1970-1971). At low monomer-to-water ratios a significant part of the monomer is dissolved in the aqueous phase (e.g., 20-30% of the total monomer is in the aqueous phase when M/W = 0.1). This monomer remains inert until phase equilibrium forces it to return to the partly polymerized polymer particles to replenish monomer that has been converted to polymer. The total monomer conversion vs the conversion in the organic phase (particles) is shown in Figure 1when it is assumed that liquidliquid equilibrium is attained at every instant. For monomer to water ratios greater than 0.25, the two conversions become almost equal, because the aqueous phase monomer is a small fraction of the total monomer (-5 % at M/ W = 0.5). Nevertheless, the polymerization of this small fraction is still an important problem from the environmental point of view. In Figure 2 the interphase transport mass-transfer coefficient for the monomer (VAc)is varied over 4 orders of magnitude (10L10-8 cm/s). With k m , a d 2 10" cm/s, liquid-liquid equilibrium is achieved for all conversions, because the intrinsic monomer mass-transport rate is significantly faster than the rate of monomer consumption by polymerization. For values of km,a+ lower than 10" cm/s the process becomes increasingly mass-transfer limited. A t the other extreme, km,a+ I 104 cm/s, monomer transport is negligible compared to monomer consumption by polymerization. If the principal mass-transfer resistance is at the boundary layer between the two phases, calculations for
.
.
.
. I
.
.
.
.
.
.
I
eo 120 160 200 240 Figure 3. Experimental conversion profiles from VAc batch suspension polymerizationsalong with simulation predictions from the two phase model. In the simulationsthe mass transfer coefficient was set to zero. 0
10
t
.
I
0.8
40
the mass-transfer coefficient,km,a+, at the boundary layer around the monomer droplet even with the most conservative estimate for the Sherwood number (Sh = 2) result in values greater than 10" cm/s for droplet diameters of the normal size in suspension polymerizations (Ranz and Marshall, 1952). Thus one would not expect mass transfer resistance for the case of vinyl acetate polymerization. Rodriguez et al. (1989) attempted to determine masstransfer coefficients experimentally for the miniemulsion copolymerization of styrene and methyl methacrylate without success. However, they observed significant masstransfer limitations and therefore used in their simulations values for k q a d between 10" cm/s and 1o-S cm/s. These workers suggested that the surfactant-cosurfactant combination could create an additional barrier and thus cause a significant resistance to the transfer of monomer. Our batch suspension experiments with vinyl acetate at low monomer to water ratios produced conversion profiles significantly different from those expected from the homogeneous polymerization model (HFR). A sharp drop of the polymerization rate when almost all the monomer in the polymer particles is consumed suggests that the transport of the dissolved monomer back to the polymerizing particles is hindered. The simulated profiles from the two-phase model (SUHFR) with the mass-transfer coefficient for the monomer set to zero and VAc water solubility of 4.0 wt '% show reasonable agreement with the experimental results at M / W = 0.2 and M/W = 0.1 (Figure 3). The solubility of vinyl acetate in water has been ' at 70 "C (Lindemann, 1967). If reported to be 3.5 wt % we assume, as has been reported for other surfactants, that PVA enhances the solubility of vinyl acetate in water then assuming a value of 4 wt % for the solubility is quite reasonable. The suspension copolymerization of styrene and acrylonitrile is another example process where one of the monomers is partially water soluble. Whereas styrene is almost totally insoluble in water, acrylonitrile may be dissolved to about 5 wt 5% based on total weight of the aqueous phase. Modeling of this process was studied also by Hagberg (1988). In Hagberg's model the partition of acrylonitrile between the two phases was assumed always at equilibrium. The copolymerization ratios (rl = 0.47, r2 = 0.03) show that styrene is incorporated faster into the copolymer and there is an azeotropic composition of the copolymer at 65 mol 5% styrene. For this system, good agreement between our two-phase model predictions and the experimental results of Mino (1956) at two temperature levels is obtained (Figure 4). To fit the profile of the weight fraction of the dissolved acrylonitrile, the mass-transfer coefficient had to be set to 5 X 10-7 cm/s, an unrealistically low value based on
Ind. Eng. Chem. Res., Vol. 32, No. 9,1993 1827
1:1
i 0.1
-
0.4
1 g
. o
0
0.2
0.01
0.01 m l m w o o 2 M x y ) 2 w o o 3 M x x )
0
I
,
104
104
Time ( a 3
0.001
,
I
I
I
I
0.01
0.1
1
lo
loo
\
d k. I6 k ,
Figure 4. Conversion profiles and weight fraction of acrylonitrile in the aqueous phase were predicted accurately by the two-phaee model in isothermal batch suspension copolymerizations of styrene and acrylonitrile. The monomer mass transfer coefficient was set to 5 X 10-7 cm/s. Experimental results by Mino (Mino, 1956). 0.650
1I
t
Figure 6. Effectiveness factor plot indicating the region where the suspension polymerization of a partially water soluble monomer will become mass transfer controlled. The points correspond to vinyl acetate suspension polymerization at 60 -C and 0.5 wt % BPO initiator in droplets of 0.2 mm in diameter and at four different values of the mass-transfer coefficient.
Let us defiie an effectivenessfactor for the polymerizing monomer droplet to be the ratio of the rate of monomer consumption with mass-transfer limitations to the rate without any transfer limitations:
0
r
n ~ 0.0
'
"
I
0.2
" ' I
" ' I 0.4
"
'
0.6
I 0.8
" . I 1.0
Total Mommer Conversion
Figure 5. Copolymer composition drift in suspension copolymerization is stronger when the less reactive monomer is the most soluble in the water.
simple boundary layer resistance at Sh = 2. In this case most of the dissolved acrylonitrile in the aqueous phase returns back to the polymer particles after 50% of the total monomer in the particles has been converted to polymer. Toward the end of the polymerization when the styrene has been completely converted, monomer transport seems to be hindered even more and the weight fraction of acrylonitrile in the water tends to level off to an asymptotic level. This is primarily the result of poor acrylonitrile solubility in the polymer particles after the styrene disappears (cf. Guillot and Guerrero (1981)). Further detailsof this model may be found in Kalfas (1992). In Figure 5 we compare the simulation results with the experimental measurements of the copolymer's composition at several values of the total monomer conversion. The agreement is not good toward the end of the reaction, but the experimental polymer compositions (about 43 % acrylonitrile) given by Mino a t high conversion are inconsistent with an initial charge to the reactor of 51.4 mol % acrylonitrile. The final polymer composition from the experiment suggests that part of the acrylonitrile remains dissolved in the aqueous phase and has not reacted. This fact again suggests the existence of masstransfer limitations from the aqueous phase to the polymer particles. A general criterion for droplet mass-transfer limitations applicable to all polymerization systems and operating conditions is now needed. The following arguments allow for an estimation of the mass-transfer coefficient value for which a certain suspension polymerization reaction will start to become mass-transfer controlled.
where ka = kP[2fkdcd(kt, + ktd)11/2 is the apparent polymerization rate constant. By equating the rate of monomer transfer to the rate of monomer consumption, we obtain Adkm,+(Ckl - CM)
VdkacM
(21)
which yields
The value of the dimensionless ratio dkJ6km,-.d is a measure of the intrinsic rate of mass transfer compared to the intrinsic polymerization rate similar to the square of the Thiele modulus for a diffusion-limited reaction in a catalytic particle. A plot of the effectiveness factor vs this ratio is shown in Figure 6. Four points corresponding to simulations of vinyl acetate suspension polymerization a t 60 "C and 0.5 w t % BPO initiator in droplets of 0.2 mm in diameter and at four different values of the mass-transfer coefficient are also shown. The efficiency factor in these cases was taken to be the minimum of the ratio of the actual rate over the rate of the VAc polymerization in bulk. The initial value of the apparent polymerization rate constant was used for the calculation of the x coordinate. Note that this general analysis confirms that there should be no significant mass-transfer limitations for the vinyl acetate polymerization until k m decreases to 1V cm/s.
Conclusions A homogeneous free-radical polymerization reactor model has been extended to consider the existence of two phases, the equilibrium partition and the transport of monomers between the two phases, in order to simulate the suspension polymerization of partially water soluble monomers. The suspension (two phase) model can be used to study effects of monomer to water ratio, mass-transfer
1828 Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993
coefficient, monomer solubility, droplet diameter, and swelling agent. Simulation results from this model are compared to experimental results from the literature and from our own suspension polymerization experiments. Mass transfer limitations were evident in vinyl acetate suspension polymerizations at low monomer to water ratios and in styreneacrylonitrile copolymerizations. The monomer mass-transfer coefficient value for which mass-transfer limitations will begin to become apparent a t a given polymerization rate and droplet diameter can be easily estimated based on a simple effectiveness factor analysis. In both the examples treated, there appears to be a strong influence of the suspending agents on the masstransfer coefficient and the degree of water solubility of the monomers. Further fundamental studies are required to more quantitatively clarify these issues. In a companion paper, we provide more extensive experimental results and further detailed tests of the model presented here.
Acknowledgment The authors are grateful to the National Science Foundation and the Industrial Sponsors of the University of Wisconsin Polymerization Reaction Engineering Laboratory for their support of this research. Donald Paquet and Pave1 Rosendorf were especially helpful with the numerical algorithm drivers linked to the POLYRED models which were created in this work.
Nomenclature A = chain-transfer agent species A, = heat-transfer area from/to the environment [cm21 Ad = interfacial surface area of the droplet [cm21 Aj = heat-transfer area from/to the jacket [cm21 Ak, = apparent rate constants C = co-initiator or special initiator species text = external heat capacitance of the reactor walls [cal/K] cc = co-initiator concentration [mol/Ll CI = initiator concentration [mol/Ll CM = monomer concentration [mol/Ll Ci,d = actual concentration of species i in the droplets [mol/ cm31 c:,! = concentration of species i in the droplets at equilibrium with the amount of species i dissolved in the aqueous phase [mol/cm31 d = droplet diameter [cml D, = dead polymer chain with n monomer units DP, = number averagechain length (degreeof polymerization) DP, = weight average chain length e = enthalpy of reacting mixture [cal/gl Einput= enrgy input to the reactor [cal/sl f = initiator efficiency Fi,,+ = rate of the monomer transport [mol/sl F M= ~ instant comonomer mixture composition (mole fractions) Fp, = cumulative copolymer composition (mole fractions) FR( = instant radical composition (mole fractions) g = co-initiator efficiency h = heat transfer coefficient [cal s-1 cm-2 K-’1 AH = heat of polymerization [cal/moll I = initiator species InstFpi = instant copolymer composition (mole fractions) k, = apparent polymerization rate constant [s-l] k d = initiator decomposition rate constant [s-11 kt = fluid thermal conductivity [cal s-1 cm-’ K-’1
ki = special initiation rate constant [s-1 or L mol-’ 8-11 ki,a-d = overall mass-transfer coefficient of monomer i from the aqueous phase “a” to the droplets “d” [cm/sl k , = propagation rate constant [L mol-’ s-13 kt, = termination by combination (coupling)rate constant [L mol-ls-l] k a = termination by disproportionation rate constant [L mol-’ 9-11
k,,, = chain transfer to agent rate constant [L mol-’ 8-11 kt, = chain transfer to monomer rate constant [L mol-’ s-ll kt, = chain transfer to solvent rate constant [L mol-’ 8-11 kb, = spontaneous termination rate constant [s-l] ktx = termination by inhibition rate constant [L mol-’ s-11 Mi = monomer species of type i r n i j = (= vi/vj)is the ratio of the molar volumes of species i and species j (equivalent segment fraction) M M = monomer molecular weight [g/moll M M = monomer mixture average molecular weight [g/mol] N = number of monomers in the polymerization scheme Nu = Nusselt number Pi = copolymer chain with n monomer units and an end radical of type i Pn = live polymer chain with n monomer units Q = stream flowrate [L/sl Q, = mass flowrate of the aqueous phase [g/sl Rx, = polymerization rate (=rate of change of the first bulk polymer moment) [mol/(L/s)] rij = propagation reactivity ratio rd = radius of monomer droplet [pm] Pr = Prandtl number R’ = species with a free radical Re = Reynold number S = solvent species SLi = average sequence lengths Sh = Sherwood number U, = heat-transfer coefficient from/to the environment [cal K-1 cm-2 s-11 Uj = heat-transfer coefficient from/to the jacket [cal K-1 cm-2 s-11 V = volume of reacting mixture [L] v d = volume of the droplet [cm3] W, = total weight of the aqueous phase [gl W,,M = weight fraction of monomer in the aqueous phase WM= weight of the monomer dissolved in the aqueous phase ws = weight fraction of the solvent X = inhibitor species xp”” = fractional monomer conversion taking into account also the monomer dissolved in the aqueous phase x p = fractional monomer conversion 2, = polydispersity of the chain length distribution (=DP,/ DPn) Greek Symbols t
Xi
= effectiveness factor = ith total polymer chain length distribution moment
= ith live polymer chain length distribution moment density of the reacting mixture [g/cm3 or kg/L] 4i.d = volume fraction of species i in phase d xij = Flory-Huggins interaction parameter of species i and species j pi
p =
Appendix: Model Derivation Homopolymerization. Applying the moment definitions in eqs 1 and 2 to the population balances of the live and total concentrations of the polymer species yields
Ind. Eng. Chem. Res., Vol. 32, No. 9,1993 1829 the kinetic rates of change of the polymer moments, which for the homopolymerization kinetics are given by flow
+ 2fk,[Il
termination
+ gki[Cla[Mlbhvc
special initiation
y ~ O spontaneous termination (Al) flow
j = 1, ...,N (A6) Under the steady-state assumption for all types of polymer radicals:
R,G) = 0, j = 1,...,N (A71 The N linear equations (A7) can be solved to give the radical concentrations [POI. The composition of the instantly formed polymer is defined by the monomer mole fractions, InstFp,, in the copolymer:
propagation
InstFp,= R M , / P M h ,
j = 1,
...,N
(AS)
=1
- (k, + ktd)PlC10 - Xk,[Jlrl
N
N
+ kp[M1ro
termination
where R M is ~ the rate of consumption of monomer j : N
transfer to M,S,A
- k,[Xl~1
-
N
regular initiation
- (k, + k,)ro 2
-k
constants are used, we need additional equations to predict the evolution for the monomer mixture and copolymer compositions and the average sequence lengths. The rates of change for the live polymer radical concentrations are
inhibition
spontaneous termination (A2)
where C ~ M [ J= I k h [ M l + k d S 1 + kk,[Al Under the RSSA (reactor steady-state assumption; all radical concentrations are constant), the equations can be solved explicitly to give the live polymer NCLD moments PO and ri. The form of the totalpolymer NCLD momenta balances is given as follows: flow
RMj = -{~kpij[P"'lj[Mjl, j = 1, ...,N
The probability of creating n sequential units of Mj or the instant fraction of all Mj sequences with length n is:
Lj["l = p;:l(l
- Pjj)
(A101
where kpjj[Mjl
-
Pii -
=-
FM,
2'
zkpj!Mjl ill
1-1
(All) rji
The instant average sequence length for Mj is m
m
InstSLj = zn=l n L j [ " ]= znp;:'(ln=l
+ 2fk,[Il
regular initiation
(A9)
r=l
pjj)
(A12)
Therefore FMi
7-
+ gki[Cla[MlbhvC + zk,[Jlpo -&= dXl
special initiation
transfer to M,S,A (A3)
ev X ~ - $ A l - v - & 1 dV
+ kp[Ml~o
flow
propagation (A4) flow
+ 2kp[MIrl + k&,2
propagation termination (A5)
The ROCs of the polymer properties are then written as a function of the total polymer moment ROCs. Copolymerization. In addition to the homopolymerization modeling equations where the apparent rate
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Receiued for reuiew December 29, 1992 Accepted May 14, 1993O
* Abstract published in Advance ACS Abstracts, August 15, 1993.
