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Improved Operating Scenarios for the production of acrylonitrile-butadiene emulsions.

INTRODUCTION

Acrylonitrile (AN)-butadiene rubber (nitrile rubber or NBR) latexes are produced on a large scale using a train of 6-12 continuous stirred tank reactors (CSTRs). In addition to the advantage of high production volume, a train of CSTRs would usually yield almost plug flow behavior. The number and size of the reactors can be chosen such that the performance and product characteristics obtained can be similar to those obtained in a batch reactor. NBR is also produced in batch reactors as well as using the semibatch mode of operation, where some of the reaction ingredients are intermittently added during the reaction. Independent of the type of reactor operation, one's main objective would be to produce NBR with the desired product properties. The properties of interest that characterize polymer product quality in this study comprise average molecular weights (both weight- and number-average), bound AN in the copolymer (cumulative copolymer composition or [[bar.F].sub.AN]), degree of chain branching, number or average size of polymer particles and, of course, conversion. Embree et al. (1), Wall et al. (2), Vega et al. (3), and Gugliotta et al. (4) have reported typical trends and profiles of some of these polymer properties obtained experimentally in the batch emulsion polymerization of NBR. Vega et al. (3) and Gugliotta et al. (4) have in addition discussed semibatch strategies used to produce NBR emulsions of desired quality. With respect to the analysis of NBR continuous emulsion polymerization in a series of CSTRs, mathematical modeling efforts have borrowed ideas from the analysis of styrene-butadiene rubber (SBR), because they both follow the same type of emulsion polymerization kinetics (Case II kinetics). The framework for most of the reported mechanistic models for both SBR and NBR continuous systems was based on early studies by Broadhead et al. (5). These models have been used for predicting both dynamic as well as steady-state (SS) behavior of the reactor train. The publications by Broadhead et al. (5), Kanetakis et al. (6), and Gugliotta et al. (7) illustrate model uses for SBR, whereas those by Dube et al. (8), Rodriguez et al. (9), and Vega et al. (3) show model trends and uses for the analysis of NBR systems. A look at some of the above literature sources on continuous operation (e.g., (5-7) and others, which will be discussed shortly, reveals that analysis of production and product properties in the literature is primarily focused on the SS behavior of these reactors/trains (and this for a good reason, since most of the published industrial data/trends are around the SS operation) rather than the dynamic counterpart. Analysis of the dynamic behavior of the continuous train is equally significant, because it would give information on such important aspects of the operation as amount of off-spec material, smoothness (stability) of the operation, and optimality issues related to time periods to reach SS (and nature of the ensuing transients). Both dynamic and SS behaviors depend on how the reactor train is started up and the accompanying operating procedures and feed stream policies. A recent publication from our group (10) has concentrated on different start-ups of the reactor train and their corresponding effect on the profiles of the different polymer properties. In this article, additional important operating policies and procedures that can be used in the case of a train of CSTRs for the production of NBR emulsions are addressed and discussed. The performance of the train of CSTRs with respect to the emulsion copolymerization of AN and butadiene (Bd) is described using a mechanistic mathematical model, whose basic structure is adapted from the work of Broadhead et al. (5), Hamielec et at. (11), Mead and Poehlein (12), and Dube et al. (8). The validity of this model and some of its traits were recently presented by Washington et al. (13) and Madhuranthakam and Penlidis (10). The current model is very detailed and can handle different start-up modes and different feed policies and operating procedures, in addition to handling different options such as desorption, monomer, and water soluble impurities, different methods of monomer partitioning, etc. More details on this model and its features can be found in Washington et al. (13) and Madhuranthakam and Penlidis (10).

In this article, after a brief discussion of model, reactor, and recipe specifications, several reactor operating scenarios are analyzed and discussed in Reactor Operating Scenarios Section. Reactor Train Feed Policies section is concerned with reactor train feed policies and their effect on product characteristics. In the interim, in Reactor Stability Considerations Section, a novel criterion is introduced, capable of describing reactor stability. Benefits from these operational strategies are highlighted throughout the article, by comparing profiles from the suggested "optimal" procedures with the corresponding relevant profiles obtained from base-case and more conventional operating modes.

MATHEMATICAL MODEL

A mechanistic model for prediction of the performance of a train of CSTRs for the emulsion copolymerization of NBR was obtained from the corresponding component and moment balances of the different reaction species. The development and details of the model can be found in Washington et al. (13) and Madhuranthakam and Penli-dis (10). Nevertheless, a brief summary of the reaction mechanism and the model equations used in the simulations are provided in Appendices B and C of this article. The model comprises of 32 ordinary differential equations, describing molar balances for the reaction components and information for the number and average diameter of polymer particles, molecular weight averages and tri- and tetrafunctional branching averages. Other model traits include handling different start-up modes, different monomer partitioning options, desorption, effect of water soluble and monomer soluble impurities, and options for other operational policies.

RECIPE AND REACTOR TRAIN SPECIFICATIONS

The base-case recipe, based on typical industrial conditions for production of the so-called "cold" NBR, is shown in Table 1. All reaction ingredients are in parts per hundred monomer (pphm).

Unless otherwise specified, simulation results were obtained using a typical production train of eight CSTRs, where a mean residence time of 60 min per reactor was used with the volume of each CSTR being 20,000 L. Recipe and reactor train specifications were chosen such that the simulations mimic typical industrial production conditions. All reactors were operated isothermally.

REACTOR OPERATING SCENARIOS

In this section, we will discuss two reactor train operational policies, which we call "Type A" and "Type B". These policies are complementary to several start-up procedures we described in Madhuranthakam and Penlidis (10) but were devised with specific property targets in mind. In what is designated as "Type A" policy, the first reactor in the reactor train (of eight reactors) is full of "batch recipe", that is, loaded with the ingredients of Table 1 in the appropriate proportions (as described in Table 1) to give the corresponding ingredient concentrations in a batch reactor, while all other reactors are "half-full" of the reaction ingredients corresponding to the batch recipe, except for the redox initiator pair components (which are fed in their entirety in reactor 1). If one compares "Type A" scenario with the reactor train operating full of "batch recipe" (i.e., all reactors with all ingredients in the proportions of Table 1), then one can perceive several advantages: "Type A" operation is relatively fast; it uses a smaller amount of reaction ingredients; and it can achieve lower molecular weight averages and branching frequencies, with a higher number of polymer particles.

TABLE 1. Typical "cold" NBR process recipe.

Ingredient (pphm)                                      Amount [8]

Acrylonitrile (AN)                                             32

Butadiene (Bd)                                                 68

Water                                                         180

p-Melhane hydroperoxide (PMHP) (initiator)                  0.223

Ferrous sulfate monohydrate (Fe[SO.sub.4] [H.sub.2]O)      0.0056
(metal source for redox system)

Sodium formaldehyde sulfoxylate (SFS) (reducing              0.12
agent)

Dresinate (emulsifier)                                       1.25

Tamol (emulsifier)                                           2.85

tert-Dodecyl mercaplau (chain transfer agent)                0.42

Temperature ([degrees]C)                                       10


Figure 1 shows a comparison of dynamic profiles for conversion obtained in the reactor train with "Type A" policy and full of "batch recipe" operation. It is evident from the figure that the conversion profiles obtained with "Type A" operation are smooth and gradual, compared with the more abrupt overshoots observed in the full of "batch recipe" case. The profiles corresponding to the first reactor are the same, since the first reactor is full of "batch recipe" in both policies. "Type A" operation is fast and smooth, hence it falls somewhere between policies that go to SS fast but exhibit overshoots and other transients, and policies that are very slow, hence producing excessive off-spec material. Overall, "Type A" is a good operational compromise.

SS values of other important latex and polymer properties obtained in all eight reactors for "Type A" operation are summarized in Fig. 2. The cumulative copolymer composition in Fig. 2a is expressed in terms of the cumulative mole fraction of AN in the copolymer chains ([[bar.F].sub.AN]), Fig. 2b shows total number of polymer particles per liter of water ([N.sub.p]), Fig. 2c shows the corresponding SS profiles for weight-average molecular weight ([[bar.M].sub.w]), while the average number of tri-functional long chain branches per molecule ([BN.sub.3]) is depicted in Fig. 2d. These property SS profiles for "Type A" operation are compared with their "batch recipe" operation counter parts. The profiles in Fig. 2 are in agreement with the conversion behavior of Fig. 1. In addition, one can observe the operational benefits cited earlier, with respect to [N.sub.p], [[bar.M].sub.w] and [BN.sub.3]. The obtained profiles are in agreement with similar considerations and trends described earlier in Minari et et al. (14)

The second operational policy, "Type B," is a design modification whereby the first reactor in the reactor train has a much smaller volume than all subsequent equal-sized reactors. The volume of the first reactor is chosen to be one-sixth of the volume of the other reactors in the train. As the mean residence time per reactor for the other reactors in the reactor train was maintained at 60 min, this would correspond to a mean residence time of 10 min for the first reactor. (This also makes an additional option available that of splitting the total feed streams between the first small reactor and the other reactors in the train). This scenario is targeted toward using the first small reactor as a polymer particle nucleating vessel. In the "Type B" operational policy, the first small reactor was allowed to overflow to the subsequent empty equal-sized reactors of the train. Figure 3 shows the comparison of the SS values obtained for [N.sub.p], conversion (X), [[bar.M].sub.w], and [BN.sub.3]. From Fig. 3a, it is evident that the total number of particles (per liter of water) nucleated with the "Type B" policy is about twice that of the case with equal reactor volumes. This confirms the role of the first small reactor as essentially a particle nucleating vessel that is "seeding" the rest of the train. Consequently, particles with a smaller diameter are obtained with the "Type B" scenario compared to the diameter of particles with "equal volume" reactors in the train. In addition to obtaining a higher number of particles, there are other benefits with the "Type B" scenario. The monomer droplets disappear earlier with "Type B", thus essentially "freeing" up reactor volume and hence increasing overall productivity. The conversion levels are higher (as shown in Fig. 3b). Subsequently, the corresponding [[bar.M].sub.w] and [BN.sub.3] levels are also higher (see Figs. 3c and d, respectively), however this can be counteracted, if so desired, by the addition of chain transfer agent. Though higher levels of branching frequencies (both tri- and tetra-functional) are not desired in the final product, an interesting result observed is that by using only seven reactors with the "Type B" scenario, one could obtain the same levels of conversion, branching frequencies and average molecular weights that are obtained using a reactor train with eight equal volume reactors, but with a smaller particle diameter.

REACTOR STABILITY CONSIDERATIONS

Operating a train of CSTRs or a single CSTR for the production of some polymers by emulsion polymerization might result in reactor oscillations. This oscillatory (and highly undesirable) behavior has been observed in the continuous emulsion polymerization of chloroprene, vinyl acetate (V Ac) and vinyl choloride (VCM) (15-18), in all latex and polymer properties such as conversion, particle size, number of particles, branching frequencies and molecular weight averages. The bottom line is that these oscillations are due to the competition between the rate of particle nucleation and that of particle growth. Especially in systems that follow CASE I kinetics emulsion polymerization, where these oscillations are predominant, the average number of radicals per particle is less than 0.5 (i.e., [bar.n] < 0.5), and after the first burst of particle nucleation, a low particle growth is observed (and, hence, a lower polymerization rate) due to the very low value of [bar.n]. This low rate of particle growth leads in turn to an increase in the available surfactant concentration in the emulsion system, thus resulting in a new particle generation, and so on. Let's now formulate the ratio of particle nucleation rate over the particle growth rate, and call this ratio our criterion [alpha], where [alpha] is given by Eq. 1:

[alpha] = Rate of particle nucleation/Rate of particle growth (1)

Then it is clear that when systems with oscillations (mainly Case I kinetics) are compared with systems without oscillations (emulsion systems described by Case II/III kinetics), the corresponding value of [alpha] would be much higher in the former case than in the latter. In other words, the ratio of [alpha] of an oscillating system to [alpha] of a nonoscillating system would be far greater than unity. This can also be shown mathematically very quickly by looking at general equations for the rate of particle nucleation and rate of particle growth.

If we consider emulsion homopolymerization for simplicity (extensions to copolymerization are similar in principle but more complicated numerically), the rate of particle nucleation, f(t), is given by Eq. 2.

f(t) = ([R.sub.I]+ [[rho].sub.des]) ([k.sub.cm][A.sub.m] + [k.sub.h])/([k.sub.cm][A.sub.m] + [k.sub.h] + [[epsilon][A.sub.p]) [N.sub.A] (2)

[R.sub.I] in [E.sub.q]. 2 is the rate of radical generation by initiator decomposition (rate of initiation); [[rho].sub.des] is the rate of radical desorption from polymer particles; the term ([k.sub.cm][A.sub.m]) represents the micellar contribution to particle nucleation, with [k.sub.cm] being the apparent rate constant for micellar nucleation and [A.sub.m] the free emulsifier area (i.e., emulsifier surface area available to form micelles and hence generate polymer particles via micellar nucleation); [k.sub.h] a specific homogeneous nucleation rate constant, represents the contribution to particle nucleation from homogeneous nucleation; [epsilon] is the product of [k.sub.ab] (the radical capture (absorption) rate constant) and [k.sub.v] (the ratio of emulsion phase volume to aqueous phase volume); [A.sub.p] is the total polymer particle surface area; the term ([epsilon][A.sub.p]) represents radicals captured (absorbed) by polymer particles, that is, radicals entering pre-existing particles or the growing particles that have been formed by micellar or homogeneous nucleation, hence the term [([k.sub.cm][A.sub.m] + [k.sub.h])/([k.sub.cm][A.sub.m] + [k.sub.h] + [epsilon][A.sub.p])] represents the fraction (of total radicals) that leads to the nucleation of polymer particles; and [N.sub.A] is Avogadro's number for unit consistency.

The total volumetric particle growth rate, g(t), is given by Eq. 3:

g(t) = [v.sub.p] * f(t) + [lambda] * [xi] * [A.sub.p], (3)

where [v.sub.p] is the volume of particles at any time t, and [lambda] and [xi] are represented by Eqs. 4, 5, respectively.

[lambda] = ([k.sub.p][d.sub.m]/[N.sub.A][d.sub.p])[(f[k.sub.d]m[k.sub.p][N.sub.A]/12[pi][D.sub.w][delta][k.sub.fm]).sup.1/2] (4)

[xi] = [phi]/(1 - [phi]) [I.sub.w.sup.1/2]/[A.sub.p.sup.1/2], (5)

where [k.sub.p] is the propagation rate constant, [d.sub.m] and [d.sub.p] are densities of monomer and polymer, respectively, f is the efficiency factor for initiator decomposition, [k.sub.d] is the rate constant for initiator decomposition, m is the partition coefficient for monomeric radicals between water and particle phases, [D.sub.w] is the diffusion coefficient of monomeric radicals in the water phase, [delta] is a lumped monomeric radical diffusion constant, [k.sub.fm] is the rate constant for transfer to monomer, [empty set] is the volume fraction of monomer in the polymer particles, and [I.sub.w] is the concentration of initiator in the water phase.

By substituting Eqs. 2-5 in Eq. 1, one can obtain the expression for [alpha] in terms of pertinent process characteristics and rate constants corresponding to the rates of nucleation and particle growth. Thus, [alpha] can be estimated for any system accordingly from the steps illustrated above. For instance, in a copolymer system, one can make use of the "pseudo-rate constant" methodology and the corresponding value of [alpha] can thus be evaluated.

If we now consider emulsion polymerization kinetics for different monomer systems, with monomers that follow Case I kinetics (VAc, VCM), the rate of radical desorption ([[rho].sub.des]) and the transfer to monomer rate constant ([k.sub.fm]) are much higher compared to systems that follow Case II or III kinetics (styrene, styrene-butadiene, and NBR). Based on this fact, evaluation of [alpha] for different systems leads to the conclusion that [alpha] is always greater in Case I systems compared to [alpha] in Case II or Case III systems. Mathematically this can be represented by Eq. 6 (with a few more details for its derivation in Appendix A):

[[alpha].sub.Case I]/[[alpha].sub.Case II/III] [much greater than] 1. (6)

The criterion of Eq. 6 was evaluated for many emulsion polymerization examples. For illustration purposes, consider VCM representing a Case I kinetics system, and NBR of the current paper being a Case II system (similar results were obtained for other Case I and Case II or III kinetics combinations, evaluated for both continuous and batch reactors; in continuous reactors, Eq. 6 was evaluated at different levels of [theta], [theta] being the ratio of real time t to the reactor mean residence time [tau], hence [theta] being a dimensionless time (number of reactor residence times elapsed from t = 0); with batch reactors, Eq. 6 was evaluated at different times during the reaction, preferably during stage 1 when polymer particles are being nucleated). The ratio of Eq. 6 ranged from 2.3 x [10.sup.5] for a CSTR (with [theta] = 2) to 3.7 x [10.sup.3] for a batch reactor (at t = 10 min), that is, overall the [alpha] criterion of Eq. 1 for an oscillatory Case I system appears to be three to five orders of magnitude larger than the corresponding [alpha] for a Case II or III system. This is a quick, practical and useful criterion to evaluate the expected performance of a certain monomer in emulsion polymerization in a CSTR. A confirmation is given in Fig. 4, where typical profiles for a single CSTR are compared for VCM (Case I) and NBR (Case II), and where the oscillatory behavior is clear for the Case I system in both conversion and number of particles.

REACTOR TRAIN FEED POLICIES

According to these feed policies, different reaction ingredients are added to the reactor/reactor train targeted toward achieving improved polymer properties. Reactor policies are also used to reduce process transients (i.e., the amount of "off-spec" material generated) whenever grade changes are made (14), (19). Since some of the most commonly specified properties of emulsion polymers are particle size (related to the number of particles), copolymer composition and average molecular weights, the effect of corresponding reaction ingredients on these (and other) properties are discussed in this section. This would give an insight into how and what reactor inflows could be manipulated to obtain desired polymer properties. Quite often finding an optimal reactor policy for a specified objective is very challenging as there would be con-fficting constraints, which are to be satisfied for achieving the target polymer properties/production (e.g., achieve a small particle size while maintaining low levels of conversion and molecular weight averages).

PARTICLE SIZE (NUMBER OF PARTICLES)

In the NBR system, under typical operating conditions, the rate of generation of particles due to micellar nucleation is at least four to five orders of magnitude greater than that of the particles generated due to homogeneous nucleation (due to the high solubility of AN in water). In general, the number of polymer particles is a function of the rate of generation of radicals in the aqueous phase and the concentration of emulsifier, as per Eq. 7.

[N.sub.p] [alpha] [R.sub.I.sup.2/5][[E].sup.3/5] (7)

[R.sub.I] is the rate of initiation and [E] denotes emulsifier concentration. The right-hand side of Eq. 7 is also a function of temperature, but a much weaker function at that, relative to changes in [R.sub.I] and/or [E]. If the objective is to influence or control the number of particles, then the concentrations of initiator and emulsifier in the feed are to be manipulated. An additional operating variable that would affect the number of particles especially in a continuous process is the mean residence time of the reactor ([tau]). It is known that (for specified initiator and emulsifier concentration levels) with an increase in the mean residence time, [N.sub.p] increases up to a certain limit after which it decreases (16), (13), for both NBR and SBR systems (the maximum in [N.sub.p] with [tau] is more pronounced in the SBR system, where both comonomers are of very low water solubility). As shown in Fig. 5, for the NBR system, Np would be maximum when the mean residence time is approximately 10 min. But this would raise questions as to the feasibility of conducting NBR emulsion polymerization in a train with a mean residence time of 10 min. Though one can obtain maximum number of particles, the conversion levels are very low with a 10 min mean residence time. However, this fact of maximum nucleation rate with a mean residence time around 10 min could be used to make a design change in the train of reactors, where the very first reactor is small compared to the other reactors in the train. This design change usually promises generation of more particles as already discussed in Reactor Operating Scenarios Section. Nevertheless, for a [tau] = 60 min the number of particles generated would be closer to the maximum [N.sub.p] (see Fig. 5).

Hence, in order to influence (control) the number of particles in a continuous NBR operation, the preferable option would be to manipulate the initiator and emulsifier flow levels in the reactor train to obtain the desired number of particles rather than manipulating temperature or mean residence time. Figure 6a and b shows the profiles obtained for [N.sun.p] and the corresponding diameter ([d.sub.p]) of the particles, respectively, for various initiator and emulsifier flow levels in a single CSTR with a full of water start-up. In Fig. 6, [F.sub.10] and [F.sub.E0]) are the base-case molar flow rates of initiator and emulsifier, respectively.

As the initiator and emulsifier molar flow rate levels are increased in Fig. 6a/6b, the number of particles increases while the corresponding diameter of particles decreases (as expected). Another method of obtaining more particles in a continuous operation, especially in a train of reactors, is to add more surfactant to downstream reactors mid-way through the train (see Fig. 6c). Though this will usually lead to a higher number of particles, it would also result in a bimodal size distribution of particles, and with an expected decrease in the final average particle size. For example, addition of extra emulsifier (four times that of the base case) to the fifth reactor in the train of CSTRs led to a burst of particles in the fifth reactor. Correspondingly, the diameter of the particles (not shown here for the sake of brevity) from the fifth reactor onward decreased. Of course, these changes will also result, accordingly, in changes in the levels of conversion, average molecular weights and average branching frequencies.

CUMULATIVE COPOLYMER COMPOSITION

Copolymerization of AN and Bd usually leads to variation in the cumulative copolymer composition. To overcome this problem of copolymer composition variation, using as a guide the cumulative mole fraction of AN bound in the copolymer chains, [[bar.F].sub.AN], controlled addition of the highly reactive monomer (AN here) in a semibatch fashion was suggested and verified by Minari et al. (20) (general concepts also described in (19)). However, if a train of CSTRs is used to produce NBR, then a good degree of freedom to use in order to achieve constant copolymer composition is by splitting the flow of the highly reactive monomer (AN) along the reactor train. Often, adding the monomer in predetermined (open-loop optimal or suboptimal) flows along the length of the train of reactors is practiced, especially when grade changes are made. The splitting of monomer should be within the first few reactors before the monomer droplet phase disappears (mainly for reactor productivity considerations, since the separate monomer phase does occupy extra volume and is carried from reactor to reactor). Minari et al. (21) have shown that the addition of predetermined amounts of AN to all reactors in the train can lead to an improvement in [[bar.F].sub.AN], though the improvement with respect to the required (desired) target property was very small. The experimental results reported showed a change in the [[bar.F].sub.AN] from 0.37 to 0.34, with the desired [[bar.F].sub.AN] being 0.34.

Aside from variation in copolymer composition due to recipe loadings/monomer flows as counterbalanced by the rate of monomer incorporation in the copolymer chains, another cause for variation in [[bar.F].sub.AN] for the NBR system is the fact that a significant amount of AN is present in the aqueous phase due to the high solubility of AN (about 8.5 g per 100 g of water at typical conditions). In contrast to AN, Bd is almost insoluble in water. In a continuous operation, even after the monomer droplets disappear in the last few reactors of the CSTR train, relatively large amounts of AN are present in the aqueous phase, which will eventually find their way into the particle phase. This transfer of AN from the aqueous phase to the polymer particles depends on the difference in monomer concentrations between particles and water phase (and not solely on the rate of reaction in the particle phase). Minari et al. (21) used predetermined flow rates to all reactors in the reactor train based on SS analysis. This would be feasible if and only if the polymer particles can completely take all of the fresh monomer being added into the reactors, after the droplets have already disappeared. Otherwise, a ternary phase is created that would lead to longer times for the (new) monomer droplets to disappear. Further addition of different reaction species (especially monomer) would lead to variations in mean residence times that would indirectly affect other polymer properties, such as conversion, particle size, average molecular weights and average branching frequencies.

Figure 7 illustrates the different pros and cons of two different options targeted toward controlling the [[bar.F].sub.AN] in the eighth reactor. The first option deals with splitting the AN stream between the first two reactors in the train (80/20), while in the second option an additional amount of AN (compared to the base-case concentrations as per Table 1) is added to the first reactor (1.2*[F.sub.AN, in]). In Fig. 7, [F.sub.AN, in] Ii refers to the base-case molar flow rate of AN.

Splitting the flow of acrylontrile (80% to the first reactor and 20% to the second reactor) would result in achieving a lower [[bar.F].sub.AN] in the eighth reactor compared to the base-case. Lower average molecular weights (and branching frequencies) are obtained compared to the base-case (see Fig. 7a), whereas the number of particles increases (see Fig. 7b) compared to the base-case due to more emulsifier being used along with other reaction ingredients in the first reactor to make up for the remaining 20% of the AN feed stream (for residence time considerations). An additional benefit with this option is that it will maintain the mean residence time at a constant level (at 60 min). An alternate way of achieving a desired [[bar.F].sub.AN] of 0.34 is to use more AN in the feed stream to the first reactor. A 20% extra molar flow rate of AN is used to show its effect on [[bar.F].sub.AN] and other properties. From Fig. 7c, a [[bar.F].sub.AN] of 0.33 is achieved compared with 0.29 with the base- case and 0.27 with the monomer split case (80/20). The number of particles decreases while the average molecular weights (and branching frequencies, not shown here) increase. Nevertheless, the increase in molecular weight can be independently handled (if so desired) by adding more chain transfer agent (discussed in the next section). This would also decrease branching frequencies.

Another interesting result observed with using the option with the higher AN level (compared to the monomer split case (80/20)) is that the monomer droplets disappeared in the sixth reactor (while they disappeared in the seventh reactor in the monomer split case), thus allowing the last two reactors to produce more polymer (see Fig. 7d). If the objective is to increase productivity, then this can be obtained by increasing the molar flows to the first reactor. While this option leads to an increase in productivity, some of the product polymer properties could be slightly off-spec, but they can be independently controlled by manipulating other feed streams. Figure 8a shows the SS conversion versus reactor number for the case where productivity is increased by increasing the molar flows of all inlet reaction species by 50%, except emulsifier (where a 125% increase is used). In Fig. 8, [F.sub.feed, in] refers to the base-case total molar flow rate entering the first reactor in the train.

Results in Fig. 8 show that a 25-30% increase in productivity can be achieved with the same copolymer composition (compared to that of the base-case) in the reactor train by manipulating the molar flows of the inlet stream to the first reactor. In general, higher conversions would lead to higher average molecular weights, which in turn would lead to an increase in branching frequencies. However, this could be controlled by adding more CTA (or predetermined amounts of CTA) to the reactor train (see next section).

AVERAGE MOLECULAR WEIGHTS AND BRANCHING FREQUENCIES

This section illustrates how average molecular weights (and branching frequencies) can be controlled by manipulating the flow rate of chain transfer agent (CTA). CTA can be added intermittently along the reactor train to control the average molecular weights that would otherwise become extremely high, especially in the last few reactors of the train. This can also reduce average branching frequencies. To demonstrate the effect of CTA on average molecular weights and branching frequencies, a step input of 0.5 mol/min in CTA molar flow is given to the last four reactors where droplets do not exist (designated as "Type C" scenario). Desired levels in the average molecular weights and branching frequencies can be obtained for an optimal inflow of CTA (into the desired reactors), which can be obtained from dynamic optimization procedures (which will be the topic of a follow-up article). Figure 9 shows the comparison of the average molecular weights and branching frequencies obtained after adding the CTA to the reactor train with respect to the base-case simulation results. It was observed that all other polymer properties such as conversion and [[bar.F].sub.AN] remained constant (not shown here for the sake of brevity), while the average molecular weights and branching frequencies were reduced significantly with the addition of more CTA.

CONCLUSION

A detailed dynamic model for the emulsion copolymerization of AN and Bd has been employed to address several improved operating scenarios in a train of eight CSTRs. In one of the scenarios, the first reactor was started full of "batch recipe" while other reactors were started half full of all other "batch recipe" ingredients, except for initiator and reducing agent. In the other scenario, a smaller first reactor was used in the reactor train while other reactors were of equal size. It was found that both scenarios led to more benefits compared to the base-case and conventional operating modes. Secondly, a novel criterion, namely the ratio of the rate of particle nucleation to the rate of particle growth, was defined and used to characterize reactor stability. The criterion was found to be a relative measure of the presence of oscillations in continuous emulsion polymerization. The criterion value in systems with Case I kinetics was observed to be far greater than that in systems with Case II/III kinetics. Finally, reactor feed policies that are targeted toward producing NBR latexes with desired properties were discussed, again employing the model as a guiding tool for comparisons. It was found that the total number of particles can be increased by increasing the molar flow levels of initiator and surfactant. On the other hand, for obtaining a constant [bar.F].sub.AN], it was observed that increasing the molar flow of AN had more benefits compared to splitting of the AN flow between the first two reactors. Further, desired molecular weights and in turn branching frequencies were obtained by manipulating the molar flow of CTA to the reactors mid-way through the train.

APPENDIX A

Let us define [alpha]' as the reciprocal of [alpha] in Eq. 1. Then using Eqs. 2 and 3, we obtain:

[alpha]' = 1/[alpha] = g(t)/f(t) = ([v.sub.p]f(t) + [lambda][xi][A.sub.p])/f(t) = [v.sub.p] + [lambda][xi][A.sub.p]/f(t) (A.1)

If we consider two systems with Case I kinetics and Case II/III kinetics, starting with the same particle volume, then Eq. A.1 becomes

[alpha]' = 1/[alpha] [approximately equal to] [lambda][xi][A.sub.p]/f(t) (A.2)

Substituting Eqs. 2, 4, and 5 in Eq. A.2. we eventually obtain:

[alpha] = ([R.sub.I] + [[rho].sub.des])([K.sub.m][A.sub.m] + [K.sub.h])/([K.sub.m][A.sub.m] + [K.sub.h] + [epsilon][A.sub.p]])[N.sub.A]/([k.sub.p][d.sub.m]/[N.sub.A][d.sub.p])[([fk.sub.d][mk.sub.p][N.sub.A]/12[pi][D.sub.w][delta][k.sub.fm]).sub.1/2] ([phi]/(1 - [phi])) [I.sub.w.sup.1/2][A.sub.p.sup.1/2] (A.3)

If the rate of initiation is considered to be the same in both systems, the ratio of [alpha] in systems with Case I kinetics to systems with Case II/III systems would be far greater than unity because of two reasons. In Case II/III systems, the rate of desorption would be negligible. Further, the rate constant corresponding to transfer to monomer ([k.sub.fm]) in Case I systems will be very high compared to that of Case II/III systems. Hence, it can be concluded that [alpha] in Case I systems would always be much greater than [alpha] in Case II/III systems. Mathematically, this can be represented by Eq. A.4, which is Eq. 6.

[[alpha].sub.Case I]/[[alpha].sub.Case II/III] [much greater than] 1 (A.4)

APPENDIX B. REACTION MECHANISMS AND SCHEME (13)

Mechanism               Reaction/Event

Redox decomposition     [S.sub.2][O.sub.8.sup.2-] + [Fe.sup.2+] [??]
                        [SO.sub.4.sup.-*] + [Fe.sup.3+] +
                        [SO.sub.4.sup.2-]

                        [Fe.sup.3+] + SFS [??] [Fe.sup.2+] +
                        [SFS.sup.+]

Thermal decomposition   [S.sub.2][O.sub.8.sup.2-] [??]
                        2[SO.sub.4.sup.-*]

Radical initiation      [SO.sub.4.sup.-*] + [M.sub.j] [??] [R*.sub.I,
                        j]

Propagation             [R*.sub.n, i] + [M.sub.j] [??] [R*.sub.n + 1,
                        j]

Termination             [R*.sub.n, i] + [R*.sub.m, j] [??] [P.sub.(m +
                        n)] Or [P.sub.n] + [P.sub.m]

Transfer to monomer     [R*.sub.n, i] + [M.sub.j] [??] [P.sub.n, i] +
                        [M*.sub.j]

Transfer to polymer     [R*.sub.n, i] + [P.sub.m, j] [??] [P.sub.n, i] +
                        [P*.sub.m, j]

Transfer to CTA         [R*.sub.n, i] + [(CTA).sub.j] [??] [P.sub.n, i]
                        + [CTA*.sub.j]

Reaction with internal  [R*.sub.n, i] + [P.sub.m.sup.i =] [??] [R*.sub.n
double bonds            + m, i]

Reaction with terminal  [R*.sub.n, i] + [P.sub.m.sup.t =] [??] [R*.sub.n
double bonds            + m, i]

Reaction with           [R*.sub.n, i] + [WSI.sub.j] [??] [P.sub.i]
water-soluble
impurities

Reaction with           [R*.sub.n, i] + [MSI.sub.j] [??] [P.sub.i]
monomer-soluble
impurities

Micelles nucleation     [R*.sub.n, i] + Micelle [??] Particle

Homogeneous             [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
nucleation              ASCII]

Capture of radical by   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
particle                ASCII]

Desorption of radical   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
from particle           ASCII]


APPENDIX C. MODEL EQUATIONS

1. Initiator Balance

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[R.sub.I] = [k.sub.I][[I].sub.a][[Fe.sup.2+].sub.a] + 2f[k.sub.d][[I].sub.a]

2. Reducing Agent Balance

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[R.sub.RA] = [k.sub.2][[RA].sub.a][[Fe.sup.3+].sub.a]

3. Oxidising Agent Balance

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

4. Particle Nucleation

[dN.sub.p] = [F.sub.p, in] - [F.sub.p] + ([R.sub.hom] + [R.sub.mic]) * [V.sub.a] * [N.sub.A]

[R.sub.hom] = [k.sub.h] [[R*].sub.a.sup.mic]

[R.sub.mic] = [[rho].sub.des.sup.mic] + [k.sub.cm][[R*].sub.a.sup.mic]/[r.sub.mic]

5. Water Balance

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

6. Monomer and Polymer Balances

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

7. Monomer Partitioning

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

8. Emulsifier Balance

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

9. Impurities Balance

Water soluble impurities

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Monomer soluble impurities

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

10. Chain Transfer Agent Balance

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

11. Molecular Weight Distribution Moments Balances

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

12. Branching Average Balances

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For more details (including values of rate constants and other parameters), see the appendices of ref. (13). Any new symbols in Appendix C are explained below.

NOTATION (FOR APPENDIX C)

[[bar.BN].sub.i]              Average number of tri- and
                              tetra-functional branches (i = 3, 4)
                              per chain (#/molecule)

[cta.sub.j]                   jth chain transfer agent

[e.sub.j]                     jth emulsifier

F                             Initiation efficiency

[[i].sub.a]                   Concentration of species i (= I, RA,
                              [Fe.sup.2+], [Fe.sup.3+]) in
                              aqueous phase

[[I].sub.a]                   Concentration of initiator in
                              aqueous phase (mol/L)

I                             Initiator

[MATHEMATICAL EXPRESSION NOT  Total molar inflow of species i = I,
REPRODUCIBLE IN ASCII]        RA, Fe, [Fe.sup.2+], [Fe.sup.3+],
                              [m.sub.j], [pol.sub.j], w,
                              [e.sub.j], [wsi.sub.j],
                              [msi.sub.j], [cta.sub.j] (mol/min)

Fe                            Iron

[F.sub.i]                     Total molar outflow of species i =
                              I, RA, Fe, [Fe.sup.2+], [Fe.sup.3+],
                              [m.sub.j], [pol.sub.j], w,
                              [e.sub.j], [wsi.sub.j],
                              [msi.sub.j], [cta.sub.j] (mol/min)

[k.sub.1], [k.sub.2]          Reaction rate constants (L/mol/min)

[k.sub.cm]                    Micelle radical capture rate
                              constant (dm/min)

[k.sub.d]                     Decomposition rate constant
                              (initiator) (L/mol/min)

[k.sub.h]                     Homogeneous nucleation rate constant
                              ([min.sup.-1])

[MATHEMATICAL EXPRESSION NOT  Partition coefficient of monomer f'
REPRODUCIBLE IN ASCII]        between aqueous and particle phase

[m.sub.j]                     jth monomer

[msi.sub.j]                   jth monomer soluble impurity

[N.sub.A]                     Avogadro's number (#/mol)

[N.sub.i]                     Total moles of species i = I, RA,
                              Fe, [Fe.sup.2+], [Fe.sup.3+],
                              [m.sub.j], [pol.sub.j], w,
                              [e.sub.j], [wsi.sub.j],
                              [msi.sub.j], [CTA.sub.j]

[N.sub.p]                     Number of particles (#)

[pol.sub.j]                   jth polymer

[r.sub.mic]                   Average micelle radius (dm)

RA                            Reducing agent

[R.sub.I]                     Rate of initiation (mol/L/min)

[MATHEMATICAL EXPRESSION NOT  Rates of redox ingredient
REPRODUCIBLE IN ASCII]        consumption (mol/L/min)

[R.sub.mic], [R.sub.hom]      Rate of micellar and homogeneous
                              nucleation (#/L/min)

[R*.sub.a.sup.hom]            Concentration of radicals in the
                              aqueous phase capable of undergoing
                              homogeneous nucleation (mol/L)

[R*.sub.a.sup.mic]            Concentration of radicals in the
                              aqueous phase capable of being
                              captured by micelles (mol/L)

[MATHEMATICAL EXPRESSION NOT  Rate of chain transfer agent
REPRODUCIBLE IN ASCII]        consumption(mol/L/min)

[MATHEMATICAL EXPRESSION NOT  Rales of polymerization in aqueous
REPRODUCIBLE IN ASCII]        and panicle phases (mol/L/min)

[MATHEMATICAL EXPRESSION NOT  Rates of impurity
REPRODUCIBLE IN ASCII]        consumption(monomer and water
                              soluble) (mol/L/min)

[MATHEMATICAL EXPRESSION NOT  Rate of moment generation for
REPRODUCIBLE IN ASCII]        moments i = 0, 1, 2 (mol/L/min)

[MATHEMATICAL EXPRESSION NOT  Rate of [[bar.BN].sub.i], generation
REPRODUCIBLE IN ASCII]        for i = 3, 4 (mol/L/min)

[V.sub.a], [V.sub.p]          Volume of aqueous and panicle phase
                              (L)

[MATHEMATICAL EXPRESSION NOT  Volume of monomer i  in aqueous and
REPRODUCIBLE IN ASCII]        particle phase (L)

[V.sub.p] [Q.sub.i]           ith moments of the molecular weight
                              distribution (mol)

[V.sub.p] [Q.sub.o]           Zeroth moments of the ith (tri- and
[[bar.BN].sub.i]              tctra-) functional branching
                              frequency distributions(mol
                              #/molecule)

w                             water

[wsi.sub.j]                   jth water soluble monomer

[[rho].sub.des.sup.mic]       Rate of recapture of desorbed
                              radicals by micelles (#/L/min)


Correspondence to: Alexander Penlidis; e-mail: penlidis@uwaterloo.ca

DOI 10.1002/pen.23231

Published online in Wiley Online Library (wileyonlinelibrary.com).

[C] 2012 Society of Plastics Engineers

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Chandra Mouli R. Madhuranthakam, Alexander Penlidis

Department of Chemical Engineering, Institute for Polymer Research (IPR), 200 Univerisity Avenue West, Waterloo, Ontario, Canada N2L 3G1
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