Modeling of industrial styrene polymerization reactors.
The bulk polymerization of styrene usually occurs with a high viscosity reaction mixture. Many industrial continuous polymerization reactors are thus motionless mixers, where the extent of backmixing is low as a result of the high viscosity, and the radially mixed flow. This results in a homogeneous product with uniform properties.
Studies of various aspects of polymerization in tubular reactors has been reported in the literature. Using simplified models for the process, the parametric sensitivity (1), stability (2), and optimization (3) have been considered. More rigorous models have been presented by Lynn and Huff (4), Sala et al. (5) and Agarwal and Kleinstreuer (6). In these studies the equations of motion and transport equations for energy and mass are coupled through variable density, viscosity and other properties of the mixture. Craig (7) has presented a simulation study, in which a motionless mixer has been modeled as a PFR. In most of the above studies, simplified models for the polymerization kinetics have been used. Previous studies have not considered polystyrene reactors with recycle.
In this work, plug flow models have been used to simulate two industrial styrene polymerization reactors which are operated in series. The first reactor is a shell and tube heat exchanger with the reactants on the shell side [ILLUSTRATION FOR FIGURE 1 OMITTED], and the second reactor is cylindrical reactor with cross-sectional stirring [ILLUSTRATION FOR FIGURE 2 OMITTED]. Part of the output of the first reactor is recycled. The detailed model of Arai et al. (8) for the kinetics of thermally initiated polymerization of polystyrene is used in the study. The main objective of the work is to develop and validate a detailed model for the reactor train which can subsequently be used for understanding the role of different operating parameters in the reactor system. In the following sections we first discuss the polymerization kinetics, followed by the reactor models. The results and discussion and conclusions are given next.
Arai et al. (8) have presented a model for bulk thermal polymerization of styrene, which accounts for diffusional limitations caused in the advanced stage of polymerization. The rate constants are related to diffusivities of the monomer and polymer in the reaction mixture. The diffusivities are estimated by a modification of Bueche's (9) free volume theory presented by Vrentas and Duda (10, 11). The chain entanglement effects in the termination step are also accounted for. A review of the main features of the reaction scheme used by Arai et al. (8) is briefly presented here.
Arai et al. (8) have used the mechanism for thermal initiation of styrene polymerization proposed by Pryor and Coco (12). The rate of initiation is given by
[Mathematical Expression Omitted] (1)
where [Mathematical Expression Omitted] is the initiator efficiency. The rate constants [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are obtained from the initial rate of polymerization and f is related to the diffusivity of monomers in the polymer solution.
The monomer-polymer mixture is assumed to have a quasi-crystalline structure, and two distinct types of radicals are assumed to exist in the reaction mixture - active and inactive. Active radicals are those which have at least one monomer molecule in the nearest neighbor cells. Each propagation step, which proceeds with a rate constant [[Kappa].sub.pc], thus results in another active radical with probability (1 - [P.sub.E](O)) or an inactive radical with probability [P.sub.E](O). The inactive radical then becomes active owing to diffusion of monomers with a rate determined by a second order rate constant [[Kappa].sub.Diff]. Chain transfer to monomer is considered in the model, with a rate constant [[Kappa].sub.fmc]. Again chain transfer may result in an active or inactive radical.
The termination is assumed to occur by combination, and the diffusion limited rate constant for termination is
[Kappa].sub.t] = [[Kappa].sub.tc] [[Kappa].sub.tD]/[[Kappa].sub.tc] + [[Kappa].sub.tD] (2)
where [[Kappa].sub.tc] is the termination rate constant in the absence of diffusional limitations ([[Kappa].sub.tD] [approaches] [infinity]). The diffusional rate constant [[Kappa].sub.tD] is predicted based on the free volume and chain entanglement theories by Arai et al. (8). The termination due to motion of active centers by propagation is also taken into account in terms of a rate constant [[Kappa].sub.tp.]
The reader is referred to the original paper by Arai et al. (8) for a detailed description of the calculation of the above rate constants. The notation used in this paper is identical to that of Arai et al. (8). The following constants have been modified to account for the presence of ethyl benzene, which is used as a diluent in the process (10),
[Mathematical Expression Omitted]
The free volumes are calculated using the correlation given by Arai and Saito (13).
The models for the two reactors are discussed in this section. The first reactor is a baffled shell-and-tube heat exchanger with the reaction occurring on the shell side [ILLUSTRATION FOR FIGURE 1 OMITTED]. The coolant flows on the tube side, which has four passes, and undergoes a temperature decrease of about 4-5 [degrees] C. A part of the product coming out of this reactor is recycled and remaining goes to the second reactor, which is a vertical cylindrical reactor with horizontal banks of heat exchanger tubes with agitator blades in between [ILLUSTRATION FOR FIGURE 2 OMITTED]. The reaction mixture flows through the shell side of this reactor as well. There are three cooling zones in the reactor, with a different coolant temperature in each zone [Ref. Makwana (14) for details]. Both reactors are modeled as plug flow reactors here. The choice of [TABULAR DATA FOR TABLE 1 OMITTED] model was based on the fact that the viscosity of the reaction mixture is high, hence little backmixing is expected to occur in the reactor.
The mass balances and energy balance for styrene polymerization are written in terms of moments of the [TABULAR DATA FOR TABLE 2 OMITTED] concentrations. Three moments each of the active radical concentration, total radical concentration, and the dead polymer concentration are required in order to calculate the conversion, number average molecular weight and weight average molecular weight. These moments are defined as
[Mathematical Expression Omitted] (3)
[Mathematical Expression Omitted] (4)
[Mathematical Expression Omitted] (5)
[Mathematical Expression Omitted] (6)
where [R.sub.j] represents the total concentration of radicals having j monomer units, [Mathematical Expression Omitted] is the concentration of active radical having j monomer units, and [P.sub.j] is the concentration of dead polymer molecules with j monomer units. The balance equations for the moments are given below.
Total polymer radical concentration:
[Mathematical Expression Omitted] (7)
[Mathematical Expression Omitted] (8)
Active polymer radical concentration:
[Mathematical Expression Omitted] (9)
[Mathematical Expression Omitted] (10)
Total polymer concentration
[Mathematical Expression Omitted] (11)
[Mathematical Expression Omitted] (12)
[Mathematical Expression Omitted] (13)
The energy balance equation for the reactors is
[Delta]T/[Delta]t = [Delta][[Rho] [C.sub.p]q(T - [T.sub.R])]/[Delta]V + (- [Delta]H) [r.sub.p] - hA(T - [T.sub.c]). (14)
In the above equations, the differential volume is calculated assuming the total volume to be apportioned uniformly along the pathlength in the reactor. The path in reactor 1 is assumed to be the midline along the flow path in the reactor, and thus comprises perpendicular segments along and perpendicular to the tubes. The conversion (X), number average chain length (J.sub.N), and weight average chain length ([J.sub.w]) are obtained from
X = [m.sub.1]q/[M.sub.0][q.sub.0], [J.sub.N] = [m.sub.1]/[m.sub.0], [J.sub.w] = [m.sub.2]/[m.sub.1]. (15)
We discuss the estimation of the heat transfer coefficients for the two reactors next. The resistance to heat transfer on the coolant side is expected to be much lower than that on the reaction mixture side, and hence is neglected.
In the first reactor, which is a baffled shell-and-tube exchanger with the reaction occurring in the shell side, the flow is partly cross-flow and partly along the tubes. We assume that the heat transfer coefficient in the parallel-flow region (for which no correlation is available) [TABULAR DATA FOR TABLE 3 OMITTED] is equal to that of the cross-flow. Gnielinski (15) has given the following correlation for cross-flow heat transfer on a single tube.
Nu = 0.75[(RePr).sup.1/3] for Re [less than] 1 (16)
where the Nusselt number (Nu) and the Reynolds number (Re) are based on the streamed length l along a tube (half of the circumference, l = [Pi]d/2). The Nusselt number is defined as
Nu= hl/[Kappa] (17)
To obtain the heat transfer coefficient for a bank of tubes, the Nusselt number is multiplied by a factor [f.sub.A], and the Reynolds number is given by
Re = wl[Rho]/[Psi][Mu] (18)
where w is the velocity in the free area, and [Psi] is a geometrical parameter defined as
[Mathematical Expression Omitted]
where d is the tube diameter, [s.sub.1] is the distance between tubes in a row, and [s.sub.2] is the distance between two rows. The factor [f.sub.A] has been defined by Gnielinski (15), but was treated as a fitting parameter in this study (Fitted value was [f.sub.A] = 5.8). All the physical properties are evaluated at the bulk temperature, but the Nusselt number is multiplied by a correction factor K given by
K = [(Pr/[Pr.sub.wall]).sup.0.25] for Pr/[Pr.sub.wall][greater than]1 (19)
K = [(Pr/[Pr.sub.wall]).sup.0.11] for Pr/[Pr.sub.wall][less than]1. (20)
For reactor 2, the heat transfer coefficient for a row of tubes as given by Gnielinski (15) is used. The Nusselt number is the same as in Eq 16, but the definition of [Psi] changes to
[Psi] = 1 - [Pi]/4[s.sub.1] (21)
The correction factor as defined in Eq 19 and 20 was employed for reactor 2 as well. The Reynolds number in this case is calculated the same way as in reactor 1 [TABULAR DATA FOR TABLE 4 OMITTED] (Eq 18) except that now the fluid velocity is calculated at half the agitator radius. The same value of the correction factor [f.sub.A] was used in this case as well. We note that the heat transfer coefficient has only a small dependence on viscosity through the correction K, since the product RePr is independent of viscosity.
The density was calculated using the correlation given by Arai et al. (8). The thermal conductivity correlation was taken from Husain and Hamielec (1), and viscosity was calculated using the correlation given by Dreval et al. (16).
RESULTS AND DISCUSSION
The differential equations in the PFR model were solved using the subroutine LSODA of ODEPACK (17). The recycle in the first reactor results in a two-point boundary value problem, the solution of which generates the profiles of variables, such as, temperature [TABULAR DATA FOR TABLE 5 OMITTED] and viscosity along the reactor. The boundary conditions at the inlet are guessed to convert it to an initial value problem, which can be easily solved. Iterations are then carried out until the guessed conditions match the calculated conditions. Convergence was obtained in 8 to 16 iterations.
Model validation was first carried out by comparing the simulation results with plant data. Representative results are shown in Figs. 3-5 for the temperature profile in the two reactors for three different operating conditions (Table 1). Both flow rate and ethyl benzene concentration have been varied. The corresponding solids fraction and molecular weights are also given in Table 1. Although the plant data for the molecular weights are not available for the above runs, values for similar conditions for the weight average molecular weight [Mathematical Expression Omitted] are in the range 1.8-2.0 x [10.sup.5]. The agreement between the model predictions and plant data is reasonably good in all cases. The only fitting parameter for the model is [f.sub.A] used in the calculation of the heat transfer coefficient, and the same value was used in both reactors for all the runs.
We discuss next the results of a study to determine the role of various parameters on the process. The set of data corresponding to the first column in Table 1 (feed flow rate of 905 kg/h) is chosen as the base case in all the simulations reported below.
Increase in the feed rate shows a significant decrease in solids fraction (conversion) and a small increase in molecular weight at the exit of reactor 1 (Table 2). Higher feed rates give lower residence times in the reactor and thus lower conversions resulting in lower solids fraction. This is reflected in the variation of viscosity in reactor 1 [ILLUSTRATION FOR FIGURE 6B OMITTED]: higher flow rates give lower viscosities. The increase in molecular weight with flow rate is primarily due to the higher monomer concentrations for the higher flow rates (lower conversions) which result in acceleration of the propagation reaction relative to the termination step. Because of high recycle ratios, the variation in flow rates causes only small changes in the heat transfer coefficient and as a result, the temperature profile in reactor 1 is not sensitive to flow rates. In the second reactor, the lower initial conversion for the higher flow rates gives higher rates of reaction and a larger exothermic heat of reaction [ILLUSTRATION FOR FIGURE 6C OMITTED]. As a result, the temperature is higher than the base case, and the differences in conversion for different flow rates become smaller (Table 2). The increased temperatures and lower conversions at higher flow rates give significantly lower viscosities [ILLUSTRATION FOR FIGURE 6D OMITTED], and thus termination rates are higher resulting in lower molecular weights at the exit of reactor 2.
The main effect of changing the ethyl benzene concentration is on the viscosity of the reacting mixture [ILLUSTRATION FOR FIGURE 7B, D OMITTED]. For example, reducing the ethyl benzene weight fraction from 7% to 1% gives a twofold to threefold increase in the exit viscosity in reactor 2. Such viscosities would result in excessive pressure drop in the reactor. Increase in the EB concentration is accompanied by a significant decrease in solids fraction (due to dilution by EB), though the fractional conversion decreases only slightly (Table 3). At the exit of reactor 2, for higher EB concentrations, there is also a slight decrease in the molecular weight because of lower viscosity. These results suggest that the reaction should be carried out at the lowest EB concentration, as determined by the maximum allowable pressure drop.
Thermic fluid temperatures have a significant effect on the operation of the reactors. Simulation results are presented in Fig. 8 and Table 4 for the case when all thermic fluid temperatures are 10 [degrees] C higher than the base case and when they are 10 [degrees] C lower than the base case. In reactor 1, higher temperatures result in higher conversions as reflected in the solids fraction values (Table 4). While viscosity at the lower temperature is significantly lower than the base case, it is nearly the same as the base case value at the higher temperature. This is because the increase in viscosity due to the higher solids fraction is compensated by the decrease in viscosity due to the higher temperature. Molecular weights at the exit of reactor 1 are higher at lower temperatures because of the reduced conversion. The temperature and viscosity profiles in the initial part of reactor 2 show trends similar to those in reactor 1. Close to the middle of the reactor, however, there is a sharp increase in the viscosity and temperatures signaling the onset of the gel effect in the case of lower thermic fluid temperature. A consequence of this is an unacceptably high viscosity at the exit of reactor 2. There is an increase in exit MW with decrease in temperature, and the polydispersity index is significantly higher for low temperature case relative to all other cases considered.
Increasing the recycle ratio beyond the base case value has little effect on the performance of the reactors as shown in Fig. 9 and Table 5. Lower recycle ratios, however, result in a significantly higher conversions (as evidenced by higher solids fraction), and lower molecular weights. The former results in high viscosities at the exit of reactor 1. The inlet temperature is lower because of the lower recycle amount, but increases sharply because of the exothermic heat of reaction from the high conversion and the lower heat transfer coefficient (lower flow rates). The high conversion in reactor 1 results from two causes: i. lower conversions at the inlet giving higher monomer concentration and thus higher reaction rates ii. longer residence times due to the lower flow rates. The molecular weights at the exit of reactor 1 are also lower for low recycle ratio. This is because of high termination rates due to the low viscosities in the initial half of reactor 1, where most of the conversion occurs. This effect propagates through reactor 2, giving a lower final MW and lower viscosity.
Plug flow models for a train of industrial styrene polymerization reactors, using detailed polymerization kinetics were presented. The predictions of the model were found to be in reasonably good agreement with temperature, solids fraction and molecular weight data from an industrial plant operating at different conditions. A single fitting parameter was used in the model.
The results of a parametric study showed the complex behavior of the process on varying some of the important process parameters (feed rate, ethyl benzene concentration, thermic fluid temperatures and recycle ratios). The temperature and viscosity profiles and the conversion and exit solids fraction and molecular weight variation with operating conditions are discussed in physical terms. Conclusions of the parametric study are summarized below:
1. The main consequences of increasing the feed rate is a dramatic reduction in the viscosity and small reductions in conversion and molecular weights.
2. Ethyl benzene concentration only affects the viscosity profile significantly. Low concentrations ([less than] 1%) may result in excessively high viscosities.
3. Decreasing the temperature of the thermic fluid results in lower conversions, and lower viscosities, initially. At later stages, however, the system may undergo the gel effect in which viscosities and temperatures sharply increase. Molecular weights and polydispersity index are high at low temperatures.
4. Recycle ratios higher than 6.67 for recycle in the first reactor, have little effect on the performance of the reactor train. Low recycle ratios, however, result in high conversion in the first reactor and lower molecular weights. There is little effect on the final conversion.
d = Tube diameter.
h = Heat transfer coefficient.
I = Initiator.
J = Chain length.
[J.sub.N] = Number average chain length.
[J.sub.W] = Weight average chain length.
k = Thermal conductivity.
[k.sub.Diff] = Rate constant for conversion of inactive monomer radicals to active monomer radical by diffusion.
[k.sub.fmc] = Rate constant for chain transfer to monomer.
[k.sub.pc] = Rate constant for propagation.
[k.sub.t] = Overall termination rate constant.
[k.sub.tc] = Rate constant for chemically controlled termination.
[k.sub.tD] = Rate constant for diffusion controlled termination.
K = Correction factor for the heat transfer coefficient.
l = Streamed length on a tube, Eq 17.
[m.sub.k] = Sum of k-th moments of polymer and polymer radical concentrations.
M = Monomer molecule and its concentration.
[M.sub.O] = Initial monomer concentration.
[M.sub.E] = Monomer molecule around a polymer chain and its concentration.
[N.sub.u] = Nusselt number.
P = Inactive polymer molecule and its concentration.
[P.sub.E](0) = Probability of finding no monomer molecules at the nearest-neighbor cells to a polymer chain.
q = Volumetric flow rate of the feed.
[r.sub.i] = Rate of initiation.
[r.sub.p] = Rate of propagation.
[P.sub.r] = Prandtl number.
[R.sub.g] = Gas constant.
R = Polymer radical and its concentration.
R = Non-growing polymer radical.
[R.sup.*] = Growing polymer radical and its concentration.
Re = Reynolds number.
[s.sub.1] = Distance between tubes in a row.
[s.sub.2] = Distance between two rows in a bank of tubes.
t = Time.
T = Temperature.
[T.sub.R] = Reference temperature.
Mathematical Expression Omitted] = Critical amount of local free volume per gram.
[Omega] = Velocity in the free area in bank of tubes.
X = Conversion of monomer to polymer.
[Rho] = Density.
[[Mu].sub.b], [[Mu].sub.w] = Viscosity of the mixture in bulk and at the wall.
[Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted], = k-th moment of size distribution of inactive polymer, polymer radical or growing chain radical.
[Psi] = Dimensionless parameter used in Reynolds number calculation.
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|Title Annotation:||International Forum on Polymers - 1996|
|Author:||Makwana, Y.; Moudgalya, Kannan M.; Khakhar, D.V.|
|Publication:||Polymer Engineering and Science|
|Date:||Jun 1, 1997|
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