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Determination of Gas-Liquid Mass-Transfer Limitations in Slurry Olefin Polymerization Pa˚ l Kittilsen,† Rune Tøgersen,† Erling Rytter,†,‡ and Hallvard Svendsen*,† Department of Chemical Engineering, Norwegian University of Science and Technology (NTNU), and Statoil Research Centre, 7005 Trondheim, Norway
Quantifying the effect of gas-liquid mass-transfer resistance is vital in studies aimed at extracting intrinsic kinetic parameters. This paper describes methods to evaluate this effect in a reactor used for slurry polymerization. Propene was polymerized using a modern ZieglerNatta catalyst dispersed in decane. The stirring rate was changed during polymerization, and the observed monomer feed rates were analyzed using methods based on steady-state and dynamic mass balances to obtain mass-transfer coefficients. It was found that the system investigated showed a 1.2 order dependency in monomer concentration and that the mass transfer increased with the stirring rate. Nonideal dynamic responses were observed when changing the stirring rate. This was qualitatively described using a nonideal mixing model with a dead and active volume. It was found that introducing baffles and sparging considerably increased the mass-transfer ability at high stirring rates. 1. Introduction The transport of a chemical component over a gasliquid boundary is of interest in many chemical processes. The present problem is focused on a heterogeneous olefin polymerization where the catalyst/polymer particles are suspended in a diluent. The monomer is introduced into a stirred tank reactor as a gas. The monomer must dissolve in the liquid phase for the reaction to take place on the solid catalyst. In our laboratory this setup has been used to investigate the kinetics of polymer reactions. To interpret the observed monomer flow into the reactor in terms of intrinsic kinetics, there is a need to understand and quantify the effect of the gas-liquid mass transfer and to provide criteria for how reactors should be operated and designed to have control of the gas-liquid mass-transfer resistance. Often problems with high gas-liquid masstransfer resistance can be solved by increasing the stirring rate. However, many systems have physical limitations in the stirring rate itself, or there may be a fragile product that limits the agitation power. For these systems, it is important to know the mass-transfer properties and choose the right loading to avoid masstransfer limitations. The main focus in this work is on the mass transfer to a free interface in an unbaffled vessel, but also baffled and sparged reactors as alternatives to increase masstransfer rates are considered. This paper mainly deals with the experimental determination of gas-liquid mass-transfer effects. A subsequent paper1 concerns the modeling of these effects. The transport of a component from gas to liquid phase in a stirred vessel is normally considered to be a transport between two (ideally) well-mixed phases, and changes in the concentration of a component are limited to a small part of the system volume near the interface. * To whom correspondence should be addressed. E-mail:
[email protected]. Fax: +47 73594080. † NTNU. ‡ Statoil Research Centre.
The transport is described in terms of mass-transfer coefficients on the gas and liquid sides (kG and kL), a proportionality constant between mass flux, and a driving force, often a concentration difference.2 In systems where the transferred component has a high mole fraction in the gas phase, convection will be important and the gas-side transfer resistance may be disregarded. In this paper we will only aim at getting experimental values for the mass-transfer factor (the mass-transfer factor is used for denoting kLA, the mass-transfer coefficient times the contact area, and the mass-transfer coefficient is used for denoting kL). The more general method is to report mass-transfer coefficients. However, experimentally one normally observes the mass-transfer factor, and an estimate for the contact area is needed to obtain the mass-transfer coefficient. At low stirring rates for an unsparged reactor, the estimate of the contact area is simple, just the cross-sectional area of the reactor. At higher stirring rates, the effects of vortex formation and rippling become important. This requires a more thorough discussion, which we leave for the subsequent paper1 where the kLA term is split. Keii et al.3 have shown how to obtain mass-transfer factors using mass balances in an analysis of the dynamic and steady-state reaction rates resulting from variations in the stirring rate in a first-order polymer reaction. There are few other papers describing experimental observations of gas-liquid mass transfer in reactors operated at polymerizing conditions. Li et al.4 investigated the absorption of hydrogen, ethylene, and propylene into a reactor without reaction, containing 0-30 wt % polymer particles. They found that the masstransfer coefficient increased and decreased up to a factor 2 at low (10 wt %) and high (30 wt %) polymer contents, respectively. This result is the same as that which Floyd et al.5 report in their study of models for mass transfer in polymerization reactors. They conclude that as long as the properties of the liquid phase are not dramatically changed from that of pure liquid, i.e., at moderate polymer contents, the absorption can be considered unaffected by the presence of polymer. In this
10.1021/ie000577p CCC: $20.00 © 2001 American Chemical Society Published on Web 02/01/2001
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catalyst (Ti on a MgCl2 support, Borealis BC-1). The reactor was operated at constant pressure. Because the monomer was consumed by the reaction, a pressure control valve was automatically opened to let the monomer in and keep the pressure constant. The feed rate of propene was recorded by a flowmeter at a frequency of 1 Hz. The reactor was operated at 70 °C and 4 bar total pressure unless stated otherwise. For details of the method, see work by Tøgersen.6 3. Methods To Determine the Mass-Transfer Factor
Figure 1. Schematic drawing of the reactor used for the experiments: (1) stirrer motor; (2) stainless steel cylinder; (3) metal cap with circulating heat media (water or oil); (4) thermoelement; (5) hollow tube stirrer.
work, we do not make such assumptions. We measure the mass-transfer characteristics directly in the reactor used for the kinetic study of polymer reactions. We have extended the steady-state method of Keii et al.3 to also include other reaction orders and discussed the influence of the reaction order and catalyst deactivation. The results from the steady-state and dynamic methods differ significantly, and an ideally mixed reactor cannot explain the observed effects. However, these discrepancies can be explained by nonideal mixing of the reactor. 2. Experimental Section A 600 mL semibatch cylindrical vessel equipped with a two-bladed, possibly self-sparging, stirrer was used for the experimental investigation. The inner diameter of the reactor was 63.3 mm, and the diameter of the impeller was 34.1 mm. The reactor is shown in Figure 1. The monomer used was propene, the solvent was 175 mL of decane, giving a liquid height of about 6 cm, and the catalyst was 40 mg of a modern Ziegler-Natta
The mass-transfer factor can be determined through separate analysis of steady-state and dynamic feed rates of the monomer as the stirring rate is changed. Figure 2 shows how the feed rate changes with time and stirring rate. The methods used in this section are based on the ideas of Keii et al.,3 who analyzed a nondeactivating, first-order, polymer reaction. In this section, we will also discuss methods for dealing with deactivating systems and non-first-order reactions. 3.1. Mass Balance. The rate of a polymerization reaction can often be described by
R ) kpGMn
(1)
where R is the total rate of the reaction, kp is the propagation constant per unit mass of catalyst, G is the total mass of the catalyst, M is the monomer concentration in the bulk of the reactor, and n is the monomer overall reaction order. In general, kp is a function of time. The gas-liquid mass-transfer rate is described by
F ) kLA(M0 - M)
(2)
where kL is the mass-transfer coefficient defined by this equation, A is the interfacial area, and M0 is the equilibrium monomer concentration. kL is assumed to be independent of the monomer concentration. Combining the reaction rate and the gas-liquid masstransfer rate with the change in monomer concentration gives the general mass balance for an ideally mixed
Figure 2. Feed rate of propene as a function of time: (a) a typical experiment (no. 97) where the stirring rate, N, has been changed during the run. The stirring rates are in rpm. The dashed line shows the approximate feed rate with negligible mass-transfer resistance. (b) Corresponding feed rates and stirrer rates schematically. F0 is the feed rate with no mass-transfer resistance, F1 and F2 are the steady-state feed rates at stirring rates N1 and N2, respectively, and F is the observed dynamic feed rate. The period in which there is a changing feed rate is called the dynamic period.
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semibatch reactor with constant volume and catalyst concentration:
dM ) kLA(M0 - M) - kpGMn V dt
(3)
where V is the liquid volume. 3.2. Steady-State Method. (a) Nondeactivating System. At steady state in a nondeactivating system (kp ) kp0), the monomer concentration is constant and the dynamic term of the general mass balance can be disregarded. The mass balance thus becomes
Fi ) kLA(M0 - M) ) kpGMn
(4)
where Fi is the steady-state feed rate at stirring rate Ni. The mass balance with no mass-transfer resistance, i.e., M ) M0, is
F0 ) kpGM0n
where
C ) ln |Fi0 - Fi|
Here Fi0 is the feed rate just after changing the stirring rate to Ni, and Fi is the steady-state feed at Ni. A plot of ln|F - Fi| versus t should thus give a linear relation with slope -Ri. The mass-transfer factor is
βi ) RiV - kpG ) RiV - F0/M0
Fi
(5)
M0[1 - (Fi/F0)1/n]
(6)
(b) Deactivating System. Catalysts used for olefin polymerization do generally deactivate, and a timedependent term has to be included in the propagation constant:
kp ) kp0 - kpd(t)
M ) M0 - F/β
A constantly deactivating system will never reach a steady state because the activity, and thus the monomer concentration (which is dependent on the activity), changes continuously. Nevertheless, these systems can sometimes be treated as quasi-steady-state systems. The criterion is that the rate of change of the monomer concentration is much smaller than the rate of the monomer feed. The mass balance will thus be the same as the one explained for the steady-state case, and the solution to the problem is the same as that discussed above (eq 6) when corresponding values of Fi and F0 (with respect to the degree of deactivation, i.e., at the same t) are used and when the deactivation is independent of the monomer concentration. These assumptions were proven valid in a kinetic study (see section 4.2) and in a numerical analysis of the deactivation. 3.3. Dynamic Method. (a) Nondeactivating System. Keii et al.3 solved the general mass balance (eq 3) analytically for the first-order nondeactivating case (n ) 1.0, kp ) kp0):
F ) e-RitFi0 + (1 - e-Rit)Fi
(8)
where
Ri )
βi + kpG V
(9)
V ln |F - Fi| ) -Rit + C
f
(10)
(13)
1 dF dM )dt β dt
(14)
The propagation term can be expressed by the intrinsic feed rate from eq 5:
kpG ) F0/M0n
(15)
Using these three equations and the equation for the feed (eq 2), the mass balance (eq 3) can be rewritten as
(7)
(12)
We call this method the first-order method. The cases where n * 1 are more complex because there exists no simple analytical solution to the differential mass balance. However, by introducing known relations into the mass balance, we can derive an implicit expression for β as a function of only known parameters. The monomer concentration can be expressed in terms of the feed rate from eq 2:
Combining these two equations and solving for the unknown mass-transfer factor β ) kLA give
βi )
(11)
(
F V dF ) F - F0 1 β dt βM0
)
n
(16)
Thus, we have an implicit expression for β, and we call this method the nth-order method. The special case n ) 1 can be solved explicitly:
V β)
dF F0F + dt M0 F0 - F
(17)
By using these expressions and corresponding values of F and F0 (at the same t), it is possible to obtain values for the mass-transfer factor β. Equation 16 is generally valid for reactions of order n and for any degree of deactivation. The difficulty might be to get an accurate value of the derivative of F. From the expression for n ) 1 (eq 17), it is obvious that the method will be uncertain near F ) F0, where the denominator approaches infinity. This also prevails for n * 1 but close to 1. (b) Deactivating System. An analytical solution exists for first-order reactions with a linear deactivation function (kpd ) constant) but is complex. Other cases must be solved numerically. As pointed out above, the nth-order method will be correct regardless of deactivation. For systems with a relatively small deactivation rate and a monomer reaction order close to 1, the solution found for first-order nondeactivation systems can be useful. To test the accuracy of this method, we made a numerical investigation. Responses of a 1.2order reaction with deactivation corresponding to the experimental system were simulated and interpreted using the first-order method.
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Figure 3. (a) Reaction rate in four different experiments. Three experiments were performed with constant pressures of 2, 4, and 6 bar, and one experiment was performed where the pressure was varied during the polymerization at the levels 6, 5, 3, 5, and 6 bar. The dashed lines are the modeled rates using a linear deactivation model. (b) Corresponding reaction rates near t ) 2000 s as a function of the monomer concentration in decane. The slope of the line equals the propene reaction order.
To compensate for the deactivation, we recalculated the feed rate using the relation between the reaction rate that would be observed if deactivation was not present and the actual reaction rate:
kp0 F0(0) )F kp F0
F′ ) F
(18)
Then |F′ - F′i| was calculated in the dynamic period, and F0(0) was used for F0 in eq 12. It was found that eq 12 gave β values with typical deviations of (0-10% from the value set in the simulation. The way we have compensated for the deactivation will not rule out the influence from the deactivation completely. Considering the mass balance (eq 3) and introducing F ) kLA(M0 M) give
F)V
dM + kpGMn dt
(19)
When the deactivation is compensated for, only the last term will be influenced (because it contains the propagation constant), leaving the first term unchanged. Thus, the deactivation will still influence the results, but now to a smaller extent. This method of compensating for the deactivation was used when analyzing the experimental data with the first-order method. 4. Results and Discussion The results of an experimental series analyzed for gas-liquid mass-transfer effects, using the methods shown above, are presented here. As pointed out, the monomer reaction order needs to be known in order to obtain an accurate result, and one subsection deals with this problem. The following subsections concern masstransfer factors from experiments with varying stirring rates and the scatter in the results from the different methods. Finally we show the effects of baffles and sparging on gas-liquid mass transfer. However, first we discuss the general use of the methods. 4.1. Discussion of the Methods. The methods derived above are generally valid also for cases with sparged reactors. They can be used for any reactor operated in such a way that the feed rate of the monomer is controlled by keeping the pressure constant. To use the dynamic methods, a high sampling rate is necessary to get reliable data at least for fast responses,
i.e., at large mass-transfer factors. The methods provide mass-transfer data directly for the reactor used in kinetic studies and thus have advantages over pure estimates of the mass-transfer factor from correlations. The methods can be used to get quick and reliable experimental estimates of the mass-transfer resistance. 4.2. Kinetics of the System. A kinetic study of the system was performed through a variation of the monomer concentration (total pressures of 2-6 bar), keeping all other parameters constant. On the basis of observed activity, a linear deactivation model was assumed (kpd(t) ) Ct) and individually fitted to the observed reaction-time curves at the different pressures. A logarithmic plot of the reaction rate as function of the monomer concentration (from SRK-EOS) revealed a monomer dependency of 1.18-1.22 (varying with time) with an average of 1.20. A further investigation using a reaction order of 1.20 revealed that all levels of reaction rates could be described well by the same simple linear deactivation model; kp ) 9.1 × 10-8 (m3/mol)1.2 mol/kg/s and kpd ) 6.4 × 10-12t (m3/mol)1.2 mol/kg/s. The deactivation seems to be independent of the monomer concentration and of the amount of polymer produced. The reaction rates at various pressures are shown in Figure 3. 4.3. Effect of the Stirring Rate. Sets of experiments were performed where the reactor was operated: (a) unbaffled and nonsparging; (b) baffled and nonsparging; and (c) unbaffled and sparging. When changing the stirring rate from 1500 to 2000 rpm, we found that the steady-state monomer feed rate changed minimally. Thus, it is assumed that at 1500 rpm there is negligible mass-transfer resistance. Every experiment started and ended with a period with a stirring rate of 1500 rpm, thereby ensuring the assumption of equilibrium between the gas and liquid to be reasonable in these periods. The response of a typical experiment is shown in Figure 2. The mass-transfer factors were calculated using the steady-state and dynamic methods as described in section 3, and factors from experiments where the reactor was operated unbaffled and nonsparging are plotted as a function of the stirring rate in Figure 4. The mass-transfer factors increase significantly with the stirring rate as expected. At higher stirring rates, the turbulence intensity increases, shortening the contact time between liquid elements and gas, thus renewing the liquid in contact with gas more often. At higher stirring rates, the mass-transfer factor is also increased
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Figure 4. Mass-transfer factor, kLA, for propene into decane at 70 °C as a function of the stirring rate. Only values obtained with the reactor operated unbaffled and nonsparging are shown. Values from three different methods of analysis are shown: steady state, dynamic first order, and dynamic nth order.
because of the increase of the interfacial area because the surface becomes cone-shaped and rippled. For a more thorough discussion of these effects, see the subsequent paper.1 When the first-order method is used and ln |F - Fi| is plotted as a function of t (eq 10), some of the low stirring rate experiments (N < 300 rpm) gave curved lines and thus ranges of mass-transfer coefficients. The coefficients plotted in Figure 3 are averages of these ranges. An example of such an experiment where the dynamic response did not show ideal behavior is shown in Figure 5a. This curvature can be explained neither by the uncertainty in Fi from the deactivation nor by a reaction order that is different from 1. These factors have all been ruled out through simulations. At higher stirring rates (N > 500 rpm), the dynamic responses look ideal. An example of such an experiment is shown in Figure 5b. Despite the linearity in the logarithmic plot, it is not possible to reproduce the dynamic responses with the β values obtained from the first-order dynamic method. There is a discrepancy between the shape of the dynamic curve and the level it reaches at steady state. In other words, an ideally mixed reactor cannot be used as a model in this case. Possible reasons are discussed in the next section. Also the nth-order method showed a variation in the β values with time when analyzing a dynamic response. So, also this method indicates the nonideality of the investigated reactor. 4.4. Effect of Nonideal Mixing. The reactor used in this study is a cylindrical vessel stirred with a T-shaped agitator made of tubes. This setup presumably
does not cause good mixing but mainly makes the fluid rotate in the vessel. Thus, there will be low axial and radial mixing in the reactor: mechanisms that are important for the transport of monomer from the gas phase to the lower parts of the reactor volume. Mass transport within such a nonideal reactor must to a greater extent rely on molecular diffusion, which is a very slow process compared to convection and turbulent diffusion. A common way of regarding a reactor with imperfect mixing is to model it as consisting of two compartments each with good internal mixing but with a restricted exchange of mass.7 The compartments are often referred to as bulk and dead volumes. Thus, there are two levels of mass transfer: one external mass transfer from gas to the bulk of the liquid and another internal mass transfer from the bulk to the dead volume. The catalyst/polymer particle may be nonhomogeneously distributed in the reactor. In the investigated system, the density of the liquid decane is about 700 kg/m3 and the density of the catalyst/polymer particles is approximately equal to the density of pure polypropene, about 900 kg/m3. At high stirring rates, the liquid swirls around in the reactor, and the centrifugal force will force the catalyst/polymer particles toward the reactor walls. At low stirring rates, the axial velocity is low, and part of the particles will normally be found close to the bottom of the reactor in zones of little mixing, i.e., in the dead volumes. We have done simulations using a two-compartment model with both external (gas to liquid) and internal (bulk to dead-volume) mass-transfer resistances and a possible catalyst concentration variation. The mass balances were taken for each compartment, and the resulting coupled differential equations were solved numerically for some sets of parameter values. Deactivation of the reaction rate was not considered. For both cases, we used a model with 50% dead volume. The results were as follows: (i) Low Stirring Rate (500 rpm). In this simulation we assumed the other extreme case with all of the catalyst being in the bulk volume. The gas-liquid masstransfer factor is increased to 5.0 × 10-6 m3/s (typically near 800 rpm), and the internal mass-transfer factor is increased to 1.5 × 10-6 m3/s. The simulation is done by assuming an initial quite low and equal monomer concentration in the two compartments (0.6M0). This set of parameters was tuned to obtain results similar to the experimental ones. The responses from this simulation are shown in Figure 7. Analysis of these steady-state and dynamic feed rates gives transfer factors of 5.0 × 10-6 and 2.0 × 10-6 m3/s, respectively. The steady-state value is now equal to the “true” one set in the simulation. However, the dynamic value is much smaller than the one set in the simulation. The dynamic value is a result of bulk to dead volume dynamic behavior, a much slower responding system than the ideal one. The results with a higher steady-state-based value compared to the dynamic one are also found experimentally. If the model simulations using the dead volume model are reasonable, the above discussion shows that both the steady-state- and dynamic-based coefficients tend to be too low compared to the “real” values. The steadystate values are too low because of the presence of catalyst in compartments with lower monomer concentration than that in the bulk, and the dynamic-based factors are too low because the dynamic response is a
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5. Conclusions
Figure 8. Mass-transfer factors as a function of the stirring rate for a reactor with baffles and with sparging. The “normal” refers to the unbaffled, nonsparging mode. The data are average values obtained by the steady-state method.
result of the dynamics of the bulk to dead volume, a slower system than the gas to ideally mixed reactor. 4.5. Effects of Baffles and Sparging. The effects of some alternative reactor designs were also tested. The variations of the mass-transfer factor as a function of the stirring rate when introducing baffles and when operating with a self-sparging agitator were measured. The steady-state values are plotted in Figure 8. The effects are discussed below. (i) Baffles. A couple of experiments were performed with four baffles inserted into the reactor vessel. Their widths were 1/12 reactor diameter. There were no gaps between the baffles and the reactor wall. Compared to the unbaffled case, the mass-transfer factors with baffles were lower at low stirring rates, little difference was found at moderate rates (500 rpm), and considerably higher factors were obtained at high stirring rates. Introducing baffles will, in general, increase the power input, thus increasing the turbulent intensity and giving an enhanced mass-transfer factor at the same stirring rate. However, there is an effect that might make the mass-transfer factor lower. At low stirring rates, the fluid can be more stagnant in a baffled tank; i.e., there can exist larger dead volumes, and/or the polymer particles might get stuck at the baffles, giving lower activity, which will be interpreted as a lower mass-transfer factor. (ii) Sparging. The agitator was of a hollow shaft type. When rotating, an underpressure is introduced at the paddle ends. This underpressure can drive gas through holes in the gas-phase part of the shaft and into the liquid, making a self-sparging agitator. The agitator could be operated self-sparging or not, changing the mode by removing a plug in the shaft end. Without the plug, the underpressure causes liquid to be sucked into the shaft from the bottom of the agitator and circulates bulk fluid in the shaft only. There is also a required minimum stirring rate to overcome the static pressure; thus, at low stirring rates (below 500 rpm), it seemed that the agitator never operated in self-sparging mode. The mass-transfer factor for a sparged reactor at a stirring rate of 1000 rpm was significantly larger than that for the nonsparging case. At lower stirring rates, there was no significant difference, probably because the agitator at this speed was not sparging. The sparging mainly increases the interface area between the gas and the liquid, A, because of the high specific area of the small bubbles. The mass-transfer coefficient, kL, is not influenced much.
An experimental investigation of the gas-liquid mass transfer in a stirred semibatch laboratory reactor for polymerization of propene using a Ziegler-Natta catalyst has been performed. We draw the following conclusions from this work: (1) A kinetic study of the system revealed that the deactivation was independent of the amount of polymer produced and that the monomer reaction order was 1.2. (2) New methods for investigating mass transfer over a gas-liquid interface in a polymerizing reactor have been developed. The methods can be used for systems having monomer dependency different from first order. (3) The dynamic and steady-state monomer feed rates were measured as a function of the stirring rate. These observations were interpreted in terms of mass-transfer factors, which were found to increase with increasing stirring rate. The results from the dynamic and steadystate methods differed significantly, and the transient responses could not be described using an ideally mixed reactor model. The discrepancy between the results can be qualitatively explained by nonideal mixing in the reactor. (4) Introducing baffles and sparging considerably decreased the gas-liquid mass-transfer resistance at stirring rates higher than 800 rpm. Baffles seemed to cause increased mass-transfer resistance at low stirring rates, explained by emphasizing the effects of nonideal mixing. Sparging increases the interfacial area and the mass-transfer factor. Acknowledgment The authors thank the Norwegian Research Council (NFR) under the Polymer Science Program for financial support of this work and Borealis AS for supplying the catalyst and chemicals. Nomenclature A ) total gas-liquid interface area, m2 Ar ) cross-sectional area of the reactor, m2 C ) constant d ) reactor diameter, m F ) monomer feed rate over the gas-liquid interface, mol/s Fi ) steady-state monomer feed rate at Ni, mol/s F0 ) monomer feed rate at M ) M0, mol/s G ) mass of the catalyst, kg kG ) gas-side mass-transfer coefficient, m/s kL ) liquid-side mass-transfer coefficient, m/s kp ) propagation constant, (m3/mol)n mol/kg/s kp0 ) propagation constant at t ) 0, (m3/mol)n mol/kg/s kpd ) deactivation term in the propagation constant, (m3/ mol)n mol/kg/s M ) monomer concentration, mol/m3 M0 ) equilibrium monomer concentration, mol/m3 n ) monomer reaction order N ) stirring rate, rpm R ) reaction rate, mol/s V ) liquid volume, m3 R ) slope, 1/s β ) mass-transfer factor ) kLA, m3/s
Literature Cited (1) Kittilsen, P.; Tøgersen, R.; Rytter, E.; Svendsen, H. Modeling of Gas-Liquid Mass Transfer Limitations in Slurry Olefin Polymerization. Ind. Eng. Chem. Res. 2000, submitted for publication.
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(2) Cussler, E. L. Diffusion. Mass transfer in fluid systems; Cambridge University Press: New York, 1984. (3) Keii, T.; Doi, Y.; Kobayashi, H. Evaluation of Mass Transfer Rate during Ziegler-Natta Propylene Polymerization. J. Polym. Sci., Part A: Polym. Chem. 1973, 11, 1881. (4) Li, J.; Tekie, Z.; Mizan, T. I.; Morsi, B. I. Gas-Liquid Mass Transfer in a Slurry Reactor Operated under Olefinic Polymerization Process Conditions. Chem. Eng. Sci. 1996, 51 (4), 549. (5) Floyd, S.; Hutchinson, R. A.; Ray, W. H. Polymerization of Olefins Through Heterogeneous Catalysis V. Gas-Liquid Mass Transfer Limitations in Liquid Slurry Reactors. J. Appl. Polym. Sci. 1986, 32, 5451.
(6) Tøgersen, R. Polymerization of Propene with an Industrial MgCl2-supported Ziegler-Natta Catalyst. MSc Thesis, The Norwegian University of Technology and Science (NTNU), Trondheim, Norway, 1998. (7) Fogler, H. S. Elements of Chemical Reaction Engineering; Prentice-Hall: London, 1992.
Received for review June 13, 2000 Revised manuscript received November 8, 2000 Accepted November 28, 2000 IE000577P
