Heat-Transfer Phenomena in Gas-Phase Olefin Polymerization Using

Ind. Eng. Chem. Res. 2004, 43, 7251-7260

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Heat-Transfer Phenomena in Gas-Phase Olefin Polymerization Using Computational Fluid Dynamics Erik J. G. Eriksson and Timothy F. McKenna* LCPP-CNRS/ESCPE-Lyon, 43 Bd du 11 Nov. 1918, Baˆ t 308F, BP 2077, 69616 Villeurbanne CEDEX, France

Computational fluid dynamics (CFD) was used to study some aspects of heat transfer from particles under conditions similar to those found during gas-phase olefin polymerization. The particles considered are initially small (diameter between 10 and 80 µm), thus making it difficult to apply classical heat-transfer correlations. The FLUENT CFD code was used to study the influence of parameters such as the maximum reaction rate, the activation time, particle interaction, and the influence of the initial catalyst particle size on particle growth and thus the production of heat. Results show that, as expected, the larger the initial catalyst particle size, the higher the temperatures inside the particle. Further it is shown that particle interactions play a large role in the heat transfer, especially when studying the effect of the bulk gas flow, both in terms of velocity and direction. It was also demonstrated that because of these interactions, both convection and conduction of heat can be important. 1. Introduction A very common way to produce polymers and copolymers from R-olefins is by using supported catalysts. These catalysts are constructed of a highly porous support material, such as silica or magnesium dichloride, with the active sites deposited on the pore surface. The particles are injected in to the reactor containing the continuous phase, where the reactants must diffuse through the continuous phase, through the pores of the support material to the active sites where they react to form the solid polymer. Generally, the catalyst particles have a diameter of 10-50 µm when entering the reactor and somewhere between 500 and 1000 µm after the reaction. Given the highly exothermic nature of the polymerization reaction (heat of reaction on the order of 100 kJ/mol) and the high rates of reaction (5-50 kg of polymer per gram catalyst per hour) currently found in industrial processes, it is clear that increasing the space-time yield in the reactor will cause problems for heat transfer. In gas-phase reactions (to which we will restrict ourselves in this paper) this problem is doubly important as the heat-transfer characteristics of gasphase reactors are typically poor.1 However, for reasons related to product quality, operability, and other constraints, gas-phase processes still find wide applications in polyolefin manufacturing. In particular, both stirred powder beds (SPB) and fluidized beds find widespread use, and a majority of the current gas-phase processes for the production of polyethylene (PE), of polypropylene (PP) and of impact copolymers use gas-phase fluidized beds (FBR) in at least one of the production steps. Because of the multiphase nature of the reactor contents, the hydrodynamics inside a FBR or SPB can be very complex. It has been demonstrated2 that it is critical to have a reliable model of the flow field inside such reactors if we are to understand the interactions between the physical and chemical phenomena that influence the reaction and thus the productivity, safety, and profitability of the process. * To whom correspondence should be addressed. Tel.: +33 4 72 43 17 75. Fax: +33 4 72 43 17 68. E-mail: [email protected].

Most attempts at modeling olefin polymerization have used a residence time distribution approach. In other words, the reactor is treated as an assembly of one or more pseudohomogeneous phases, with mass and heat transfer occurring between them. Recent comprehensive discussions can be found in the works of Hatzantonis et al.3 and Kim and Choi.4 While these works allows understanding of how polymerization occurs and what the general influence of reaction conditions on kinetics and polymer properties are, they rely on engineering correlations to predict bubble size, average velocities of gas, and particle phases in the reactor. By their very nature, correlations impose an averaged, or pseudohomogeneous, view on the entire reactor, which means that it is not easy to use such an approach for modeling hotspot formation, the effect of particle interaction, etc. It should be clear that heat-transfer problems related to particle-particle interactions5 require a full scale modeling approach that allows us to account for localized phenomena. In other words, the ideal model for a polyolefin reactor will couple a good particle growth model6 with a full scale hydrodynamic model of the reactor. This would allow us to account for local variations in flow rates, concentrations, and temperature on both particle growth and overall reactor behavior. However, it is a difficult task to develop a full hydrodynamic model of a gas-phase polymerization for a number of reasons, not the least of which being that it is extremely onerous and costly to validate such models on full-scale reactors, and laboratory-scale reactors have very different hydrodynamics due to scaling problems. This is not to say that it cannot be done at all. Recent attempts at modeling polyolefin reactors and in particular the influence of the heat transfer on the reaction kinetics have demonstrated some of the difficulties inherent in this problem. For instance, a discrete element method (DEM) was used7,8 to simulate the polymerization of ethylene and propylene in a small fluidized bed reactor. These two papers describe an interesting attempt to combine single-particle mass and energy balances6 with simplified two-dimensional mass, momentum, and energy balances of an FBR near the

10.1021/ie034328n CCC: $27.50 © 2004 American Chemical Society Published on Web 06/04/2004

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Figure 1. Example of particle mesh. Upper figure showing the mesh inside the solid particle and the bottom the meshing of the fluid.

distributor plate. This approach is interesting but limited for a number of reasons. First of all, DEM simulations require very large CPU capacity due to three reasons. One is the usage of an Euler-Lagrange approach, known as a very time-consuming approach, especially if a large number of particles is to be tracked. The second is that one is obliged to use small time steps with this approach, either to get proper particleparticle interaction or to guarantee the numerical stability. Finally, it is not clear how to extend the DEM approach to multiple particle sizes. This is not to say that DEM is not useful from an academic point of view. These two papers clearly serve to illustrate the need for more accurate descriptions of FBR hydrodynamics in the development of a full-scale integrated reactor model and to provide useful information about local heat transfer and the relationship between particle growth and fluidization conditions. Other approaches to modeling FBRs are based on CFD studies. For instance, van Wachem et al.9 used an Eulerian-Eulerian approach to model a monodispersed fluidized bed to calculate the time-averaged fluctuations in the pressure and bed density. Marschall and Mleczko10 also used CFD to look at the porosity and velocity distributions in an FBR. These two studies provide useful information on the hydrodynamics of small scale FBRs, but they cannot practically be extended to an industrial study. Another CFD study11 solves the reactor population balance using the direct quadrate method of moments. The void fraction and temperature distribution is shown for laboratory-scale fluidized bed. The results show a rapid heat transfer between the solid and gas phases and the fact that bigger particles overheat easily, potentially causing reactor hot spots. On the particle level, there are several models for describing the heat transport occurring during gas-

phase polymerization. One of the most widely used models is the multigrain model (MGM)1,12 where the polymer particle is modeled as an agglomeration of concentric layers of microparticles, in which the actual reaction occurs. Several important conclusions can be drawn from modeling studies using the MGM:1,6 (1) in most cases, intraparticle temperature gradients are negligible; (2) external heat-transfer resistances are negligible, except for highly active catalysts or at short times; and (3) heat-transfer resistances are much more important at early stages of polymerization. In an initial study by McKenna et al.,13 some of the shortcomings of this approach were discussed. These include the incapability to predict changes in rapidly evolving particles as well as physically unrealistic predictions such as melting of particle cores. Further, they showed that classical heat-transfer models, using correlations, such as the Ranz-Marshall one, can occasionally predict physically unrealistic results when applying them to modern and highly active polymerization processes. For this reason they chose to use CFD to study heat-transfer effects when length and time scales characteristic of olefin polymerization in the gas phase make experimental studies very difficult. Their main conclusion was to show that particle-particle interaction could play an important role in the evacuation of heat from small, highly active particles in gasphase processes. A solution to the problems of high initial heat-transfer resistance can be prepolymerization. Prepolymerization is usually carried out at low reaction rates and low temperatures, in order to make the fresh catalyst particles grow before being injected in the polymerization reactor. It has been observed14,15 that catalyst prepolymerization at reduced temperatures results in a significant reduction of the particle overheating and

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at the same time an increase of the overall polymerization rate. In this paper the effects of heat transfer have been studied and modeled in several different ways using the CFD approach. The base case used in this work is homopolymerization of ethylene using static polymer particles suspended in a moving gas in order to simplify the calculations. The main objective is to study problems related to particle overheating as a function of a number of important parameters. The conditions here are chosen in order that they be representative of both FBRs and SPBs, where the latter will have much lower relative gas/particle velocities and much longer contact times.

Table 1. Parameter Values

2. Model Setup The model presented in this paper was solved using FLUENT, a computational fluid dynamics (CFD) code. When creating the simulation geometry and grid, it is essential that there are neither boundary effects nor influence of the number of cells used in the modeling. The boundary effects are avoided by extending the geometry to at least 10 times the particle diameter around the largest particle. As for the number of cells, the grid is centered inside and around the particles, to make the solution grid independent. Detailing the grid in the particle vicinity allows for more data to be extracted in this interesting zone. On the other hand, moving away from the particles, the grid is made courser and courser, since the computational time is highly dependent on the number of cells in the mesh (Figure 1). Mesh construction and verification of the grid’s independence of the calculations were performed as described in a previous paper.5 The simulations were carried out in two dimensions using an axisymmetric domain. The number of used cells in the different simulations varies from about 12 000 to 20 000. When the influence of heat transfer occurs in the radial position, as is the case here, the two-dimensional simulations allows a higher precision in the form of more cells then what would be possible using three dimensions for the same computational time and power. The model is based on the two-dimensional form of the general conservation (transport) equations: The momentum equation for the gas (eq 1) and the energy balances for the gas (eq 2) and the solid particle (eq 3).

[(

)] [ ( )]

∂ui ∂uj ∂ ∂ ∂p ∂ (Fu )+ (Fu u ))- + µ + x ∂t i ∂xj j i ∂xi ∂xj ∂xj ∂ i

(1)

∂Tg ∂ ∂∂x ∂ (Fcp,gTg)+ (Fujcp,gTg)) ke,g j ∂t ∂xj ∂xj

(2)

[ ( )]

∂Ts ∂ ∂ (Fcp,sTs)) ke,s +Q ˆ vol ∂t ∂xi ∂xi

RpFeffR03 R1

3

( [

∆Hp exp -

Ea 1 1 R T Tref

parameter

value Solid 900 kg/m3 2150 J/(kg K) 0.117 W/(m K)

density heat capacity thermal conductivity Fluid density heat capacity thermal conductivity viscosity inlet gas temperature

40 kg/m3 2230 J/(kg K) 0.0242 W/(m K) 1.1 × 10-5 kg/(m s) 350 K

Reaction Parameters deactivation constant 5 × 10-5 1/s activation time 1-300 s heat of reaction 100 kJ/mol

from our previous work,13 since incorporating the influence of the exothermic nature of the reaction allows modeling of thermal runaway. Equations 1-4 are solved, in each cell of the mesh, for the different simulation setups. The simulation parameters are presented in Table 1. A generalized presentation of the particle setup is shown in Figure 2. The bulk gas flow is defined as positive when going from left to right and negative when going from right to left. The smallest particle is always placed on the left side of the large particle. In the simulations, the solid particles are considered as stationary, being suspended in a moving gas stream. Further, the following assumptions are made in this paper: (i) all particles are spherical; (ii) there exists axisymmetric geometry in two dimensions; (iii) particles are static in a moving gas flow; and (iv) identical growth rates occur for all particles. All simulations are carried out using a gas inlet temperature of 350 K. For the polymer particles, the temperature is limited to 385 K in order to avoid the influence of an eventual change of the physical properties (melting point of a typical linear low-density polyethylene). If the temperature inside the particle reaches higher than 385 K, the particle is considered overheated, and the simulation stops.

(3) 3. Simulation Results

When studying the heat-transfer effects we are especially interested in the last term in eq 3, the volumetric heat flux. In eq 4, this term is calculated.

Q ˆ vol )

Figure 2. Generalized particle setup.

])

(4)

Using a dynamic expression and not a constant value of the volumetric heat flux is a significant difference

This paper focuses on how the heat transfer is affected by the activation time, particle interaction effects, and effects of initial particle size, all for “fresh” catalyst particles (also referred to as “small” particles). 3.1. Single-Particle Simulations. The effects of catalyst activation time were investigated for a single particle. In all the simulations concerning the effect of the activation time, the catalyst particle has an initial diameter of 20 µm. The reaction rate was calculated using eq 5.

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[

( )]

Rp)kp,max 1-exp -

t exp(-kdt) ta

(5)

The effects of the reaction rates and the particle growth rate are shown for the different activation times in Figure 3. The simulated activation times (ta) are 1, 30, 50, 100, and 300 s. The area of main interest is the beginning of the polymerization up until about 300 s, since this is the interval where the particles experience the highest temperature. Beyond this point, the particles have grown to a point where the evacuation of heat is less problematic.16 Changing the activation time obviously changes the rate of polymerization and thus the growth rate of the particles. Both of these changes will have an influence on the development of temperature profiles inside the particles. On one hand, it is more difficult to evacuate the heat from smaller particles. On the other hand, one has to evacuate less heat at lower reaction rates. The particle temperatures for the different activation times have been simulated for gas velocities of 0, 0.05, and 0.1 m/s in Figures 4-6, respectively. As expected, the maximum particle temperatures are reached more or less at a time corresponding to the activation time. It can be seen from these figures that the bulk gas velocity plays an important role in the development of the temperature inside the fresh particles. Under stagnant flow conditions, which might be relatively common in an SBR (Figure 4), the temperature rises quickly and higher than under nonstagnant conditions. However, comparing Figures 5 and 6 shows that once the gas velocity is nonzero (more representative of an FBR), the temperature is essentially regulated by the activation time (and, although not shown here for the sake of brevity, the reaction rate as well). In fact, a simulation was run with an activation time of 1s, applying the reaction rate shown in Figure 3, and it was observed that the particles overheat almost immediately, regardless of the gas velocity considered. To demonstrate the effect of the reaction rate, the same simulations (activation time of 1 s) were carried out at two lower activities, 25% and 75% of the activity in Figure 3. The temperature profiles obtained are shown in Figure 7. As expected, the simulations with the lowest reaction rate only experience a small temperature increase, whereas increasing the reaction rate quickly increases the problem of particle overheating. These results show that particles can overheat in stagnant zones in the reactor, as soon as convective heat transfer is no longer viable. In the rest of this paper, an activation time of 30s was used in the simulations. Experience shows that values of much less than this are not particularly realistic for Ziegler catalysts. Thus, a relatively rapid activation time allows us to really put an upper bound on situations where we are likely to encounter problems of particle overheating. The effect of the initial particle size for two different cases of single particles were simulated: initial particle diameters of 20 and 40 µm, with the reaction rate shown in Figure 3 and an activation time of 30s. The results show that the particles reach their maximum temperatures at a time corresponding more or less to the activation time. For a single particle with an initial diameter of 20 µm (Figure 8), there is no overheating problem; for the simulated gas velocities the maximum particle temperature increases less than 8 K compared

Figure 3. Reaction rate and particle size at different activation times.

Figure 4. Maximum particle temperature at zero bulk gas velocity.

to inlet gas temperature. For the particle with an initial diameter of 40 µm (Figure 9), the situation is very different. The maximum temperature inside is substantially higher than for the 20 µm particle, and at the lowest gas flow the particle experiences problems with overheating. The simulations shown here are cut off at 385 K, but at zero bulk gas velocity, the 40 µm particle experiences thermal runaway (i.e., the solution of the balance equations never converges to a steady state). Of course, in a real polymerization the situation could be more complex, since catalysts can be thermally deactivated. Nevertheless, it can be seen here that overheating can be a problem. One way of overcoming particle overheating is prepolymerization, i.e., polymerizing the catalyst particles under very gentle conditions to some small degree of advancement before they are introduced in to the reactor. Simulating the same particle as in Figure 9, but assuming that it was prepolymerized to a degree of


Figure 5. Maximum particle temperature at bulk gas velocity 0.05 m/s. Figure 8. Maximum temperatures in a growing particle (initial diameter 20 µm) as a function of time.

Figure 6. Maximum particle temperature at bulk gas velocity 0.1 m/s.

Figure 7. Maximum temperature at lower reaction rate at activation time 1s.

1 g of polymer/g of catalyst will decrease the produced volumetric heat flux by 50%. The results in Figure 10 show that even at this low degree of prepolymerization, the problems of particle overheating are significantly reduced. For the case of zero gas velocity the balance equation never converges and the particle still experiences thermal runaway, whereas for the other cases a significant reduction in particle temperature is observed compared to a nonprepolymerized particle. Figure 11 summarizes the result of the single-particle simulations for a number of cases. It can be seen here that a catalyst particle with a diameter less than 30 µm shows only small increases in the temperature. As the catalyst particle size increases (40 µm in diameter and upward), the maximum temperatures increases significantly, as does particle overheating. As the initial catalyst particle size increases, the particle sensitivity

Figure 9. Maximum temperatures in a growing particle (initial diameter 40 µm) as a function of time.

Figure 10. Maximum temperatures in a growing particle (initial diameter 40 µm), prepolymerized to 1 g/g as a function of time.

toward the gas velocity also increases. The smallest catalyst particles have no overheating problems, even at the lowest gas velocities, whereas the larger particles overheat easier as the bulk gas velocity decreases. 3.2. Particle Interaction Simulations. To study the effects of interactions, a small particle (initial particle diameter 40 µm) was placed beside a large particle. The large particle is considered to have grown from an initial diameter of 40-150 µm. At the lowest gas velocities, the particle system shows overheating problems. In Figure 12, it is interesting to note the fact that the maximum temperature is reached faster by applying a negative gas flow than a positive one. The crossover of the temperature profiles for the positive and negative bulk gas flow occurs due to the fact that the

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Figure 14. Interaction effects of two growing particles of the same size, prepolymerization 1 g/g.

Figure 11. Single particle.

Figure 15. Volumetric heat flux, calculated under isothermal conditions and particle diameter.

Figure 12. Interaction effects of growing particles.

Figure 13. Interaction effects of two growing particles of the same size.

small particle grows and becomes bigger than the initially largest particle (after about 27 s, corresponding to the crossover points), thus reversing the effects of the bulk gas flow. In Figure 13, the case of two growing particles of the same size, with an initial particle diameter of 40 µm, is simulated. The results show a high tendency for the particles to overheat at bulk gas velocities lower than 0.1 m/s. Comparing these results with the results in Figure 9, where we used the same simulation conditions for a single particle, it is clear that the problems of

overheating are more pronounced for the case of two small particles. Repeating the same simulation at a prepolymerization degree of 1 g/g, there is a large reduction in the maximum temperatures. As can be seen in Figure 14, only the particle in a stagnant gas flow overheats. 3.3. Temperature Distribution at a Maximum Value of Qvol. In this section we will look at the maximum temperature inside the particles as a function of fresh catalyst particle size, the large particle size, and the distance between adjacent particles (in a twoparticle situation), as well as the gas velocity and direction. If one calculates the isothermal volumetric heat flux (i.e., the volumetric heat flux of a growing particle with no heat-transfer resistance), it can be seen from Figure 15 that the maximum values of the heat generation rate are reached very quickly, but drop off very quickly as well. The maximum temperature in the fresh catalyst particles (the left-hand one in Figure 2) are simulated at the given diameters (10, 20, 30, 40, 50, 60, 70, or 80 µm) and the large particles are considered to have an initial diameter of 20 µm, having grown to the final value (80, 150, 250, or 500 µm). All the particles have the intrinsic reaction rate shown in Figure 3, with an activation time of 30 s. Simulations of two-particle configurations begin to reveal some of the complexity that particle-particle interactions can have in a fluidized bed reactor. The results of a series of simulations where a fresh catalyst particle touches a larger one and where the particle centers are both aligned with the bulk gas flow are shown in Figures 1622. Before considering the results of the simulations in detail, it should be pointed out that the ensemble of the


Figure 16. Particle interaction effects, large particle diameter 80 µm.




results suggests that both conduction and convection will play a role in heat transfer. Let us recall that in an earlier work McKenna et al.5 showed that small, very hot particles could be cooled by much larger particles if they touched at a point, even if these small particles were inside the inertial boundary layer of the larger ones. In this case conduction was the major route for particle cooling. The simulations shown in the current paper cover a wider range of particle sizes and therefore bring into play additional factors, in particular the interaction between conduction and convection, and the role of fluid dynamics in establishing particle temperatures. First consider the three-dimensional plots in Figure 16-19, where the maximum temperature (at the isothermal maximum value of the volumetric heat flux, as shown in Figure 15) inside a growing small particle touching a larger one is shown for different large particle sizes (80, 150, 250, and 500 µm). As expected from other studies (e.g., Floyd et al.16), as well as the

single-particle simulations shown in Figure 11, when the fresh (small) particle is larger than 70 µm, it will overheat. Nevertheless, in other situations, the fact that there are two particles in the simulation changes things. For instance, let us consider Figures 16-18, where the large particle diameter varies from 80 to 250 µm. In terms of the combinations of small (fresh) particle size and velocity, the small particle overheats in almost the same way at positive velocities. In this instance one would reasonably suspect that convection plays a major role in the evacuation of energy. It can be seen from Figure 19 (500 µm large particle) that fresh particles on the order of 60 µm in diameter will overheat, even at the highest bulk velocity considered, whereas this is not the case in the three other figures. It appears that the reason for this is that the small particles on the leading edge are inside the large particle’s inertial boundary layer when the latter becomes large enough. This means that the convection component of the heattransfer mechanism is severely reduced and that the

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Figure 20. Particle diameters 20 and 500 µm. Top figure bulk gas flow 0.2 m/s and bottom figure bulk gas flow -0.2 m/s.

ability of the large particle to evacuate heat from the smaller ones is not enough when the small particles are too big. As the velocity drops to zero, it can be seen that in the cases of the 250 and 500 µm particles, particles smaller than 50 µm do not overheat, whereas fresh particles on the order of 40 µm will overheat when the large particle is a bit smaller (80 or 150 µm). This means that, as the large particle gets larger, it can remove heat by conduction from the smaller ones, but at 80 and 150 µm, the large particles cannot yet absorb enough heat to change the temperature inside the smaller ones. Finally, one begins to see noticeable differences in the heat transfer as the bulk velocity becomes negative (i.e., the big particle shields the smaller one from the leading edge of the flow). In these cases, one observes less overheating for the 80 µm particle,and the most overheating for the 150 µm case, with the 250 and 500 µm examples being somewhere in between. These examples begin to demonstrate the importance of the flow field generated by the particles and the boundary layer effects. To illustrate this point, consider Figures 20 and 21, where the flow fields around the particles are plotted for the case of a 20 µm particle in contact with a 150 and 500 µm particle, respectively. In the case of the 20/ 150 µm pair, the flow field around the small particle is relatively strong (the local velocity of the gas is indicated by the length of the arrows) when the gas flow is in the positive direction but significantly weaker in the other direction. This shows that the convective transport near the surface of the small particle is very weak when the

Figure 21. Particle diameters 20 and 150 µm. Top figure bulk gas flow 0.2 m/s and bottom figure bulk gas flow -0.2 m/s.

Figure 22. Particle interaction effects; large particle diameter 250 µm and particle distance 5 µm.

gas flow is negative. On the other hand, in the case of the 500 µm particle, the formation of an eddy at the trailing edge of the large particle leads to an increase in the velocity near the small particle and thus a more effective convective heat transfer. In addition, as mentioned above, the conductive contribution is more significant in the case of the larger particle. Finally, we consider that the results shown in Figure 22 reinforce the idea that conductive heat transfer


cannot be neglected for the case of the larger particles. In this event, the smaller particles are separated from the larger 250 µm particle by a distance of 5 µm. This does not change the flow fields in any observable manner, but if one compares Figures 22 and 18, it can be seen that the smaller particle will overheat more in the absence of contact than when it touches the larger particle. In addition, the spike at zero velocity, present in Figures 16 and 17 but absent in Figure 18, can be seen in Figure 22. Similar results can be seen for the 500 µm large particle (but these are not shown here for reasons of brevity). 4. Conclusions We have shown that for a single growing particle the activation time and reaction rate have a strong influence on the particle maximum temperature of fresh catalyst particles and that this influence is less important as the bulk gas velocity increases. Simulations of the maximum particle temperature at the maximum (isothermally calculated) volumetric heat flux, for a single particle, reveal that the smallest catalyst particles (less than 30 µm in diameter) have no heat-transfer problems, but as the size of the initial catalyst particle increases, heat transfer becomes more difficult. Particle interaction effects between a fresh catalyst particle and larger polymer particles can be quite complex. The bulk gas velocity and direction and the presence of a contact point can change whether particles overheat. Fresh catalyst particles have a significantly higher tendency to overheat when being shielded behind a larger polymer particle or touching other fresh particles. This might be a fairly frequent occurrence in a stirred powder bed reactor. However, this effect is slightly less important when the small particle touch a larger one because of conductive heat transfer. Furthermore, the larger particle needed to be at least 250 µm in diameter for this to be important under the conditions considered here. This limit will of course depend on the rate of polymerization and bulk gas properties to an extent not studied here. The frequency and duration of particle collisions and/or interactions will vary widely from one type of reactor to another. In the FBR, the particles will (in principle) move rapidly, with collisions lasting fractions of a second. In an SPB, this will be different, and particle-particle interactions will have a greater role to play in determining the rate of heat transfer from the growing particles. Depending on the bulk gas flow, different ways of evacuating the heat produced can be traced. At a stagnant gas flow, the conductive effects are more important than the convective ones and are more efficient when the larger of the two particles is over 250 µm in diameter. As expected, prepolymerization is a very effective way of decreasing the problems of particle overheating. Even at very low degrees of prepolymerization (1 g of polymer/g of catalyst) there is a large impact on the maximum particle temperature. Finally, it should be noted that the results presented here are full steady-state solutions. While this will not be entirely realistic in terms of real reactor operation, it has been shown by McKenna et al.5,13 that it takes only a few hundredths of a second for the particles to reach their steady-state values, so this approximation will not have an effect on the conclusions drawn here.

Acknowledgment The authors gratefully acknowledge the financial support of the European Community [FP5 Project: “POLYolefins: Improved PROPerty control and reactor operability (POLYPROP)”, Contract G5RD-CT-2001200597 and Project GRD2-2000-30189]. We are also grateful to Feng Liu and Davor Cokljat at FLUENT Europe Ltd. for their technical expertise. Nomenclature cp,g ) heat capacity of gas, J/(kg K) cp,s ) heat capacity of solid, J/(kg K) Ea ) activation energy, J/mol ke,g ) effective thermal diffusivity of gas, m2/s ke,s ) effective thermal diffusivity of solid, m2/s Q ˆ vol ) volumetric heat flux, W/m3 R ) gas constant R0 ) initial particle radius, m R1 ) instantaneous particle radius, m Rp ) rate of reaction, kg polymer/(g catalyst h) t ) time, s T ) temperature, K ui,j ) velocity components in the i and j directions xi,j ) spatial coordinates in the i and j directions Greek Letters ∆Hp ) heat of reaction, J/mol µ ) gas mixture viscosity, Ns/m2 F ) density, kg/m3 Subscripts g ) gas ref ) reference value s ) solid

Literature Cited (1) Floyd, S.; Choi, K. Y.; Taylor, T. W.; Ray, W. H. Polymerization of Olefins through. Heterogeneous Catalysis III: Polymer Particle Modelling with an Analysis of Intraparticle Heat and Mass Transfer Effects. J. Appl. Polym. Sci. 1986, 32, 2935. (2) Zacca, J. J.; Debling, J. A.; Ray, W. H. Reactor residence time distribution effects on the multistage polymerization of olefinssI. Basic principles and illustrative examples, propylene. Chem. Eng. Sci. 1996, 51, 4859. (3) Hatzantonis, H.; Yiannoulakis, H.; Yiagopoulos, A.; Kiparissides, C. Recent developments in modelling gas-phase catalyzed olefin polymerization bed reactors: The effect of bubble size variation on the reactor’s performance. Chem. Eng. Sci. 2000, 55, 3237. (4) Kim, J. Y.; Choi, K. Y. Modeling of particle segregation phenomena in a gas phase fluidized bed olefin polymerization reactor. Chem. Eng. Sci. 2001, 56, 4801. (5) McKenna, T. F.; Cokljat, D.; Spitz, R. Heat transfer from heterogeneous catalysts: An exploration of underlying mechanisms using CFD. AIChE J. 1999, 45, 2392. (6) McKenna, T. F.; Soares, J. B. P. Single particle modelling for olefin polymerization on supported catalysts: A review and proposals for future developments. Chem. Eng. Sci. 2001, 56, 3931. (7) Kaneko, Y.; Shiojima, T.; Horio, M. DEM simulation of fluidized beds of polyolefin synthesis. AIChE Symp. 1999, 321, 71. (8) Kaneko, Y.; Shiojima, T.; Horio, M. DEM simulation of fluidized beds for gas-phase olefin polymerization. Chem. Eng. Sci. 1999, 54, 5809. (9) van Wachem, B. G. M.; Schouten, J. C.; Krishna, R.; van den Beck, C. M. Validation of the Eulerian simulated dynamic behaviour of gas fluidised beds. Chem. Eng. Sci. 1999, 54, 2141. (10) Marschall, K. J.; Mleczko, L. CFD modelling of an internally circulating fluidized bed reactor. Chem. Eng. Sci. 1999, 54, 2085. (11) Fan, R.; Marchisio, D. L.; Fox, R. O. CFD Simulation of Polydisperse Fluidized-Bed Polymerization Reactors. Presented at

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the AIChE Fall Meeting, Session 40: Computational and Numerical Approaches in Particle Flows, 2003. (12) Hutchinson, R. A.; Ray, W. H. Polymerization of Olefins through Heterogeneous Catalysis VII: Particle Ignition and Extinction phenomena. J. Appl. Polym. Sci. 1987, 34, 657. (13) McKenna, T. F.; Cokljat, D.; Wild, P. CFD Modelling of Heat Transfer during Gas-Phase Olefin Polymerisation. Comput. Chem. Eng. 1998, 22, Suppl., S285. (14) Yiagopoulos, A.; Yiannoulakis, H.; Dimos, V.; Kiparissides, C. Heat and mass transfer phenomena during the early growth of a catalyst in gas-phase olefin polymerization: The effect of prepolymerization temperature and time. Chem. Eng. Sci. 2001, 56, 3979.

(15) Pater, J. T. M.; Weickert, G.; Swaaij, W. P. M. Propene Bulk Polymerization Kinetics: Role of Prepolymerization and Hydrogen. AIChE J. 2003, 49, 180. (16) Floyd, S.; Choi, K. Y.; Taylor, T. W.; Ray, W. H. Polymerization of olefins through heterogeneous catalysis. IV. Modelling of heat and mass transfer resistance in the polymer particle boundary layer. J. Appl. Polym. Sci. 1986, 31, 2231.

Received for review December 19, 2003 Revised manuscript received April 1, 2004 Accepted April 23, 2004 IE034328N

Heat-Transfer Phenomena in Gas-Phase Olefin Polymerization Using

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