Brazilian Journal of Chemical Engineering
Print version ISSN 01046632Online version ISSN 16784383
Braz. J. Chem. Eng. vol. 15 no. 3 São Paulo Sept. 1998
http://dx.doi.org/10.1590/S010466321998000300006
FLUIDDYNAMIC ASPECTS OF GASPHASE ETHYLENE POLYMERIZATION REACTOR DESIGN
R. Guardani*, A.C.S.C. Teixeira, E.L. Casella and A.M.F.I. Souza
University of São Paulo, Escola Politécnica, Chemical Engineering Department,
Cidade Universitária, 05508900 São PauloSP, Brazil, fax:+55118132380,
email: guardani@usp.br
(Received: December 15, 1997; Accepted: July 27, 1998)
Abstract  The relative importance of design variables affecting the fluiddynamic behavior of a fluidized bed reactor for the gasphase ethylene polymerization is discussed, based on mathematical modeling. The threephase bubbling fluidized bed model is based on axially distributed properties for the bubble, cloud and emulsion phases, combined with correlations for population balance and entrainment. Under the operating conditions adopted in most industrial processes, the reactor performance is affected mainly by the reaction rate and solids entrainment. Simulation results indicate that an adequate design of the freeboard and particle collecting equipment is of primary importance in order to produce polymeric particles with the desired size distribution, as well as to keep entrainment and catalyst feed rates at adequate levels.
Keywords: Ethylene polymerization, fluidized bed reactor.
INTRODUCTION
In recent years the production capacity of fluidized bed polymerization products has experienced considerable growth, particularly in the cases of polyethylene and polypropylene. A review on the state of the art of commercially available technologies, including the relevant macroscopic and microscopic aspects, has been presented by Xie et al. (1994). The increase in the industrial importance of gas phase polymerization has been accompanied by intensive studies on fundamental aspects of the process, focusing on kinetic models for predicting the molecular weight distribution and chain structure of polymer molecules (Galvan and Tirrell, 1986, de Carvalho et al., 1989), as well as on particlefluid interactions, including heat and mass transfer aspects (Galvan and Tirrell, 1986, Floyd et al., 1986, 1987, Hutchinson and Ray, 1987).
Design aspects of polymerization reactors were first presented by Choi and Ray (1985), who studied the dynamic behavior of fluidized bed reactors for polymerization of ethylene and propylene. The authors discussed the occurrence of temperature runaways as a function of operating conditions such as fluidizing gas velocity, feed gas temperature, catalyst activity, catalyst injection rate, etc. A comparison of two model types (twophase and wellmixed) for polyethylene reactors was presented by McAuley et al. (1994), showing that, due to the low reaction rates under conditions of industrial interest, a simplified model assuming CSTR flow for the gas and solid phases is accurate enough for design purposes, which clearly simplifies the calculations involved in reactor design. Choi et al. (1994) studied the effects of feed catalyst particle size distribution and the catalyst site deactivation by means of a multigrain solid core kinetic model which was incorporated into a population balance for the solid particles. Simulations showed that the fraction of larger polymer particles in the product decreases when site deactivation increases. In a recent paper by Khang and Lee (1997), the particle size distribution of the polymer product was studied in the cases of CSTR solids flow and nonideal mixing. The study was based on a kinetic model including particle diffusion and catalyst deactivation limitations. The effect of such assumptions on the product particle size distribution is discussed.
In the present study, fluiddynamic aspects of the reactor design are discussed, based on a mathematical model comprising three fluiddynamic phases (bubble, cloud and emulsion), bubble size distribution along the reactor volume and solids entrainment. The use of a relatively detailed model enabled the study of the relative importance of design variables on reactor performance, evaluated in terms of the product particle size distribution, catalyst feed rate, entrainment rate and reactant conversion, for a given production capacity.
MATHEMATICAL MODEL
The fluiddynamic model is based on the assumptions presented by Kunii and Levenspiel (1991) for the volume fractions of bubble, cloud and emulsion, as well as for the mass transfer coefficients between the fluiddynamic phases. The solution algorithm is based on the "bubble assemblage model" described by Wen and Fan (1975), which divides the bed into a series of compartments and performs the calculations along the reactor height, starting from the initial conditions near the distributor.
Table 1: Summary of the fluiddynamic model expressions
Minimum fluidization velocity (Nakamura et al., 1985)  (1)  
Bubble rise velocity (Davidson and Harrison, 1963) Bubble growth rate (Wen and Fan, 1975)  (2)
(3)  
Bubble fraction (Kunii and Levenspiel, 1991) 
with b = 2; 1; 0 and 1, for < 1; 1 < < 5; @ 5  (4)

Maximum stable bubble size (Geldart, 1986)  (5)  
Cloud volume (Grace, 1986)  with corresponding to a particle with 2.7 d_{p} with 
(6)
(7) 
Mass transfer coefficients (Kunii and Levenspiel, 1991)  (8) (9)  
Bed voidage (Wen and Fan, 1975)  , for , for  (10)
(11)
(12) 
Assumptions of the Fluiddynamic Model
a) The reactor operates at steadystate, under uniform temperature.
b) The gas flows in plug flow, with gas interchange between phases. The solids flow in CSTR (no particle segregation within the bed is considered).
c) The bed is axially divided into N compartments, each one consisting of the bubble, cloud and emulsion phases. Each compartment height, , is equal to the bubble diameter corresponding to the reactor axial coordinate, h.
d) Bubbles grow continuously along the reactor height from their initial diameter until they reach the maximum stable bubble diameter which was estimated according to Geldart (1986). Although different correlations can usually be adopted for the bubble growth rate, in this paper a simple linear correlation was used (Wen and Fan, 1975) since the resulting bubble diameters at different heights reasonably compare with results by other correlations and is accurate enough for the study considered here. The bed volume fraction occupied by bubbles, clouds and emulsion depends on the flow regime and changes along the reactor height.
e) The solids are polymer particles with a size distribution determined from the balance calculations.
f) Solids entrainment and carryover are taken into account by the model. However, particle agglomeration and breakage are not considered.
g) No chemical reaction occurs in the bubble phase which is free of solids. No chemical reaction is considered in the freeboard since the particle volume fraction in this part is considerably smaller than in the bed.
h) Bed voidage is constant from the distributor until H_{mf}. Above this height, increases linearly until the top of the bubbling bed. This approximation is included in order to account for the increasing turbulence in the upper part of the expanded bed (Wen and Fan, 1975).
Table 1 summarizes the expressions of the fluiddynamic model.
Particle Entrainment and Elutriation
A review of the literature concerning particle carryover from different beds has shown that there are considerable differences in the results of carryover rates predicted by different correlations. A recent paper by Milioli and Foster (1995) showed that the use of phenomenological models to predict entrainment and elutriation demands a series of experiments in order to fit model parameters to specific situations. In this paper, the elutriation constant approach (Geldart, 1986, Kunii and Levenspiel, 1991) was adopted, and a correlation originally proposed by Geldart and collaborators in 1979, listed in Geldart (1986), has been included in the model since it was tested in conditions similar to the ones adopted in this work. For a particle with radius r, the elutriation constant is expressed as shown in equation 13:
(13)
The flowrate of solids of all sizes removed from the bed by the gas flow is expressed as:
(14)
Kinetic Model
A relatively simple rate expression for the ethylene polymerization was adopted, based on the model used by Choi and Ray (1985) and McAuley et al. (1994). The kinetic model is expressed as firstorder in relation to the gas phase monomer concentration and the catalyst mass fraction in the polymer particle. The model can describe the growth of polymeric particles in the fluidized bed reactor, but does not include a rigorous description of the polymerization reactions. Thus, it is not adequate for predicting polymer properties, such as the molecular weight distribution or the polydispersity index. The reaction rate in the ith compartment for particles of radius r is:
(15)
with the catalyst mass fraction in the polymer particle, X_{cat}, dependent on the particle size. For the whole solid phase, the reaction rate is expressed as:
(16)
The assumptions used by McAuley et al. (1994) were also adopted here to explain the inclusion of Henrys law constant in k_{p}. In this model, the monomer diffusion rates both of the polymer particles and inside them are high compared to the reaction rate, and the ethylene dissolved in the polymer phase can be considered to be in equilibrium with the emulsion gas. The temperature effect on Henrys law constant is neglected, and k_{p} is assumed to vary with the temperature according to the Arrhenius model.
In this work, catalyst deactivation is included in the kinetic model, but is limited to its effects on the particle size distribution of the polymer product. The multigrain particle approach, as described by Khang and Lee (1997), is adopted. This is based on the fact that the catalyst particle breaks up into many small fragments (microparticles) which are not segregated from the original catalyst particle as the polymeric macroparticle starts to grow. These microparticles are distributed along the volume of the macroparticle during the growth process. Based on the multigrain model and on the assumptions of the kinetic model, an apparent catalyst deactivation rate constant, k_{d}, is adopted for a given polymer particle with radius r (macroparticle), such that:
(17)
is the growth time for a polymer particle with radius r:
(18)
where A_{i} for a polymer particle with radius r in each compartment is expressed as:
(19)
The expressions above correspond to an exponential decayment in the polymer particle growth rate as a function of the particle radius r.
Mass Balance
The bubble assemblage model divides the bed axially into a series of compartments having height equal to the bubble diameter. In each compartment, ideally mixed flow is assumed for the fluiddynamic phases (bubble, cloud and emulsion) with mass and energy interchange between them. For the whole reactor the solids are ideally mixed, and there is no gas recycle. Since most of the heat capacity in the reactor is contained in the polymer particles, the reactor can be considered isothermal. Simulation results by McAuley et al. (1994) with a twophase model have shown that the temperature difference between the emulsion and bubble phase is less than 5 K under normal operating conditions and when the maximum stable bubble diameter criterion (equation 5) is adopted. Small differences are also observed for the concentrations in both phases. However, since the volume fraction of bubbles and clouds directly affects the average residence time and reactor performance, the mass balance presented below takes the three fluiddynamic phases into consideration.
For the gas phase the mass balance is expressed as:
a) Bubbles:
(20)
with
(21)
b) Clouds:
(22)
with
(23)
c) Emulsion:
(24)
with
(25)
The mass balance for the solids is expressed as:
(26)
Population Balance
The population balance is based on a mass balance for particles of size between r and r + dr, as presented by Kunii and Levenspiel (1969, 1991):
(27)
In the expression above which is valid for spherical particles, denotes the rate of increase in particle radius and k(r) is an elutriation constant, defined as:
(28)
The overall solids mass balance, equation 26, can be written as:
(29)
Since the solids in the fluidized bed are assumed to be ideally mixed, is equivalent to . The solution of the population balance is obtained by a finite differences method consisting of the discretization of equations 27 and 29, according to the procedure presented by Overturf and Kayihan (1979). This consists of choosing a convenient number of size intervals, D r_{j}, starting from the smallest size of feed solid (catalyst), and solving equations 30 and 31, which are equivalent to equations 27 and 29 in discrete form:
(30)
(31)
Equation 30 can be used to calculate the particle size distribution of stream F_{0} or F_{1} from an initial guess of F_{0}, F_{1} or W (whichever is unknown), starting at the minimum radius, r_{min}, until r_{max}. The consistency is checked in equation 32:
(32)
After adjusting the equations above, stream F_{2} is calculated by equation 31. The particle size distribution of stream F_{2} is calculated by equation 33:
(33)
ALGORITHM
The sequence for solving the mathematical model is illustrated in Figure 1. The algorithm is based on the following input data:
Design Variables
.Reactor diameter and height;
.Initial bubble diameter (from the distributor design);
.Solids carryover control.
The simulation of solids carryover was done either by adopting different diameters for the bed and the freeboard, or by inserting a cyclone for collecting particles from the output stream, F_{2}.
Operating Variables
.Gas velocity, inlet temperature and composition;
.Reactor operating temperature and pressure;
.Catalyst particle size and initial mass fraction;
Physical Properties
.Gas viscosity, diffusion coefficient, density and molecular weight;
.Solid density;
.Minimum fluidization voidage;
.Polymerization reaction constant parameters;
.Apparent catalyst deactivation rate constant.
For a given set of design variables and operating conditions and for a desired value of the polymer production rate, F_{1}, the algorithm provides the following information:
.Axial bubble size profile;
.Axial gas phase concentrations in the bubble, cloud and emulsion;
.Required catalyst feed rate, F_{0};
.Solids carryover rate, F_{2};
.Particle size distributions of streams F_{1} (equal to the bed particle size distribution) and F_{2}.
Thus, the reactor volume and operating conditions can be adjusted in order to achieve the desired production rate and product particle size distribution. However, the prediction of polymer properties, such as the molecular weight distribution and the polydispersity index, would only be possible through the combination of the present model with rigorous mathematical models to describe the polymerization reactions which take place in the system. An overall energy balance, necessary for the conceptual reactor design, can also be performed.
Figure 1: Simplified diagram for the computation sequence adopted.
The algorithm first calculates the overall parameters of the fluiddynamic model for an initial guess concerning the particle size distribution in the bed. Then, the calculation of the axial profiles along the reactor is performed starting from the initial conditions at the distributor until the top of the bed. The third step consists of the calculation of streams F_{0} and F_{2} as well as the particle size distributions, p_{1}(r) and p_{2}(r), which was performed according to the procedure suggested by Overturf and Kayihan (1979) and which involves first fitting F_{1 }and p_{1}(r). The criterion for convergence is based on the average Sauter diameter and standard deviation of p_{1}(r).
SIMULATIONS
The values of the process parameters and the ranges of the variables adopted in this work, listed in Table 2, are based on the papers by Choi and Ray (1985) and McAuley et al. (1994). In order to minimize the number of runs, a preliminary plan was carried out aimed at evaluating the individual effect of each input variable on the set of output variables computed. A fractional factorial design of type 2_{IV}^{84} (Box et al., 1978) was adopted, based on a plan of runs generated by the software package Statgraphicsâ . The results of these preliminary runs indicated that the variables which mainly affect the reactor performance are: solid product flowrate, F_{1}, reactor diameter, D, gas velocity, u_{g}, catalyst particle size, d_{p},0, and temperature. Although the simulation results are focused on the effect of these variables, the effect of the solids carryover rate was also included, first by keeping the ratio D_{F }/ D equal to 2.0, then by varying this ratio.
The results of simulations with the model are presented in Figures 2 to 7. At first it was observed that the rates of mass transfer between the three fluiddynamic phases (bubble, cloud and emulsion) are much higher than the rate of ethylene consumption by chemical reaction, resulting in small differences in the gas concentrations in the three phases along the reactor. Figure 2 shows axial profiles for bed voidage, bubble diameter and converted fraction of ethylene under conditions which are typical of many commercial plants. The bubbles reach their maximum stable diameter in the lower part of the bed, while X_{A} increases steadily along the reactor height. This is in accordance with previous observations presented by McAuley et al. (1994). However, as shown in Figure 3, the gas velocity affects the solids residence time, (expressed as W/F_{1}), since it affects the volumetric bubble fraction in the bed. Also the values of d_{b},max, d_{p},av and the catalyst flowrate, F_{0}, are affected by the superficial gas velocity. The increase in F_{0} is caused by the increase in the solids carryover rate, which is negligible for small values of u_{g}, but increases significantly as u_{g} increases.
Table 2: Values adopted for the process parameters
(Choi and Ray, 1985, McAuley et al., 1994)
Parameter  Value 
1.15´ 10^{5}  
24  
950  
2370  
0.5  
4160  
37260 
Figure 4 illustrates the effect of the reactor operating temperature on some of the output variables, for conditions representative of industrial scale. Since the process is limited by the reaction rate, the temperature has a considerable effect on the size distribution of the product particles, as can be observed from the increase in the value of d_{p},av. The increase in the average diameter corresponds to a decrease in the degree of dispersion of the particle size distribution, as expressed by the ratio d_{p},av/S_{d}. This is a consequence of the increase in particle growth rates, which causes a significant decrease in the rates of particle carryover (F_{2}) and catalyst feed (F_{0}) for a given production rate. The same tendency is observed in Figure 5 which shows the effect of the particle size of the feed solid catalyst, d_{p},0. It can be observed that, as d_{p},0 increases, the particle carryover rate decreases markedly, resulting in larger average size of the product polymer particles and narrower particle size distribution. Thus, prepolymerization of the feed solid, as adopted in some processes, brings considerable benefits to the process.
Figure 2: Typical profiles of bubble diameter (Curve A), bed voidage (Curve B) and converted fraction of ethylene (Curve C) along the reactor height. Conditions: D = 4.0 m, D_{F }/ D = 2.0, u_{g} = 0.5 m/s, F_{1} = 3.0 kg/s, p = 25 bar, T = 378 K, d_{p},0 = 50 µm, X_{cat},0 = 1.0, k_{d }= 0.
Figure 3: Effect of the superficial gas velocity on the average particle residence time, maximum stable bubble size, average particle size and catalyst flow rate. Curve A: (= 11111 s); Curve B: d_{b},max / d_{b},max,in (d_{b},max,in = 0.75 m); Curve C: d_{p},av / d_{p},av,in (d_{p},av,in= 837 µm); Curve D: F_{0} / F_{0,in} (F_{0,in}= 0.00253 kg/s). Conditions: D = 4.0 m, D_{F} / D = 2.0, H = 6.0 m, F_{1} = 3.0 kg/s, p = 25 bar, T = 378 K, d_{p},0 = 50 µm, X_{cat},0 = 1.0, k_{d} = 0.
Freeboard design is also of prime importance in controlling particle carryover, as can be seen in Figure 6, which shows the effect of the freeboardtobed diameter ratio on some output variables for a bed operating under typical conditions of industrial scale. D_{F }/ D ratios of at least approximately 1.4 are necessary in order to keep particle carryover under acceptable values. The effect of the D_{F }/ D ratio on the particle size distribution of the polymer is illustrated in Figure 7. For ratios smaller than about 1.4 the polymer product presents bimodal particle size distributions. This occurs due to the high carryover rates under the conditions adopted in the simulations. Thus, in order to achieve a specified production rate, F_{1}, high values of F_{0} are necessary, resulting in an increase in the fraction of smaller particles in the product. For higher D_{F }/ D ratios the distribution tends to a lognormal function. In order to minimize catalyst and product loss by carryover, an adequate D_{F }/ D ratio should be adopted in the reactor design, depending on the operating conditions, otherwise one or more cyclones should be added to the design.
Figure 4: Effect of the reactor temperature on output variables of the model. Curve A: d_{p},av (µm); Curve B: d_{p},av / S_{d} ; Curve C: X_{A} (%); Curve D: 1000F_{2} (kg/s); Curve E: 1000F_{0} (kg/s); Curve F: u_{g} / u_{mf}. Conditions: D = 4.0 m, D_{F} / D = 2.0, H = 6.0 m, F_{1} = 3.0 kg/s, u_{g}= 0.5 m/s, p = 25 bar, d_{p},0 = 50 µm, X_{cat},0 = 1.0, k_{d} = 0.
Figure 5: Effect of the feed catalyst particle size on output variables of the model. Curve A: d_{p},av (µm); Curve B: d_{p},av / S_{d}; Curve C: X_{A} (%); Curve D: 1000F_{2} (kg/s); Curve E: 1000F_{0} (kg/s); Curve F: u_{g} / u_{mf}. Conditions: D = 4.0 m, D_{F} / D = 2.0, H = 6.0 m, F_{1} = 3.0 kg/s, u_{g}= 0.5 m/s, p = 25 bar, T = 378 K, X_{cat},0 = 1.0, k_{d} = 0.
Figure 6: Effect of the freeboard to bed ratio (D_{F}/D) on output variables of the model. Curve A: d_{p},av (µm); Curve B: d_{p},av / s_{d}; Curve C: X_{A} (%); Curve D: 1000F_{2} (kg/s); Curve E: 1000F_{0} (kg/s); Curve F: u_{g }/ u_{mf}. Conditions: D = 4.0 m, H = 6.0 m, F_{1} = 5.0 kg/s, u_{g}= 0.5 m/s, p = 25 bar, T = 378 K, d_{p},0 = 50 µm, X_{cat},0 = 1.0, k_{d} = 0.
Figure 7: Effect of the freeboard to bed diameter ratio (D_{F}/D) on the particle size distribution of the polymer product. Conditions: D = 4.0 m, H = 6.0 m, F_{1} = 5.0 kg/s, u_{g} = 0.5 m/s, p = 25 bar, T = 378 K, d_{p},0 = 50 µm, X_{cat},0 = 1.0, k_{d }= 0.
The effect of catalyst deactivation on the size distribution of the polymer particles is illustrated in Figure 8, for different values of k_{d}. The simulations were carried out under the same conditions of Figure 7 and for a D_{F} / D ratio of 1.375. By increasing the value of k_{d}, a decrease in the mean particle size of the product is observed, associated with a decrease in the dispersion of the distribution curves, caused by the much lower particle growth rates for the larger particles. Values of k_{d} higher than about 5x10^{6} s^{1} resulted in too small growth rates, and the particle size of the bed solids was smaller than the minimum necessary to keep a stable bubbling fluidized bed. The increase in the value of k_{d} (and decrease in the mean particle size of the bed solids) results also in a change of the fluiddynamic conditions of the bed, with smaller bubble fractions the bed height. This corresponds to higher volumetric fractions for the emulsion phase, resulting in higher gas converted fractions, which increased from about 0.05 to about 0.15 as k_{d} increased from 0 to 1x10^{6}. Thus, although the catalyst deactivation has an undesired effect in terms of the mean size of the polymer particles, the change in the fluiddynamic state of the bed can result in higher gas converted fractions and less dispersion in the particle size distribution of the product. However, more reliable predictions of this effect on the reactor performance depend on the use of rigorous kinetic models for the polymerization reactions.
Figure 8: Effect of the apparent catalyst deactivation rate constant (k_{d}) on the particle size distribution of the polymer product. Conditions: D = 4.0 m, H = 6.0 m, F_{1} = 5.0 kg/s, u_{g} = 0.5 m/s, p = 25 bar, T = 378 K, d_{p},0 = 50 µm, X_{cat},0= 1.0, D_{F}/D = 1.375.
CONCLUSIONS
By means of a relatively detailed model of a fluidized bed reactor, comprising three fluiddynamic phases (bubble, cloud and emulsion) and distributed parameters, some important aspects of the gas phase ethylene polymerization reactor design could be studied.
First of all, it became clear that the interphase mass transfer occurs at a considerably higher rate than the ethylene consumption rate by chemical reaction. Thus, it is not necessary to deal with sophisticated fluiddynamic models in order to obtain adequate results. Simplified models such as twophase integral models, comprising CSTR flow for gas and solids should be adequate. However, since the reactor volume fraction occupied by bubbles has a direct influence on the solids and gas average residence time, these parameters should be adequately computed in the model, especially due to the large production rates typically adopted.
Special attention should be dedicated to the influence of particle carryover in the process since this affects the reactor performance in terms of consumption of catalyst and product quality. For a given set of design variables (production rate, gas properties and catalyst activity), the main factors affecting the carryover rate, which can be controlled at the design stage, are the superficial gas velocity, the reactor geometry and the size of the catalyst particles. After definition of the adequate range for u_{g}, simulations should be carried out for different values of reactor diameter, freeboard height and freeboard diameter. Since there is presently a rather poor agreement among different published correlations for particle carryover, this is a subject which demands extensive study, by testing different correlations in simulations and, if possible, comparing with experimental data. As a general rule, however, a study of the carryover rate for different values of the D_{F} / D ratio, as previously discussed, could be adopted.
Concerning the size of the catalyst particles, for a given catalyst, the size of the particles fed to the reactor, d_{p},0 , can be increased by adding a prepolymerization reactor to the design, an alternative adopted by some commercial processes (Xie et al., 1994). Different configurations can be compared at the design stage by simulations.
The present study has evidenced some of the most important design variables affecting reactor performance. Based on the results of simulations, it is shown that reactor modeling can be based on simplified twophase correlations. However, the mathematical models should include the effect of bubble fraction and especially of particle carryover.
The combination of the model here presented with rigorous models describing the polymerization reactions that take place in the process, the essential informations concerning polymer properties and reactor performance can be obtained at the design stage.
NOMENCLATURE
Ar Archimedes number,
b Coefficient in equation 4
C_{b,i} Monomer concentration in bubble phase, ith compartment, mol m^{3}
C_{c,i} Monomer concentration in cloud phase, ith compartment, mol m^{3}
C_{e,i} Monomer concentration in emulsion phase, ith compartment, mol m^{3}
d_{b} Bubble diameter, m
d_{b},0 Bubble diameter near the distributor, m
d_{b},av Average bubble diameter, m
d_{b},max Maximum stable bubble diameter, m
d_{b},max,in Maximum stable bubble diameter for initial gas velocity in Figure 3, m
d_{p} Polymer particle size, m
d_{p},0 Catalyst particle size, m
d_{p},av Average polymer particle size, Sauter, m
d_{p},av,in Average polymer particle size for initial gas velocity in Figure 3, m
D Reactor diameter, m
D_{A} Diffusion coefficient for ethylene, m^{2} s^{1}
D_{F} Freeboard diameter, m
Activation energy, J mol^{1}
F_{0} Catalyst feed rate, kg s^{1}
F_{0,in} Catalyst feed rate for initial gas velocity in Figure 3, kg s^{1}
F_{1} Flowrate of polymer product, kg s^{1}
F_{2} Flowrate of solids removed by elutriation, kg s^{1}
g Gravitational acceleration, m s^{2}
h Bed axial coordinate
H Bed height, m
H_{mf} Bed height at minimum fluidizing condition, m
D h Compartment height, m
k Elutriation constante, equation 28, s^{1}
k* elutriation constant, equation 13, kg m^{2} s^{1}
k_{bc} Coefficient of gas interchange between bubble and cloud phase, s^{1}
k_{ce} Coefficient of gas interchange between cloud and emulsion phase, s^{1}
k_{d} Apparent catalyst deactivation rate constant, s^{1}
k_{p} reaction rate constant, m^{3 }kg _{cat}^{1 }s^{1}
k_{p}0 Preexponential factor, m^{3 }kg_{ cat}^{1 }s^{1}
L_{max} Maximum reactor height, m
M_{e} Ethylene molecular weight, kg mol^{1}
N Number of compartments
p Reactor pressure, bar
p_{0} Particle size distribution of feed catalyst, m^{1}
p_{b} Particle size distribution in the reactor, m^{1}
p_{1} Particle size distribution of polymer product, m^{1}
r Particle radius, m
r_{min} Minimum particle radius, m
r_{max} Maximum particle radius, m
r_{p,i} Reaction rate in the ith compartment for particles with radius r, kg s^{1}
D r_{j} Particle size interval, m
Rate of increase in particle radius, m s^{1}
Re_{mf} Reynolds number at minimum fluidizing condition,
R_{p,i} Reaction rate for the whole solid phase, ith compartment, kg s^{1}
R_{p,c,i} Reaction rate in the cloud phase, ith compartment, kg s^{1}
R_{p,e,i} Reaction rate in the emulsion phase, ith compartment, kg s^{1}
S_{d} Standard deviation of polymer particle size distribution, m
S Crosssectional area of fluidized bed, m^{2}
T Temperature, °C
u_{g} Fluidizing gas velocity, m s^{1}
u_{b} Velocity of a bubble rising up through the bed, m s^{1}
u_{br} Velocity defined by equation 7, m s^{1}
u_{f} Fluidization velocity defined as , m s^{1}
u_{mf} Minimum fluidizing velocity, m s^{1}
u_{T} Terminal velocity of a falling particle, m s^{1}
V_{b,i} Volume of bubble phase, ith compartment, m^{3}
V_{c,i} Volume of the cloud phase, ith compartment, m^{3}
V_{e,i} Volume of the emulsion phase, ith compartment, m^{3}
V_{i} Volume of the ith compartment, m^{3}
W Weight of solids in the bed, kg
X_{A} Ethylene conversion
X_{cat} Catalyst mass fraction in the polymer particle
X_{cat},0 Initial catalyst mass fraction in the polymer particle
Greek letters
b _{b,i} Volumetric fraction of the bubble phase, ith compartment
b _{c,i} Volumetric fraction of the cloud phase, ith compartment
b _{e,i} Volumetric fraction of the emulsion phase, ith compartment
d Bubble fraction, equation 4
e Bed voidage
e _{mf} Bed voidage at mininum fluidizing conditions
m _{g} Gas viscosity, kg m^{1 }s^{1}
r _{cat} Catalyst density, kg m^{3}
r _{g} Gas density, kg m^{3}
r _{p} Solid density, kg m^{3}
Solids residence time, s
Solids residence time for initial gas velocity in Figure 3, s
ACKNOWLEDGEMENT
The authors wish to thank CNPq for the financial support to the study.
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