Chemical Engineering Science 56 (2001) 4069–4083
www.elsevier.com/locate/ces
Modeling of particle segregation phenomena in a gas phase %uidized
bed ole)n polymerization reactor
Ju Yong Kim, Kyu Yong Choi ∗
Department of Chemical Engineering, Institute for Systems Research, University of Maryland, College Park,
MD 20742-2111, USA
Abstract
In a gas phase %uidized bed ole)n polymerization reactor, it is generally assumed that polymer particles are well mixed and
near isothermal reaction conditions prevail. When a highly active Ziegler–Natta catalyst or other type of high activity supported
catalyst is used in a gas phase %uidized bed reactor, a certain heterogeneity in the polymer properties is often observed in di4erent
size particles. Moreover, particle agglomeration and sheeting phenomena may also occur due to irregular particle growth, internal
particle segregation or nonisothermal e4ect. In many of the past reports, particle residence time distribution has been commonly
used as a tool to calculate polymer particle size distribution in a %uidized bed polyole)n reactor. In this paper, a multi-compartment
population balance model is presented to directly model the particle segregation phenomena and particle size distribution in a
gas phase ole)n polymerization reactor. To model the particle segregation e4ects, size dependent particle transfer constants are
employed. The e4ects of various %uidized bed operating conditions on the particle size distribution are investigated through model
simulations. ? 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction
Fluidized bed reactors are widely used in the polymer industry for the production of -ole)n homo- and
co-polymers with high activity transition metal catalysts
such as Ziegler–Natta catalysts, chromium oxide catalysts, and supported metallocene catalysts. One of the
major advantages of the %uidized bed reactor process is
that solid polymer particles are intensely mixed in the reactor and that reaction heat can be removed e4ectively
by fast %owing %uidizing gas.
In a %uidized bed polyole)n process, a small amount
of high activity catalyst particles (30–50 m) is supplied
continuously or semi-continuously to the reactor. As these
catalyst particles are exposed to monomer or monomer
mixture in the reactor, polymerization occurs almost immediately and the catalyst particles are quickly encapsulated by the newly-formed polymers to a size of around
300–1000 m. The reaction heat is dissipated from the
∗ Corresponding author. Tel.: +301-405-1907;
fax: +301-3149126.
E-mail address: [email protected] (K. Y. Choi).
growing polymer particles by a fast %owing gas stream.
Fully-grown polymer particles are withdrawn continuously or intermittently from the bottom portion of the reactor (above distributor plate) while keeping the bed level
approximately constant. Since very high %uidizing gas
velocity is used for heat removal purpose, the monomer
conversion per pass is quite low (¡ 5%) and a large
amount of unreacted gas containing an inert gas leaving
the reactor is cooled, compressed, and recycled back to
the reactor for additional reaction heat removal. An inert hydrocarbon liquid may also be added to the recycle
gas stream to increase the reactor heat removal capacity
(condensed mode operation) and hence to increase the
polymer throughput.
Although polymer particles in a %uidized bed polymerization reactor are generally assumed to be very well
mixed, particle segregation may occur to some extent in
a large industrial scale %uidized bed ole)n polymerization reactor. For example, the size distribution of polymer particles removed from the bottom of the reactor
may di4er from the polymer particle size distributions in
other locations of the reactor. Axial temperature gradients often observed in industrial %uidized bed polyole)n
0009-2509/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 0 9 - 2 5 0 9 ( 0 1 ) 0 0 0 7 8 - 1
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J. Y. Kim, K. Y. Choi / Chemical Engineering Science 56 (2001) 4069–4083
reactors are believed to be a strong function of the axial
solids mixing (Meier, Weichert, & van Swaaji, 2000).
Also, the feed catalyst particles are not always of uniform size but have a certain size distribution. Since the
size of a polymer particle produced in the reactor is
determined by the particle’s residence time (or reaction time) in the reactor, the polymer particles in a gas
phase ole)n polymerization reactor exhibit a broad particle size distribution. If the control of polymer particle
size distribution is important for post-reaction particle
treatment, or if the polymer properties vary with size,
it would be important to understand the non-uniform
mixing of polymer particles in the reactor under various reactor operating conditions. It is also believed that
particle agglomeration and polymer sheet formation phenomena (often regarded as a major cause of ‘headache’
to %uidized bed reactor operators) are interrelated to
particle segregation and concomitant non-isothermal
e4ects.
Several workers have investigated polymer particle
size distribution in a %uidized bed polyole)n reactor in
recent years (Choi, Zhao, & Tang, 1994; Zacca, Debling,
& Ray, 1996; Zacca, Debling, & Ray, 1997; Khang &
Lee, 1997; Hatzantonis, Goulas, & Kiparissides, 1998).
Table 1 summarizes some highlights of recent reports
concerning the particle size distribution in a gas phase
ole)n polymerization reactor process. In most of these
studies, perfect back mixing of solids was assumed and
particle residence time distribution function was used
to calculate steady state particle size distribution in the
reactor. Choi et al. (1994) used a steady state population
balance model with a simpli)ed multigrain model to investigate the e4ects of feed catalyst size distribution and
catalyst deactivation on the particle size distribution and
the average molecular properties. Zacca et al. (1996) incorporated the concept of size selection factor proposed
by Kang, Yoon, and Lee (1989) into the catalyst residence time distribution model to calculate the particle
size distribution in a product stream. By using the size
selection factor, they developed a method to model the
preference for bigger particles being removed from the
bottom of the reactor. Hatzantonis et al. (1998) developed a population balance model in which the e4ects
of particle growth, attrition, elutriation and agglomeration were included. In their work, perfect back mixing
of solids was assumed in the main body of a %uidized
bed.
In a %uidized bed, density di4erence is considered as the most powerful cause for particle segregation. Many researchers have studied particle segregation phenomena for non-reactive binary particle mixture systems in the past (e.g., Gibilaro &
Rowe, 1974; Naimer, Chiba, & Nienow, 1982; Ho4mann, Janssen, & Prins, 1993). In the model by
Gibilaro and Rowe (1974), it is assumed that the
amount of segregation occurring at any point for a
binary mixture of particles of di4erent densities is
proportional to the concentration of jetsam at that point.
The down-%ow of segregating jetsam is also compensated by an equal volumetric up%ow of bulk material.
Solids circulation rate, bulk/wake exchange rate coef)cient, axial mixing coeNcient, and segregation coeNcient are the major model parameters used to describe
the equilibrium concentration of jetsam in the bulk and
wake phases as a function of bed height. Naimer et al.
(1982) extended Gibilaro and Rowe model by linking
these parameters directly to the physics of a bubbling
%uidized bed and to the preferential downward movement of the jetsam particles relative to the %otsam.
Wu and Baeyens (1988), introduced a mixing index
which can be correlated in terms of the bubble %ow
rate and the particle size ratio for a binary mixture of
particles.
Although these particle segregation models for a mixture of jetsam and %otsam particles have been generally
successful in predicting the particle segregation phenomena for non-reactive and binary particle mixtures, few experimental and modeling studies have been reported on
the particle segregation in a %uidized bed of multi-size
or continuous particle size distribution (Nienow, Naimer,
& Chiba, 1987; Ho4mann & Romp, 1991). In the model
proposed by Chen (1981), a %uidized bed is divided into
a static phase which is segregated at the bottom of the bed
and a %uidized bed phase which is a well mixed %uidized
region above the segregated static phase. Although this
model simulations show a reasonable agreement with experiments the division of a bed into two parts is an oversimpli)cation of the rather continuous nature of segregation through the vertical location of the bed. According to
Ho4mann and Romp (1991), a %uidized bed of powder
of a continuous size distribution may exhibit severe axial particle size distribution up to velocities considerably
above its minimum %uidization velocity. In our previous
work (Kim and Choi, 1999), we presented a simpli)ed
model where size dependent particle transfer parameters
were used to simulate a seed-bed and a reactive %uidized
bed.
In a gas phase ole)n polymerization reactor, the polymer particles of di4erent sizes are almost of the same density. Thus, if particle segregation occurs in a gas phase
polyole)n reactor, the particle size di4erence is thought
to be the primary cause of particle segregation. There
have been a few reports on the e4ects of reactor operating conditions on polymer particle size distribution in
gas phase polyole)n reactors. Yet, little has been reported
on the particle segregation phenomena inside a gas phase
%uidized bed polymerization reactors.
In this paper, we shall present a multi-compartment
population balance model using the concept of sizedependent absorption/spillage model to investigate the
e4ects of %uidization and reaction conditions on the
reactor performance.
J. Y. Kim, K. Y. Choi / Chemical Engineering Science 56 (2001) 4069–4083
4071
Table 1
Reported work on particle size distribution in %uidized bed polyole)n reactors
Authors
Systema
Modeling/method of analysis
Remarks
Caracotsios (1992)
PP
Horizontal stirred bed reactor
Choi et al. (1994)
PE
Zacca et al. (1996)
PP
Perfectly back-mixed n-CSBRs;
Use of mean residence times; Steady state
Steady state population balance
model; Perfectly back-mixed reactor
Steady state population balance model;
Perfectly back-mixed reactor;
Catalyst residence time as main coordinate
Khang and Lee (1997)
PE
Hatzantonis et al. (1998)
PE
This work (2000)
PE
a PE:
Steady
model;
Steady
model;
Steady
model;
state population balance
Perfectly back-mixed reactor
state population balance
Perfectly back-mixed reactor
state population balance
Seed-bed and reactive bed
Inclusion of multigrain
solid core model
Size selection factor;
(r) = exp[b(r − rcut )];
Use of heterophasic multigrain model
of particle growth (Debling & Ray, 1995);
Comparison with other reactor types
Size selection factor;
(r) = 0 exp[ − ar]
Inclusion of particle attrition
and agglomeration e4ects
Multi-compartment model;
Size-dependent particle
transfer rate constants;
PSD inside a reactor
polyethyelene, PP: polypropylene.
2. Size-dependent particle transfer parameters
Several authors presented the %uidized bed polymerization reactor models (e.g., two-phase bubbling bed model
(Choi & Ray, 1985) or simpli)ed continuous stirred bed
reactor model (McAuley, Talbot, & Harris, 1994). In
these reactor models, perfect back-mixing of solids was
assumed. Although these models are not capable of predicting the exact particle size distribution in the reactor, they are useful to analyze the reactor stability and
steady state and transient behaviors of a %uidized bed
polyole)n reactor and to design advanced reactor control
systems.
In a %uidized bed, solid mixing is induced by fast
moving gas bubbles. When a bubble rises through the
bubbling %uidized bed, the exchange of solid particles
occurs between the wake and the surrounding bulk emulsion phase. The wake material dragged upward by rising
bubbles is splashed onto the bed surface.
In our %uidized bed reactor modeling, the objective
is to develop a model to predict the e4ects of reactor
operating conditions on the particle size distribution in
the reactor. In this work, we postulate that the segregation of a particle mixture of equal density occurs due
to size-dependent particle transfer between the bulk and
the wake phases as gas bubbles rise in the reactor. In a
bubbling %uidized bed, a rising bubble drags a wake of
solids up the bed. The wake sheds and leaks solids as it
rises, indicating that there is a continuous interchange of
solids between wake and emulsion phases. As a rising
bubble with wake solids reaches the bed surface, it bursts
and the wake solids are thrown as a clump into the freeboard. Two bubbles may also coalesce as they break the
bed surface, ejecting wake solids into the freeboard. The
solids thrown into the freeboard are a representative sam-
ple of the bed solids (Kunii & Levenspiel, 1991). These
particles then descend in the bulk phase and are mixed
with other particles in the bed. It has been reported that the
amount of entrained small particles in the freeboard zone
increases as %uidizing gas velocity is lowered. Although
the behaviors of particles in the main bed and in the
free-board zone are di4erent, little has been reported on
the e4ect of particle size or particle size distribution on the
particle interchange rate between wake and bulk emulsion
phases.
In our model, we adopted the following exponential
correlation (Kunii & Levenspiel, 1991), which has been
developed for the particle entrainment process, for the
particle transfer constant from bulk to wake:
k(rp ) = Bg e−ut =u0 ;
(1)
where g is the density of %uidizing gas, u0 is the input
gas velocity, ut is the terminal velocity, and B is an adjustable parameter. The following correlations are also
used to calculate the terminal velocity of particles (Kunii
& Levenspiel, 1991):
−1=3
2g
∗
;
(2)
u t = ut
(p − g )g
−1
2:335
18
−
1:744
s
ut∗ =
+
;
(3)
(d∗p )2
(d∗p )0:5
p (p − g )g 1=3
∗
;
(4)
dp = d p
2
where dp is the particle diameter. Notice that the particle
transfer (absorption) rate constant (k) becomes size dependent through Eqs. (2) – (4). Here, we also assume that
the particle spillage is not a4ected by particle size. As will
be shown later in this paper, the particle size-dependent
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J. Y. Kim, K. Y. Choi / Chemical Engineering Science 56 (2001) 4069–4083
Fig. 1. Size-dependent particle transfer rate constant at di4erent %uidizing gas velocities (p = 0:4 g=cm3 ; g = 0:02 g=cm3 ;
= 1:5 × 10−4 g=cm s; s = 1; B = 0:0658).
Fig. 2. Multi-compartment %uidized bed reactor model.
transfer rate constants are the main model parameters in
calculating the particle segregation e4ects. Fig. 1 shows
the calculated particle transfer rate constant values at different %uidizing gas velocities. Notice that the transfer
rate constant is more particle size dependent at lower %uidizing gas velocity.
3. Multi-compartment steady state population balance
model
To model the particle segregation phenomena in a
%uidized bed reactor, we divide a %uidized bed into N
equally sized virtual compartments. Fig. 2 illustrates the
schematic of the proposed multi-compartment model.
Each vertical unit consists of a wake compartment and
a bulk emulsion phase compartment. As gas bubbles
rise through the emulsion phase, polymer particles are
dragged or absorbed from the bulk phase to the wake
phase and a leakage (or shedding) of particles from wake
to bulk also occurs. At the top of the bed, bubbles burst
and wake solids are splashed as a clump onto the top
of the %uidized bed and they descend in the bulk phase.
The volume of the wake compartment relative to the
bulk phase compartment is determined by the volume
fraction of the bubble phase and the fraction of wake in
the bubble using appropriate correlations. For 100–600 particles, the wake fraction (Vwake =Vbubble ) is generally in
the range of 0.2– 0.5.
In the reactor model, one can assign any bulk
phase compartment to which high activity catalyst is injected. It is then possible to investigate
the e4ect of catalyst injection point on the particle size distribution, which has never been modeled
in previous reported works. We make the following
assumptions:
(1) Catalysts are injected only to the emulsion (bulk)
phase;
(2) No reaction occurs in the wake phase (short bubble/wake residence time);
(3) Bubble size is constant throughout the bed;
(4) Solids are spherical;
(5) Solids in each compartment are homogeneously
mixed;
(6) The solid mass in each compartment remains
constant;
(7) Particles are removed only from the bottom
compartment of the bulk phase;
(8) No particles are lost by elutriation;
(9) Particle agglomeration and attrition are absent;
(10) The reactor is operated isothermally.
(11) No intraparticle and interfacial mass and heat transfer limitations are present and therefore, the rate of
polymerization in each particle is identical.
At steady state a mass balance on particles of size between rp and rp + Srp in the bulk phase compartment
gives
Catalyst
Solids leaving
−
entering in feed
to lower compartment
Solids entering
+
from upper compartment


Solids leaving
Solids entering


−  from bulk phase 
+
from wake phase
to wake phase
J. Y. Kim, K. Y. Choi / Chemical Engineering Science 56 (2001) 4069–4083

Solids growing

+  into the interval



from a smaller size

Solids growing
Solid generated


−  out of the interval  +
= 0:
by reaction
to a larger size
(5)

polymer density, respectively. By dividing Eq. (6) by
hSrp and for Srp → 0; we obtain
dPe (rp ; 1)
1
(ub Aw Pw (rp ; 1)
=
drp
hAe Re (rp )
− (ub Aw + Ge )Pe (rp ; 1))
−
Or in symbols, the above equation is represented as
follows.
4. Top compartment (n = 1)
4.1. Bulk phase
qc P0 (rp ; 1)Srp + (ub Aw Pw (rp ; 1)Srp
− Fe; 1 Pe (rp ; 1)Srp ) − Aw hSrp
Ae
×
k(rp )Pe (rp ; 1) − k1 Pw (rp ; 1)
Aw
drp drp + Ae h Pe (rp ; 1)
− Pe (rp ; 1)
dt rp
dt rp +Srp
3Ae hPe (rp ; 1) drp
Srp = 0;
(6)
rp
dt
where qc is the catalyst feed rate, P0 (rp ; 1) is the catalyst size distribution function at the top compartment. If
no catalyst is injected to the top compartment, as practiced in industry, P0 (rp ; 1) = 0. ub is the bubble rising
velocity, Aw and Ae are the e4ective cross-sectional areas of the wake and the bulk phase, respectively, and h
is the height of each compartment. Fe; 1 is the solid %ow
rate from compartment 1 to compartment 2, Pe; (rp ; 1) and
Pw (rp ; 1) are the particle size distribution functions in the
bulk emulsion phase and the wake phase, respectively,
in the top compartment. k is the size-dependent particle
absorption rate (or particle transfer rate) constant from
the bulk emulsion phase to the wake phase and k is the
spillage rate constant from the wake phase to the bulk
emulsion phase. The mass balance equations for other
compartments are derived similarly.
In practice, feed catalyst particles have a certain size
distribution. In an ideal case where the feed catalyst particles are of uniform size, P0 (rp ; 1)=0 for all rp ¿ rc where
rc is the catalyst particle radius. The particle growth rate
is calculated from the rate of polymerization by (Choi
et al., 1994)
rc3 c Rp
drp
=
;
(7)
Re (rp ) ≡
dt
3(1 − ")rp2 p
+
where Rp is the polymerization rate, " is the particle
voidage, c and p are the catalyst density and the
4073
Aw
Ae Re (rp )
Ae
k(rp )Pe (rp ; 1)
Aw
5Pe (rp ; 1)
− k (rp )Pw (rp ; 1) +
:
(8)
rp
4.2. Wake phase
ub Aw (Pw (rp ; 2)Srp − Pw (rp ; 1)Srp )
+hSrp (Ae k(rp )Pe (rp ; 1) − Aw k1 Pw (rp ; 1)) = 0:
(9)
Recall that no reaction is assumed to occur in the wake
phase. Eq. (9) can also be expressed as
ub Pw (rp ; 2) + h(Ae =Aw )k(rp )Pe (rp ; 1)
: (10)
Pw (rp ; 1) =
hk1 + ub
Ge is the polymer production rate, which is assumed to be
same in each compartment. Other symbols in the modeling equations are de)ned in Notation section. The downward volumetric solid %ow rate from the )rst bulk phase
compartment is given by Fe; 1 = ub Aw + Ge . Recall that
the wake phase volume remains constant and the bubble/wake rising velocity (ub ) is constant. Thus, the forward and reverse particle exchange rates between the bulk
and the wake phases are same.
Since wake solid volume is constant, the overall mass
balance for wake compartment is expressed as
Aw ub (Pw (rp ; 2) − Pw (rp ; 1))
rp
+h
rp
(Ae k(rp )Pe (rp ; 1) − Aw k1 Pw (rp ; 1)) = 0: (11)
Note that the )rst summation term in the above equation
is zero.
5. nth compartment
5.1. Bulk phase
qc P0 (rp ; n)Srp + (Fe; n−1 Pe (rp ; n − 1)Srp
− Fe; n Pe (rp ; n)Srp )
− Aw hSrp
Ae
k(rp )Pe (rp ; n) − kn Pw (rp ; n)
Aw
4074
J. Y. Kim, K. Y. Choi / Chemical Engineering Science 56 (2001) 4069–4083
drp drp + Ae h Pe (rp ; n)
− Pe (rp ; n)
dt rp
dt rp +Srp
+
3Ae hPe (rp ; n) drp
Srp = 0;
rp
dt
(12)
where P0 (rp ; n) is the particle size density function in
the catalyst feed stream. For the catalyst feed of uniform
particle size, P0 (rp ; n)=0 for all rp ¿ rc (catalyst particle
radius). Thus, by dividing the above equation by Srp and
letting Srp → 0, we obtain the following equation:
dPe (rp ; n)
1
=
{(ub Aw
drp
hAe Re (rp )
Fig. 3. Spillage rate constant in each compartment, u0 = 20 cm=s.
+(n − 1)Ge )Pe (rp ; n − 1)
− (ub Aw + nGe )Pe (rp ; n)}
Aw
Ae Re (rp )
Ae
k(rp )Pe (rp ; n)
Aw
5Pe (rp ; n)
− k (rp )Pw (rp ; n) +
:
rp
−
(13)
For a catalyst feed of uniform particle size (rc ); where
P0 (rc ; n)Srp = 1; we assume that catalyst particles grow
instantly to a larger size as soon as they enter the reactor
and thus at rp = rc , Pe (rc ; n) = 0 and all other terms except for the grow-out-of-size-cut term in Eq. (12) vanish.
Then, the following equation is obtained:
qc − Ae hPe (rc ; n)Re (rc ) = 0:
(14)
balance (Eq. (16)) and if it is not satis)ed, kn is updated
using the following equation (Eq. (16)):
Ae =Aw rp (k(rp )Pe (rp ; n))
:
(17)
kn =
rp Pw (rp ; n)
This procedure is repeated until kn value converges to
a constant value. It was observed that the )nal value of
kn was quite independent of the initially guessed value
(e.g., k(rp ) value of 100–500 and 1500 ). Fig. 3 illustrates the kn values thus obtained for 10 compartments.
Notice that kn decreases slightly toward the bottom of the
%uidized bed. Other correlations used for the calculation
of cross-sectional areas for the bulk and the wake phases
are listed in Table 2.
The downward solid %ow rate from the nth compartment
is expressed as Fe; n = Fe; n−1 + Ge = ub Aw + nGe .
5.2. Wake phase
ub Pw (rp ; n + 1) + h(Ae =Aw )k(rp )Pe (rp ; n)
:
Pw (rp ; n) =
hkn + ub
(15)
The overall solid mass balance for the wake phase is
given by
Aw ub
(Pw (rp ; n + 1) − Pw (rp ; n))
rp
+h
rp
(Ae k(rp )Pe (rp ; n) − Aw kn Pw (rp ; n)) = 0: (16)
The )rst summation term in Eq. (16) is zero. The spillage
rate constant for the nth compartment wake phase (kn ) is
determined as follows. First, an initial guess of kn value
is assumed. For example, k(rp ) value of 500 is taken
as an initial value of kn . Then, the model equations are
solved. The results are applied to the wake phase mass
6. Bottom compartment (n = N )
6.1. Bulk phase
dPe (rp ; N )
1
=
{(ub Aw + (N − 1)Ge )
drp
hAe Re (rp )
Pe (rp ; N − 1) − ub Aw Pe (rp ; N )}
Q
Aw
Pe (rp ; N ) −
hAe Re (rp )
Ae Re (rp )
Ae
×
k(rp )Pe (rp ; N ) − kN Pw (rp ; N )
Aw
−
+
5Pe (rp ; N )
;
rp
(18)
where Q is the product withdrawal rate from the bottom
bulk phase compartment.
J. Y. Kim, K. Y. Choi / Chemical Engineering Science 56 (2001) 4069–4083
4075
Table 2
Correlations for %uidized bed modeling

d3p g (p − g )g

umf =
33:72 + 0:0408
dp g
2
ub = u0 − umf + 0:711(gdb )1=2
u0 − umf
(=
ub
u
0 − umf
ab =
ub (1 − Fwb )
a − (
Vw
= b
fw =
(=0:4 for glass sphere)
Vb
(
a − (
Vw
= b
Fwb =
Vw + Vb
1−(
ab
ab − (
Fw = Fwb
=
1−(
1−(
)
)
Ae = d2t (1 − ab );
Aw = d2t (ab − ()
4
4
0:3z
dbm − db
= exp −
dbm − db0
dt
H
db A d z
dt
0:3H
dTb = 0 H
= dbm +
exp −
H
dt
0 A dz
) 2
dbm = 0:65[ dt (u0 − umf )]0:4
4
2:78
db0 =
(u0 − umf )2
g
1=2

− 33:7
(Wen & Yu, 1966)
(Davidson & Harrison, 1963)
(Naimer et al., 1982)
(Kunii & Levenspiel, 1991)
(Mori & Wen, 1975)
− 1 (dbm − db0 )
6.2. Wake phase
Pw (rp ; N ) =
h(Ae =Aw )k(rp ) + ub
Pe (rp ; N ):
hkN + ub
(19)
To calculate the product withdrawal rate (polymer production rate), the following overall mass balance equations are used.
Total solid balance:
dW
= qc − Q + WRp Xcat ≈ −Q + WRp Xcat = 0: (20)
dt
Catalyst balance:
d(WXcat )
= qc − QXcat = 0:
dt
(21)
Active site balance:
d(WC ∗ Xcat )
= qc C0∗ − QXcat C ∗ − WXcat kd C ∗ = 0; (22)
dt
where W is the total bed weight, Xcat is the catalyst fraction in a solid particle, C ∗ is the active site concentration,
C0∗ initial active site concentration, Rp is the polymerization rate, kp is propagation rate constant, M is monomer
concentration, Mw is monomer molecular weight, kd is
catalyst deactivation rate constant. Since the reaction is
isothermal and the volume of each bulk phase compartment is identical, the polymer production rate from each
compartment is Q=N .
From these equations, the production rate, Q can be
explicitly expressed as
(Wkd )2 + 4WRp0 qc − Wkd
;
(23)
Q=
2
where Rp0 = kp MC0∗ Mw is the initial polymerization rate.
6.3. Catalyst feed with size distribution
For a catalyst feed of non-uniform particle size distribution, the polymer particles with a particular size rp
consist of particles originally from a variety of smaller
sizes, which can be described as
fraction of particles
of size rp
rc;max
fraction of particles of
=
size rp grown from size rc
rc =rc;min
fraction of catalyst of size
×
(24)
rc in the feed
or in symbols
rc;max
Pn (rp )Srp =
pn (rp ; rc )Srp P0 (rc )Src ;
(25)
rc =rc;min
where Pn (rp ; rc ) is the particle size distribution function
in the nth compartment for the particles of size rp grown
4076
J. Y. Kim, K. Y. Choi / Chemical Engineering Science 56 (2001) 4069–4083
Table 3
Reaction kinetic scheme
Initiation
k
i1
C ∗ + M1 →
P1; 0
k
i2
C ∗ + M2 →
P0; 1
Propagation
kp11
Pn; m + M1 → Pn+1; m
kp12
Pn; m + M2 → Qn; m+1
kp21
Qn; m + M1 → Pn+1; m
kp22
Qn; m + M2 → Qn; m+1
Fig. 4. Bimodal catalyst size distribution—left distribution average
particle size—dTc = 4:5 ; 0 = 0:257; right distribution average particle
size—dTc = 14:9 ; 0 = 0:257.
from the initial catalyst of size rc . rc; min is the smallest
catalyst size, and rc; max is the largest catalyst size. In the
simulation, the size distributions calculated for uniform
catalyst feed system, Pn (rp ; rc ) is summed with Eq. (25)
over the entire range of catalyst sizes. The overall distribution can also be expressed in a continuous form by
taking a limit on particle size as
rc;max
Pn (rp ; rc )P0 (rc ) drc :
(26)
Pn (rp ) =
Chain transfer
kfm11
Pn; m + M1 → Dn; m + P1; 0
kfm12
Pn; m + M2 → Dn; m + Q0; 1
kfm21
Qn; m + M1 → Dn; m + P1; 0
kfm22
Qn; m + M2 → Dn; m + Q0; 1
kfh1
Pn; m + H2 → Dn; m + C ∗
kfh2
Qn; m + H2 → Dn; m + C ∗
kf1
Pn; m → Dn; m + C ∗
kf2
Qn; m → Dn; m + C ∗
Deactivation
k
d
C∗→
D∗
rc;min
The following log-normal distribution function is used
for the catalyst size distribution.
1
[ln(rc ) − ln(rTc )]2
exp −
;
(27)
P0 (rc ) = √
2(ln 0g )2
2)rc ln 0g
where rTc is the average radius of the catalyst particle and
0g is the geometric standard deviation. In our simulations, a bimodal catalyst size distribution (Fig. 4) which
is a composite of two log-normal size distributions is
used. This particular catalyst particle size distribution is
similar to that used in an industrial gas phase ethylene
polymerization process. Of course any other catalyst
particle size distributions can be used but in our work
we have chosen the bimodal catalyst size distribution
for illustration purposes. In the catalyst size distribution shown in Fig. 4, the mean particle diameter in the
left-side curve is 4:5 and that in the right-side curve
is 14:9 . The standard deviation used in each curve
is 0:257 . A standard copolymerization kinetic model
has been used in our model calculations (see Table 3).
The kinetic parameters used in the model simulations
are shown in Table 4. The base case model simulation
◦
conditions are: temperature = 70 C; reactor pressure =
20 atm; M1 (ethylene concentration) = 0:112 mol=l;
M2 (1-butene concentration) = 0:00355 mol=l; H2
(hydrogen concentration) = 0:00142 mol=l; bed weight
(W )=140 kg; reactor diameter=40 cm; reactor height=
5:1 m; catalyst injection rate (qc ) = 1 g=min.
7. Results and discussion
The proposed model is simulated to investigate
the e4ects of various reactor operating conditions on
the particle size distribution in the reactor. Fig. 5 shows
the e4ect of %uidizing gas velocity on the particle size
distribution in the reactor. We converted the particle mass
density distribution function to the weight fraction curve
because the particle size distribution is measured in terms
of weight fractions or size cuts in practice. For standard
reactor simulation conditions, the minimum %uidization velocity is 3:34 cm=s. For illustration purposes, the
particle size distributions in the top and the bottom compartments are shown. As expected, particle segregation
e4ect becomes pronounced as the %uidizing gas velocity
is lowered. It is seen that at low gas velocity, the amount
of small particles in the top compartment is substantially
larger than in the bottom compartment. The amount of
small particles in the top compartment should be taken
as overestimated because particle transfer to freeboard
region is not included in our model. If the transfer of
particles to freeboard is included, the amount of small
particles in the top compartment will be smaller than
what is shown in Fig. 5. It should also be recalled that no
particle attrition and agglomeration e4ects were included
in the model. The particle size distributions in the bulk
and the wake phases are shown in Fig. 6. Notice that the
di4erence in the particle size distribution curves in both
J. Y. Kim, K. Y. Choi / Chemical Engineering Science 56 (2001) 4069–4083
4077
Table 4
Kinetic parameters: k = k0 exp(−Ea =(RT )) (Ref.: Zacca, 1995)
Parametera
k0
Unit
xTi∗
wTi
Propagation (kp ) = Initiation(ki )
Ethylene
1-Butene
Chain transfer
Hydrogen (kfh )
Spontaneous (kf )
Monomers (kfm )
Site deactivation
Spontaneous (kd )
40.0
2.0
mol%
wt%
1.87E10
1.33E8
(l/mol/s)
(l/mol/s)
10
10
2.22E10
1.72E6
2.76E7
(l/mol/s)
(1/s)
(l/mol/s)
14
14
14
7.92E3
(1/s)
12
a Where
Ea
(kcal/g mol)
∗ is fraction of titanium atoms which are catalytic sites.
wTi is titanium wt%=mol% fraction in the catalyst, xTi
Fig. 6. Particle size distribution in bulk and wake phases.
Fig. 5. E4ect of %uidizing gas velocity.
phases is not quite signi)cant. It is in agreement with a
general observation that the particles thrown from bursting bubbles to freeboard are a representative sample of
bed solids.
The e4ect of catalyst activity is illustrated in Fig. 7.
Here, C0∗ is the standard catalytic site concentration used
in our model simulations. As the catalyst activity (or site
concentration) is increased by ten times the standard activity (Fig. 7b), polymer generation rate increases and the
amount of large polymer particles increases. On the other
hand, the catalyst residence time decreases (from 2 h at
C0∗ to 0:63 h at 10C0∗ ), as the bed volume is held constant.
Fig. 8 shows the e4ect of catalyst injection rate. Increased
catalyst feed rate decreases particle residence time due to
increased production rate and hence the polymer particle size distribution shifts to the left (smaller sizes). The
particle size distribution also becomes narrower as catalyst feed rate is increased.
The e4ect of catalyst deactivation is shown in Fig.
9. Here, t1=2 represents the catalyst half-life. When catalyst deactivation occurs ()rst-order deactivation mechanism is assumed), the particle size distribution shifts
to smaller sizes. For a rapidly deactivating catalyst, the
particle size distribution becomes quite broad. Fig. 9(a)
indicates that the shape of product particle size distribution is quite di4erent from the feed catalyst particle size
4078
J. Y. Kim, K. Y. Choi / Chemical Engineering Science 56 (2001) 4069–4083
Fig. 9. E4ect of catalyst deactivation: (a) 6 = 6:1 h; (b) 6 = 2:4 h.
Fig. 7. E4ect of catalyst activity: (a) 6 = 2:0 h; (b) 6 = 0:63 h.
Fig. 8. E4ect of catalyst injection rate: (a) 6 = 2:0 h; (b) 6 = 0:63 h.
distribution (Fig. 4) when catalyst deactivates rapidly.
It is also observed that the particle size distributions in
the top and bottom compartments are signi)cantly di4erent for such catalysts. The particle residence time for the
rapidly deactivating catalyst (Fig. 9a) is 6:1 h but polymer
particles do not grow as much as slowly deactivating catalyst with shorter residence time (2:4 h; Fig. 9b).
Fig. 10 illustrates various particle size distributions and
segregation patterns with di4erent types of catalyst size
distribution (note: no deactivation is assumed). We can
observe that the shape of product particle size distribution is qualitatively similar to that of feed catalyst size
distribution but not quite a replica of the catalyst size
distribution.
In an industrial %uidized bed ole)n polymerization reactor, a small amount of high activity catalyst is injected
to the reactor such that catalyst particles are uniformly
mixed with existing polymer particles without being lost
by elutriation. Fig. 11 shows how the polymer particle
size distribution is a4ected by the location of catalyst injection point. In this simulation, the bed height is constant. Notice that the catalyst injection to an upper part
3
) gives rise to a larger amount
of the reactor (e.g., n = 10
of smaller particles in the top compartment. Although
elutriation e4ect is not included in this model, it is not
diNcult to expect in an industrial %uidized bed polymerization reactor that a large amount of small particles may
be transferred to freeboard region. Such )ne particles may
cause some operational problems.
In the multi-compartment model developed in this
work, the number of compartments is one of the model
parameters. N = 10 was used throughout model simulations. Fig. 12 shows the e4ect of number of compartments. It is seen, as expected, that the use of small
N value does not provide a strong particle segregation
e4ect. The increase in N values over 10 does not necessarily increase the segregation e4ect either.
J. Y. Kim, K. Y. Choi / Chemical Engineering Science 56 (2001) 4069–4083
4079
Fig. 10. Polymer particle size distributions for various feed catalyst particle size distributions.
Bubble size is expected to have some e4ect because
it determines the quantity of particles thrown into the
freeboard. Fig. 13 shows the e4ect of bubble size. Bigger bubbles promote increased mixing by the increased
circulation of particles with its larger wake phase volume;
however, once the size reaches a limiting value (e.g.,
db ¿ 9 cm) the bubble size e4ect seems to diminish.
Fig. 14 shows the descending solid %ow rate in each
compartment at two di4erent catalyst injection rates. The
solid %ow rate increases slightly as N (compartment number) increases (i.e., toward bottom compartment) but for
practical sense, the solid %ow rate in bulk phase can be
considered as nearly constant. The increase in the catalyst
injection rate has only a small e4ect on the descending
solid %ow rate.
In the above model simulations, it is clear that the concentration of large particles is higher at the bottom part
of the reactor. In other words, the particle size distribution of polymers withdrawn from the reactor is not quite
same as that in the reactor. Zacca and coworkers (1996)
used a size selection factor approach for a %uidized bed
ole)n polymerization reactor to account for the particle
size dependent catalyst residence time distribution. The
particle selection factor ( (rp )) relates the probability of
a polymer particle of radius rp in the exit stream to the
probability of that size of particle existing in the reactor.
The use of size selection factor, albeit empirical, allows
one to calculate the particle size distribution in the product stream based on the particle size distribution in the
main body of the %uidized bed reactor in which perfect
back mixing of particles is assumed. Zacca and coworkers (1996) used the following exponential correlation to
relate the selection factor and particle radius:
(rp ) = exp[b(rp − rp; cut )];
(28)
where b is a parameter that depends mainly on %uidization conditions, and rp; cut is the particle cutting radius
corresponding to a selection factor of (rp ) = 1. In our
4080
J. Y. Kim, K. Y. Choi / Chemical Engineering Science 56 (2001) 4069–4083
Fig. 11. E4ect of catalyst injection point.
Fig. 13. E4ect of bubble size.
Fig. 12. E4ect of number of compartments.
work, size selection factor is not introduced but corresponding size selection factor can be directly calculated
by comparing the particle size distribution in the product stream (i.e., bottom compartment) and the overall
particle size distribution in the reactor (insets). Fig. 15 (a
Fig. 14. Downward bulk solid %ow rate in each compartment for
di4erent catalyst injection rates, u0 = 20 cm=s.
J. Y. Kim, K. Y. Choi / Chemical Engineering Science 56 (2001) 4069–4083
4081
conditions indicate that the model provides qualitatively
reasonable results. The proposed model is quite a simpli)ed view of extremely complex particle behaviors in a %uidized bed gas phase polyole)n reactor. For example, the
e4ects of elutriation, agglomeration, and non-isothermal
reaction are not included in the model. If these e4ects
as well as non-isothermal reaction e4ects can also be incorporated into the model, more realistic predictions of
particle size distribution can be studied. The proposed
model can be used as an alternative to a steady state residence time distribution model to estimate the particle segregation e4ects in a %uidized bed ole)n polymerization
reactor.
Notation
A
ab
Ae
Af
Aw
C∗
C0∗
D∗
Dn; m
Fig. 15. (a) Size selection factors for bimodal particle size distribution system; (b) size selection factors for unimodal particle size
distribution system.
and b) illustrates the calculated size selection factor pro)les for di4erent %uidizing gas velocities with two different overall particle size distributions (i.e., for bimodal
and unimodal feed catalyst size distributions). Notice that
the particle size distribution in the bottom compartment
is shifted toward the larger particles. It is also interesting
to observe that the size selection factor values are quite
similar to those used by Zacca and coworkers (1996).
8. Concluding remarks
In this work, a multi-compartment steady state population balance model has been developed to model particle segregation phenomena in a gas phase %uidized bed
polyole)n reactor. Due to the lack of reported data or
model for the e4ect of particle size on the particle transfer
rate between the bulk and the wake phases, we adopted
an empirical correlation proposed for particle elutriation
process to represent the size-dependent transfer constants
between the bulk and the wake phases. Although no experimental data are available to validate the proposed
model, the simulation results for various reactor operating
db
dT b
dp
dt
Fe
fw
Fw
Fwb
g
Ge
h
k
k
ki
kfm
kfh
kpij
kd
M
Mw
%uidized bed cross-sectional area, cm2
bubble and wake volume fraction
bulk phase cross-sectional area (= )4 d2t (1 −
ab )); cm2
ratio of cross-sectional area of wake to bulk,
dimensionless
wake phase cross-sectional area (= )4 d2t (ab −
ab )); cm2
active site concentration, mol-site/g-cat
initial active site concentration, mol-site/g-cat
deactivated catalytic site
dead polymer with n units of M1 monomer
and m units of M2 comonomer
bubble diameter, cm
mean bubble diameter, cm
particle diameter, cm
bed diameter, cm
downward bulk solid %ow rate, cm3 =s
wake fraction in bubble, dimensionless
volume fraction of wake solid in all solids,
dimensionless
volume fraction of wake in bubble/wake
phase, dimensionless
gravitational acceleration, cm=s2
polymer production rate in each compartment,
g/s
compartment height, cm
size-dependent absorption rate constant, 1/s
spillage rate constant, 1/s
initiation rate constant, l/mol/s
chain transfer rate constant to monomer,
l/mol/s
chain transfer rate constant to hydrogen,
l/mol/s
rate constant for propagation of monomer i at
a chain ending with monomer j; l/mol/s
catalyst deactivation rate constant, 1/s
monomer concentration, mol=cm3
monomer molecular weight, g/mol
4082
Pn; m
Q
Qn; m
qc
P0
Pe
Pw
Re
Rp
Rp0
rc
rTc
rp
ub
umf
u0
ut
uw
Vb
Vw
W
wTi
X
Xcat
xTi∗
z
J. Y. Kim, K. Y. Choi / Chemical Engineering Science 56 (2001) 4069–4083
live polymer with n units of M1 monomer and
m units of M2 comonomer, M1 at active site
polymer withdrawal rate from the bottom
compartment, production rate, g/s
live polymer with n units of M1 monomer and
m units of M2 comonomer, M2 at active site
catalyst injection rate, cm3 =s
catalyst size distribution function, 1/cm
particle size distribution function in bulk
phase, 1/cm
particle size distribution function in wake
phase, 1/cm
particle growth rate, cm/s
rate of polymerization, g-solid/g-catalyst/s
initial
polymerization
rate,
g-solid/
g-catalyst/s
catalyst particle radius, cm
mean particle radius, cm
polymer particle radius, cm
bubble rise velocity, cm/s
gas velocity at minimum %uidization, cm/s
super)cial inlet gas velocity, cm/s
terminal velocity, cm/s
wake rise velocity = ub ; cm=s
bubble volume, cm3
wake volume, cm3
total bed weight, g
titanium weight fraction in the catalyst,
dimensionless
weight fraction, dimensionless
catalyst fraction in a solid particle, dimensionless
fraction of titanium atoms which are catalytic
sites, dimensionless
vertical location in the bed, dimensionless
Greek letters
(
s
"
0g
bubble fraction in the bed, dimensionless
density, g=cm3
spherisity, dimensionless
particle voidage, dimensionless
size selection factor, dimensionless
viscosity, g/cm-s
geometric standard deviation, dimensionless
Subscripts
b
c
g
n
s
w
bulk or emulsion phase
catalyst
gas
compartment number
solid phase
wake phase
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