Journal of Environmental Treatment Techniques  
2020, Volume 8, Issue 1, Pages: 272-283  
J. Environ. Treat. Tech.  
ISSN: 2309-1185  
Journal web link: http://www.jett.dormaj.com  
CFD Modeling of Polypropylene Fluidized Bed  
Reactor  
1
1
2
2
Hossein Esmaeili *, Salar Azizi , Seyyed Mojtaba Mousavi , Seyyed Alireza Hashemi  
1
Department of Chemical Engineering, Bushehr Branch, Islamic Azad University, Bushehr, Iran  
2
Department of Medical Nanotechnology, School of Advanced Medical Sciences and Technologies, Shiraz University of Medical Sciences, Shiraz, Iran  
Received: 02/08/2019 Accepted: 16/11/2019 Published: 20/02/2020  
Abstract  
Poly propylene is one of the famous polymers with great application. In this study, CFD model on dynamics of the fluidized bed  
polyethylene production process has been investigated. A detailed CFD model for sticky poly propylene fluidized bed was formulated  
in this work. As a result, detailed information on the PSD and hydrodynamic fields of the gas and solid phases can be obtained from  
the simulations. Defluidization due to particle aggregation also can be simulated. For modeling plant-scale poly propylene reactors, a  
chemical look-up table should be used to solve efficiently the solid species equations. In order to address all the issues in FB  
polymerization, models for simplified polymerization kinetics, polydisperse multiphase flow, and mass and heat transfer between the  
gas and solid particles are combined together. As a result, physically aggregation because of a tactic polypropylene is more important  
than other aggregation reasons. Finally, as mentioned earlier, after DQMOM is applied to the multi-fluid CFD model, new terms  
accounting for the effect of aggregation and breakage need to be added on the right-hand sides of the solid-phase momentum, energy,  
and species equations.  
Keywords: Polypropylene, Reactor, Fluidized bed, Kinetic theory, CFD  
1
At PP polymerization reactors, propylene as a monomer  
1
Introduction  
contact with Ziegler-Natta catalyst which activated with co  
catalyst and forms poly propylene. PP collected to three types  
as isotactic, atactic and syndiotactic. Purpose is isotactic PP  
production and avoids great amount atactic PP production.  
Atactic poly propylene percentage specification is by means  
xylene solubility percentage. Atactic PP is liquefied and a  
small percentage of its made special application of PP, but a  
great amount of atactic PP make sticky powder and ultimately  
cause choking bed and effect to PSD at PP polymerization  
reactor. A stereo modifier as an external donor used to avoid  
unwanted xylene solubility of PP. donor usage is small as a  
catalyst. Partially a special Ziegler-Natta catalyst used with  
the internal donor. Normally industrial PP reactors haven’t  
syndiotactic PP production. In this paper, a cold PP gas  
fluidized bed when PP powder is sticky (xylene solubility is  
upper) studied which could give valuable information about  
this matter and help to design, optimization and scale up with  
avoiding choking.  
Poly propylene (PP) is one of the famous polymers with  
great application. Recent PP production reactor technology is  
a gas-phase fluidized bed and stirred bed. In gas-phase  
polymerization, small particles (e.g., 20-80 μm) are  
introduced at a point above the gas distributor, and when  
exposed to the gas flow containing the monomer,  
polymerization occurs (1-3). At the early stage of  
polymerization, the catalyst particles fragment into a large  
number of small particles, which are quickly encapsulated by  
the newly-formed polymer and grow continuously, reaching a  
typical size of 200-3000 μm. Due to the differences in the  
polymer particle size, segregation occurs and fully-grown  
polymer particles migrate to the bottom where they are  
removed from the reactor. The smaller pre-polymerized  
particles and fresh catalyst particles tend to migrate to the  
upper portions of the reactor and continue to react with  
monomers. Because polymerization is exothermic, the  
temperature of polymer tend to rise and sometimes it will  
exceed the melting point of the polymer, then polymer particle  
can become sticky and during collisions can form large  
agglomerates that can possibly undergo sintering and cause  
defluidization. In the opposite situation, if the bed is too cold,  
the particles can become brittle and may fracture forming  
unwanted small fragments that elutriate with the gas. Hence,  
heat and mass transfer to particle surface controls the local  
particle temperature and the rate of agglomeration and  
breakage (4).  
2 Modeling equations  
Two methods improved for CFD modeling of gas-solid  
flows, discrete element model (DEM) and two fluid model  
(TFM). In the DEM gas phase is described by locally  
averaged Navier-Stokes equations. Newtonian equations of  
motion for individual particles are then solved and individual  
particle trajectories are traced, taking into account effects of  
particle collisions and forces acting on the particle by flowing  
gas. In such models, the computational demand rises strongly  
with the number of traced particles, which limits its  
applicability. In the TFM model, based on the momentum,  
two phases are mathematically treated as interpenetrating  
continua. The success of TFM depends on the proper  
description of the interfacial forces and the solid stress. The  
interfacial forces are used to describe the momentum transfer  
Corresponding author: Hossein Esmaeili, Department of  
Chemical Engineering, Bushehr Branch, Islamic Azad  
University,  
Bushehr,  
Iran.  
E-mail:  
&
esmaeili.hossein@gmail.com  
esmaeili.hossein@iaubushehr.ac.ir.  
2
72  
Journal of Environmental Treatment Techniques  
2020, Volume 8, Issue 1, Pages: 272-283  
between the two phases which has the primary effect on the  
hydrodynamic behavior. The stress which represents the solid  
phase force due to particle-particle interactions has only  
secondary effect. By introducing the concepts of solid  
Where, is the rate of granular energy dissipation due to  
inelastic collisions and  is the diffusive flux of granular  
energy. The term accounts for the transfer of granular  
th  
energy between the gas phase and the m solids phase,  
―pressure‖ and ―viscosity‖, the well-known granular kinetic  
whereas accounts for the transfer of granular energy  
theory has been employed for computation of the solid stress.  
In the TFM models, the conservation equations for each of the  
two phases are derived to obtain a set of the equation with the  
similar mathematical structure for both phases, which makes  
the mathematical manipulation of the system relatively easier  
and minimizes the computation cost (5, 6).  
Governing equations detailed for more reliability.  
Conservation equations and related terms have shown in this  
section. Meanings of the symbols used are listed in the  
Nomenclature section.  
th  
mt  
between the m and l solids phases. Supplying constitutive  
relations for granular energy equation and numerically solving  
the M coupled partial differential equations it represents is an  
onerous task.  
The granular energy equation is still under development. An  
algebraic expression for granular temperature, Θ , obtained  
m
from the energy equation of Lun (9), by assuming that the  
granular energy is dissipated locally; neglecting the  
convection and diffusion contributions; and retaining only the  
generation and dissipation terms (10-12). The resulting  
algebraic granular energy equation is:  
2
.1 Conservation of Mass  
The continuity equation for the gas phase is (7, 8):  
ꢒ  
ꢌꢍ  
ꢁ ꢃ )ꢄꢅꢆ(ꢁ ꢃ ꢇ)ꢈ∑ ꢉ  
ꢊꢎꢏ ꢂꢊ  
ꢩꢷ ꢥ ꢤꢸꢐꢹꢓ  
ꢏꢒ ꢑꢒ  
ꢢꢥꢑꢒ  
(
(1)  
(2)  
ꢈꢵ  
ꢂ ꢂ  
ꢂ ꢂ ꢂ  
ꢺꢒ  
ꣁꢛ  
There are M solids-phase continuity equations as follows:  
ꢤꢸ ꢐꢹꢓꢥ ꢄꢻꢷ  
ꢼꢷꢤꢸ ꢐꢹꢓꢄꢢꢷꢤꢸꢐꢹꢓꢽꣀ  
ꢏꢒ  
ꢑꢒ  
ꢺꢒ ꢑꢒ  
ꢐꣂꢓ  
ꢔꢕ  
ꢐꢁ ꢃ ꢓꢄꢅꢆꢐꢁ ꢃ ꢇ̅ ꢓꢈ∑ ꢉ  
ꢢꢥ  
ꢑꢒ ꢺꢒ  
ꢑꢒ ꢑꢒ  
ꢑꢒ ꢑꢒ ꢑꢒ  
ꢊꢎꢏ ꢑꢒꢊ  
ꢢꢐꣃꢩ꣄ꢓꢃ  
ꢒꢒ ꢑꢒ ꣆ꢒꢒ  
ꢺꢒ  
ꣁꣁꣁꢲꢲꢲꢲꣁꣁꣁꣁꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꣁꣁꣁꢐ꣋ꢓ  
The first term on the left in equations (1) and (2) accounts for  
the rate of mass accumulation per unit volume, and the second  
term is the net rate of convective mass flux. The term on the  
right accounts for inters phase mass transfer because of  
chemical reactions or physical processes, such as evaporation.  
ꢒ  
꣉꣊  
2
.3 Conservation of Momentum  
The gas-phase momentum balance is expressed as (7, 8):  
→ꢝ  
→ꢝ →ꢝ  
ꢂ ꢂ ꢂ ꢂ  
2
.2 Granular energy conservation  
The kinetic theory describing the flow of smooth, slightly  
ꢼꢁ ꢃ ꢇꢽꢄꢧꢆꢼꢁ ꢃ ꢇ ꢇꢽ  
ꢂ ꢂ ꢂ  
ꢣꢤ  
꣍ꢝ →ꢝ  
ꢈꢧꢆꢖ ꢄꢁ ꢃ ꣅꢩ ꢮ꣌ ꢄ꣎ ꣁꢲꢲꢲꢐ꣏ꢓ  
inelastic, spherical particles were used in the derivation of the  
constitutive relation describing the stress tensor in the m  
th  
ꢂ ꢂ  
ꢂꢒ  
ꢒꢎꢏ  
solids phase, . The resulting constitutive relations contaitnh  
the quantity Θ , called the Granular temperature of the m  
→꣍ꢝ  
m
where  is the gas-phase stress tensor, is an interaction  
solids phase. The granular temperature is proportional to the  
granular energy of the continuum, where granular energy is  
defined as the specific kinetic energy of the random  
fluctuating component of the particle velocity:  
force representing the momentum transfer between the gas  
th  
phase and the m solids phase, and is the flow resistance  
offered by internal porous surfaces. The momentum equation  
th  
for the m solids phase is  
ꢗ ꢈ ꢙꢚ ꢜ  
(3)  
→꣍ꢝ  
→꣍ꢝ→꣍ꢝ  
ꢑꢒ ꢑꢒ ꢑꢒ ꢑꢒ  
ꢁ ꢃ ꢇ ꢽꢄꢧꢆꢼꢁ ꢃ ꢇ ꢇ ꢽ  
ꢑꢒ ꢑꢒ ꢑꢒ  
ꢣꢤ  
Where ꢙꢚ ꢜis the fluctuating component of the  
→꣍ꢝ  
ꢈꢧꢆꢖ ꢄꢁ ꢃ ꣅꢄ꣌  
ꢑꢒ ꢑꢒ ꢑꢒ ꢂꢒ  
th  
instantaneous velocity  of the m solids phase defined by:  
ꢑꢒ  
꣍ꢝ  
ꢩꢮ꣌ ꣁꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꣁꢐ꣐ꢓ  
̇
ꢈꢇ ꢄꢚ  
ꢒꢯ  
̅
(4)  
ꢑꢒ  
ꢯꢎꢏ  
ꢯꢱꢒ  
th  
th  
Where  is the stress tensor for the m solids phase. The  
th th  
→꣍ꢝ  
The transport of granular energy in the m solids phase is  
governed by following equation:  
term is the interaction force between the m and l solids  
phases. The first term on the left in these momentum  
equations represents the net rate of momentum increase. The  
second term on the left represents the net rate of momentum  
transfer by convection. The first term on the right represents  
normal and shear surface forces, while the second term  
represents body forces (gravity in this case). The next term in  
equation (8) represents the momentum transfer between the  
fluid and solids phases; the final term represents the  
momentum transfer between the fluid and a rigid porous  
structure. The last two terms in equation (9) represent the  
momentum exchange between the fluid and solids phases and  
between the different solids phases, from left to right.  
ꢡꢣ  
ꢥ ꢃ ꢦ ꢄ ꢧꢆꢥ ꢃ ꢦ ꢇ  
ꢑꢒ ꢑꢒ ꢒ  
ꢑꢒ ꢑꢒ ꢒ ꢑꢒ  
ꢢꢣꢤ  
ꢈ[ꢖ ꢨꢧꢇ ꢩꢧꢆꢪ ꢩꢬ ꢄꢭ  
ꢑꢒ  
ꢑꢒ  
ꢕ  
ꢂꢒ  
ꢕ  
ꢄꢮꢭ ]ꢆꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢐꢳꢓ  
ꢯꢒ  
ꢯꢎꢏ  
ꢯꢱꢒ  
2
73  
Journal of Environmental Treatment Techniques  
2020, Volume 8, Issue 1, Pages: 272-283  
2
.3.1 Fluid-Solids Momentum Transfer  
A simplified version of the kinetic theory was used by  
Syamlal [7], to derive an expression for the drag coefficient  
Fsml,  
The interaction force, or momentum transfer between the  
gas and the  solid phase, is modeled by:  
꣍ꢝ  
→꣍ꢝ →ꢝ  
ꢈꢩꢁ ꢧ꣒ ꢩ꣓ ꢼꢇ ꢩꢇꢽꣁꣁꣁꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꣁꣁꣁꢐꣃ꣔ꢓ  
ꢈꢡꢐꣃꢄ꣄ ꢓꢐ꣊꣥ꢢꢄꢚ ꣊ ꣥꣏ꢓꢁ ꢃ ꢁ ꢃ  
ꢯꢒ  
꣦ꢯꢒ  
ꢑꢯ ꢑꢯ ꢑꢒ ꢑꢒꢲꢲꢲꢲꢲꢐꢏ꣧꣨꣩ꢓ  
ꢂꢒ  
ꢑꢒ  
ꢂꢒ  
ꢑꢒ  
ꢐ꣇ ꢄ꣇ ꢓ ꣅ |ꢇ ꢩꢇ |  
꣈ꢯ ꣈ꢒ ꣆ꢯꢒ ꢑꢯ  
ꣁꣁꣁꣁꢐꣃ꣋꣪ꢓ  
ꢝ →꣍ꢝ  
Where the first term on right side describes the buoyancy  
force, the second term describes the drag force.  
Syamlal and O'Brien (1987) derived the following formula for  
converting terminal velocity correlations to drag correlations  
13):  
ꢑꢒ  
ꢑꢒꢯ  
ꢢ꣊ꢐꢃ ꣇ ꢄꢃ ꣇ ꢓ  
ꢑꢯ ꣈ꢯ  
ꢑꢒ ꣈ꢒ  
Where elm and Cflm are the coefficient of restitution and  
(
coefficient of friction, respectively, between the  and  
solids-phase particles. The radial distribution function at  
contact is that derived by Lebowitz (24) for a mixture of  
hard spheres:  
ꢡꢁ ꢁ ꢃ  
ꢉ꣄ →꣍ꢝ →ꢝ  
ꢑꢒ ꢂ  
ꢑꢒ ꢂ ꢂ  
ꢂꢒ  
ꢚ ꣗ ꣘|ꢇ ꢩꢇ|ꣁꣁꣁꣁꣁꣁꢐꣃꣃꢓ  
꣖ꢑ  
ꢇ ꣇  
꣕ꢒ  
ꢒ ꣈ꢒ  
Where Vrm is the terminal velocity correlation for the mth  
solids phase. Vrm can be calculated from the Richardson  
ꢡ꣇꣈ꢯ꣈ꢒ  
ꢑ꣭  
ꢕ  
ꢈ ꢄ  
ꢮ ꣁꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꣁꣁꢐꣃ꣏ꢓ  
(1954) (14), correlation only numerically; an explicit formula  
ꢁ ꢁ ꢐ꣇ ꢄ꣇ ꢓ ꣇  
꣈ꢯ  
꣈ꢒ  
꣈꣭  
ꢎꢏ  
cannot be derived. However, a closed formula for Vrm can be  
derived from a similar correlation developed by Garside (15),  
2
.3.3 Fluid-Phase Stress Tensor  
The stress tensor for the fluid phase, either gas or liquid, is  
given by:  
ꢇ ꢈ  
ꢆꢳꢼ꣙ꢩ  
̿
̿
ꢖ ꢈꢩ꣒꣌ꢄ꣮̿ ꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢐꣃ꣐ꢓ  
ꢂ ꢂ ꢂ  
ꢆ꣔ꣂꢉ꣄ ꢄ꣚ꢐ꣔ꢆ꣔ꣂꢉ꣄ ꢓ ꢄ꣔ꢆꣃꢢꢉ꣄ ꢐꢢ꣛ꢩ꣙ꢓꢄ꣙ ꢽꢲꢲꢲꢐꣃꢢꢓ  
ꢆꢏꢺ  
Where Pg is the pressure. The viscous stress tensor, gτ, is  
assumed to be of the Newtonian form.  
ꢈꢁ ꢲꢲ꣜ꢲꢲꢲꢲꢲꢲ꣛  
ꢏꢆꢛ꣝  
ꢆ꣏ꢁ ꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲ꣞꣎ꢲꢲꢲꢲꢲꢲꢲꢲꢲꢁ ꣟꣔ꢆ꣏ꢳ  
ꢈ{  
ꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢐꣃꢡꢓꢲꢲꢲ  
ꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲ꣞꣎ꢲꢲꢲꢲꢲꢲꢲꢲꢲꢁ ꢜ꣔ꢆ꣏ꢳ  
ꢛꢆ꣠꣡  
ꢈꢢꢁ ꣤ ꢹ ꢄꢁ ꣯ ꢤꢸ(ꢹ )꣌ꣁꣁꣁꣁꣁꢲꢲꢲꢲꣁꣁ꣰꣰ꣁꣁꢐꢢ꣔ꢓ  
ꢂ ꢂ ꢂ  
ꢂ ꢂ  
And the Reynolds number of the solids phase is given by  
Where I is the identity tensor and Dg is the strain rate tensor  
for the fluid phase, given by:  
̅
꣣ꢇ ꢩꢇ꣣ꢃ  
ꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢐꣃꢻꢓ  
̅
꣈  
ꢒ ꢑꢒ  
ꢂ  
ꢂ ꢂ  
ꢉ꣄ ꢈ  
ꣃ →ꢝ  
→ꢝ  
  *ꢧꢇ ꢄꢐꢧꢇꢓ +ꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꢐꢢꣃꢓ  
Here, CDs (Re /V ) is the single-sphere drag function. Of the  
numerous expressions available for CDs (16), we chose the  
following simple formula proposed by Dalla Valle (17):  
m
rm  
2
.3.4 Solids-Phase Stress Tensor  
The theories are combined by introducing a "switch" at a  
critical packing, ε , the packed-bed void fraction at which a  
granular flow regime transition is assumed to occur:  
g
ꢆ꣏  
ꢚ ꢐꢉ꣄ꢓꢈ꣗꣔ꢆꣂꢡꢄ  
꣘ ꢲꢲꢲꢲꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꢲꢲꢲꢲꢐꣃꢳꢓ  
ꢉ꣄  
ꢩ꣒ ꣌ꢄ꣮ ꣁ꣞꣎ ꢁ ꣟ꢁ  
ꢑꢒ  
ꢑꢒ  
ꢈꢩꣲ  
ꣁꣁꣁꣁꣁꣁꣁꢐꢢꢢꢓ  
To use this formula in equation (8), note that Re must be  
replaced with Rem/Vrm.  
ꢩ꣒ ꣌ꢄ꣮ ꣁ꣞꣎ ꢁ ꢜꢁ  
ꢑꢒ  
ꢑꢒ  
th  
Where Psm is the pressure and is the viscous stress in the m  
solids phase. The superscript p stands for plastic regime and v  
2
.3.2 Solids-Solids Momentum Transfer  
Compared to fluid-solids momentum transfer, much less is  
known about solids-solids momentum transfer. It is safe to  
assume that the major effect is the drag between the phases  
because of velocity differences. Arastoopour (18), observed  
that such a term is necessary to correctly predict segregation  
among particles of different sizes in a pneumatic conveyor.  
Arastoopour (19) studied this effect experimentally in a  
pneumatic conveyor. Equations to describe such interactions  
have been derived or suggested by several researchers: Soo  
for viscous regime. In fluidized-bed simulations, ε is usually  
g
set to the void fraction at minimum fluidization. The granular  
pressure is given by:  
ꢈꢷ ꢁ ꢦ ꣁꢲꢲꢲꢲꢲꢲꢲꢲꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꢐꢢꢡꢓ  
ꢯꢒ ꢑꢒ ꢒ  
ꢑꢒ  
 ꢈꢢꢐꣃꢄ ꢓꢃ ꣅ ꣁꢲꢲꢲꢲꢲꢲ꣰꣰ꣁꣁꣁꣁꣁꣁꣁ꣰ꣁꢐꢢꢻꢓ  
ꢒꢒ ꢑꢒ ꣆ꢕꢕ  
(
20), Nakamura (21), Syamlal (10, 22), and Srinivasan (23).  
The granular stress is given by  
In the present work the solids-solids momentum transfer, Iml,  
is represented as  
ꢈꢢ꣤ ꢹ ꢄ꣯ ꢤꢸ(ꢹ )꣌ꣁꣁꣁꣁꢲꢲꢲꢲꢲꢲꢲꣁꣁꣁꣁꢐꢢꢳꢓ  
ꢑꢒ ꢑꢒ ꢑꢒ  
ꢑꢒ  
ꢑꢒ  
꣍ꢝ  
→ꢝ →꣍ꢝ  
ꢈꢩ꣓ ꢼꢇ ꢩꢇ ꢽꣁꣁꣁꣁꣁꣁꣁꢲꢲꢲꣁꣁꣁꣁꣁꢐꣃꣂꢓ  
ꢒꢯ  
ꢑꢒꢯ  
ꢑꢯ  
ꢑꢒ  
Where ꣯  
ꢑꢒ  
, the second coefficient of viscosity for the  
solids phase is given by:  
2
74  
Journal of Environmental Treatment Techniques  
2020, Volume 8, Issue 1, Pages: 272-283  
ꢈꢷ ꢁ ꣚ꢦ ꣁꣁꣁꣁꣁꣁꣁꢲꢲꢲꢲꢲꢲꢲꢲꣁꣁꣁꣁꣁꣁꢐꢢꣂꢓ  
ꢛꢒ ꢑꢒ ꢒ  
ꢑꢒ  
ꢈꢃ ꢚ ꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꣁꣁꣁꢐꢡꣂꢓ  
ꢂ ꣼ ꢂ  
꣇ ꢃ ꢐꣃꢄ꣄ ꢓꢁ ꣅ  
ꢒ ꢑꢒ  
ꢒꢒ ꢑꢒ ꣆ꢕꢕ  
ꢒ  
ꢩ ꢷ ꣁꣁꢲꢲꢲꢲꢲꣁꢐꢢ꣋ꢓ  
ꢘꢒ  
ꢡ꣉꣊  
Where  is the turbulence dissipation rate and ꢈ꣔ꢆ꣔꣐.  
Turbulence predications of continuous phase are obtained  
from the following equations of modified k-epsilon model:  
ꢘꢒ  
꣈  
꣉꣊  
ꣵꣃꢄ꣔ꢆꢻꢐꣃꢄ꣄ ꢓꢐꢡ꣄  
ꢒꢒ ꢒꢒ  
ꢡꢐꢡꢩ꣄ ꢓ  
ꢒꢒ  
ꢒ ꢑꢒ  
{
ꢁ ꣅ ꢐꣃꢄ꣄ ꢓ  
ꢌ  
ꢑꢒ ꣆ꢕꢕ  
ꢒꢒ  
(ꢥ ꢃ ꣻ )ꢄꢧꣽꢼꢥ ꢃ ꢇ ꣻ ꢽꢈꢧꣽꢼꢥ ꢧꣻ ꢽꢄ  
ꢂ ꢂ ꢂ ꢂ ꢂ ꢂ ꢂ ꢂ ꢂ  
ꢀ ꣿ꤀  
ꢩꣃꢓꢁ ꣅ ꣶꢄ  
}ꣁꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢐꢢ꣏ꢓ  
ꢑꢒ ꣆ꢕꢕ  
ꢳ꣉꣊  
ꢥ ꤁ ꢄ∏  
ꢩꢥ ꢃ ꢁ  
ꢂ ꢂ ꢂ  
ꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲ ꢐꢡ꣋ꢓ  
ꢂ ꤂꤃ꢂ  
꤂ꢂ  
The factor , the shear viscosity for the solids phase is  
given by:  
ꢑꢒ  
ꢂ ꢂ ꢂ ꢂ  
(
ꢥ ꢃ ꢁ )ꢄꢧꣽꢼꢥ ꢃ ꢇ ꢁ ꢽ  
ꢂ ꢂ  
ꢣꢤ  
ꢷ ꢁ ꣚ꢦ ꣁꣁꣁꣁꣁꣁꣁꢲꢲꢲꢲꢲꣁꣁꣁꣁꣁꣁꢲꢲꢲꣁꢐꢢ꣐ꢓ  
ꢑꢒ ꢘꢒ ꢑꢒ  
ꢈꢧꣽ꤄ꢥ ꢧꢁ ꤇  
The strain rate tensor,ꢑꢒ is given by  
ꢄꢥ (ꢚ ꤁ ꢩꢚ ꢃ ꢁ )  
ꢏ꤆ ꤂꤃ꢂ  
ꢛ꤆ ꢂ ꢂ  
→꣍ꢝ  
→꣍ꢝ  
ꢑꢒ  
ꢄ꤈  
ꣁꢲꢲꢲꢲꢲꢲꣁꣁꣁꣁꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꣁꢐꢡ꣏ꢓ  
꤆ꢂ  
  ꣵꢧꢇ ꢄꢐꢧꢇ ꢓ ꣶꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꢐꢡ꣔ꢓ  
ꢑꢒ  
Similar to the functions typically used in plastic flow theories  
25), an arbitrary function that allows a certain amount of  
The influences of the dispersed phase on the continuous phase  
are given by:  
(
compressibility in the solids phase represents the solids  
pressure term for plastic flow regime:  
ꢂ  
ꢂ  
ꢈ꤉(ꣻ ꢩꢢꣻ )ꣁꣁꣁꣁꣁꣁꣁꣁꣁꢲꢲꢲꢲꢲꢲꢲꣁꣁꣁꢐꢡ꣐ꢓ  
ꢂꢒꢑ ꢂ  
ꢈꢁ ꣒ ꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁ꣰ꣁꣁꣁꣁꣁꣁꢐꢡꣃꢓ  
ꢑꢒ  
ꢑꢒ  
ꢈꢚ ꤄ ꤇꤈  
ꣁꣁꣁꣁꢲꢲꢲꢲꢲꢲꣁꣁꣁꣁꣁꣁꣁꢐꢻ꣔ꢓ  
꤂ꢂ  
ꢐꢢꣻꢄꢡꤋꢒꢑꢂꢒꢑꢓꣁꣁꣁꣁꢐꢻꣃꢓ  
ꢘ꤆  
Where is represented by an empirical power law  
ꢀ  
ꢄꢐꣃꢄꤋ ꢓꤊ  
ꢈ꣙ꢐꢁ ꢩꢁ ꢓ ꣁꣁꣁꢲꢲꢲꢲꢲꢲꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꢐꢡꢢꢓ  
ꢂꢒꢑ ꢈ  
ꢒꢑꢂ ꢀ  
Typically, values of A=1025 and n=10 have been used. These  
stresses are calculated only for solids phase-1, even when  
multiple solids phases are specified:  
The production of turbulence kinetic energy, Gk,g, is computed  
from:  
꣹꣞꣺ꢭ  
ꣁꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꣁꢐꢡꢡꢓ  
ꢢ꣚꣌꣖  
꤂꤃ꢂ ꢈ꣤ꢼꢧꢇꢄꢐꢧꢓ ꢽꢨꢧꢇꣁꣁꣁꢲꢲꢲꣁꣁ꣰ꣁꣁꢐꢻꢢꢓ  
ꢑꢯ  
ꢑꢯ  
ꢈꢢ꣤ ꢹ ꣁꣁꣷ꣸꣄ꢸ꣄ꣁꣁ꣤ ꢈ  
ꢑꢯ ꢑꢯ  
2
.6 Turbulence in the dispersed phase  
Predictions for the turbulence quantities of the dispersed  
The second invariant of the deviator of the strain rate tensor is  
phase are obtained using the Tchen theory of dispersion of  
discrete particles by homogeneous turbulence (27, 28). The  
turbulence quantities include:  
ꢑꢏꢏ  
  ꣵꢐꢹ ꢩꢹ ꢓ ꢄꢐꢹ ꢩꢹ ꢓ ꢄꢐꢹ ꢩꢹ ꢓ ꣶ  
ꢑꢏꢏ  
ꢑꢛꢛ  
ꢑꢛꢛ  
ꢑꢘꢘ  
ꢑꢘꢘ  
ꢄꢹ ꢄꢹ ꢄꢹ ꣁꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꣁꣁꢐꢡꢻꢓ  
ꢑꢏꢛ  
ꢑꢛꢘ  
ꢑꢘꢏ  
ꢑꢂ  
2
.4 Turbulence model  
The effects of turbulent fluctuations of velocities and  
ꣻ ꢈꣻ ꤄  
꤇ꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꢐꢻꢡꢓ  
ꢒꢑ  
ꣃꢄꤊꢂ  
scalar quantities in the gas phase are described using the  
dispersed turbulence model. In this turbulence closure model,  
turbulence predictions for the continuous phase are obtained  
using the standard k-epsilon model(26) supplemented with  
extra terms dealing with interphase turbulent momentum  
transfer while predictions of the turbulence quantities for the  
particulate phase are obtained using the Techen-theory  
correlations.  
ꢂ  
ꢄꤊꢂ  
ꢄꤊꢂ  
ꢈꢢꣻ꤄  
꤇ꣁꣁꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꣁꣁꣁꣁꣁꢲꣁꣁꢐꢻꢻꢓ  
ꢂ  
ꢈ ꣻ ꣮ ꣁꣁꣁꣁꣁꣁꣁꣁꣁꢲꢲꢲꣁꣁꣁꣁꣁꣁ ꢻꢳ  
ꢐ ꢓ  
ꢑꢂ ꢀ꤃ꢑꢂ  
ꢑꢂ  
 ꢈꢹ ꢄ꣗ ꣻ ꢩ꣪ ꣻ ꣘꣮ ꣁꣁꢲꢲꢲꣁꣁ꣰ꣁꣁꣁꢐꢻꣂꢓ  
ꢀ꤃ꢑꢂ  
ꢒꢑ  
ꤌ꤃ꢑꢂ  
2
.5 Turbulence in the continuous phase  
The Reynolds stress tensor for the continuous phase takes  
the following form:  
꣨ꢏ  
ꢒꢑ  
ꢈꢐꣃꢄꢚ ꢓ꤄ ꢄꢚ ꤇ ꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꢐꢻ꣋ꢓ  
ꢂ ꢂ ꢂ ꢂ  
 ꢈꢩ ꢼꢃ ꢥ ꣻ ꢄꢃ ꢥ ꣤ ꢧꢆꢇ ꢽ꣌ꢄꢢꢃ ꢥ ꣤ ꢹ ꣁꢲꢲꢐꢡꢳꢓ  
ꢂ ꢂ ꢂ  
ꢂ ꢂ ꢂ  
Where C is the added-mass coefficient, equal to 0.5. and  
V
 is the characteristic particle relaxation time connected  
with inertial effects acting on a dispersed phase. is the  
The turbulence viscosity  is written in terms of the turbulent  
kinetic energy of gas phase as:  
2
75  
Journal of Environmental Treatment Techniques  
2020, Volume 8, Issue 1, Pages: 272-283  
Lagrangian integral time scale calculated along particle  
trajectories.  
ꤓ  
ꢉꤔꢂ  
ꢈ  
ꣁꣁꣁꣁꣁꢲꢲꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁ꣰꣰ꣁꣁꢐꢳꢢꢓ  
Or as an incompressible fluid with a constant density. The  
user may specify any other equation of state by modifying the  
equation of state subroutine (EOSG). These volume fractions  
are assumed to be continuous functions of space and time. By  
definition, the volume fractions of all of the phases must sum  
to one:  
2
.7 Aggregation, breakage and growth  
In order to account for the particle size distribution (PSD), a  
population balance must be solved simultaneously with the  
other equations(29). In this work, the direct quadrature  
method of moments (DQMOM) is combined with the multi-  
fluid CFD model to describe polydisperse solids undergoing  
aggregation, breakage, and growth. The detailed derivation of  
the DQMOM equations is given by Fan (30). Neglecting  
changes in momentum due to aggregation and breakage, the  
DQMOM equations are:  
  ꢮꢁ ꢈꣃꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꢲꢲꢲꣁꢐꢳꢡꢓ  
ꢑꢒ  
ꢒꢎꢏ  
Where M is the total number of solids phases  
ꢌ  
ꢔꢕ  
ꢒꢎꢏ  
ꤏ ꢑ ꣈ꢒ ꢒ ꢒ  
ꢄꢧꢆ(ꢁ ꢃ ꤎ )ꢈꢩ∑ ꢡꣻ ꢃ꣇ ꤐ ꤁ ꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢐꢻ꣏ꢓ  
ꢂ ꢂ ꢂ  
3
Results and discussion  
The two dimensional model is considered with width and  
ꢔꢕꢔꢕ  
ꢄꢧꢆꢐꢁ ꢃ ꤎ ꢓꢈꢡꣻ ꢃ ꣇ ꢐ꣪ ꢄꤐ ꤁ ꢓꢩ  
ꢑꢒ ꢑꢒ ꢑꢒ  
ꤏ ꢑꢒ ꣈ꢒ ꢒ  
ꢒ ꢒ  
height of 5 and 30 cm, respectively. The properties of solid  
and gas are shown in Table 1. This calculating area was  
discreet to 20*90 cells and the time of modeling (physical  
time) was considered 10 sec. The following assumptions were  
made for the simulations.  
ꢢꣻ ꢃ ꣇ ꤑ  
ꢲꢲꢲ  
(49)  
ꤏ ꢑꢒ ꣈ꢒ ꢒ  
ꢣꢁ ꢃ ꣇  
ꢑꢒ ꢑꢒ ꣈ꢒ  
ꢄꢧꢆ(ꢁ ꢃ ꣇ ꤎ )  
ꢑꢒ ꢑꢒ ꣈ꢒ ꢑꢒ  
ꢣꢤ  
ꢈꢻꣻ ꢃ ꣇ ꢐ꣪ ꢄꤐ ꤁ ꢓ  
ꢩꢡꣻ ꢃ ꣇ ꤑ ꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꣁꣁꣁꣁꣁꣁꣁꣁꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢲꢐꢳ꣔ꢓ  
ꤏ ꢑꢒ ꣈ꢒ ꢒ  
ꢒ ꢒ  
1) Physical properties such as temperature, density and  
viscosity of the gas and solid are assumed to be constant.  
ꤏ ꢑꢒ ꣈ꢒ ꢒ  
2
3
) Mass transfer between solid phases is ignored.  
) The gas phase is composed of pure air.  
Where the particle number density  is related to the solids  
volume fraction by  
4) Two solid phases with different initial diameters and  
volume fractions are used to represent aggregation and  
breakage.  
5) The aggregation and breakage efficiencies are independent  
of particle size.  
ꢒ  
ꤐ ꢈ  
ꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꣁꢐꢳꣃꢓ  
ꣻ ꣇  
ꤏ ꣈ꢒ  
In the present study, by designing optimum gas  
distributor, it is tried to reduce the required time for lump  
formation in the polymerization reactor. Accordingly, three  
types of gas distributor with different spacing in inlet vents  
were considered and the results were investigated in three  
case studies. Figure 1 shows the features of gas distributor in  
lower section of the polymerization reactor. It is obvious that  
difference in inlet vents spacing is a function of fluidized bed  
reactor. But the sectional velocities of all distributors were  
equal to 0.25 m/s (see Table 1). Gas velocity in inlet vents of  
distributors for case studies 1 to 3 are 0.25, 0.535 and 1.25  
m/s, respectively.  
The shape factor k depends on the particle morphology and  
v
has a value for k = π/6 for spherical particles. The rates of  
v
aggregation and breakage determine the rate constants  and  
ꢒ  
.
2
.8 Equation of state and other equations  
The fluid phase can be modeled as a gas obeying the ideal  
gas law,  
Table 1: The CFD domain and parameters used in the simulation  
Property  
units  
value  
Number of phases of solid particles, N  
-
2
Initial diameter of particles, dpm  
Density, ρms  
Friction factor, e  
μm  
Kg/m  
-
408 & 168  
2530  
0.8  
3
Internal friction angle, φ  
rate of particle attachment  
Particle breaking rate  
Hold up of compact bed, ε  
Reactor pressure  
degree 30.0  
-
-
-
0.001  
0.0001  
0.48  
20  
*
g
atm  
K
Inlet gas temperature  
316  
Gas density, ρg  
Kg/m3 0.1  
Gas viscosity, μg  
Average velocity of inlet gas on the cross-platform, U  
Pa.s  
m/s  
0.0000114  
0.25  
2
76  
Journal of Environmental Treatment Techniques  
2020, Volume 8, Issue 1, Pages: 272-283  
Figure 1: Schematic of the gas distributor for the three case studies  
3
.1 Distribution of gas hold-up  
Contours of gas hold-up in different times and for all three  
2. Reactor recovery as a result of a decrease in the  
produced number of polymer particles which should  
gas distributors are shown in Figures (2) to (4). Because of  
high rate of particle attachments inside the reactors compared  
with particle cracking, the number of particles inside fluidized  
bed reactor increases. The sectional fluidizing velocity is  
constant for all case studies; therefore, as the velocity of  
particles decreases in these beds the size of bubbles increases  
until the beds velocity move toward zero. The conditions in  
which fluidizing disappears inside the reactors are shown in  
Figures 2 to 4. As it is seen in these figures, as time increases  
gravity force will also increase due to enlargement of size and  
mass of particles and drag force cannot move the particles to a  
higher height. Since particle diameter increases, the height of  
bed decreases (reduction in bed expansion) and finally the bed  
be in desired level (a market grade with appropriate  
standards);  
The more steady gas distribution inside the bed, the more  
suitable stay time and gas contact with active solid particles  
(with active continuum for reaction with gas monomer). A  
good indicator of this point is the expansion of fluid bed or  
average volume fraction of gas for initial moments of  
fluidizing just before critical state and bed alleviation.  
Accordingly, the bed of gas distributor with larger spacing is  
not suitable for continuous reactor polymerization.  
In order to investigate the distribution of apparent density  
and mass fraction of fine and coarse particles inside the case  
studies, Figures (6) to (8) obtained from numerical modeling  
will be discussed. Considering the contours shown for coarse  
particles, the accumulation of these particles in the lower  
section of fluidized beds was observed; meaning that coarse  
particle separation happens in the lower section and transfer  
of fine solid particles will be toward upper fluidized beds(31).  
As time increases and enhancement occurs in fluidized  
alleviation, in all three case studies decrease in the size of gas  
bubbles in the lower section up to the free surface of beds was  
observed. Designing fluidized bed applicable in  
polymerization reactors, discharge of produced powder takes  
place from the lower section of the bed. As a result, the  
diameter increases and accumulates in the lower section of the  
bed leading to flocculation and shutting gas inlet. Regarding  
Figures (6) to (8), a gas distributor with larger spacing has a  
better function for prevention of coarse particle accumulation  
on the gas distributor.  
loses the fluidizing state.  
Figures (2) to (4) show that  
fluidizing state of distributors with bigger spacing happens  
later and fluid bed with steady gas distributor loses fluid state  
faster.  
3
.2 Distribution of apparent density and mass fraction of  
particles  
According to time delay applied for preventing  
flocculation and fluidizing state in previous section, selection  
of gas distributor with larger spacing would be a better choice.  
But uncontrolled conditions and tendency to flocculation  
happen in fluidized bed reactors. Two points should be  
considered in these reactors:  
1
.
Gas distribution in fluid bed for longer contact time  
of gas phase (reaction monomers) for optional stay  
time in reactors;  
2
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Figure 2: Distribution contours of gas hold-up for case study 1  
Figure 3: Distribution contours of gas hold-up for case study 2  
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Figure 4: Distribution contours of gas hold-up for case study 3  
Figure 5: Average spatial distribution of gas hold-up in the fluidized bed versus time for three type of different gas distributors  
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Figure 6: Apparent density distribution contours of solid particles for case study 1  
Figure 7: Apparent density distribution contours of solid particles for case study 2  
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Figure 8: Apparent density distribution contours of solid particles for case study 3  
3
.3 Particle Size Distribution  
Particle size distribution is affected by two factors: 1)  
beds versus time is observed. But this difference is not  
significant between the steady gas distributor and medium  
spacing gas distributor. As time passes, increase in the  
number of coarse particles on the beds with steady distributor  
stops 6 seconds after fluidizing which is a function of  
fluidized alleviation and settling coarse particles. This time is  
equal to 8 seconds for the medium spacing gas distributor. On  
the third bed the diameter of coarser particles increases which  
implies fluidizing behavior.  
particle attachment; 2) physical and chemical cracking of  
particles(32). In fluidized bed reactors, one of the physical  
factors is particle collision. In lower temperatures the  
possibility of cracking in particles increases and otherwise,  
the possibility of flocculation increases. On the other hand,  
reaction with monomer increases the size of particles and in  
case of cohesion presence in the system (polymer eutectic) the  
number of coarse particles increases.  
Studies on chemical effects need an investigation of  
catalysts and operational conditions in the reactor. But  
physical factors depend on flow regime and reactor designing  
parameters like a gas distributor. In the present study, the size  
distribution of a poly-ethylene polymerization reactor was  
calculated using numerical modeling for a typical reactor in  
normal conditions. In other words, some parameters  
considered constant including growth rate, solid particle  
cracking from chemical reactions, particle cohesion and  
particle cracking due to the physical collision of powder  
particles. The only variable was the difference in gas and solid  
distribution which is produced by different gas distributors. In  
Figure (9), the changes in the number of fine particles  
(averaged with inside volume of reactor) versus time is  
shown. As it is seen, the number of particles increases as time  
passes. In addition, the diameter of fine particles for the  
steady gas distributor is larger and for long spacing gas  
distributor is shorter.  
Figure 9: Average size of the fine solid particles (first step of solid  
particles) versus time for different gas distributors  
In Figure (10) the average diameter of coarse particles  
versus time is shown similar to Figure (9) and the different  
trend is observed for these particles. In the beginning, with  
similar initial conditions, the diameter of coarse particles  
increases and the difference between size distributions in the  
In order to determine the average number of fine and coarse  
particles inside the fluidized bed, it should be determined the  
mass percentage of fine and coarse particles. In Figure (11),  
modeling time distribution of fine solid particles for different  
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gas distributors is shown. It is obvious that mass fraction of  
coarse particles is equal to unity minus this value which is  
shown in Figure (12).  
the average diameter of the particles. This result is due to  
changes in flow regime and convection term (Equations (155)  
to (161)) inside fluidized reactors.  
Figure 10: Average size of the coarse solid particles (second step of  
solid particles) versus time for different gas distributors  
Figure 13: The average size of solid particles (Average of first and  
second steps of solid particles) versus time for different gas  
distributors.  
Mass fraction of coarser particles increases with time and  
mass fraction of finer particles decreases. According to the  
results, it is clear that mass fraction of coarser particles for gas  
distributors with larger spacing is less than other distributors.  
More fine particles attend inside the bed. Although it is a  
negligible amount, it promotes to a larger extend to fluidizing  
and delay in the alleviation of the bed.  
4
Conclusions  
A detailed CFD model for sticky poly propylene fluidized  
bed was formulated in this work. As a result, detailed  
information on the PSD and hydrodynamic fields of the gas  
and solid phases can be obtained from the simulations.  
Defluidization due to particle aggregation also can be  
simulated. For modeling plant-scale poly propylene reactors, a  
chemical look-up table should be used to solve efficiently the  
solid species equations. The aggregation and breakage  
efficiencies should also be related to the particle velocity and  
xylem solubility.  
As a result, physically aggregation  
because of atactic polypropylene is more important than other  
aggregation reasons. Finally, as mentioned earlier, after  
DQMOM is applied to the multi-fluid CFD model, new terms  
accounting for the effect of aggregation and breakage need to  
be added on the right-hand sides of the solid-phase  
momentum, energy, and species equations. These  
modifications and detailed simulation results for plant-scale  
fluidized bed polymerization reactors will be reported in  
future communications. During simulation, the growth rate of  
the number of particles has a good match with gas distribution  
in the fluidized bed. Increasing mass and particle size, gravity  
force increases and drag force cannot move the particles to a  
higher height. On the other hand, for the larger diameter of the  
particles, the height of the bed decreases (reduction in the  
expansion of the bed) and finally the bed lose its fluidizing  
state. If solid particles were not expelled from the reactor in a  
definite time or an unpredictable growth takes place, primary  
conditions for formation of flocculation would be provided  
and as a result the whole reactor would be filled with melted  
polymer and after shutting down, a polymer cast would be  
created that needs about six months for cleaning and  
discharge. Obviously, significant losses would be created and  
this confirms the importance of this study for optimization.  
According to the results, more steady gas flow in the gas  
distributors of polymer fluidized beds, the higher will be the  
growth rate. But for sharper gas distribution (longer spacing  
between the gas inlets), the fluidizing state of the bed tends to  
flocculation later. Therefore, an optimized option should be  
considered for a gas distributor in designing polymerized  
reactor. It is suggested that a steady gas distributor should be  
selected for fluidized bed and this is done by dividing  
distributor area into a number of sections and inlet gas is  
designed into the reactor with tree diagram pattern. In case of  
all open inlets, the flow would be steady and in case of some  
Figure 11: The average of the fine solid particles hold-up (first step)  
versus time for different gas distributors  
Figure 12: The average of the coarse solid particles hold-up (second  
step) versus time for different gas distributors  
Figure (13) shows the average particle diameter inside the  
fluidized beds for different distributors that is produced by  
merging Figures (10) to (12). This figure shows that  
increasing unsteadiness in the gas distributor inlets decreases  
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2020, Volume 8, Issue 1, Pages: 272-283  
closed inlets, the spacing between inlets will increase. Such  
flexibility leads to the suitable function of the normal reactor  
and prevents flocculation in critical conditions. As a result,  
without fundamental changes in polymerized reactor structure  
that costs a huge amount, optimization would be performed.  
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