2pb

VRMa

8b

c

d

Ue

U

a

A

R

R

A

A

K

C

P

F

r

D

m

1

GftiT

h0

chemical engineering research and design 1 0 4 ( 2 0 1 5 ) 53–67

Contents lists available at ScienceDirect

Chemical Engineering Research and Design

j ourna l h omepage: www.elsev ier .com/ locate /cherd

D CFD-PBM simulation of hydrodynamic andarticle growth in an industrial gas phase fluidizeded polymerization reactor

ahid Akbaria,b, Tohid Nejad Ghaffar Borhania, Ahmad Shamiri c,d,oya Arameshc,e, Mohamed Azlan Hussainc,ohd. Kamaruddin Abd. Hamida,∗

Process Systems Engineering Centre (PROSPECT), Faculty of Chemical Engineering, Universiti Teknologi Malaysia,1310 UTM Johor Bahru, Johor, MalaysiaDepartment of Process Engineering, Razi Petrochemical Company, P.O. Box 161, Bandar Imam, IranDepartment of Chemical Engineering, Faculty of Engineering, University of Malaya, 50603 Kuala Lumpur, MalaysiaChemical & Petroleum Engineering Department, Faculty of Engineering, Technology & Built Environment,CSI University, 56000 Kuala Lumpur, MalaysiaDepartment of Computer and Communication Systems Engineering, Faculty of Engineering,niversiti Putra Malaysia, 43400 Selangor Darul Ehsan, Malaysia

r t i c l e i n f o

rticle history:

eceived 21 December 2014

eceived in revised form 13 July 2015

ccepted 17 July 2015

vailable online 26 July 2015

eywords:

omputational fluid dynamics

opulation balance model

luidized bed polymerization

eactor

irect quadrature method of

oments

a b s t r a c t

In an industrial fluidized bed polymerization reactor, particle size distribution (PSD) plays a

significant role in the reactor efficiency evaluation. The computational fluid dynamic (CFD)

models coupled with population balance model (CFD-PBM) have been extensively employed

to highlight its potential to analyze the industrial-scale gas phase polymerization reactor

(FBRs) utilizing ANSYS Fluent software. The predicted results reveal an acceptable agree-

ment with the observed industrial data in terms of pressure drop and bed height. Courant

number independent study has been carried out to record the mesh and time step indepen-

dent results for large scale FBRs. Furthermore, the minimum fluidization velocity (Umf) and

size-dependent particle growth rate have been assessed to emphasize the impact of PSD

along the reactor. The results show transient regime in the case of minimum fluidization

velocity. The simulation results signify that in order to improve the polymerization yield,

the amount of gas velocity can be increased without change in the fluidization regime, i.e.

segregation. Hence, the 2D CFD-PBM/DQMOM coupled model can be used as a reliable tool

for analyzing and improving the design and operation of the gas phase polymerization FBRs.

© 2015 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

to the particle attrition or breakage. The developed small

. Introduction

as-phase fluidized bed reactor (FBR) is extensively appliedor olefin polymerization due to its high rate of heat/massransfer, simple construction, and considerable particles mix-ng rate (Akbari et al., 2014; Shamiri et al., 2013, 2010, 2012).

o review briefly the reaction mechanism, at the early stage of

∗ Corresponding author. Tel.: +60 7 5535517; fax: +60 7 5536165.E-mail address: kamaruddin@cheme.utm.my (Mohd.K.Abd. Hamid)

ttp://dx.doi.org/10.1016/j.cherd.2015.07.016263-8762/© 2015 The Institution of Chemical Engineers. Published by

polymerization, the catalyst breaks down into a large numberof smaller particles which are encapsulated by the grow-ing semi-crystalline polymer. During their residence in thereactor, the size of the catalyst particles grows due to poly-merization (70–1600 �m). However, it might decrease owing

.

particles might leave the reaction zone with the fluidizing

Elsevier B.V. All rights reserved.

54 chemical engineering research and design 1 0 4 ( 2 0 1 5 ) 53–67

List of symbols

u velocity (m/s)n(V,t) the number density functionmk the kth moment of the PSDN the specific number of momentsε volume fraction� density (kg/m3)Umf minimum fluidization velocity

List of acronymsPBE population balance equationPBM population balance modelKTGF kinetic theory of granular flowPMLM polymeric multilayer model

gas. Therefore, real polymerization systems are character-ized by a broad particle size distribution (PSD) that changescontinuously due to the fluid–particle and particle–particleinteractions (Ahmadzadeh et al., 2008; Ashrafi et al., 2012;Shamiri et al., 2012). The PSD can directly affect the quality offinal product, mixing/segregation, and hydrodynamic param-eters. Thus, providing a precise prediction of the PSD is crucialnot only for understanding and optimizing the performanceof gas-phase polymerization fluidized bed reactors (FBRs), butalso for safe operation of process.

Advanced computational methods and computer program-ming have made it possible for computational fluid dynamic(CFD) to emerge as an effectual technique to understandthe influence of the fluid dynamics on the performance ofchemical reactors. For gas–solid fluidization systems, state-of-the-art CFD models are modifying rapidly. This improvementleads to more reasonable and precise prediction of the flowcharacteristics of gas–solid reactors to quantify the impactof flow behavior on the reactor performance. However, thereare still some limitations on the application of CFD on anindustrial scale (Mazzei, 2013; Mazzei et al., 2012) due to differ-ent reactor scales requiring different operating conditions anddictating various levels of interaction complexity between thephases, which control the efficiency of polymerization reac-tion and product quality. Recently, the CFD models have beenemployed to analyze the scale-up performance. However, dueto the computational constraints, most CFD studies have beenperformed in the laboratory.

Generally, two different categories of CFD models canbe applied for modeling multiphase gas–solid flow, namelyEulerian–Eulerian and Eulerian–Lagrangian formulations(Behjat et al., 2011; Chen et al., 2011; Reddy and Joshi, 2010;Zimmermann and Taghipour, 2005). The Eulerian–Eulerianmodel assumes both phases as a fluid and it takes the inter-penetrating effect of each phase into consideration by usingdrag models, While, the Eulerian–Lagrangian model exploitsNewton’s equations of individual particles motion and usesa continuous interpenetrating model (Eulerian framework)for the gas phase. The Eulerian–Lagrangian model providesa more reliable and detailed representation of the fluidizedbed since it tracks each particle–particle collision. This modelperforms more efficiently when the number of particles aresmall (typically <1 × 106), and dispersed volume fraction doesnot exceed 10% of the mixture in any region (i.e., in academic

researches on a laboratory scale), due to high computationalcost. For industrial simulation with large geometries and high

mass loading, the Eulerian–Eulerian model is more preferablesince it is significantly less computationally expensive andreadily provides the required information (average pressuredrops, velocity fields, void fraction profiles, etc.).

To describe the PSD in a multiphase flow, the popula-tion balance equation (PBE) needs to be solved along withcontinuity, momentum, and energy equations (Fan, 2006). Sev-eral methods have been proposed to solve the PBE, such asdiscretized population balance (DPB) or class method (CM)(Kumar and Ramkrishna, 1996), standard method of moment(SMM) (Marchisio et al., 2002), the method of moments (MOM)(Diemer and Olson, 2002; Hutton et al., 2012), quadraturemethod of moments (QMOM) (Buffo et al., 2013; Silva et al.,2010), sectional quadrature method of moments (SQMOM)(Attarakih et al., 2009), direct quadrature method of moments(DQMOM) (Fan et al., 2004; Mazzei et al., 2009) and fixedpivot quadrature moments (FPQMOM) (Yao et al., 2015). Thedetails of the mathematical and numerical issues involved inthese methods have been discussed by Ramkrishna (2000) andRigopoulos (2010). A brief comparison on various PBE solvingmethods is illustrated in Table 1.

QMOM has been extensively validated for a wide rangeof problems with different internal coordinates (Buffo et al.,2013; Claudotte et al., 2010; Silva et al., 2010; Yan et al., 2012).One of the main restrictions of QMOM approach is that thesolid phase is represented through the moments of the distri-butions and the phase average velocity of the different solidphases must be known to solve the transport equations forthe moments. Yan et al. (2012) suggested CFD-PBM-PMLMintegrated model of a gas–solid fluidized bed polymerizationreactor on a laboratory scale. The QMOM and CFD have beencombined to investigate the particle growth rate. It should benoted that they have solved the particle growth rate outsidethe CFD framework using polymeric multilayer model (PMLM).Initially, the authors decoupled the growth rate from the flowequations and PBE. The major drawback of this approach isthat when the flow field is decoupled from the PBE, the averageparticle diameter is used to solve the momentum equations.This means that the particle growth in each numerical cell issolved disregarding the drag force (Ahmadzadeh et al., 2008).

Mazzei et al. (2012) developed QMOM by considering thedifferent particle velocity fields, which were fulfilled in anEulerian framework to solve PBE. Their new QMOM modelrelies on the volume instead of the number of density func-tion. Their results were reasonably good, either QMOM wassolved with the first-order upwind discretization scheme orit tracked two quadrature nodes. Although, their simulationfailed either by using higher-order discretization schemesor by tracking higher quadrature nodes, they suggested afew strategies that might be employed to overcome this. Forinstance, they did not consider the impact of growth, aggrega-tion, and breakage phenomena.

In order to address these issues, the direct quadraturemethod of moments (DQMOM) has been formulated and val-idated by Fan et al. (2004) in laboratory scale. In contrast tothe QMOM, the advantage of the DQMOM is that it deals withquadrature nodes and weights instead of moments which ismore convenient due to calculation of the quadrature nodes(Mazzei, 2013). In addition, the moments are tracked with anaverage solid velocity, each node (different solid phase) rep-resents the solid phase properties (Fan et al., 2004). Thus,the DQMOM suggests a promising capability describing PSD

bearing particle kinetics in the context of CFD-PBM gas–solidmodeling.

chemical engineering research and design 1 0 4 ( 2 0 1 5 ) 53–67 55

Table 1 – Comparison of several population balance equation.

Model Finding/remarks Reference

SMM The population balance equation is transformed into a set of transport equationsfor the moments of distribution.

(Marchisio et al., 2003a)

Reduction of the dimensionality of the problem and relatively simple utilization forsolving transport equations for lower order moments.Limited only in case of size-independent aggregation, breakage and growth rate.Therefore, it is not possible to model size dependence kernels.

MOM The PSD is tracked through its moments via integrating the internal coordinates. (Hutton et al., 2012;Passalacqua et al., 2010)

The number of necessary scalars is very small (i.e., usually 4–6), which makes itfeasible to be implemented in CFD simulations.Due to the difficulty of expressing the transport equations in terms of momentum,this method has not been widely used.

QMOM It is based on the approximation of the unclosed terms by using an ad-hocquadrature formula.

(Claudotte et al., 2010; Silvaet al., 2010; Yan et al., 2012)

The quadrature approximation (its abscissas and weights) can be determined usingthe lower-order moments by resorting to the product-difference (PD) algorithm.QMOM has been extensively validated for a wide range of problems with differentinternal coordinates.As drawback, the phase average velocity of the different solid phases must beknown to solve the transport equations for the moments.

SQMOM Using advantage of CM and QMOM and minimize their drawback to the inversion oflarge sized moment problems as required by QMOM.

(Attarakih et al., 2009;Drumm et al., 2008)

The SQMOM can track accuracy of any set of low-order moments with the ability ofthe shape of the distribution reconstruction.This method becomes ill conditioned by using a large number of moments.

DQMOM DQMOM is based on the directly integrating the transport equations for weights andabscissas of the quadrature approximation.

(Mazzei, 2013)

It deals with quadrature nodes and weights instead of moments which is moreconvenient due to calculation of the quadrature nodes.Each node (different solid phases) represents the solid phase properties.The DQMOM suggests a promising capability describe PSD bearing particle kineticsin the context of CFD-PBM gas–solid modeling.

FPQMOM Compare to QMOM and DQMOM, suitability of the FPQMOM for modelingpolymerization FBRs with simultaneous polymerization particle growth andaggregation

(Yao et al., 2015)

ndust

retmflyibcaai

2m

AmamVaPPc

It needs more investigation in breakage, 3D and i

Since the DQMOM and other moment methods are stillelatively new, they have not been thoroughly validated,specially in industrial FBRs. To the best of our knowledge,he application of an existing 2D CFD-PBM/DQMOM coupled

odel to simulate particle growth phenomena in industrialuidized bed polymerization reactor has not been reportedet. This work attempts to simulate a 2D-transient of thempact of PSD on minimum fluidization velocity and dynamicehavior of size dependent growth rate in industrial FBRs. Theurrent study further focuses on finding the suitable time stepnd mesh size to ensure the independent results of the gridnd time step in these types of industrial reactors. The models embodied in the commercial CFD code ANSYS Fluent 14.

. Generic multiphasic CFD-PBM coupledodel

2D CFD-PBM coupled model based on the Eulerian–Eulerianodel incorporating the kinetic theory of granular flow (KTGF)

nd the PBM has been applied to describe the gas–solidultiphase flow in the fluidized bed polymerization reactor.

ariations of the particle diameter over the course of timend along the bed have been considered together with theBM to accomplish the coupling of particle kinetics model and

BM. In addition, the DQMOM was used to solve the PBE andompleted the combination of the CFD model and PBE. More

rial scale.

details of this description can be found in Fan et al. (2004) andother investigations (Marshall Jr et al., 2011; Mazzei et al., 2009;Selma et al., 2010; Zucca et al., 2007). In the following sections,only a brief review of the equations is provided.

2.1. The population balance model and DQMOM

A polydisperse solid-phase can be modeled by the followingmultivariate distribution function (Fan et al., 2004).

∂n (L, us; x, t)∂t

+ ∇. [usn (L, us; x, t)] + ∇us . [Fn (L, us; x, t)]

= S (L, us; x, t) (1)

where x is the spatial coordinate, and t is time. In Eq. (1),S (L, us; x, t) is a source term that represents discontinuouschanges in property space due to kinetic particles (aggrega-tion, breakage, etc.), whereas F is the acceleration force ofeach solid phase. The mean force F, conditioned over the par-ticle diameter L and defined as F|L = Lk acts to accelerate theparticles of solid phase k. us|L is the mean velocity condi-tioned on the particles of diameter L. By definition, us|L = Lk

equals usk to evaluate average velocity of kth solid phase par-ticles. The DQMOM approach considers a Gaussian quadrature

56 chemical engineering research and design 1 0 4 ( 2 0 1 5 ) 53–67

approximation to the integral of the number density functionn (L; x, t) by a summation of N Dirac delta functions:

n (L, us; x, t) =N∑

˛=1

ω˛ (x, t) ı [L − L˛ (x, t)] ı [us − us˛ (x, t)] (2)

where ω˛ is the weight of the delta function centered on thecharacteristic length, L˛. By inserting Eq. (2) in Eq. (1), andapplying a moment transformation, the population balancein terms of the presumed finite mode PSD can be shown asEq. (3) (Dutta et al., 2012):

N∑˛=1

ı(L − L˛(x, t))

[∂ω˛(x, t)

∂t+ ∇ · (ω˛(x, t)us˛)

]

−N∑

˛=1

ı′(L − L˛(x, t))[

ω˛

(∂L˛

∂t+ us˛ · ∇L˛

)]= S(L; x, t) (3)

where ı′ (L − L˛ (x, t)) is the first derivative of the Dirac deltafunction. The transport equations for the weights ω˛ andweighted abscissas L˛ (L˛ = L˛ω˛) are as follows:

∂ω˛

∂t+ ∇. (us˛ω˛) = a˛ (4a)

∂L˛

∂t+ ∇. (us˛L˛) = b˛ (4b)

where a˛ and b˛ are defined through a linear system foundfrom the first 2N moments (i.e., mk with k = 0, . . ., 2N − 1). Thereader may refer to Fan et al. (2004) and Marchisio et al. (2003b,2003a) for detailed solution of this linear system. In Eqs. (4a)and (4b), the functional forms of a˛ and b˛ can be determinedby using lower-order moment and moment of transformation.The lower-order moment can be applied by using as few as twoor three nodes (=N) defined as (Dutta et al., 2012; Fan et al.,2004):

Mk (x, t) =∞∫0

n (L; x, t) LkdL ≈N∑

i=1

ω˛ (x, t) Lk˛ (x, t) (5)

It is noteworthy that the nodes do not give any physicalmeaning (Dutta et al., 2012; Fan et al., 2004). However, thiscan be considered as different solid phases with characteristicparticle diameter and velocity. The weights and abscissas aresimply the quadrature approximation for the moments deter-mining the shape of the underlying PSD. By using DQMOMvariable in multi-fluid model, the weights and abscissas canbe correlated with the solid volume fraction and the effectiveparticle diameter (length) for each solid phase. The transportequations for each weights and abscissas undergoing growth,aggregation and breakage rate can be written as follows (Fanet al., 2004):

∂εs˛�s

∂t+ ∇. (us˛εs˛�s˛) = 3kv�s˛L2

˛ (b˛ + w˛G˛) − 2kv�s˛L3˛a˛ (6a)

∂εs˛L˛�s 3 4

∂t+ ∇. (us˛εs˛L˛�s˛) = 4kv�s˛L˛ (b˛ + w˛G˛) − 3kv�s˛L˛a˛

(6b)

where εs˛ = kvL3˛ω˛ and εiLi = kvL4

˛ω˛ are the volume fractionand the effective particle length, respectively. wi is the num-ber of particles per unit volume and Gi is the growth rateat quadrature point. ai and bi can be computed through alinear system resulting from the moment transformation ofparticle number density transport equation using N quadra-ture points (Fan et al., 2004). kv is a volumetric shape factor(equal to �/6 for spherical particles). It should be noted that thepresent study does not consider the aggregation and break-age phenomena, except particle growth phenomena which isdescribed in Section 3.4.4. Eq. (6a) represents the conserva-tion equation for the volume fraction of the kth solid phase inthe presence of growth rate. Eq. (6b) is a conservation equa-tion presenting the effective particle diameter. Further detailson the derivation and solution of these equations can befound elsewhere (Fan et al., 2004; Marchisio and Fox, 2005;Mazzei et al., 2009). DQMOM directly tracks the nodes andweights by integrating transport equations (Eqs. (4a) and (4b))that govern their evaluation. Finally, in order to couple thePBM/DQMOM model with the multi-fluid CFD framework, Eq.(6a) and (6b) and the momentum equation of solid consti-tuting the set of DQMOM multi-fluid model equations arecombined.

2.2. The Eulerian–Eulerian CFD model

In the multi-fluid continuum model, different phases aretreated as interpenetration continuum. The conservationequations are solved simultaneously for each phase in theEulerian framework. Each particular phase is characterizedby the unique diameter, density, and other individual fea-tures. Several drag models are considered to calculate thegas–solid inter-phase exchange coefficient (Yao et al., 2014).In this study, the interphase momentum exchange was closedusing Gidaspow drag model (Akbari et al., 2015). Kinetic theoryof granular flow (KTGF) was employed to describe the rheologyand characterize the motion of particles. In addition, consti-tutive equations derived by Chen et al. (2011) were extensivelyutilized (Ahuja and Patwardhan, 2008; Chen et al., 2011; deSouza Braun et al., 2010; Esmaili and Mahinpey, 2011; Sunet al., 2011; Wei et al., 2011). The radial distribution functionbetween two solid phases was modeled according to (Syamlalet al., 1993). Furthermore, a standard k–ε per phase turbulencemodel was used to solve the turbulent kinetic energy k and itsdissipation rate ε (Chen et al., 2011). Further relevant discus-sion can be found in our previous studies (Akbari et al., 2014,2015).

2.3. Numerical solution procedure

The 2D simulations relying on the CFD-PBM/DQMOM inte-grated model with the Eulerian–Eulerian approach werecarried out with the commercial CFD package, ANSYS/FLUENT14, in double precision mode. The proposed particle growthrate has been defined using User Defined Function (UDF).

For Eulerian–Eulerian multiphase simulation, the PhaseCoupled SIMPLE (PC-SIMPLE) algorithm (Vasquez and Ivanov,2000) was applied to handle pressure–velocity coupling. PC-SIMPLE algorithm is an extension of the SIMPLE algorithm(Patankar, 1980) to multiphase flows. The velocities are cal-culated coupled by phases, but in a segregated fashion. Theblock algebraic multigrid scheme used by the coupled solver

described by Weiss et al. (1999) was applied to solve a vec-tor equation formed by the velocity components of all phases

chemical engineering research and design 1 0 4 ( 2 0 1 5 ) 53–67 57

sbtisddogwsu

dgeutetitwdCEt

3

Ttttdtstcmptcs

4

Ttcw02tt0tiwfbrt

Fig. 1 – Sketch of the gas-phase LLDPE polymerizationreactor used in simulation.

imultaneously. Then, a pressure correction equation wasuilt based on total volume continuity rather than mass con-inuity. The convergence criterion of 10−3 with around 20terations per time-step was chosen to ensure that numericaltability was achieved. Since it was impossible to intro-uce diffusion in ANSYS Fluent, the convergence study forrag monitor, mass flux residuals, and residual convergencef continuity equations was performed to secure conver-ence. In addition, the under relaxation factors equal to 0.3ere adopted for all the variables with the exception of

olid DQMOM phases in which a factor equal to 0.5 wassed.

The impact of first order upwind and second order upwindiscretization scheme on numerical diffusion was investi-ated by Cooper and Coronella (2005) and de Souza Braunt al. (2010). To reduce the numerical diffusion, second orderpwind discretization scheme was employed to discretizehe momentum, volume fraction, and population balancequations, as recently suggested by Mazzei et al. (2012). Theemporal discretization is achieved by using the first ordermplicit scheme in ANSYS Fluent 14. The conservation equa-ions were integrated in aspect of space and time. The reactoras simulated for about 100 s to settle quasi steady state con-itions. The simulations were executed on High Performanceomputing (HPC) machines at Universiti Teknologi Malaysia.ach simulation required 199,759 iteration, which correspondso 250 h of CPU time.

. Simulation objective

he simulated gas phase fluidized bed reactor in this study ishe same as the industrial scale LLDPE polymerization reac-or in Amirkabir Petrochemical Company (2012). The width ofhe square domain considered in the simulation is equal to theiameter of the actual reactor (D = 5 m). The reactors consist ofhree zones: fluidization zone, disengagement zone and bulbection. The disengagement zone has a larger diameter (oneo two times of diameter) than the fluidization zone (D). As aonsequence, a proper gas velocity is achievable which mini-izes the entrainment and elutriation of catalyst and polymer

articles. The height of the reactor is 33.9 m. The fluidiza-ion medium includes a gas mixture of monomer (ethylene),o-monomer, hydrogen, and inert gases. Fig. 1 illustrates theketch of the reactor.

. Initial and boundary conditions

he top of the bed was set as constant pressure outlet, andhe uniform inlet velocity was designed as inlet boundaryondition. The industrial reactor has cylindrical geometryith the operational superficial gas velocity range of between

.5 m/s and 0.75 m/s (Amirkabir Petrochemical Company,012). Therefore, from a numerical point of view, to matchhe pressure drop and the bed height with available indus-rial data, the superficial gas velocity range (0.5, 0.55, 0.65 and.75 m/s) was analyzed in rectangular geometry (2D) owingo the high computational cost. Finally, the inlet gas veloc-ty has been considered to be 0.5 m/s, whereas the bed heightas over-predicted by 0.55, 0.65 and 0.75 m/s. The effects of

ront and back walls were neglected. No-slip and free-slip walloundary conditions were employed for gas and solid phases,

espectively. Dirichlet boundary condition was employed athe bottom of the bed to create a uniform gas inlet velocity in

the absence of solid phases. The bed was considered to be inthe initial well-mixed condition and all velocities were set tozero at t = 0. The static bed height is 10 m and the value of voidfraction is 0.51. The outlet pressure boundary condition wasset to 24 bar.

In this study, one gas phase and three quadrature points(particle phases) were assumed. The gas phase was consid-ered as the primary phase, whereas the particle phases werecharacterized by a specific length, volume fraction, densityand other properties (i.e., particle shape factor). In DQMOM(x, t0), the initial conditions of weights, ωi, and abscissas, Li,are listed in Table 4. On all boundaries the quadrature weightsand weighted nodes fluxes have been set to zero. The particledensity is 850 kg/m3. The inlet gas densities and viscosity is20 kg/m3 and 12 × 10−5 Pa s, respectively. These data representthe real conditions of the industrial scale gas-phase polymer-ization reactor. Moreover, since the smaller particles fill theinterstitial spaces among the larger particles, the maximumpacking fraction was considered to be 0.74. The restitutioncoefficient is 0.8. Also, it was assumed that an efficient removalof the reaction heat had been achieved and isothermal condi-tion could be maintained throughout the bed (Ahmadzadehet al., 2008). The simulation and wall boundary conditions areshown in Tables 2 and 3.

58 chemical engineering research and design 1 0 4 ( 2 0 1 5 ) 53–67

Table 2 – Operate and boundary condition.

Parameter Value

Inlet boundary condition Velocity inletOutlet boundary condition Pressure outletWall boundary condition No slip for gas,

free-slip for solidphase

Superficial gas velocity (m/s) 0.5Initial bed height (m) 10Initial void fraction 0.49Operating pressure (bar) 24Gas density (kg/m3) 20Gas viscosity (Pa s) 1.2 × 10−5

Particle density (kg/m3) 850coefficient of restitution 0.8angle of internal fraction 30Maximum solid packing volume fraction 0.74Time step (s) 0.01Activation energy, E (J mol−1) 5.04 × 104

Active site of catalyst (mol m−3) 1.88 × 10−4

Feed monomer concentration (mol m−3) 0.45Pre-exponential factor, kp0 (m3 mol−1 s−1) 1.2 × 104

Table 3 – Initial conditions of weights and abscissas inEqs. (4a) and (4b).

Parameter Value

Number of nodes 3Weights

ω1 0.34ω2 0.324ω3 0.32

Abscissas (m)L1 1.57 × 10−3

L2 1.142 × 10−3

L3 1.356 × 10−3

5. Sensitivity analysis of grid, time stepand model validation

In order to validate the model, grid and time step sensitivityanalysis was conducted by comparing the simulation resultswith the industrial data gathered from an industrial LLDPEgas-phase polymerization reactor (Akbari et al., 2014). Thesimulation condition is displayed in Tables 2 and 3. Averageproduct size depends on the PSD which is affected by the oper-ating condition, catalyst properties, particle residence time,

catalyst injection, product withdrawal position and productseparation devices. Therefore, the distribution of the particle

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 10 20 30 40 5

pres

sure

dro

p (b

ar)

Tim

Nodes= 52007 Nodes= 2

min press ure drop max pres

Fig. 2 – Comparison the effect o

sizes inside the reactor was considered to be similar to theproduct particle size distribution for validation purpose.

Pressure drop and bed height are important fluidizationcharacteristics which indicate the gas turbulence, bubblehydrodynamic, and operating conditions of the bed. Moreover,in order to neglect the distributor effect, two representativepressure points were considered at 0.5 m above the distribu-tor. Figs. 2–5 show the transient behavior of pressure drop andbed height together with their comparison with the industrialdata. As can be seen, the whole simulation process can beidentified as two stages: a start-up stage and a quasi-steadyfluidization stage. The steady state bed height in the gas–solidflow is achievable after 73 s, while the pressure drop usuallyoscillates inside its operational range due to the fluidizationcharacteristics.

5.1. Grid independent analysis

By using the technique of boundary and gradient adaption,a 2D analysis was carried out to verify that the results wereindependent of grid as the resolution increases. By using thismesh refining technique, most of the added mesh points couldbe located in the regions of high gradient in the fluidizationand the inlet regions.

Figs. 2 and 3 display the transient pressure drop and bedheight at three mesh resolutions, respectively. The simulationwas conducted for 100 s real time for the first 10 m of the bedheight considering the value of 0.5 m/s for the superficial gasvelocity. The three different exploited grids for discretizing the2D flow domain into square cells are reported in Table 4.

Fig. 2 shows that the pressure drop is not really sensitivetoward the grid resolution. It can be concluded that the aver-age pressure drop varied only by 0.55% when the number ofnodes changed from 21,464 to 52,007. However, when the gridresolution increased from 52,007 to 102,343, the pressure dropimproved around 3.5%. It is evident that the pressure drop isnot sensitive to grid.

In addition, the results shown in Fig. 3 illustrate that thehigher number of nodes predict lower bed height. Accord-ingly, similar fluctuations in the bed height are observed for allthree cases, at the steady state fluidization condition, bubblesare generated continuously during the gas flow and particlesmove vigorously inside the bed. This regime is characterizedby an almost constant bed height and mean pressure drop.When 21,464 nodes are used, the bed expansion is overes-

timated by 7.2%. This number of grids is too coarse so thatit results in the continuous fluctuation of the bed height.

0 60 70 80 90 100

e (s)

1464 Nodes= 102343

sure drop

f nodes on pressure drop.

chemical engineering research and design 1 0 4 ( 2 0 1 5 ) 53–67 59

10

12

14

16

18

20

22

0 10 20 30 40 50 60 70 80 90 100

Bed

Heig

ht (m

)

Time (s)

Nodes= 52007 Nodes=21464 Nodes=102343 actual bed height

Fig. 3 – Comparison the effect of number of nodes on bed height.

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0 10 20 30 40 50 60 70 80 90 100

Pres

sure

Dro

p (B

ar)

Time (s)

�me step=0.01 s �me ste p=0.001 s �me step =0.1 s

Maximum Press ure Dro p Minim um Pressure Dr op

Fig. 4 – Effect of time step on pressure drop (52,007 nodes).

10

12

14

16

18

20

0 20 40 60 80 100

Bed

heig

ht (m

)

�me (s)

�me step=0.01 s �m e st ep=0.0 01 s actual bed height �me step=0.1 s

Fig. 5 – Effect of time step on the bed height (52,007 nodes).

Table 4 – Grid and time step analysis.

Time step(s)

Number ofnodes

Averagegrid size (m)

Courantnumber

Pressuredrop (bar)

Bed height(m)

% pressuredrop error

% bed heighterror

Case 1 0.01 21,464 0.12 0.041 0.599 21.55 1.95 7.2Case 2 0.01 52,007 0.05 0.1 0.596 20.15 1.25 0.75Case 3 0.01 102,343 0.025 0.2 0.616 19.5 4.89 2.5Case 4 0.1 52,007 0.05 1 0.566 20.6 3.75 3Case 5 0.001 52,007 0.05 0.01 0.591 19.8 0.56 1

60 chemical engineering research and design 1 0 4 ( 2 0 1 5 ) 53–67

Table 5 – Comparison between industrial data and CFD simulated results.

Parameter Simulated result Industrial data % bed height error % pressure drop error

Final bed height 20.15 20 0.75 –Average pressure drop 0.578 0.588 – 1.24

However by using 52,007 and 102,343 nodes, the bed expan-sion can be predicted accurately. The numerical simulationsfor 52,007 and 102,343 nodes result in the bed height of 20.15 mand 19.5 m, respectively, whereas the industrial value for thefinal bed height is about 20 m. Both grids provide the stablepredictions (±5% variation) once the bed is fully expanded. Thesmall difference in the bed height and pressure drop between52,007 and 102,343 nodes indicates acceptable grid conver-gence. Therefore, it is necessary to find a tradeoff betweenthe required accuracy and the computational time. However,102,343 nodes predict the acceptable results, but the maindrawback is the increase in computational time. Therefore,it can be justified performing the following simulations basedon the 52,007 nodes.

5.2. Time step analysis

Since 52,007 nodes gave a satisfactorily mesh-independentresult, this number of nodes was used for time step analy-sis. Usually, a time step of 0.001 s has been used for gas–solidfluidized beds (Chen et al., 2011; Yan et al., 2012). Gobinet al. (2003) used a time step of 0.01 s. In the current study,three fixed time step sizes were investigated (see Table 4).Figs. 4 and 5 demonstrate the pressure drop and bed height,respectively, as a function of time for different time steps.Additionally, 52,007 nodes, employing time steps of 0.01 and0.001 s, gave very similar steady state results 20.15 m and19.8 m, which are close to the industrial values. A time stepof 0.1 s results in different transient behaviors leading to theprediction of a higher bed height. Moreover, the bed heightfor the time step of 0.001 s was less stable. As clearly shownin Fig. 5, the finer time step (0.001 s) predicts less bed height.From the numerical perspective, the unsteady-state problemsare solved in each time step in a realm iteration way to reach aspecific convergence. In this study, the value 10−3 of absoluteconvergence criteria was used. If the time step is too small,

the result predicts large overall error due to the small changebetween two time steps compared with fixed convergence

10

12

14

16

18

20

22

0 10 20 30 40 5

Bed

Heig

ht (m

)

Time

Nodes= 52007

Fig. 6 – Comparison of the CFD predicted

criteria. On the other hand, too large time steps predict a sharpchange of cell properties and lead to less accurate results.

In addition, the dimensionless Courant number, Nc, givesan idea of the prediction of cell property, which depends ontime step, grid size in direction of flow and the gas velocity. Itis defined as:

Nc = U�t

�x(7)

The same grid size at different time steps and the sametime step for different grid sizes were simulated at the samegas velocity (0.5 m/s). Table 4 shows that the Courant numberless or greater than 0.1 leads to the decrease of the accuracy ofthe predictions. This observation was confirmed by (Coroneoet al., 2011) who proposed the Courant number between 0.03and 0.3. On the one hand, further decreasing the time stepto 0.0001 s did not show realistic physical behavior due to theprediction of the less bed height than larger time step. There-fore, these time steps were not illustrated in Fig. 5. It can beseen that the simulations show less realistic physical behaviorwhen using excessively large time steps. Since the proposedgrid (0.05 m) and time step (0.01 s) give the independent resultswith respect to pressure drop and bed height, these values canbe used as the base case setting. Furthermore, it can be con-cluded from Figs. 2–5 and Table 4 that in order to get moreaccurate results, the larger grid size and time step (30 dp and0.01 s) should be used in the large scale FBRs. In contrast, thesmaller grid size and time step (10 dp and .001 s) are requiredin the laboratory scale FBRs (Chen et al., 2011; Yan et al., 2012).

5.3. Model validation

The predicted pressure drop and the bed height profile alongthe time imply a very good agreement with the industrialdata. The error values for the prediction of pressure drop and

bed height were calculated 1.25% and 0.75%, respectively (seeFigs. 6 and 7 and Table 5).

0 60 70 80 90 100

(s)

actual bed height

bed height data with industrial data.

chemical engineering research and design 1 0 4 ( 2 0 1 5 ) 53–67 61

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 10 20 30 40 50 60 70 80 90 100

pres

sure

dro

p (b

ar)

Time (s)

Nodes= 52007 min pres sure drop max pressure drop

ed p

pdCsi

6

Wrig

6

TpoiTpb

atBus

Fp

Fig. 7 – Comparison of the CFD predict

However, the pressure drop and bed height are importantarameters to monitor the fluidization structure. The vali-ation of the model is also conducted by comparing both theFD simulation and industrial data of the bed height versusuperficial gas velocity. It can be seen that the bed height isncreased by increasing the superficial gas velocity (see Fig. 8).

. Results and discussion

hen the CFD-PBM coupled model provides the satisfactoryesults compared with the operational data, it can be used tomplement a parametric and optimization study to developas-phase FBRs design and better operational conditions.

.1. Minimum fluidization velocity

he minimum fluidization velocity is a crucial hydrodynamicarameter in the design of a FBR. It determines the transitionf fixed bed into a fluidized bed and the quantification of the

ntensity of the fluidization regime at higher velocity levels.he definition of the minimum fluidization velocity of polydis-erse systems is not straightforward and has to be discussedased on the specific features of each system.

Fig. 9 shows the simulation results of pressure drop fluctu-tions for various superficial gas velocities to define Umf inhe reactor, using the average product size (1300–1600 �m).y increasing the gas velocity, the bed pressure drop steps

p along the bed, where all small and large particles aretill stagnant. Subsequently the pressure drop deviates from

6

8

10

12

14

16

18

20

0.50 1.51 2.52 3.53 4.54

Bed

hei

ght (

m)

Superficial gas velocity (m/s)

Industrial data CFD predictions

ig. 8 – Comparison of the steady state bed height CFDrediction with industrial data.

ressure drop with the industrial data.

the curve which this deviation represents the performanceof the fixed bed reactor. This is due to the segregation ofsmall particles and partial de-fluidization of solids at thebottom of the bed. This period reaches a peak at Umf,j (seeFig. 9). By further increasing the gas velocity, the pressuredrop decreases slightly owing to the small particles whichtend to fill the coarse inter-particle voids, although the coarseparticles remain unchanged. At the point shown by Uff, thebed is steadily fluidized and effective bed mixing overshad-ows the de-fluidization at the bottom of the bed. Also, at Uff,the total pressure drop is equal to the weight of the wholebed. In other words, �p − u diagram demonstrates a transitiondomain between the point where the first particles begin tofluidize at Umf,f to the point where all the particles are fluidizedat Uff. Furthermore, the following sequence of PSD fluidizationstructure and transition is observed with increasing the gasvelocity:

Regime I: (U < Umf.f) The two initially well-mixed phasesshould not be separate and remain in the fixed bed conditiondue to the absence of bubbles. The pressure drop increaseslinearly until it reaches the minimum fluidization velocity ofsmall particles at Umf,f

Regime II: (Umf,f < U < Uff) Because of varying drag force onthe particle with different characteristic properties, transientfluidization structure started with the enhancement of thevelocity. In addition, due to the generation of the small bub-bles, segregation of small particles at the bottom of the bedstarted up to the bed surface. By further increasing the veloc-ity, the pressure gradient tends to be a uniform value throughthe bed, and bubble induced mixing progressing overtakessegregation until the bed becomes uniform.

Regime III: (U > Uff) Vigorous bubbling is established and thebed is in a well-mixed condition.

Fig. 9 – Variation of simulated bed pressure drop withsuperficial gas velocity.

62 chemical engineering research and design 1 0 4 ( 2 0 1 5 ) 53–67

Fig. 10 – Particle diameter counter with their solid volumefraction in a FBR, superficial gas velocity of 0.10 m/s.

This fluidization behavior is also in good agreement withthe proposed fluidization structure of Gera et al. (2004), Olivieriet al. (2004), and Tagliaferri et al. (2013). In a mono-dispersedsystem, the Umf condition of FBRs is defined at point A (seeFig. 9) where the rising pressure drop line describing the stateof fixed bed meets the constant pressure drop line, whichdepicts the status of fluidization state along with the varia-tion of the gas velocity. However, in a polydispersed system,it can be assumed that both small and large particles are notfully fluidized at point A. This is an intermediate state betweenthe minimum fluidization velocity of small particles and theminimum velocity at which the bed is completely fluidized.Accordingly, Umf,f is the maximum velocity for the fixed bedregime, while Uff implies the minimum velocity for the fullyfluidized condition. As regards the reactor under study, thebed is fluidized at Umf = 0.14m/s, which is 1.3 of the minimumfluidization velocity of a mono-dispersed bed with the sameaverage particle diameter (point A).

Here, the computed values of Umf from Eq. (8) (Chen et al.,2011) are used to verify the CFD prediction. According to Eq.(8), Umf for small and large particles are calculated 0.1 and 0.15(m/s), respectively. The comparison of these data proves thatthe CFD predictions are in good agreement with the resultsof Eq. (8). The difference between the values predicted by twomodels in the present study is lower than 6.6%. Table 2 sum-marizes the simulation condition to define Umf.

Umf = �

dp�g

⎧⎨⎩

[33.72 + 0.408

d3p�g (�p − �g) g

�2

]1⁄2− 33.7

⎫⎬⎭ (8)

Therefore, in this study, the subsequent results areobtained with the superficial gas velocity of 0.5 m/s (3.57 Umf),except for the case where the effect of particle growth rate hasbeen investigated.

6.2. Particle growth rate

Figs. 10 and 12 monitor size dependent particle growth rateon the fluidization structure, which can be used significantlyfor operational problems (i.e., de-fluidization, hot spot forma-tion, etc.), especially during the start-up process. In the currentstudy, the polymerization reaction rate (Yao et al., 2015) can beexpressed as follows:

G(Li) = d (Li)dt

= RpL30

3�sL2i

(9)

Rp = Kp0 exp(

− E

R (273.15 + t)

)[M][C∗]

The solid particle was placed in the initial 10 m of the bedheight with the initial volume fraction of 0.5. The simulationprocess started with the uniform particles of 70 �m as the ini-tial catalyst particle growth up to the final polymerization size(0.0012 m), in 100 s of the simulation time. This assumptionenables acquiring the fundamental knowledge of the flow pat-tern in any kind of gas fluidization system where the particlesize is changing. The effect of size dependent particle growthrate on the PSD and, consequently, on the flow pattern canbe predicted using the value of drag force, which is calculated

based on the average particle size in each time step. Moreover,it has been assumed that the growth of size-distribution of the

supported catalyst particle (L0) continues until the particlesreach the maximum allowable size of the product (L).

Fig. 10 represents the variation of particle diameter alongthe time by using the contour-plot of solid volume fraction.The bed started to fluidize with a gas velocity 0.1 m/s, whichis higher than the minimum fluidization velocity of initial cat-alyst particles in FBRs (Fig. 10a). In the initial phase of thegrowth process, the bed particles are not large enough to beseparated by the classifier. Later, the particles including thealready grown particles started to be separated. Nevertheless,

in this situation, the particles in the middle or on the top of the

chemical engineering research and design 1 0 4 ( 2 0 1 5 ) 53–67 63

Fig. 11 – Particle diameter with their solid volume fractionin a gas phase FBR: (a) superficial gas velocity of 0.25 m/s,(b) superficial gas velocity of 0.35 m/s, (c) superficial gasvelocity of 0.45 m/s.

flqdl

Fwtosotnmdtltb5t

0.5

0.55

0.6

0.65

0.7

0.75

0.8

012345 Gas

vol

ume

fra

Letral distance (m )

t= 90 sec (0.35 m/s)

y=10 m y=15 m y=5 m

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

012345

Gas

vol

ume

frn

Leteral distance (m)

t= 100 sec (0.45 m/s )

y=15m y=10 m y=5m

0

0.2

0.4

0.6

0.8

1

1.2

012345

Gas

vol

ume

frn

Leteral distance (m )

t= 30 sec (0.1 m/s)

y= 10m y= 15 m y= 5 m

0.5

0.55

0.6

0.65

0.7

0.75

012345

Gas v

olum

e fr

a

Leteral distance (m)

t= 75 sec (0.25 m/s)

y= 10 m y= 15 m y=5 m

Fig. 12 – Radial variation of gas volume fraction at differentheights and superficial gas velocities.

the drag force increased and the bed becomes fluidized which

uidized bed are still in the fluidized state (Fig. 10b–d). Conse-uently, particle segregation should take place due to higherrag force on small particles than on large particles. Therefore,

arge solids tend to accumulate at the bottom of the bed.The small particles move faster than the big particles in the

BRs on account of their smaller inertia and higher drag force,hile the large particles tend to appear at the bottom and near

he side walls of the bed (see Fig. 10b and c). After elapsing 40 sf fluidization, due to the particle growth, already establishedize-distribution, and various drag forces particle segregationccurs (Fig. 10d). Moreover, smaller particles tend to reside onhe upper part of the bed, while the bigger particles tend to stayear the distributor. By forming the large particles, the move-ent of the larger particles becomes more difficult and the gas

oes not exert enough drag force on the formed larger particleso balance their weight and keep them fluidized. In addition,arger particles hinder the movement of smaller particles inhe bed, and, as a result, the big portion of the gas entering theed passes through the channels formed by large particles. At0 s, a significant portion of the particles is accumulated near

he bottom of the bed and forms a de-fluidized bed zone above

the distributor. Therefore, the established height and weightin the region of bubbling shifts to the top of the de-fluidizedlayer (Fig. 10d).

Fig. 11 presents the continuation of particle growth to mon-itor particle diameter with their solid volume fraction. Sincethe particle size increased, in order to reach the fluidizationstate, the gas inlet velocity increased to 0.25 m/s after 50 s(Fig. 11a). At this point, the whole mass of particles is sup-ported by the following gas and therefore bubbling take places,

promotes the growth of bubble flow. Accordingly, mixing of

64 chemical engineering research and design 1 0 4 ( 2 0 1 5 ) 53–67

0

5

10

15

20

25

30

0.20.40.60.811.2

Bed

heig

ht (m

)

Gas volume f rac�on

t= 30 sec (0 .10 m/s )

t= 50 sec (0 .25 m/s )

t= 75 sec (0.35 m/s )

t=100 sec (0.45 m/s)

Fig. 13 – Gas volume fraction variation along the centralaxis at different superficial gas velocities.

00.511.522.533.544.55

012345

Axia

l vel

ocity

of s

olid

pha

se (m

/s)

Leteral dista nce (m )

t= 100 sec (0.45 m/s)

y=10 m y=15 m y=5 m

00.511.522.533.544.5

012345 Axia

l vel

ocity

of s

olid

(m/s

)

letral distance (m)

t= 75 sec (0.25 m/s)

y=10 m y=15 m y=5 m

012345678

012345

Axia

l ve

/s)

Lateral distance (m)

t= 90 sec (0.35 m/s)

y=10 m y= 15 m y=5 m

012345678

012345

Axia

l vel

ocity

of s

olid

Leteral distance (m)

t= 30 sec (0.1 m/s)

y=10 m y=15m y=5m

Fig. 14 – Radial variation of axial particle velocity atdifferent bed heights and superficial gas velocities.

the particle enhanced and the bed height increased. In orderto keep the fluidization state, the inlet gas velocity increasesthree times higher than the minimum fluidization velocity oflarge particles (Fig. 11b and c). After 90 s, the particles diame-ter became as big as 0.0012 m (see Fig. 11b) with the inlet gasvelocity of 0.35 m/s. In general, the final particle size distri-bution confirms the observed PSD in product (Fig. 11c). Theparticles grow with time along the bed, while larger particlestend to move toward the distributor and wall due to less dragforce. In addition, the PSD in the bed become more homoge-neous by increasing the gas velocity. An important point tomake is that to keep the well-mixed condition and preventthe segregation, de-fluidization, hot spot formation, and elu-triation of small particles the gas velocity should be increasedstep by step.

Fig. 12 illustrates the comparison of time averaged gas vol-ume fraction profiles in the radial direction at different bedheights of 5, 10 and 15 m. The predicted gas volume fraction isflat through the radial position at low superficial gas velocity,while the cure-annular structure is predicted, carried up thesolid particles and annular region where particles fall downalong the wall (see Figs. 12 and 15). In addition, numericalresults show increase of gas volume fraction at the bottom ofthe bed and decrease w through the bed height at low superfi-cial gas velocity. Increasing the gas velocity leads to fluidizingthe coarse particles, while the fine particles are already fullybalanced by gas drag. This is followed by the mixing of par-ticles with the exception of two regions located above of thedistributor and at the top of the bed, where coarse and smallparticles are found (Fig. 13).

The axial profile velocity of solid particles is depicted inFig. 14. It shows that the particle vertical velocity at the bottomof the bed is higher, though slightly, than at the top of the bed.In addition, the vector plot of axial particle velocity is shownin Fig. 15. There is growth of small particles at the top of thebed, accelerated by gravity when falling particles would collidewith the rising particles that change the motion direction ofthe particles and produce the wiggle phenomenon throughthe bed.

Accordingly, the successful predicted results can be usedto monitor the flow structure in the gas-phase polymerizationreactor:

– To reduce the possibility of polymer particle sintering andpoor fluidization that leads to poor heat/mass transfer rate,

poor heat removal, and hot spot formation (Ahmadzadeh

et al., 2008; Hutchinson et al., 1992), hence, to increase thereactor efficiency.

– To achieve good control of the PSD under a broad range ofoperation conditions to ensure well-mixed operation con-dition.

chemical engineering research and design 1 0 4 ( 2 0 1 5 ) 53–67 65

Fig. 15 – Solid velocity vector at different superficial gas velocities.

7

IggpgppmMguwtpopt

mation and Communication Technology (CICT) in Universiti

. Conclusion

n this study, a CFD-PBM coupled model was applied in aeneric framework for the detailed modeling of polydisperseas–solid FBRs including fully coupling between CFD multi-hase simulation, PBM/DQMOM and size dependent particlerowth rate. The application of the model was highlighted asredicting the behavior of industrial scale gas-phase LLDPEolymerization reactor. The accuracy and capability of theodel was validated using the industrial data obtained fromahshahr special economic zone, Iran, which have the same

eometry in terms of bed height and pressure drop. The sim-lation results showed that for industrial scale FBRs, 0.05 mith 0.01 s are sufficient values to capture the mesh and

ime step independent results, respectively. The CFD-PBM cou-led model was implemented to illustrate the impact of PSDn minimum fluidization velocity as well as size dependent

article growth rate. In the case of Umf, the �p − u showedhree regions of fluidization structure different from the

fluidization structure with monodisperse PSD. The simulatedresults showed that the PSD varies due to particle growth andlarger particle exit at the bottom of the bed while smaller par-ticles tend to migrate to the top of the bed. The CFD-PBMcoupled model applied in a generic simulation framework isa suitable tool to provide a realistic estimation of the effect ofoperating conditions used for enhancing the performance ofgas-phase polymerization reactors.

Acknowledgments

We would like to thank the Research Council of the Uni-versity of Malaya under High Impact Research Grant(UM.C/625/1/HIR/MOHE/ENG/25), Postgraduate ResearchGrant (PPP) project No. PG131-2014A and Center for Infor-

Teknologi Malaysia for supporting and providing facilitiesand services of high performance computing.

66 chemical engineering research and design 1 0 4 ( 2 0 1 5 ) 53–67

References

Ahmadzadeh, A., Arastoopour, H., Teymour, F., Strumendo, M.,2008. Population balance equations’ application in rotatingfluidized bed polymerization reactor. Chem. Eng. Res. Design86, 329–343.

Ahuja, G., Patwardhan, A., 2008. CFD and experimental studies ofsolids hold-up distribution and circulation patterns ingas–solid fluidized beds. Chem. Eng. J. 143, 147–160.

Akbari, V., Borhani, T.N.G., Godini, H.R., Hamid, M.K.A., 2014.Model-based analysis of the impact of the distributor on thehydrodynamic performance of industrial polydisperse gasphase fluidized bed polymerization reactors. Powder Technol.267, 398–411.

Akbari, V., Nejad Ghaffar Borhani, T., Shamiri, A., KamaruddinAbd. Hamid, M., 2015. A CFD-PBM coupled model ofhydrodynamics and mixing/segregation in an industrialgas-phase polymerization reactor. Chem. Eng. Res. Design 96,103–120.

Amirkabir Petrochemical Company, M., Khoozestan, Iran, 2012.LLDPE polymerization reactor data. WWW.npc-akpc.com

Ashrafi, O., Mostoufi, N., Sotudeh-Gharebagh, R., 2012. Two phasesteady-state particle size distribution in a gas-phase fluidizedbed ethylene polymerization reactor. Chem. Eng. Sci. 73, 1–7.

Attarakih, Drumm, C., Bart, H.-J., 2009. Solution of the populationbalance equation using the sectional quadrature method ofmoments (SQMOM). Chem. Eng. Sci. 64, 742–752.

Behjat, Y., Shahhosseini, S., Marvast, M.A., 2011. CFD analysis ofhydrodynamic, heat transfer and reaction of three phase riserreactor. Chem. Eng. Res. Design 89, 978–989.

Buffo, A., Vanni, M., Marchisio, D., Fox, R.O., 2013. Multivariatequadrature-based moments methods for turbulentpolydisperse gas–liquid systems. Int. J. Multiphase Flow 50,41–57.

Chen, X.-Z., Shi, D.-P., Gao, X., Luo, Z.-H., 2011. A fundamentalCFD study of the gas–solid flow field in fluidized bedpolymerization reactors. Powder Technol. 205, 276–288.

Claudotte, L., Rimbert, N., Gardin, P., Simonnet, M., Lehmann, J.,Oesterlé, B., 2010. A multi-QMOM framework to describemulti-component agglomerates in liquid steel. AIChE J. 56,2347–2355.

Cooper, S., Coronella, C.J., 2005. CFD simulations of particlemixing in a binary fluidized bed. Powder Technol. 151, 27–36.

Coroneo, M., Mazzei, L., Lettieri, P., Paglianti, A., Montante, G.,2011. CFD prediction of segregating fluidized bidispersemixtures of particles differing in size and density in gas–solidfluidized beds. Chem. Eng. Sci. 66, 2317–2327.

de Souza Braun, M.P., Mineto, A.T., Navarro, H.A.,Cabezas-Gomez, L., Da Silva, R.C., 2010. The effect ofnumerical diffusion and the influence of computational gridover gas–solid two-phase flow in a bubbling fluidized bed.Math. Comp. Model. 52, 1390–1402.

Diemer, R.B., Olson, J.H., 2002. A moment methodology forcoagulation and breakage problems: Part 3—Generalizeddaughter distribution functions. Chem. Eng. Sci. 57,4187–4198.

Drumm, C., Attarakih, M., Tiwari, S., Kuhnert, J., Bart, H.,Johansen, S., Olsen, J., Ashrafian, A., 2008. Implementation ofthe sectional quadrature method of moments in a CFD code.In: Proceedings of the 6th International Conference on CFD inOil & Gas, Metallurgical and Process Industries, Trondheim,Norway, June 10–12. SINTEF Materials and Chemistry,Trondheim, Norway.

Dutta, A., Constales, D., Heynderickx, G.J., 2012. Applying thedirect quadrature method of moments to improve multiphaseFCC riser reactor simulation. Chem. Eng. Sci. 83, 93–109.

Esmaili, E., Mahinpey, N., 2011. Adjustment of drag coefficientcorrelations in three dimensional CFD simulation of gas–solidbubbling fluidized bed. Adv. Eng. Softw. 42, 375–386.

Fan, R., Marchisio, D.L., Fox, R.O., 2004. Application of the direct

quadrature method of moments to polydisperse gas–solidfluidized beds. Powder Technol. 139, 7–20.

Fan, R., 2006. Computational Fluid Dynamics Simulation ofFluidized Bed Polymerization Reactors. Iowa State University,Ames, IA (PhD thesis).

Gera, D., Syamlal, M., O’Brien, T.J., 2004. Hydrodynamics ofparticle segregation in fluidized beds. Int. J. Multiphase Flow30, 419–428.

Gobin, A., Neau, H., Simonin, O., Llinas, J.R., Reiling, V., Sélo, J.L.c.,2003. Fluid dynamic numerical simulation of a gas phasepolymerization reactor. Int. J. Numer. Methods Fluids 43,1199–1220.

Hutchinson, R., Chen, C., Ray, W., 1992. Polymerization of olefinsthrough heterogeneous catalysis X: modeling of particlegrowth and morphology. J. Appl. Polym. Sci. 44, 1389–1414.

Hutton, K., Mitchell, N., Frawley, P.J., 2012. Particle sizedistribution reconstruction: the moment surface method.Powder Technol. 222, 8–14.

Kumar, S., Ramkrishna, D., 1996. On the solution of populationbalance equations by discretization—I. A fixed pivottechnique. Chem. Eng. Sci. 51, 1311–1332.

Marchisio, Barresi, A.A., Garbero, M., 2002. Nucleation, growth,and agglomeration in barium sulfate turbulent precipitation.AIChE J. 48, 2039–2050.

Marchisio, D.L., Vigil, R.D., Fox, R.O., 2003a. Quadrature method ofmoments for aggregation-breakage processes. J. ColloidInterface Sci. 258, 322–334.

Marchisio, D.L., Fox, R.O., 2005. Solution of population balanceequations using the direct quadrature method of moments. J.Aerosol Sci. 36, 43–73.

Marchisio, D.L., Pikturna, J.T., Fox, R.O., Vigil, R.D., Barresi, A.A.,2003b. Quadrature method of moments forpopulation-balance equations. AIChE J. 49, 1266–1276.

Marshall Jr., C.L., Rajniak, P., Matsoukas, T., 2011. Numericalsimulations of two-component granulation: comparison ofthree methods. Chem. Eng. Res. Design 89, 545–552.

Mazzei, L., 2013. Segregation dynamics of dense polydispersefluidized suspensions modeled using a novel formulation ofthe direct quadrature method of moments. Chem. Eng. Sci.101, 565–576.

Mazzei, L., Marchisio, D.L., Lettieri, P., 2009. Direct quadraturemethod of moments for the mixing of inert polydispersefluidized powders and the role of numerical diffusion. Ind.Eng. Chem. Res. 49, 5141–5152.

Mazzei, L., Marchisio, D.L., Lettieri, P., 2012. Newquadrature-based moment method for the mixing of inertpolydisperse fluidized powders in commercial CFD codes.AIChE J. 58, 3054–3069.

Olivieri, G., Marzocchella, A., Salatino, P., 2004. Segregation offluidized binary mixtures of granular solids. AIChE J. 50,3095–3106.

Passalacqua, A., Fox, R.O., Garg, R., Subramaniam, S., 2010. A fullycoupled quadrature-based moment method for dilute tomoderately dilute fluid–particle flows. Chem. Eng. Sci. 65,2267–2283.

Patankar, S.V., 1980. Numerical Heat Transfer and Fluid Flow.Taylor & Francis.

Ramkrishna, D., 2000. Population Balances: Theory andApplications to Particulate Systems in Engineering. AcademicPress.

Reddy, R.K., Joshi, J.B., 2010. CFD modeling of pressure drop anddrag coefficient in fixed beds: wall effects. Particuology 8,37–43.

Rigopoulos, S., 2010. Population balance modelling ofpolydispersed particles in reactive flows. Prog. EnergyCombust. Sci. 36, 412–443.

Selma, B., Bannari, R., Proulx, P., 2010. Simulation of bubbly flows:comparison between direct quadrature method of moments(DQMOM) and method of classes (CM). Chem. Eng. Sci. 65,1925–1941.

Shamiri, A., Azlan Hussain, M., Sabri Mjalli, F., Mostoufi, N.,Hajimolana, S., 2013. Dynamics and predictive control of gas

phase propylene polymerization in fluidized bed reactors.Chin. J. Chem. Eng. 21, 1015–1029.

chemical engineering research and design 1 0 4 ( 2 0 1 5 ) 53–67 67

S

S

S

S

S

T

V

hamiri, A., Hussain, M.A., Mjalli, F.S., Mostoufi, N., 2010. Kineticmodeling of propylene homopolymerization in a gas-phasefluidized-bed reactor. Chem. Eng. J. 161, 240–249.

hamiri, A., Hussain, M.A., Mjalli, F.S., Mostoufi, N., 2012.Improved single phase modeling of propylene polymerizationin a fluidized bed reactor. Comp. Chem. Eng. 36, 35–47.

ilva, L., Rodrigues, R., Mitre, J., Lage, P., 2010. Comparison of theaccuracy and performance of quadrature-based methods forpopulation balance problems with simultaneous breakageand aggregation. Comp. Chem. Eng. 34, 286–297.

un, J., Zhou, Y., Ren, C., Wang, J., Yang, Y., 2011. CFD simulationand experiments of dynamic parameters in gas–solidfluidized bed. Chem. Eng. Sci. 66, 4972–4982.

yamlal, M., Rogers, W., O’Brien, T.J., 1993. MFIX Documentation:Theory Guide. Technical Note, DOE/METC-94/1004,NTIS/DE94000087. National Technical Information Service,Springfield, VA.

agliaferri, C., Mazzei, L., Lettieri, P., Marzocchella, A., Olivieri, G.,Salatino, P., 2013. CFD simulation of bubbling fluidizedbidisperse mixtures: effect of integration methods andrestitution coefficient. Chem. Eng. Sci. 102, 324–334.

asquez, S., Ivanov, V., 2000. A phase coupled method for solvingmultiphase problems on unstructured meshes. In:

Proceedings of ASME FEDSM’00: ASME 2000 FluidsEngineering Division Summer Meeting, Boston.

Wei, L.-H., Yan, W.-C., Luo, Z.-H., 2011. A preliminary CFD studyof the gas–solid flow fields in multizone circulatingpolymerization reactors. Powder Technol. 214,143–154.

Weiss, J.M., Maruszewski, J.P., Smith, W.A., 1999. Implicit solutionof preconditioned Navier–Stokes equations using algebraicmultigrid. AIAA J. 37, 29–36.

Yan, W.-C., Luo, Z.-H., Lu, Y.-H., Chen, X.-D., 2012. ACFD-PBM-PMLM integrated model for the gas–solid flow fieldsin fluidized bed polymerization reactors. AIChE J. 58,1717–1732.

Yao, Y., He, Y.-J., Luo, Z.-H., Shi, L., 2014. 3D CFD-PBM modeling ofthe gas–solid flow field in a polydisperse polymerization FBR:the effect of drag model. Adv. Powder Technol. 25,1474–1482.

Yao, Y., Su, J.-W., Luo, Z.-H., 2015. CFD-PBM modelingpolydisperse polymerization FBRs with simultaneous particlegrowth and aggregation: the effect of the method ofmoments. Powder Technol. 272, 142–152.

Zimmermann, S., Taghipour, F., 2005. CFD modeling of thehydrodynamics and reaction kinetics of FCC fluidized-bedreactors. Ind. Eng. Chem. Res. 44, 9818–9827.

Zucca, A., Marchisio, D.L., Vanni, M., Barresi, A.A., 2007.

Validation of bivariate DQMOM for nanoparticle processessimulation. AIChE J. 53, 918–931.