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Transcript
This article appeared in a journal published by Elsevier. The attachedcopy is furnished to the author for internal non-commercial researchand education use, including for instruction at the authors institution
and sharing with colleagues.
Other uses, including reproduction and distribution, or selling orlicensing copies, or posting to personal, institutional or third party
websites are prohibited.
In most cases authors are permitted to post their version of thearticle (e.g. in Word or Tex form) to their personal website orinstitutional repository. Authors requiring further information
regarding Elsevier’s archiving and manuscript policies areencouraged to visit:
CFD simulation of hydrodynamics and heat transfer in gas phase ethylenepolymerization reactors☆
Mohammad A. Dehnavi a, Shahrokh Shahhosseini a,⁎, S. Hassan Hashemabadi a, S. Mehdi Ghafelebashi b
a Simulation and Control Research Laboratory, School of Chemical Engineering, Iran University of Science and Technology, P.O. Box 16765-163, Tehran, Iranb Petrochemical Research and Technology Co., Iran
The hydrodynamics and temperature of a two-dimensional gas–solid fluidized bed of gas phase olefinpolymerization reactor had been studied. A two-fluid Eularian Computational Fluid Dynamics (CFD) modelwith closure relationships according to the kinetic theory of granular flow has been applied in order tosimulate the gas–solid flow. Fluidization regime and gas–solid flow pattern were investigated using threedifferent drag models. Model predictions of bed pressure drop were compared with correspondingexperimental data reported in the literature to validate the model. The predicted values were in reasonableagreement with the experimental data. The temperature behavior of fluidized bed with various drag modelswas investigated. The temperature gradient in the primary section of the bed was much larger than thegradient in other sections and the effect of all drag models on temperature gradient along the bed wasapproximately similar.
Gas phase olefin polymerization fluidized bed reactors have beenlong recognized as one of the main technologies for polyolefinproduction. Compared to other reactors, fluidized beds have severaladvantages such as the capability of continuous operation and transportof solids in andoutof thebed; highheat andmass transfer rates fromgasto particles leading to fast reaction and uniform temperature in the bed[1,2]. In gas phase polymerization, small catalyst particles (e.g., 20–80 µm) are fed to the reactor, at a point above the gas distributor, wherethey are exposed to the gas flow containing the monomers andpolymerization occurs. At the early stage of polymerization, the catalystparticles fragment into a large number of small particles, which arequickly encapsulated by the newly-formed polymer and grow contin-uously, reaching a typical size of 100–5000 µm. Due to the differences inthe polymer particle sizes, segregation occurs and fully-grown polymerparticles migrate towards the bottom of the bed, where they areremoved from the reactor. The smaller pre-polymerized particles andfresh catalyst particles tend to migrate to the upper portions of thereactor and continue to react with the monomers [3–6].
The modeling of polyethylene production in fluidized bed reactorshas received considerable attention in recent years. Significant progresshas been made in the area of hydrodynamic modeling of gas-fluidizedsuspensions. Broadly speaking, two different classes of models can be
distinguished; Eulerian (continuum) models and Lagrangian (discreteparticle) models. The two major computational modeling techniquesthat are widely used for modeling of multiphase flow are Eulerian–Lagrangian and Eulerian–Eulerian formulations [7,8].
In the Eulerian–Lagrangian approach, the continuous phase ismodeled using conventional Eulerian approach, whereas the dispersedphase is simulated by solving Newton's equations of motion for eachdispersed phase particle considering particle–particle as well asparticle–wall interactions. The particle–particle interactions are mod-eled by either hard sphere or soft sphere approaches or Monte Carlotechnique [9].
The particles interact through binary collision in the hard sphereapproach. The key parameters that are used in the hard sphere modelsare the coefficient of restitution, coefficient of dynamic friction andcoefficient of tangential restitution. In the soft sphere approach theparticles are allowed to overlap slightly and contact forces are calculatedbased on a spring-dashpot model [7–9]. When the dispersed phaseoccupies more than 10% of continuous phase Eulerian–Eulerianapproach should be used as the other approach becomes computation-ally expensive [9].
During last threedecades, extended researches havebeen conductedin order to simulate gas phase ethylene polymerization of fluidized bedreactors and different phenomena inside them. Kinetic models havebeen developed to predict molecular weight distribution as well asactivation and deactivation kinetics. For an individual polymer particle,mass and heat transfer resistances inside the particle and around theboundary layer have been analyzed. Particle size growth rate and theeffects of temperature and reactant concentrations on it are alsomodeled. Macro approach simulation has been applied to simulate heat
International Communications in Heat and Mass Transfer 37 (2010) 437–442
☆ Communicated by W.J. Minkowycz.⁎ Corresponding author.
and mass transfer between the phases, hydrodynamic behavior andpressure and temperature profiles inside the fluidized bed [10–16].
Most of these researches are related to what occurs inside thereactor and very few models have been reported to describe overallbehavior of the reactor. These models consider the rector as a wellmixed single phase, two phases of emulsion and bubbles or threephases of emulsion, bubbles and particles [10,17–20].
All these models use several simplifying assumptions such asoccurrence of the reactions only in the emulsion phase. There are afew reported researches about CFD simulation of individual particlesand their interactions in the reactor [21–24]. In this research,hydrodynamics of the reactor and heat transfer between polyethyleneparticles and the fluid phase are simulated applying two-fluid CFDmodels, based on Eulerian–Eulerian approach.
2. Two-fluid model equations
The CFD method used in this work is based on a two-fluid model(TFM), derived from the kinetic theory of gases. In a TFM both phasesare considered to be continuous and fully interpenetrating. In thisapproach the usual thermodynamic temperature is replaced by thegranular temperature of granular flow. The particle viscosity andstress are functions of this granular temperature, which varies withtime and position in the fluidized bed. Both phases are described interms of some separate sets of conservation equations, consideringthe coupling between the phases. Physical characteristic of solidparticles such as shape and size are included in the continuumrepresentation through the empirical relations for the interfacialfriction. However, these models do not recognize the discretecharacter of the solid phase [25,26].
By definition, the volume fractions of two phases must sum tounity.
εg + εs = 1 ð1Þ
2.1. Conservation equations
The conservation equations used in two-fluid model (TFM), aregeneralized forms of single phase mass, momentum and energyequations. The accumulation of mass in each phase is equal to theconvective mass flow [7–9,26,27]:
For gas phase:
∂∂t εgρg
� �+ ∇⋅ εgρg
→vg� �
= 0 ð2Þ
For solid phase:
∂∂t εsρsð Þ + ∇⋅ εsρs
→vs� �
= 0 ð3Þ
where, ε, ρ and v→ are volume fraction, density and velocity vector ofeach phase, respectively. For the given control volume the first termon the left hand side is mass change rate and the second termrepresents the net flow of the mass. The momentum balance for thegas phase is given by the Navier–Stokes equation, modified to includean interphase momentum transfer term.
∂∂t εgρg
→vg� �
+ ∇⋅ εgρg→vg
→vg� �
= −εg∇P + ∇⋅ τ��g + εgρg→g + Kgs
→vg−→vs
� �
ð4Þ
The following equation can be used for the solid phase.
∂∂t εsρs
→vs� �
+ ∇⋅ εsρs→vs
→vs� �
= −εs∇P−∇Ps + ∇⋅ τ��s + εsρs→g + Kgs
→vs−→vg
� �
ð5Þ
where, the pressure of solid phase is Ps and P is the gas phase staticpressure, g is the gravity acceleration, Kgs is the gas–solid momentumexchange coefficient, and τ−− g and τ−− s are gas and solid phase stresstensors respectively. The first term of each equation describesmomentum change rate per unit volume. The second term is changein momentum per unit volume caused by convection. The third termis the pressure force per unit volume. The fourth term is the viscousforce and the fifth term represents the gravitational force per unitvolume. The solid pressure, bulk viscosity and shear viscosity can beobtained from the kinetic theory of granular flow, as given below.
The energy balance for the gas phase is:
∂∂t εgρgcpgTg
� �+ ∇⋅ εgρgcpg
→vgTg� �
= −∇⋅qg−Hgs ð6Þ
where, qg is the gas phase conductive heat flux, and Hgs is theinterphase heat transfer between gas phase and solid phase.
The energy balance for the solid phase is:
∂∂t εsρscpsTs
� �+ ∇⋅ εsρscps
→vsTs� �
= −∇⋅qs + Hgs−ΔHrs ð7Þ
where, qs is the solid phase conductive heat flux, and ΔHrs is the heatof reaction.
2.2. Kinetic theory of granular flow
The kinetic theory of granular flows is based on similaritiesbetween the flow of a granular material and gas molecules. Thistreatment uses classical results from the kinetic theory of gases topredict the form of transport equations for a granular material. Usingthis theory it is possible to describe the behavior of molecules orparticles with well-defined properties and interactions. This methodintroduces a granular temperature, which represents the fluctuationof the particle velocity. Analogous to the thermodynamic temperature
Nomenclature
v→ Phase velocity vectorv′ Particles velocity fluctuatingP Gas phase static pressurePs Solid phase pressureg→ Gravity accelerationKgs Gas–solid momentum exchange coefficientJs Dissipation or creation of granular energy from the
working of the fluctuating forceT Phase temperaturecpg Heat capacity of gas phasecps Heat capacity of solid phaseq Phase conductive heat fluxΔHrs Heat of polymerizationHgs Heat transfer from gas phase to solid phase
for gases, the granular temperature (Θ) of solid phase is defined asone-third of the mean square velocity of particles' random motion[8,9,28–30].
Θ =13
v02D E
ð8Þ
where, v′ is the fluctuating particle velocity.Granular temperature is different from solid phase temperature
and proportional to the granular energy. The following solid phasegranular energy equation must be solved to calculate the granulartemperature. Since the solid phase stress depends on the magnitudeof these particle velocity fluctuations, a balance of the granular energy32Θ� �
associated with these particle velocity fluctuations is required tosupplement the continuity and momentum balance for both phases.This balance is given as:
32
∂∂t εsρsΘð Þ + ∇⋅ εsρsΘ
→vs� �� �
= −∇Ps→I + τ��s
� �: ∇→vs−∇⋅ κs∇Θð Þ−γs−Js
ð9Þ
where the first term on the right hand side represents the generationof fluctuating energy due to shear in the particle phase, the secondterm represents the diffusion of fluctuating energy along gradients inΘ, γs represents the dissipation due s to inelastic particle–particlecollisions, and Js represents the dissipation or creation of granularenergy resulting from the working of the fluctuating force exerted bythe gas through the fluctuating velocity of the particles.
3. CFD simulations
The governing equations were solved applying finite volumemethod, using appropriate initial and boundary conditions. Thecalculation domainwasdivided into afinite number of non-overlappingcontrol volumes. Volume fraction, density and granular temperaturewere stored at themain grid points, placed in the center of each controlvolume. In order to reduce numerical instabilities, a staggered grid wasused for discretization of the governing equations. Scalars such aspressure and volume fraction were stored at the cell centers and thevelocity components were stored at cell surfaces. The conservationequations were integrated in space and time. The integration wasperformed using first order upwind differencing in space and fullyimplicit methods in time. The phase-coupled SIMPLE (PC-SIMPLE)algorithm, which is an extension of the SIMPLE algorithm tomultiphaseflows with partial elimination of interphase coupling, was employed tosolve the discretized equations. Due to the strong interactions betweenthe phases through the drag forces the two-phase partial eliminationalgorithm (PEA) Spaldingwas generalized for multiple phases and usedto decouple the drag forces. The IPSA (interphase slip algorithm) wasused to take care of the couplingbetween thecontinuity and the velocityequation [9,25,31].
The model was used to simulate two-dimensional gas–solidfluidized bed at different superficial gas velocities. Several simulationswere performed in order to investigate the effects of different operatingconditions, and to obtain an adequate description of the gas–solid flowpattern in the bed. The reactor was simulated for about 10 s with anaverage time step of 0.001 swith20 iterationsper timestep. The relativeerror between two successive iterations was specified by using aconvergence criterion of 10–3 for each scaled residual component.Sensitivity analysis was conducted through a case study to inspect theeffects of time step, discretization schemes, and convergence criterionon the final modeling results. The simulated fluidized bed was two-dimensional at unsteady condition. The continuous phase was ethylenegas. The dispersed phase consists of spherical particles whose meandiameterwas supposed tobeuniformand constant. Table 1 summarizesmodel parameters applied for the CFD simulation of the 2-D fluidized
bed. The two-dimensional computational domain used in the fluidizedbed simulations is shown in Fig. 1.
4. Results and discussion
Several simulations were performed in order to investigate theeffect of different operating conditions, and to get an adequatedescription of the gas–solid flow pattern in the reactor. Fig. 2 shows acontour plot of solid phase volume fraction for the Gidaspow dragmodel at the times of 0.005, 2, 8.5, 10, 20 and 40 s. All simulationswere continued for 40 s of real simulation time. Time-averageddistribution of variables was then computed considering the last 20 sof simulation. The figure indicates that after an initial period of about10 s, the flow hydrodynamics in the bed reaches a state, characterizedby a strong non-stationary. It also displays the distributions ofinstantaneous porosity in the bed at the superficial gas velocity of0.12 m/s. This figure shows that as the time increases, the bed isimpulsively fluidized at uniform velocity of 0.12 m/s. The drag forcebetween the gas and solid particles is one of the dominant forces in afluidized bed. The drag laws to model the interphase momentumexchange are usually developed empirically. Therefore, their applica-bility to model a fluidized bed of specific particle size and flow regimeneeds to be evaluated. Fig. 3 shows a comparison between Gidaspow,Syamlal-O'Brien, and Wen Yu drag functions to predict time-averagebed expansion ratio. As indicated in this figure, solid phase volumefraction predations by CFD simulation are similar for all of the dragmodels and gas–solid flow patterns.
The pressure drop along the reactor is depicted in Fig. 4. It showspressure drop is nearly linear in the bed. Moreover, it can be seen thatpressure drop values, predicted by Syamlal-O'Brien and Gidaspow
Table 1Model parameters.
Parameter Value
Height (cm) 150Initial bed height (cm) 75Width (cm) 50Minimum fluidization velocity (m/s) 0.12Solid density (kg/m3) 950Bulk density (kg/m3) 24Particle diameter (µm) 500
Fig. 1. Computational domain in the simulations.
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drag models, reasonably agree with those values of experimentalmeasurements, reported in the literature [32].
Fig. 5 shows the solid volume fraction of the particle phase as afunction of bed height using Syamlal-O'Brien drag model at the
superficial gas velocities of 12, 18 and 24 cm/s. This figure indicatesthat when superficial gas velocity increases, fluctuations of solidphase volume fraction in the axial direction of the bed are greater. It isclear that an escalation of superficial gas velocity leads to an increase
Fig. 2. Contours of solid phase volume fraction.
Fig. 3. Contours of solid phase volume fraction for various drag models at t=10 s.
440 M.A. Dehnavi et al. / International Communications in Heat and Mass Transfer 37 (2010) 437–442
in the bed expansion. The values of the bed expansion were 0.84, 1.02and 1.18 m when superficial gas velocities were 12, 18 and 24 cm/srespectively. At the lower superficial gas velocities, the particles wereaccumulated in the lower portion of the bed. As superficial gasvelocity was raised, solid particle volume fraction at the bottom of thebed declined.
The effects of Gidspaw, Syamlal-O'Brien and Wen Yu drag modelson temperature of the gas phase polyethylene reactor fluidized bedwere also investigated. The temperatures of ethylene and polyethyl-ene particles at the entrance were 350 K and 380 K, respectively. Thesimulation results for bed temperature gradient are shown in Fig. 6.This figure indicates that the temperature gradient in the primarysection of the bed is much larger than the other section. The meantemperature of the bed is in the range of 375–379 K. Fig. 6 indicatesthat the predictions of temperature gradient along the bed areapproximately similar using any of the three drag models. However,the profiles produced using Gidspaw and Syamlal-O'Brien dragmodels are closer.
5. Conclusions
In this article, unsteady state behavior of gas phase polyethylenereactor gas–solid fluidized beds has been investigated using CFDmethods. Simulation results indicate that Eulerian–Eulerian model issuitable for modeling of industrial polyethylene fluidized bedreactors. The effect of Gidspaw, Syamlal-O'Brien and Wen Yu dragmodels on hydrodynamics and temperature of the gas phasepolyethylene reactor fluidized bed was investigated. The pressuredrop values predicted by Syamlal-O'Brien and Gidaspow drag modelreasonably agree with those values from experimental measurementsthat are reported in the literature. Simulation results for the effects ofsuperficial gas velocities on solid volume fraction of particle phaseindicated that increasing superficial gas velocity cause greater solidvolume fraction fluctuations in the axial direction, leading to anincrease in the bed expansion. At the lower superficial gas velocities,the particles were accumulated in the lower portion of the bed. Assuperficial gas velocity was raised, solid particle volume fraction at thebottom of the bed declined. The bed temperature simulation resultsshowed that the temperature gradient in the primary section of thebed is larger. The mean temperature of the bed was in the range of375–379 K. From the simulation results, it could also be deduced thatapplying the drag models leads to similar profiles of temperaturegradient along the bed approximately.
Acknowledgements
The authors are grateful to acknowledge financial and technicalsupports of Iranian Petrochemical Research and Technology Companyto conduct this research.
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