A Lagrangian-Eulerian hybrid model for the

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A Lagrangian-Eulerian hybrid model for the simulation of poly-disperse fluidized beds: Application to industrial-scale olefin polymerization Simon Schneiderbauer* , a, b , Stefan Pirker b , Stefan Puttinger b , Pablo Aguayo c , Vasileios Touloupidis c , Alberto Martínez Joaristi d a

Christian Doppler Laboratory for Multi-Scale Modeling of Multiphase Processes, Johannes Kepler University, Altenbergerstraße 69, Linz 4040, Austria Department of Particulate Flow Modelling, Johannes Kepler University, Altenbergerstraße 69, Linz 4040, Austria c Borealis Innovation Process Technology, Linz 4040, Austria d Borealis Innovation Process Technology, Schwechat-Mannswörth 2320, Austria b

A R T I C L E

I N F O

Article history: Received 12 May 2016 Received in revised form 12 November 2016 Accepted 19 December 2016 Available online xxxx Keywords: Two-fluid model (TFM) Olefin polymerization Gas-phase fluidized bed reactor Sub-grid drag modification Discrete phase model (DPM)

A B S T R A C T We present a generalization of the Lagrangian-Eulerian hybrid model for the numerical assessment of polydisperse gas-solid flows (Schneiderbauer et al., Pow. Tech., 2016) to olefin polymerization fluidized beds. The main idea of such a modeling strategy is to use a combination of a Lagrangian discrete phase model (DPM) and a coarse-grained two-fluid model (TFM) to take advantage of the benefits of those two different formulations. On the one hand, the DPM model unveils additional information such as the local particle size distribution, which is not covered by TFM. On the other hand, the TFM solution deflects the DPM trajectories due to the inter-particle stresses. Moreover, sub-grid drag corrections are applied to account for the impact of the small unresolved scales on the gas-solid drag force. This hybrid approach further enables the efficient evaluation of gas-solid reactions at a particle level using DPM. In particular, at each DPM trajectory we consider the olefin polymerization accounting for the catalyst profile (activity over time), the pressure driven solubility of the reaction gases in the polymer particles, the particles crystallinity and the corresponding reaction masses and reaction heat. These, in turn, are mapped to the TFM approach, where they appear as additional mass and energy source terms. The predictive capability and numerical efficiency of this reactive hybrid modeling approach is demonstrated in the case of (i) an inert bi-disperse fluidized bed and (ii) in the case of an industrial-scale olefin polymerization fluidized bed. The results show that the model is able to correctly predict segregation in poly-disperse gas-solid flows. In addition, the model is able to predict the particle growth in a fluidized bed reactor as well as its impact on the hydrodynamics of the bed. The results further give a closer insight about the temperatures and the crystallinity of the polymer particles. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Polyolefins (POs), including polyethylene (PE), polypropylene (PP), ethylene-propylene elastomer (EPR), etc., have found widespread use in many modern human-life applications due to their excellent chemical, physical and mechanical properties, superior processability, good recyclability and relatively low cost [2] One common approach to produce polyolefins is employing gas-phase fluidized bed reactors, which are recognized as economically attractive [3] In those catalytic fluidized bed reactors (FBR), catalyst or

* Corresponding author at: Christian Doppler Laboratory for Multi-Scale Modeling of Multiphase Processes, Johannes Kepler University, Altenbergerstraße 69, Linz 4040, Austria. E-mail address: [email protected] (S. Schneiderbauer).

prepolymerized particles are continuously fed into the reactor at a point above the gas distributor and react with the incoming fluidizing monomer(s) to form a broad distribution of polymer particles (e.g., 100 to 5000 lm) [4] The particulate flows inside most practically relevant gas-phase Olefin polymerization reactions manifest a broad particle size distribution [2, 5, 6, 7] Thus, it is important to understand the mixing and segregation of the particles in the process to evaluate its efficiency [8, 9] One of the most straight forward numerical methods to account for poly-disperse mixtures is CFD-DEM (CFD: computational fluid dynamics; DEM: discrete element method), where the primary fluid phase is governed by computational fluid dynamics and the secondary dispersed phase is pictured by a discrete element model [10, 11] However, since the total number of particles in fluidized bed reactors is extremely large, it may be impractical to solve the equations of motion for each particle. It is, therefore, common to investigate particulate

http://dx.doi.org/10.1016/j.powtec.2016.12.063 0032-5910/© 2016 Elsevier B.V. All rights reserved.

Please cite this article as: S. Schneiderbauer et al., A Lagrangian-Eulerian hybrid model for the simulation of poly-disperse fluidized beds: Application to industrial-scale olefin polymerization , Powder Technology (2016), http://dx.doi.org/10.1016/j.powtec.2016.12.063

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flows in large process units using averaged equations of motion, i.e. two-fluid models (TFM), which include the inter-particle collisions statistically by kinetic theory based closures of the particle stresses [12, 13, 14] However, each representative particle diameter requires an additional momentum and continuity equation, which considerably raises the computational demand with increasing number of particle diameters [15, 16] One may restrict the calculations to spatially constant particle size distributions to evaluate the gas-solid drag force [2] Furthermore, the TFM approach requires considerably fine grids since the minimum stable sizes of clusters are around ten particle diameters [17] Thus, due to computational limitations a fully resolved simulation of industrial-scale gas-solid flows is still unfeasible. It is, therefore, a frequent practice to use coarse grids to reduce the demand on computational resources. However, such a procedure inevitably neglects the small (unresolved) scales, which leads, for example, to a considerable overestimation of the bed expansion in the case of fluidized beds of fine particles [18, 19, 20] Recently, several authors have applied TFMs to polymerization reactors [5, 6, 7, 21, 22, 23, 24]. However, only a few studies are available dealing with a fully coupled simulation of the hydrodynamics of the fluidized bed and the polymerization reactions including reaction heat [25] and particles growth [7] For example, Yan et al. [7] coupled the TFM approach with a population balance equation (PBE) to account for transport of the different particle sizes as well as a polymeric multilayer model (PMLM) to include particle growth. Such a modeling approach inevitably demands significantly enhanced computer resources to solve the additional transport equations for the PBE as well as the PMLM locally for a huge number of particles. Thus, most of the studies dealing with the chemistry in gas-phase reactors restrict themselves to global balance equations [26, 27, 28, 29] To sum up, the current understanding of the numerical simulation of industrial-scale gas-phase Olefin Polymerization reactors exhibits three computational demanding limitations. First, resolving each particle trajectory and each particle-particle collision requires on the one hand unaffordable computational resources (with respect to time and costs); on the other hand, using an averaged model (such as TFM) does not account for poly-disperse particle flows in first place. In particular, standard TFM approaches require a separate set of transport equations for each particle fraction under consideration. This inevitably leads again to excess computational costs in particle laden flows, which are characterized by broad particle size distributions. Second, the TFM approach additionally requires very fine numerical grids, which additionally increases the computational demands in the case of industrial-scale applications considerably. Third, an efficient and resource-saving computational approach is required to account for the effect of the polymerization heat and the related particle growth on the hydrodynamics of gas-phase reactors. To address the first concern, we follow our previous works [1, 16, 30, 31, 32] and present a hybrid model for the numerical assessment of poly-disperse gas-solid fluidized beds. The main idea of such a modeling strategy is to use a combination of a Lagrangian discrete phase model (DPM) and a TFM to take advantage of the benefits of those two different formulations. On the one hand, the local degree of poly-dispersity (i.e. the local particle size distribution), which is essential for the evaluation of the gas-solid drag force, can be obtained by tracking statistically representative particle trajectories for each particle diameter class. On the other hand, the computationally demanding tracking of the inter-particle collisions can be obtained from the inter-particle stresses, which are deduced from the TFM solution. These then appear in addition to the gas-particle drag as a body force in the equation of motion of each DPM-trajectory. Thus, the hybrid model represents a TFM simulation with additional DPM particles, which are used, for example, to provide a closure for the poly-disperse drag law. The second concern, is addressed by applying sub-grid modifications to the drag force and the kinetic theory stresses to account for the small unresolved scales

in the case of coarse grids. We refer this approach to as coarse grained TFM (cgTFM). Here, we employ sub-grid modifications presented in previous studies [2, 18, 19] Finally, by employing the above hybrid Lagrangian-Eulerian hybrid model, the polymerization rate as well as the corresponding reaction heat can be computed based on the representative Lagrangian particles. This, in turn, includes particle growth, since the diameter of those particles can be determined from the polymerization rate, which affects the particle size distribution (and thus the mean diameter) locally. In this paper, we therefore demonstrate the predictive capability of such a hybrid model for the numerical assessment of poly-disperse gas-solid fluidized beds. First, we study segregation in an inert bidisperse fluidized bed [9] The results show that the hybrid model is able to correctly predict the segregation of large particles as well as the pressure distribution in the bed. Second, we extend the hybrid approach to the numerical assessment of industrial-scale gas-phase Olefin Polymerization reactors by accounting for reacting enthalpies as well as particle growth by employing appropriate formulas for the reacting kinetics. The results are discussed with respect to temperature measurements in the corresponding real plant and the evolution of the particle size distribution with time. The following section discusses the sub-grid drag corrections [19] of the coarse grained TFM, the concept of the additional Lagrangian tracer particles and the polymerization model. Next, implementation issues are discussed in Section 4. In Section 5 results for the bi-dispersed fluidized bed and the polymerization fluidized bed are discussed. A conclusion section ends this paper. 2. Modeling of non-reactive poly-disperse gas-solid flows 2.1. Coarse grained two-fluid model (cgTFM) Similar to our earlier study, Schellander et al. [16] we used a kinetic-theory based two-fluid model (TFM) to study gas-phase olefin polymerization fluidized beds. Since these equations have been extensively discussed in our previous works [13, 18, 19] , we do not repeat all the details here and solely present the continuity and momentum equations for the solid phase below:

∂ 4 q + ∇ • (4s qs us ) = Rs , ∂t s s

(1)

  ∂ kc fr (4s qs us)+∇ • (4s qs us us ) = −4s ∇p−∇ • Ss + Ss +b(ug −us)+4s qs g . ∂t (2) Here, qs , 4s and us denote density, volume fraction and local-average velocity of the solid phase, respectively; Rs denotes the rate of generation of polymer due to chemical reactions; p is the gas phase pressure; ug is the local-average velocity of the gas phase; b is the microscopic drag coefficient, which will be discussed later; g is the kc fr gravitational acceleration; finally, Ss and Ss are the stress tensors associated with the solids phase, where the frictional contribution, fr Ss , arises from enduring or multi-particle collision events in dense kc areas. The kinetic-collisional part, Ss , is closed using kinetic theory [12, 13] which requires an additional equation for the granular temperature. Since we thoroughly discussed the closures for the solid stresses and the granular temperature equation, we do not discuss them here. For further details we refer to our previous study [13] In the case of coarse grids it is common to use balance equations for the filtered counterparts of the local-average solids volume fraction 4s and solids velocity us reading as [18, 19, 20, 33]

∂ ˜ s, q 4¯ + ∇ • (qs 4¯ s u˜ s ) = R ∂t s s

(3)

Please cite this article as: S. Schneiderbauer et al., A Lagrangian-Eulerian hybrid model for the simulation of poly-disperse fluidized beds: Application to industrial-scale olefin polymerization , Powder Technology (2016), http://dx.doi.org/10.1016/j.powtec.2016.12.063

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for the solids continuity equation and

∂ ¯fi • S +beff (u ˜ g − u˜ s )+qs 4¯ s g (4) ¯ (q 4¯ u˜ )+∇ • (qs 4¯ s u˜ s u˜ s ) = −¯4s ∇ p−∇ s ∂t s s s for the solids momentum equation with

3

Table 2 Summary of microscopic poly-disperse drag coefficient of Beetstra et al. [37], which has been adapted in our previous work [2]. Here, 4¯ g denotes the filtered gas volume fraction, ds  the Sauter diameter, 4s,i the volume fraction of particle size class i and Nsp the number of particle size classes.    N sp ¯ xi Fpoly (yi ) ˜ d  , b˜ = 18lg 4¯ s 4¯ 2 F 4¯ s , 4¯ g , Re g

s

i=1

d2s,i

with

˜ beff = HD b,

(5)

4¯ g qg ds u˜ g −u˜ s  , lg 4¯ g yi + 4¯ s y2i + 0 064¯4g y3i ,

˜ d  = Re s Fpoly (yi ) =

˜ d  ) = F(¯4s , 4¯ g , Re s

¯kc ¯fr ¯fi ¯R Ss = Ss + Ss + Ss ,

(6)

  ¯R Ss = qs 4s u¯s us − 4¯ s u˜ s u˜ s .

(7)

Eqs. (3) and (4) have the same form as the microscopic TFM Eqs. (1) and (2), with the phase velocities and other variables now representing filtered 4¯ s (or Favre averaged u˜ s = 4s¯us /4¯ s ) values. Additional terms appear that represent the unresolved part of the buoyancy, the unresolved part of the drag, HD , and a Reynolds stress-like con¯R tribution coming from the particle phase velocity fluctuations, Ss . It has to be noted that in case of coarse grids the contribution coming from the filtered kinetic theory stresses can be neglected [19, 34, 35, ¯R 36] while in case of fine grids the contribution of Ss vanishes and ¯kc ¯R kc Ss approaches Ss . Constitutive relations for HD and Ss are given in Table 1, where the non-negligible contribution stemming from fric¯R tional contacts is substituted into Ss [19] Finally, in Eq. (5)b˜ indicates that the drag coefficient is evaluated from filtered variables. In this work we employ the poly-disperse drag coefficient of Beetstra et al. [37] which is summarized in Table 2.

.

10¯4s 4¯ g3

  ˜ 0.413Re 1 /2 ds  + 4¯ g 1 + 1.5¯4s + 3 24¯4g



˜ −0.343 4¯ g−1 +3¯4g 4¯ s +8.4Re ds  −(1+4¯4s )/2 3¯ 4 ˜ s 1+10 Reds 



and the dimensionless parameters 4¯ d x¯ i = 4¯s,is , yi = dsi 

which is connected to the volume fraction 4¯ s,i by

4¯ s,i = ni p

d3s,i 6

(9)

yielding

x¯ i = p

d3s,i ni 6 4¯ s

(10)

In Eq. (8), Dg denotes the grid spacing of the Eulerian grid and the set Pi contains all parcels of particle size class i. We further obtain the local Sauter diameter, which is required for the evaluation of the drag force and the kinetic theory stresses, from Beetstra et al. [37] ⎡

⎤−1 Nsp  x¯ i ⎦ , ds,i

2.2. Lagrangian discrete phase model (DPM)

ds  = ⎣

The local volume fraction of the different particle size classes 4¯ s,i = xi 4¯ s , which is required for the evaluation of the gas-solid drag force (compare with Table 2), can be obtained by tracking statistically representative particle trajectories for each particle diameter class along the solids flow obtained from the coarse-grained TFM. In particular, such a trajectory k represents ak real particles and we, therefore, those trajectories refer to parcels [38] Thus, we obtain for the number density of size class i ni (x) =



ak GDg (x − xp,k ),

(8)

k∈Pi

Table 1 Summary of filtered sub-grid modifications of Schneiderbauer and Pirker [19]. 1. Filtered drag coefficient:   ˆ f ) Hu (¯4s , uˆ˜gs , D ˆ f ), HD = 1 − h4 (4¯ˆ s )hD (D max −¯4 )D ˆ5 ˆ5 s f /(Df +1024)

ˆ f ) = (uˆ˜gs )−9¯4s (4s Hu (¯4s , uˆ˜gs , D h4 (4ˆ¯ s ) = ˆ f) = hD (D



2

−6.7434ˆ¯ s +6.7284ˆ¯ s

3 2 4ˆ¯ s −7.2474ˆ¯ s +6.2894ˆ¯ s +0.384 ˆ 2.664 D f

ˆ 2.664 +25.89 D f

4ˆ¯ s = 4¯ s /4smax

s

.

s

with Ss = Ds − (1/3)tr(Ds ) and Ds = (∇us + (∇us )T )/2. (a) Filtered particle phase normal stress: ¯ pR ˆ −31/21 ˆ 16/7  s  S¯ s 2 Cp = 0.17¯4s + Df 2 = 4Cp Fr qs ut

(b) Filtered particle phase shear viscosity: ¯ lsR g ˆ −4/3 ˆ 2  Cl = 0.105¯4s + Df  S¯ s  3 = 4Cl Fr qs ut

0.0235¯4s 4smax (4smax −¯4s )3/4

0.001¯4s 4smax (4smax −¯4s )

3. Dimensionless variables: u˜ −u˜  (a) Slip velocity: uˆ˜gs = gut s , where ut is the terminal settling velocity. ˆ f = Df with Lch = u2 /g Fr−2/3 and Fr = u2 /(ds g). (b) Filter size: D Lch

t

where Nsp the number of particle size classes. It remains to discuss the equation of motion of such a tracer parcel k, which reads [1, 31]  dup,k 1  poly u˜ s − up,k + F k + g, = dt tc,k

(12)

where up,k denotes the velocity of the Lagrangian tracer parcel k, u˜ s the filtered solids velocity, g the gravitational acceleration and tc,k is a collisional time scale required to accelerate a single particle to the average solids velocity [1, 31, 39]  2 Nsp  x¯ j ds,k + ds,j g0,kj 1 3(1 + e)  4¯ s up,k − u˜ s  = . tc,k 4 d3s,k + d3s,j j

(13)



2. Reynolds-stress like contribution: ¯R S = p¯R I − 2l¯R S¯ s , s

(11)

i=1

t

Here, e ≈ 0.9 is the coefficient of restitution, Nsp is defined in Eq. (11), ds,j the particle diameter of class j and x¯ j is defined in Table 2. g0,jk denotes the radial distribution function, which accounts for the poly-disperse mixture of hard spheres [15] Note that Eq. (13) accounts for the contribution coming from the inter-particle stresses, i.e. inter-particle collisions. These are determined by the coarsegrained TFM solution and affect the trajectories of the tracer parcels by the collisional time scale tc,k . poly is the Since the tracers show different particle diameters F k acceleration of a single particle of diameter ds,k within the local poly-disperse mixture of particles (units force per unit parcel mass, i.e. m s −2 ) due to the gas-solid drag force. Thus, the acceleration

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of parcel k due to the effective drag force (i.e. including sub-grid heterogeneities) can be written as [1] poly

Fk

=

  1 ˜ bk HD u˜ g − up,k , x¯ k 4¯ s qs

(14)

where HD is given from Table 1 and b˜ k is presented in Table 2. Note that here b˜ k is computed based on the Reynolds number computed from the local velocity of the tracer instead of the local velocity of the solid phase as in Eq. (5).

of species i by chemical reactions. Finally, the diffusion flux J˜ i is written as ∇ T˜ g , J˜i = −qg Dm,i ∇ Y˜ i − DT,i T˜ g

(16)

where Dm,i is the mass diffusion coefficient for species i and DT,i is the thermal (Soret) diffusion coefficient [40] To describe the conservation of energy in fluidized bed reactors, a separate transport equation is solved for the specific enthalpy, hq , of each phase:   ∂ 4¯ q qq h˜ q ¯R + ∇ • 4¯ q qq u˜ q h˜ q = Ss : ∇ u˜ q − ∇ • qq + Sq + Qgs , ∂t

3. Reaction kinetics considerations The following reaction kinetics considerations are required to determine the production/consumption rates of the individual gas and solid species. This, in turn, enables the prediction of the polymerization rate as well as the corresponding heat of reaction and different polymer properties. However, coupling such a kinetic model with a pre-existing CFD model should not add significant CPU effort, which would lead to a considerable increase of simulation time. Nevertheless, this model should still offer a convenient connection of the catalyst parameter values and major thermodynamic effects to the polymerization rate. More specifically, the aim of this work is to develop a simplified reaction kinetics model, which is able to capture the effects of (i) temperature, (ii) pressure (directly affecting monomer solubility to the solid amorphous polymer phase), (iii) catalyst system in use (connected to the kinetic rate constants), and, (iv) crystallinity (affecting polymerization rate) on process performance variables, including: (i) monomer(s) consumption rate, (ii) polymer production rate and heat of reaction, and, (iii) co-polymer content.

(17)

where the heat flux qq is modeled by using Fourier’s law qq = kq ∇ T˜ q and Sq accounts for the reaction heat. In case of the gas phase the heat conductivity kg is computed employing a weighted average of the individual heat conductivities of the monomers. For the heat exchange between the gas and the solid phase, Qgs , we employ the correlation proposed by Gunn [41] Assuming constant specific heats cp,q the filtered phase temperature and phase enthalpy are correlated as follows h˜ q = cp,q T˜ q .

(18)

Note that in a first step we do not account for unresolved heterogenous structures in the transport equations for the gas species as well as the phase enthalpies as reported in, for example, recent studies [42, 43] However, in contrast to the species transport Eq. (15) the unresolved heterogenous particle structures affect the net rate of the generation of the solid phase enthalpy due to the meso-scale particle stresses. In the following the proposed model correlations for Ri and Sq are presented [28, 29] 3.2. Rate of polymerization (RP )

To conclude, the proposed hybrid methodology provides an efficient technique to compute the polymerization rates and reaction heats. First, we are able to deduce the polymerization rate and reaction heat for each tracer trajectory. Second, by mapping these data to the Eulerian grid, we are able to determine the source terms for the volume fraction, species and energy transport equations. Thus, it has to be emphasized again that the DPM parcels provide a particle size dependent closure for the particle based olefin polymerization without requiring a continuum based closure for the TFM approach. Finally, it has to be noted that the prediction of the polymer microstructure (e.g., molecular weight distribution, chemical composition distribution, comonomer sequence length distribution) is out of scope in this study.

In this study, we assume that each catalyst particle can be modeled as a single batch reactor [26] Thus, the particles are exposed to different but constant monomer concentrations and temperatures during each simulation time-step. In order to simplify reaction kinetics, the catalyst polymerization behavior is approximated by a single-site catalyst. This assumption is then included to the kinetic parameters. The rate of polymerization, Rp (kg m −3 ) of the monomers, [M1 ] and [M2 ], is therefore calculated from ∞

RP =

 d([M1 ] + [M2 ]) = (kP1 [M1∗ ] + kP2 [M2∗ ]) [Pr ] dt r=0

= (kP1 [M1∗ ] + kP2 [M2∗ ]) [Y0 ],

(19)

The local concentration of monomer i is described by a transport equation for a corresponding species Yi of the gas phase, which reads

where kpi is the propagation rate constant (l mol −1 s −1 ) corresponding to the monomer i active concentration at the active sites [Mi∗ ] (–) and [Y0 ] (kg m −3 ) is the total molar concentration of living chains in the reactor (i.e. particle) Pr (mol l −1 ). The potential catalyst concentration, [C], can be described as follows

  ∂ 4¯ g qg Yi g,i , J i + 4¯ g R + ∇ • 4¯ g qg Yi = −∇ 4¯ g ug  ∂t

d[C] = −ka [A][C], dt

3.1. Species transport and heat transfer

(15)

where qg is the density of the gas phase given by the equation of state for ideal gases and Ri accounts for net rate of generation/destruction

(20)

where ka is the kinetic constant for potential catalyst activation by cocatalyst A (l mol −1 s −1 ) with the corresponding concentration

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[A] (mol l −1 ). Assuming that the cocatalyst concentration remains practically constant, Eq. (20) can be analytically solved as [7] [C] = [C0 ]e

−ka t

.

(21)

Here [C0 ] is the initial catalyst concentration and ka = ka [A]. Thus, the time evolution of the living chains concentration can be described as d[Y0 ]  = ka [C] − kd [Y0 ] = ka [C0 ]e−ka t − kd [Y0 ], dt

[Y0 ] =

  k −ka 1− d t ka

kd ka

1−

[C0 ]e−kd t .

(23)

It should be noted that all kinetic rate constants, kl (l ∈ {P1, P2, a, d}), follow an Arrhenius temperature (T˜ s ) law dependence described by the form [7] kl (T˜ s ) = Kl e

E − ˜l

RTs

,

(24)

where Kl and El are the corresponding pre-exponential constants and activation energies, respectively. 3.3. Mass of polymer produced (MPi ) Since the time steps in a CFD simulation are relatively short compared to the time scale of the chemical reactions involved, the polymerization rate can be considered as constant during a single CFD time step, which yields for the production rate of polymer for tracer parcel k dMPi,k ∗ = ak kPi [Mi∗ (xp,k )][Y0,k ]Vam,k . dt

3.4. Comonomer content (cck ) The content of comonomer (we denote gas species 2 as comonomer) on a particle can be calculated based on the amount of produced polymer t cck (t) =  t

0

MP2,k dt

0 (MP1,k + MP1,k ) dt

(29)

(22)

where kd is the kinetic constant for spontaneous deactivation. Finally, using the initial condition [Y0 ](0) = 0, Eq. (22) can further be simplified 1−e

5

(25)

3.5. Active monomer concentration ([Mi∗ ]) The produced polymer consists of two phases. First, an amorphous polymer phase swells with the sorbed reaction species and second, a crystalline polymer phase acts as a barrier for the gas species transport to the active catalyst sites [44] The local/active monomer concentration in the particle is basically given by species concentration of the monomers qg Yi . However, the maximum amount of monomer available for polymerization is given by the solubility of the monomer into the amorphous phase, Si (≡ ratio between the monomer mass and the amorphous polymer mass). Thus, we have   [Mi∗ (xp,k )] = min qg Y˜ i (xp,k ), qam Si,k ,

(30)

where qg is the density of the gas phase. Typically, monomer solubility values can be estimated by using appropriate equations of states (e.g., Sanchez-Lacombe [45]). Alternatively, this information can be included into the model as solubility functions of pressure and temperature, which are based on experimental data. For the present case study, ethylene (species 1) and propylene (species 2) solubility partial pressure profiles can be found in literature [27] Thus, ethylene solubility, S1,k , is described by a linear correlation, which depends on the ethylene partial pressure, p1,k = 4¯ g (xp,k )Y˜ 1 (xp,k )p(xp,k ), and temperature, T˜ s,k = T˜ s (xp,k ) S1,k = Asol,1 (T˜ s,k )p1,k + Bsol,1 ,

(31)

∗ Here, Vam,k denotes the volume of the amorphous polymer on a tracer particle, which belongs to parcel k. Thus, this reveals the following source terms for species i (Eq. (15)) as well as for the solid phase (Eq. (3))

where Asol,1 (T˜ s,k ) is the temperature dependent slope and Bsol,1 is the intercept. The solubility of propylene, S2,k , is described similarly to Eq. (31)

 dMPi,k ˜ g,i (x) = − 1 GDg (x − xp,k ), R 4¯ g dt

S2,k = Asol,2 (T˜ s,k )p2,k + Bsol,2 ,

˜ s (x) = R



k∈P

GDg (x − xp,k )

k∈P

where P =

 dMPi,k , dt

(26)

i

N sp 

Pi (i.e. P contains all tracer parcels). The heat of reac-

(32)

with the temperature dependent slope Asol,2 (T˜ s,k ) and the constant intercept Bsol,2 . ∗ ) 3.6. Amorphous polymer volume (Vam

i

tion, HPi,k , can be further calculated as the product of mass of polymer produced and heat of polymerization, DHi HPi,k = MPi,k DHi .

(27)

Finally, it remains to calculate the amorphous polymer volume, which is determined as follows ∗ = Vam,k

pd3s,k 6

(1 − cf ,k ),

(33)

Therefore, the corresponding source terms in Eq. (17) read S˜g (x) = −



GDg (x − xp,k )

k∈P

S˜s (x) =



k∈P

GDg (x − xp )

 i



cp,g T˜ g

i

DHi

where cf is the volume fraction of the crystalline polymer phase. Typically, this fractional crystallinity can be correlated to the content of comonomer cc(described by the slope, Acf , and intercept, Bcf ) [27]

dMPi,k . dt

dMPi,k . dt

(28)

cf ,k = Acf cck + Bcf .

(34)

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To sum up, the proposed model includes the propagation rate constants for monomers, living chain concentration profiles (connected to the catalyst system and initial catalyst concentrations) and solubility as well as crystallinity correlations for pressure and comonomer content. The parameter values used by the model are summarized Table 3 [27] The kinetic parameters involved (referring to the specific catalyst system in use every time) were estimated based on the simplified kinetic model presented and a series of polymerization experiments in bench-scale.

4. Implementation Since the motion equation of the Lagrangian particles (Eq. (12)) does only account for collision implicitly by using Eq. (13) the total Nsp volume fraction of the tracer particles, 4¯ s,p = n pd3s,i /6 (comi=1 i pare with Eq. (8)) may exceed the maximum packing locally. This, in turn, may yield an unphysical accumulation of tracer particles in dense regions. Thus, we introduce an additional repulsive mechpack anism F k (units m s −2 ), which prevents the Lagrangian tracer particles from forming dense aggregates exceeding the maximum packing fraction. For more details the reader is referred to our previous study [1] Finally, it has to be noted that in the case where no tracer particle is in a specific numerical cell we apply a diffusive smoothening approach to the exchange fields locally (i.e. to the Sauter mean diameter) [46] For the numerical simulation we use the commercial finite volume CFD-solver FLUENT (version 16), whereby we implemented the poly-disperse drag force (Table 2), the sub-grid modifications of Schneiderbauer and Pirker [19] (Table 1), the momentum balance of the tracer trajectories (Eq. (12)) as well as the equations required for the reaction kinetics (Eqs. (25), (26) and (28)) by using user defined functions. For the discretization of all convective terms the QUICK (Quadratic Upwind Interpolation for Convection Kinematics) [47] scheme is used. The derivatives appearing in the diffusion terms are computed by a least squares method and the pressure-velocity coupling is achieved by the phase coupled SIMPLE algorithm [48] The trajectories of the Lagrangian tracer particles (Eq. (12)) are integrated after each fluid flow time step using a third-order Runge-Kutta method. Further it has to be noted that the gas velocity and the solid phase velocity in Eq. (12) are linearly interpolated to the particle positions by using a first order Taylor approximation. For the bi-dispersed fluidized bed we used a fluid time step size of 0.0002 s and for industrial-scale fluidized simulations a time step size of 0.001s.

5. Numerical simulations 5.1. Inert bi-disperse fluidized bed In our previous studies [1, 9, 31], the proposed hybrid model has been validated quantitatively on the one hand in the case of an industrial-scale hot gas cyclone separator; on the other hand, we have shown that this model is qualitatively able to predict poly-disperse gas-solid flows in fluidized bed reactors. This, in turn, implies that a rigorous quantitative verification of this hybrid approach in case of fluidized beds is missing, which we present in the following. We follow Puttinger et al. [9] who experimentally studied the degree of segregation in bi-disperse fluidized beds. The fluidized bed test rig consists of a rectangular bed with a cross-section of 0.15 × 0.04 m. The authors studied different compositions of sodalime glass spheres at different superficial gas velocities and recorded segregation using a high-speed camera in combination with color filters [9] The particle composition was then extracted from these images by employing a data processing method similar to Goldschmidt et al. [49]. Additionally, the pressure drop was recorded at different heights. In this paper, we focus on a configuration, where small particles (0) and large particles are present in equal initial fractions, i.e. x1 = (0) x2 = 0.5, where we studied four different superficial gas velocimf mf ties Wgin /Wg,2 ∈ {0.66, 0.8, 1, 1.11}. Here, Wg,2 denotes the minimum fluidization velocity of the large particles (k = 2). We employed a noslip boundary for the gas phase (air with a density of qg = 1.225 kg m −3 and a viscosity of l g = 1.78 • 10 −5 Pa s) and a partial slip boundary condition for the solid phase following our previous studies [13, (0) 18, 19] The initial bed height was 0.2 m with 4s = 4smax = 0.6 and the corresponding particle properties are summarized in Table 4. We, further, used cubic grid cells with a grid spacing of 0.005 m (≈10ds,1 ), which ensures grid independent cgTFM solutions [19]. Finally, it has to be noted that a Lagrangian tracer particle represents 32 particles in case of the small particles and 4 particles in the case of the large particles. This, in turn, yields an increase of the computational time of 15% compared to a mono-disperse two-fluid simulation, which is much smaller than using, for example, poly-disperse kinetic theory [15] In Fig. 1 the normalized extent of segregation, S, is plotted. According to Goldschmidt et al. [49] and Puttinger et al. [9] S can be computed from

S= Table 3 Parameter values used for reaction kinetics modeling. Property

Units

Method

Value

Dm,i DT,i KP1 EP1 KP2 EP2 Ka Ea Kd Ed R DH1 DH2 Asol,1 Bsol,1 Asol,2 Bsol,2 qam Acf Bcf

m2 s −1 m2 s −1 l mol −1 s −1 kcal mol −1 l mol −1 s −1 kcal mol −1 l mol −1 s −1 kcal mol −1 l mol −1 s −1 kcal mol −1 kcal mol −1 K −1 kJ g −1 kJ g −1 Pa −1 – Pa −1 – kg m −3 – –

Constant Constant Constant Constant Constant constant Constant constant Constant constant Constant Constant Constant f(T) Constant f(T) Constant Constant Constant Constant

2.9 • 10 −5 2.9 • 10 −5 2 • 108 12 1.1 • 108 12 108 12 104 12 0.001987 2.9 2.9 10 −7 T2 − 2 • 10 −5 T + 0.0018 0 3.41 • 10 −6 T −1 0 880 −4.2 55.6

Sh − 1 , Smax − 1

(35)

where the degree of segregation, Sh , is given by  he Sh = 0h e 0

x1 4¯ s z dz x2 4¯ s z dz

,

(0)

Smax =

2 − x1

(0)

1 − x1

.

(36)

Table 4 Particle properties for bi-disperse fluidized bed cases. For remaining physical properties the reader is referred to Schneiderbauer et al. [13]. (0)

k

xk

ds,k

qs,k

max 4s,k

mf Wg,k

1 2

0.5 0.5

500 lm 1.15 mm

2500 kg m −3 2500 kg m −3

0.6 0.6

0.3 m s −1 0.72 m s −1

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Fig. 1. Dimensionless extent of segregation, S, as a function of the dimensionless mf superficial gas velocity, Wgin /Wg,2 .

Here, he denotes the height of the expanded bed. Note that in case of the images recorded from the experiments computing S according to Eq. (35) requires a calibration curve, which maps x1 and x2 to grey scale values. In a second step the images are subdivided to a regular grid. For each of these grid windows the average greyscale level is calculated and then converted into a mass fraction using the calibration curve [9] Fig. 1 shows that at low superficial gas velocities (i.e. mf mf Wg,1 < Wgin < Wg,2 ) the extent of segregation increases with increasing superficial gas velocity. This can be explained by faster and larger bubbles in the regions of the small particles, which enhances the segmf regation of the large particles. At approximately Wgin /Wg,2 = 0.8, S reaches a maximum. However, further increasing the superficial gas velocity reduces the extent of segregation although Wgin is still below mf Wg,2 , since the settled large particles are re-entrained by the colliding highly fluidized small particles. Finally, S tends towards zero as mf Wgin exceedsWg,2 since both particle fractions are fluidized. The figure further presents the numerically predicted S, which is computed by averaging Eq. (35) for 10 s after S reached a statistical steady state. Remarkably, the numerically predicted S reveals an identically dependency on Wgin as observed from experiments. Furthermore, the absolute values of S are in fairly good agreement with experiments when employing the presented hybrid approach mf (denoted by ×). Especially, the maximum of S at Wgin /Wg,2 = 0.8 observed in the experiments is reproduced appropriately by the

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numerical simulation. Figs. 2 and 3 further show comparisons of snapshots of the solids volume fraction and the mass fraction of the large particles with photographs of the experiment for two different mf superficial gas velocities. At Wgin /Wg,2 = 0.8 (Fig. 2) the sedimentation of the large particles above the distributor plate is observed, while above the bed is fluidized since here the average diameter is much smaller. The snapshots further indicated that the simulation correctly pictures this transition between these defluidized and flumf idized regions. At Wgin /Wg,2 = 1.11 (Fig. 3) nearly no sedimented particles are observed since the superficial gas velocity is larger than the minimum fluidization velocities of both particle fractions. The figure clearly shows that nearly the whole bed is fluidized and that the bed is well mixed. However, Fig. 1 unveils that the bed is still not perfectly mixed (i.e. S > 0). This can be explained by Fig. 3. Here, numerical simulation as well as experiment indicate that there are still small defluidized regions near the wall due to the inhomogenous gas flow through the bed. Fig. 4 shows the pressure profiles through the bed for different superficial gas velocities. These profiles unveil on the one hand that the presented model is able to correctly predict the pressure drop in bi-disperse fluidized beds and on the other hand, provides an appropriate measure of the bed expansion. It remains to discuss the sensitivity of the simulation results to different numerical settings. We, therefore, repeated the simulation of mf the second case, where Wgin /Wg,2 = 0.8, with (i) a different integration algorithm for the tracer trajectories (first order implicit Euler instead of third order Runge Kutta), (ii) a reduced the time step size (Dt = 0.0001 s instead of Dt = 0.0002 s) and (iii) an increased number of tracer trajectories, where a tracer parcel represents 16 (instead of 32) particles in case of the small particles and 2 (instead of 4) particles in the case of the large particles. Fig. 1 clearly shows that neither using a lower order integration method, employing a smaller time step size nor increasing the number of tracer trajectories considerably affects the computed extent of segregation. In fact, the numerically predicted values for S are nearly indistinguishable, which can also be observed for the computed pressure profiles (Fig. 5). Thus, the numerical results can be considered as converged with respect to these three parameters. It has to be emphasized that a strict evaluation of the sensitivity of the results to the numerical setting also includes a grid sensitivity analysis. Here, we refer to our previous studies, where we have clearly shown grid independence of the cgTFM method for the grid spacings used in the present study [2, 19] Finally, as outlined in the implementation section the volume fraction computed from the tracer parcels, 4¯ s,p (which is basically

mf Fig. 2. Snapshots of a) solids volume fraction (blue: 4¯ s = 0, red: 4¯ s = 4smax ), b) the fraction of the large particles (blue: x¯ 2 = 0, red: x¯ 2 = 1) and c) experiment for Wgin /Wg,2 = 0.8. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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mf Fig. 3. Snapshots of a) solids volume fraction (blue: 4¯ s = 0, red: 4¯ s = 4smax ), b) the fraction of the large particles (blue: x¯ 2 = 0, red: x¯ 2 = 1) and c) experiment for Wgin /Wg,2 = 1.11. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

determined by a filter operation like in Eq. (8)), may get considerably different to the real volume fraction, 4¯ s , stemming from the cgTFM simulation. In particular, it is not necessary for the present hybrid method that there is a perfect match between both fields, since the tracer trajectories are used to obtain additional information not considered by the cgTFM approach (e.g. particle size distribution). However, considerable deviations between both volume fraction fields may cause an incorrect estimation of these additional

data. Thus, we evaluated the relative deviation of 4¯ s,p from 4¯ s as follows    4¯ s 4¯ s − 4¯ s,p  d3 x e4 = Y  , (37) ¯ s d3 x Y4 where Y denotes the simulation domain. Fig. 6 shows that the relative error of 4¯ s,p is less than 10% during the whole simulation, which

Fig. 4. Time averaged dimensionless pressure, p/(4smax qs gh0 ), as a function of the dimensionless vertical coordinate, z/h0 , for different superficial gas velocities.

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mf Fig. 5. Time averaged dimensionless pressure, p/(4smax qs gh0 ), as a function of the dimensionless vertical coordinate, z/h0 , for different numerical settings (Wgin /Wg,2 = 0.8).

is acceptable for the present applications. More importantly, e4 does not increase with time after it has reached a plateau region at the beginning. Finally, e4 appears insensitive to the numerical settings discussed above.

by one tracer parcel. This choice of parcel factors leads to approximately 2 million tracer parcels. Further important process details, as for example, mass flow rate of fluidization gas as well as the gas coming from the loop and the initial amount of HDPE are given in Table 6. For the numerical simulation we use cubic cells with a grid

5.2. Industrial-scale olefin polymerization plant In gas-phase olefin polymerization fluidized bed, the polymer powder particles (HDPE; high density poly-ethylene) are suspended in an upward flowing reaction gas. The gas velocity is high enough to fluidized the particles, but not high enough to entrain the particles out of the reactor. A continuous gas flow enters the reactor at the bottom through a gas distributor plate which evenly distributes the gas throughout the area from the reactor base (see Fig. 7). The gas in contact with the catalyst and growing polymer reacts in the dense fluidized bed. The unreacted gas leaves the reactor at the top to a recycle gas compressor. In Fig. 7 a schematic view of the pilot scale gas-phase reactor (GPR) is provided. The total volume of the reactor is 1.5 m3 with a H over D ratio of H/D = 8. The cylindrical part has a volume of 0.85 m3 and a Hc over D ratio of Hc /D = 5. The GPR is operated at approximately 85 ◦ C with an operating pressure of 15 bars, where we assume that the walls can be considered as adiabatic. The initial particle size distribution and the corresponding parcel factors are given in Table 5. Similar to the previous section the parcel factor determines the number of “real” particles represented

Fig. 7. Schematic drawing of the olefin polymerization fluidized bed with H/D = 8 and a volume of 1.5 m3 .

Table 5 Initial particle size distribution of HDPE material and parcel factor, which is the number of particles represented by one tracer parcel.

mf Fig. 6. Time evolution of normalized deviation between 4¯s and 4¯ s,p , e4 , for Wgin /Wg,2 = 0.8.

ds [mm]

Mass fraction [%]

Parcel factor

0.025 0.053 0.105 0.180 0.35 0.5 0.75 1

0.03 0.67 2.18 14 14.34 32.17 20 16.61

1.6 • 107 1.68 • 106 2.16 • 105 4.29 • 104 5830 2000 593 250

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Table 6 Physical properties and principal flow conditions in the olefin polymerization gas-phase reactor. For remaining physical parameter the reader is referred to Schneiderbauer et al. [2] the gas properties are given at standard conditions. Property/condition

Symbol

Value

Unit

Particle density Monomer density (ethylene) Monomer viscosity (ethylene) Comonomer density (propylene) Comonomer viscosity (propylene) Solids volume fraction at maximum packing Initial solids inventory Mass inflow fluidization gas Mass inflow loop gas Mass inflow 2nd gas Inflow temperature main in Inflow temperature loop in Inflow temperature 2nd gas in Initial temperature Monomer main in Comonomer main in Monomer loop in Comonomer loop in Monomer 2nd gas in Comonomer 2nd gas loop in Operating pressure Initial comonomer content Initial catalyst concentration

qs qmo l mo qco l co 4smax

855 1.18 10 −5 1.81 0.78 • 10 −5 0.52 223.6 2.2 0.25 0.1 350 300 350 350 0.59 0.41 0 1 0 1 15 0.5 0.15

kg m −3 kg m −3 Pa s kg m −3 Pa s – kg kg s −1 kg s −1 kg s −1 K K K K – – – – – – bars – mol l −1

T˜ g,0 = T˜ s,0 Y˜ 1in Y˜ 2in in,loop Y˜ 1 ˜Y in,loop 2 in,loop Y˜ 1 in,loop Y˜ 2

po cc0 [C0 ]

spacing of D = 100ds  [2] where ds  denotes the Sauter mean diameter of the HDPE material given in Table 5. This grid spacing is large enough to ensure reasonable computation times (in particular, this yields a computational time of approximately 1 h for 1 s real time employing 24 AMD 2.3 GHz AMD CPUs) and small enough to guarantee grid independent results of the cgTFM [19, 50] In the case of such coarse grids, it is common to employ no-slip boundary conditions for both phases [36]. Finally, it has to be noted that first we performed a numerical simulation without tracer parcels (i.e. assuming constant of particle size distribution throughout the reactor) and without chemical reactions to let the bed expand. Then, we inserted

Fig. 9. Time evolution of the solids inventory as well as the mean voidage.

the tracer parcels according to the particle size distribution given in Table 1 and according to the local solids volume fraction. Fig. 8 shows snapshots of the solids volume fraction in the middle plane. The figure shows that bubbles (voids) are formed directly above the distributor plate, where the fluidization gas is injected. These, bubbles coalescence and grow with respect to height. In general, many characteristic features of gas-solid FBRs are related to the presence of bubbles and dominated by their behaviors. For example, an adequate modeling of the bubble behavior (bubble rise velocity, bubble number density, bubble size distribution) is inevitable to obtain reasonable results for solids mixing and gas-solid contact [19] It has to be noted that these voids unveil a high solids volume fraction, which is coming from small unresolved solid structures within the bubbles. This is generally observed in coarse grid simulations of fluidized beds [18,19] Finally, the figure reveals that the gas voidage appears to decrease with time, which is indicated by the dense regions without bubbles in the bottom area of the bed at t = 50 s. This decrease of the mean voidage can be explained by the growing particles, where the growth rate is given by Eq. (25). Further justification is given in Fig. 9. While during the first 50 s the bed

Fig. 8. Snapshots of the solids volume fraction in the middle plane.

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Fig. 10. Time evolution of the cumulative particle size distribution. t0 corresponds to the size distribution given in Table 5.

mass increases by approximately 7 kg (≈3% of initial solids inventory), the mean voidage in the bed decreases from 0.64 to 0.62. As clearly indicated by the figure the decrease of the mean voidage is on the one hand triggered by the increase of mass through Eq. (26) since the bed requires time to adapt to the increasing solids inventory; on the other hand, the underlying particle size distribution affects the gas-solid drag force (compare with Table 2) since the particles are growing. In particular, Fig. 10 shows how the particle size distribution evolves with time. The figure clearly unveils that the particle size distribution is shifted towards larger particles showing a larger mean diameter. This, in turn, implies that the gas-solid drag force decreases when the fluidization velocity is kept constant. The figure further reveals that the intermediate sized particles show the largest growth rate, which is connected to the “catalyst profile”. Eq. (23) yields that HDPE particles of intermediate age show the largest growth rate, which, in turn, implies that the growth of the intermediate sized particles is more pronounced than the growth of the fine and large particles.

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Fig. 11 shows snapshots of the monomer concentrations for different times. It can be clearly seen that the monomer concentration decreases along the vertical direction, since the monomer is consumed by the chemical reaction leading to particle growth. Thus, in regions, where no monomer is consumed by the polymerization reaction, such as in the freeboard, the monomer concentration remains constant. Furthermore, comparing the different snapshots shows that the monomer concentration stays nearly constant in time, which implies a nearly constant reaction rates within the investigated time span. Small fluctuations are induced through the inhomogeneous solids volume fraction, such as bubbles. In Fig. 12 the snapshots of the gas-phase temperature are plotted. Due to the heat of reaction the solid phase is heated up, which in turn, yields similar heating of the gas phase through inter-phase heat transfer (Eq. (18)). Furthermore, the gas-phase temperature increases with time until the reactor reaches its equilibrium temperature (i.e. where the incoming flux plus generated heat through reactions balances the outgoing heat flux). Fig. 13 unveils that the equilibrium temperature is reached approximately after 40 s within the fluidized bed and after approximately 50 s in the freeboard. The latter is delayed in time since the heat is generated within the fluidized bed and has to be convected to the freeboard afterwards. A comparison with measurements of the gas-phase temperature (Fig. 13) shows that the computed equilibrium temperatures are in fairly good agreement with the real plant observations. However, there are still minor differences, which may be explained by the insufficient assessment of the heat losses through the reactor walls. In particular, the temperatures are slightly overpredicted. Currently, we do not assess the heat transfer through the reactor walls, which could have a considerably impact on the equilibrium temperatures. Furthermore, the unresolved heterogeneous structures may affect the reaction rates, which is not considered in this work. Similar, to the filtered drag coefficient the filtered reaction rate may be significantly smaller than the reaction rate computed from the coarse grid simulation and thus, reduce the reaction heat [43]

Fig. 11. Snapshots of the mass fraction of the monomer Y˜ 1 .

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Fig. 12. Snapshots of the gas phase temperature T˜ g in Kelvin.

Finally, in Fig. 14 the probability distribution functions of the comonomer content, cc, and the crystallinity, cf , of the polymer particles are plotted. Here, we used a 400 bins with a bin size of 1/400 to compute the probability distribution. The figure unveils that the comonomer content decreases with time, which is a consequence of the smaller kinetic constant of the comonomer, KP2 , compared to the kinetic constant of the monomer, KP1 (compare with Table 3). In contrast, the overall crystallinity of the polymer particles increases as cf decreases as cc increases. However, these effects appear to be very small within the considered time interval since the growth of the polymer particles is very slow compared to the characteristic time scale of the fluidized bed.

To conclude, modeling polymerization reactions appears to inevitably affect the hydrodynamics of the gas-phase reactor due to particle growth and reaction heat. 6. Conclusions and outlook We have presented a generalization of our previously published hybrid-TFM [1,16,31,32] by (i) additionally considering sub-grid modification to account for the unresolved scales when using coarse grids and by (ii) including a growth as well as heat transfer model to account for the chemical reactions occurring in a gas-phase polymerization plant. Such a modeling strategy enables the efficient numerical analysis of reactive poly-disperse gas-phase reactors without requiring computationally demanding multi-fluid models, which are coupled to population balance approaches. To conclude, the results clearly show that the reactive hybridTFM is able to picture segregation in bi-disperse fluidized bed as well as the global behavior of a industrial-scale olefin polymerization fluidized bed. Further main findings are 1. Segregation in fluidized beds can be effectively analyzed by employing the presented hybrid model. The increase of computational time is approximately 15% compared to TFM simulation with one solid phase for the presented cases. However, the increase of computation time is strongly connected to the number of tracer particles. This, in turn, implies that doubling the number of tracer parcels yields an increase of computational time of about 30%. 2. The particle growth has a direct impact on the bed expansion as well as local fluidization, which directly influences the gassolid contact and mixing.

Fig. 13. Time evolution of gas-phase temperature at the different probe locations (compare with Fig. 7).

Nevertheless, the polymerization model has to be verified further against more detailed plant data. Future efforts will concentrate on the numerical analysis of different process conditions and their detailed evaluation against experimental data. Finally, the impact of sticking in very hot regions on the flowability of the polymer will be considered.

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Fig. 14. Probability distribution function of a) comonomer content cc and b) the crystallinity cf at t = 100 s.

Nomenclature

Acknowledgments

For additional nomenclature regarding the kinetic constants of the polymerization model see Table 3.

This was funded by the Christian-Doppler Research Association, the Austrian Federal Ministry of Economy, Family and Youth, and the Austrian National Foundation for Research, Technology and Development. Furthermore, the authors want to thank the K1MET center for metallurgical research in Austria, which is partly funded by the Austrian government (www.ffg.at), for its financial contribution.

Operators () ( )

Favre average Reynolds average

References Latin symbols ak ds cf cp cc k e poly Fk g HD hq ni [Mi ] MPi,k N sp Rq Rp S Tq uq u p,k Wg Yi xi x p,k

parcel factor particle diameter (m) fraction crystallinity (–) specific heat (J kg −1 K −1 ) content of monomer (–) particle restitution coefficient acceleration of parcel k due to gas-solid drag force (m s −1 ) gravitational acceleration (m s −2 ) heterogeneity index specific enthalpy of phase q (kg m −1 s −2 ) tracer number density of size class i (m −3 ) concentration of monomer i (kg m −3 ) amount of polymer produced at tracer k due to monomer i (kg) number of particle size classes mass source term of phase q due to chemical reactions (kg m −3 s −1 ) rate of polymerization (mol l −1 s −1 ) extent of segregation temperature of phase q velocity of phase q (m s −1 ) velocity of tracer parcel k (m s −1 ) superficial gas velocity (m s −1 ) mass fraction of gas-species i (–) fraction of particle size class i position of tracer k

Greek symbols beff Df 4q qq Sq

effective drag coefficient (due to sub-grid heterogeneities) filter/grid size volume fraction of phase q density of phase q (kg m −3 ) stress tensor of phase q

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Please cite this article as: S. Schneiderbauer et al., A Lagrangian-Eulerian hybrid model for the simulation of poly-disperse fluidized beds: Application to industrial-scale olefin polymerization , Powder Technology (2016), http://dx.doi.org/10.1016/j.powtec.2016.12.063