Multi-fluid Eulerian modeling of limestone particles’ elutriation from a binary mixture in a gas–solid fluidized bed

Journal of Industrial and Engineering Chemistry 17 (2011) 229–236

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Multi-fluid Eulerian modeling of limestone particles’ elutriation from a binary mixture in a gas–solid fluidized bed Mehdi Azadi * Department of Chemical Engineering, College of Engineering, Shahid Bahonar University of Kerman, Kerman, Iran

A R T I C L E I N F O

Article history: Received 27 January 2010 Accepted 27 July 2010 Available online 2 March 2011 Keywords: CFD Elutriation Simulation Particle Eulerian Fluidized bed

A B S T R A C T

The elutriation of limestone from a binary mixture of particles in a gas–solid fluidized bed was simulated via computational fluid dynamics (CFD) modeling. A multiphase Eulerian model incorporating kinetic theory of granular flow was used. The effect of superficial gas velocity and particle diameter was investigated using the computational fluid dynamics model. The effects of different modeling parameters on the model predictions were evaluated. The results of simulations showed that elutriation of particles increased with increasing the superficial gas velocity and decreasing particle size. The best model predictions were achieved, using Syamlal–O’Brien drag model. The effect of restitution coefficient on the simulation results was negligible. The computed results of the simulations are compared with the experimental findings. ß 2011 The Korean Society of Industrial and Engineering Chemistry. Published by Elsevier B.V. All rights reserved.

1. Introduction Elutriation is a process which can be used as a separation method in gas–solid fluidized beds to generate solid aerosols from a mixture of particles. The importance of elutriation arises, noticing several industries require a method of separating particles according to their sizes and densities. However, complexity of the flow behavior during elutriation in fluidized beds results in the challenge of studying and analyzing elutriation’s hydrodynamics. Computational fluid dynamics (CFD) has proved to be a promising tool in studying multiphase flows, with rapid growth of numerical simulations in the recent years [1,2]. Computational fluid dynamics provides new approach to study the complicated hydrodynamics of gas and particulate flows. CFD modeling has shown its reliable accuracy to predict the flow behavior of fluidized beds [3,4]. Two different classifications of CFD models have been applied to fluidized beds; Eulerian–Lagrangian and Eulerian–Eulerian. The Eulerian–Lagrangian models are based on particle trajectories and solve the equation of motion for each particle, considering collision of particles and forces acting on the particles [5]. The particle trajectories are computed individually at specified intervals during the fluid phase calculation [6]. However, the Eulerian–Eulerian models are more appropriate to be applied to fluidized beds [6,7]. In this approach, all solid phases are treated as continuum as the

* Tel.: +98 917 304 4801; fax: +98 3412118298. E-mail address: [email protected].

fluid phase and averaged equations of motion are utilized [8–18]. Different averaging methods have been used. Ishii [19] and Drew and Lahey [20] used time averaging while Harlow and Amsden [21], Rietema and Van den Akker [22], and Ahmadi [23] used a volume averaging method. Thus, the averaged equations can be used to simulate multiphase flows containing a significant volume fraction of solid particles, in this approach. Eulerian–Eulerian and Eulerian–Lagrangian approaches were compared by Gera et al. [24]. In order to describe the rheology of particulate phases, many authors have used the kinetic theory of granular flow to obtain the constitutive equations [25–35]. In this study, a multiphase Eulerian computational fluid dynamics model using the kinetic theory of granular flow was developed to study the elutriation of limestone particles in a binary mixture of fluidized bed, with silica sand as the coarse bed particles. The effect of gas velocity and particle size was investigated using the CFD model. The influences of some modeling parameters such as drag models, restitution coefficient, time step size, and convergence criterion were observed with the results of simulations. The results of the simulations are compared with experimental findings [36] to demonstrate the effectiveness of the model. 2. Computational fluid dynamics multi-fluid model A multi-fluid Eulerian model was used to model the hydrodynamics of the elutriation in the gas–solid fluidized bed. The model involved kinetic theory of granular flow to obtain constitutive equations. Viscous forces and solid pressure of the solid phases can

1226-086X/$ – see front matter ß 2011 The Korean Society of Industrial and Engineering Chemistry. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jiec.2011.02.011

230

M. Azadi / Journal of Industrial and Engineering Chemistry 17 (2011) 229–236

kgs is gas–solid momentum exchange coefficient of gas and solid phase, r  t g is gas phase tensor. Solid phase (s = 1, 2, . . ., N):

Nomenclatures CD Cfr,i Di Ess G g0,ss I I2D Kgs kQs mi ni p Ps Re T

vi vt

drag coefficient [] friction coefficient [] diameter [m] restitution coefficient [] gravity [m/s2] radial distribution coefficient [] stress tensor [] second invariant of the deviatoric stress tensor [] gas–solid momentum exchange coefficient [] diffusion coefficient of granular energy [kg/s m] mass of particles [kg] number of particles [] fluid pressure [Pa] solid pressure [Pa] Reynolds number [] time [s] velocity [m/s] terminal velocity [m/s]

! ! ! ! ! @ ðe r n Þ þ r  ðes rs n s n s Þ ¼ r  t s þ es rs ~ g þ kgs ð n g  n s Þ @t s s s N X

þ

!

!

kns ð n n  n s Þ

where kns, is solid–solid exchange coefficient and ts is solid phase stress tensor. Gas phase stress tensor: 2 3

! T

!

!

t g ¼ eg mg ½r  n g þ r  n g   eg mg r  n g Solid phase stress tensor of phase s,   ! ! T ! 2 t s ¼ Ps I þ es ms ðr  n g þ r  n g Þ þ es ls  ms r  n s I 3

(5)

(6)

where ms and ls are solid shear viscosity and bulk viscosity, respectively. Solid shear viscosity:

ms ¼ ms;col þ ms;kin þ ms;fr

Greek letters ei volume fraction [] Qi granular temperature [m2/s2] li bulk viscosity [kg/s m] mi shear viscosity [kg/s m] ri density [kg/m3] ti stress tensor [Pa] gQm collision dissipation of energy [kg/s3 m] fgs transfer rate of kinetic energy[kg/s3 m] f angle of the integral friction [] h effectiveness factor []

(4)

n¼1;s 6¼ n

(7)

Collisional viscosity: 

4 5

ms;col ¼ e2 rs ds g 0;ss ð1 þ ess Þ

Qs p

1=2

(8)

where g0,ss is radial distribution, ess is restitution coefficient, and Qs is granular temperature. Kinetic viscosity [38]: pffiffiffiffiffiffiffiffiffiffi   e r d Qs p 2 1 þ ð1 þ ess Þð3ess  1Þes g 0;ss ms;kin ¼ s s s (9) 6ð3  ess Þ 5 Frictional viscosity [39]:

be described as a function of granular temperature, by taking advantage of the kinetic theory of granular flow [3,37]. No mass transfer was allowed between phases. The model used in this study is described as follows.

P sin f ms;fr ¼ spffiffiffiffiffiffiffi

(10)

2 I2D

where Ps is the solid pressure, f is the angle of the integral friction and I2D is the second invariant of the deviatoric stress tensor Solid bulk viscosity [40]:

2.1. Continuity equations Gas phase: ! @ ðe r Þ þ r  ðeg rg n g Þ ¼ 0 @t g g

(1)

Solid phase: ! @ ðe r Þ þ r  ðes rs n s Þ ¼ 0 @t s s

(2)

where e, r and n are volume fraction, density and velocity respectively. 2.2. Momentum equations

! ! ! @ ðe r n Þ þ r  ðeg rg n g n g Þ ¼ r p þ r  t g þ eg rg ~ g @t g g g

s¼1

where p and ~ g are gas pressure and gravity, respectively.

Qs p

1=2

(11)

Solid pressure: A general model for the solids pressure in the presence of several solid phases by Gidaspow [3] P s ¼ es rs Qs þ

N X p n¼1

3

g 0;ns d3sn ns nn ð1 þ esn Þ f ðmn ; ms ; Qn ; Qs Þ

(12)

where dns = dn + ds/2 is the average diameter, nn and ns are the number of particles, mn and ms are the masses of particles in the phase phases n and s, and f is a function of the masses of the particles and their granular temperatures.

The momentum equation for gas and solid phases Gas phase:

N X ! ! kgs ð n s  n g Þ þ



4 3

ls ¼ e2 rs ds g 0;ss ð1 þ ess Þ

However, this equation can be simplified to give the following form [6] (3) ps ¼ es rs Qs þ

N X d3 2 ns ð1 þ ens Þg 0;ns en es rs Qs d3n p¼1

(13)

M. Azadi / Journal of Industrial and Engineering Chemistry 17 (2011) 229–236

Radial distribution for n solid phases g o;ss

  N es 1=3 1 1 X en ¼ 1ð Þ þ ds 2 n¼1 dn es;max

(14)

where (n are solids phases)

es ¼

N X

en

where ens is the restitution coefficient, Cfr,ns is the friction coefficient between phases n and s, dn is the diameter of the particles of the phase n and go,ns is the radial distribution coefficient. The kinetic fluctuation energy which is derived from kinetic theory of granular flow can be described as the following expression [28]

(15)

  3 @ ðrs es Qs Þ þ r  ðrs es~ vs Qs Þ ¼ ð ps I þ t s Þ 2 @t

n¼1

Several drag models can be applied to obtain the gas–solid momentum exchange coefficient. Three drag models which were investigated with the model are as follow: The Syamlal–O’Brien drag model [38]    3es eg rg Res  ~ kgs ¼ C v ~ vg  (16) D vr;s s 4v2r;s ds

: r~ vs þ r  ðkQs rQs Þ  g Qs þ fgs

!2 4:8 0:63 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Res =vr;s

[()TD$FIG]

(17)

and vr;s is a correlation for the terminal velocity of the solid phase

vr;s ¼ 0:5ðA  0:06Res þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0:06Res Þ2 þ 0:12Res ð2B  AÞ þ A2 Þ

(18)

with

A ¼ e4:14 and B ¼ 0:8e1:28 for eg  0:85 g g and

and B ¼ e2:65 for eg  0:85 A ¼ e4:14 g g The Gidaspow drag model [42] When eg > 0.8   es eg rg ~ vs  ~ vg  2:65 3 eg K gs ¼ C D 4 ds

(19)

where CD ¼

i 24 h 1 þ 0:15ðeg Res Þ0:687 eg Res

(20)

and when eg  0.8 

K gs ¼ 150



rg es ~ vs  ~ vg  es ð1  eg Þmg þ 1:75 ds es d2s

(21)

The Wen and Yu drag model [43]   es eg rg ~ vs  ~ vg  2:65 3 eg K gs ¼ C D 4 ds

(22)

where CD ¼

i 24 h 1 þ 0:15ðeg Res Þ0:687 eg Res

(23)

For the particle–particle momentum exchange coefficient Syamlal–O’Brien-symmetric model [44] was used, which my be written as 2

kns ¼

3ð1 þ ens Þððp=2Þ þ c fr;ns ðp2 =8ÞÞes rs en rn ðdn þ ds Þ g o;ns 2pðrn d3n þ rs d3s Þ   vs  ~ vg   ~

(24)

(25)

vs is the energy generation by solid stress where ð ps I þ t s Þ : r~ tensor, kQs rQs is the energy diffusion, gQs is the collision dissipation of energy, and fgs is the transfer of kinetic energy.

Where according to Dalla Valle [41], CD is defined as CD ¼

231

Fig. 1. Schematic drawing of the fluidized bed.

M. Azadi / Journal of Industrial and Engineering Chemistry 17 (2011) 229–236

232

The diffusion coefficient of granular energy, kQs , can be described by Syamlal et al. model [45] pffiffiffiffiffiffiffiffiffiffi 15ds rs es Qs p kQs ¼ 4ð41  33hÞ   12 2 16  1þ h ð4h  3Þes g 0;ss þ ð41  33hÞhes g 0;ss (26) 5 15p

4. Results and discussion

where 1 h ¼ ð1 þ ess Þ 2 The dissipation energy of particles by collisions, gQs, which presents the rate of energy dissipation of the sth solid phase because of particle–particle collisions, is described with the following expression by Lun et al. [40]

g Qm ¼

approaching to the walls was created to take the wall effect more accurately into the account. The useful information was extracted until a quasi-steady-state condition was reached for the integral outlet solid flux. The time averaged variables were calculated between 15 s and 35 s. A summary of simulation settings is described in Table 1.

12ð1  e2ss Þg 0;ss pffiffiffiffi rs e2s Q3=2 s ds p

(27)

The kinetic energy transfer of random fluctuations in particle velocity is represented by fgs [42]

fgs ¼ 3K gs Qs

(28)

3. Simulation procedure The governing equations of the model were solved by a finite volume method. The computational domain was divided into a finite number of control volumes. The second order upwind discretization method was used. Discretization is a process which changes the partial differential equations of the model to algebric equations for numerical solution. The set of algebric equations were solved using a CFD based code (Fluent 6.3.26). The Phase Coupled SIMPLE (PCSIMPLE) algorithm by Vasquez and Ivanov [46], which is an extension of SIMPLE algorithm by Patankar [47] was utilized, for pressure–velocity coupling. Two solid phases were used to describe the binary mixture of particles. A two dimensional geometry was considered, as shown in Fig. 1, to study the elutriation of particles. Finer mesh

The CFD model was used to study the elutriation of limestone particles in a gas–solid fluidized bed of limestone–silica sand. Mixture of limestone–silica sand particles is a case which may represent a binary mixture of small-coarse particles and can be used to study the elutriation process of particles in a gas–solid fluidized bed [36,48]. Significance of limestone–sand system seems to be inevitable, as further applications of fluidized bed of limestone–sand particles have been reported in many publications [49–53]. The results of simulations were compared with the experimental data of Ma and Kato [36] to evaluate the performance of the model. 4.1. Sensitivity analysis of the model parameters Investigation of modeling parameters was carried out to determine the best configuration of the CFD model. The effects of various parameters including mesh resolution, time step size, coefficient of restitution, convergence criterion, and drag model were studied and presented as follow. 4.1.1. Mesh resolution Sensibility of the computed results to the mesh resolution was studied using different mesh cases. Three types of coarse (50  200), medium (70  300), and fine (90  400) grids were utilized to investigate the grid dependency of the modeling results. Fig. 2 offers a comparison of limestone particle axial velocities at z = 0.75 m with different mesh resolutions. As shown in Fig. 2, the medium (70  300), and fine (90  400) grids showed no significant change in results with respect to the results achieved with the coarse (50  200) grid. Increasing the grid resolution did not show important variation in the results. Thus, the 70  300

Table 1 Simulation setting for the multi-fluid model. Description

Value/setting

Compared parameters/additional comments

Bed height Bed width Static bed height

1.02 m 0.071 m 0.11 m

Fixed value Fixed value Fixed value

Small particles Mean diameter Density Volume fraction

35 mm 2350 kg/m3 0.07

58, 88 mm Limestone Fixed

Large particles Mean diameter Density Volume fraction Superficial gas velocity Gas density Gas viscosity Acceleration of gravity Restitution coefficient Drag law Grid Time step Maximum iterations per time step Convergence criteria Inlet boundary condition Outlet boundary condition Discretization method

331 mm 2600 kg/m3 0.4 0.6 m/s 1.225 kg/m3 1.85  105 kg/m s 9.81 m/s2 0.99 Syamlal–O’Brien 70  300 0.001 s 50 103 Velocity inlet Pressure outlet Second order upwind

Uniform distribution Silica sand Fixed value 0.3, 0.4, 0.5, 0.7, 0.8, 0.9 m/s Air Fixed value Fixed value 0.85, 0.9 Gidaspow, Wen and Yu 50  200, 90  400 0.0005 s, 0.0001 s 100 for convergence criterion of 104 104 Superficial gas velocity Fully developed flow Fixed

[()TD$FIG]

[()TD$FIG]

2.5

a

2.5

Particle axial velocity (m/s)

M. Azadi / Journal of Industrial and Engineering Chemistry 17 (2011) 229–236

2

233

∆t=0.001 s

Particle axial velocity (m/s)

70×300 90×400

2

50×200

1.5

1

0.5

∆t=0.0005 s ∆t=0.0001 s

1.5

1

0.5 0

Fig. 2. Comparison of particle axial velocities at z = 0.75 m with different mesh resolutions.

(radial  axial) grid was used as a grid independent case for the computations. 4.1.2. Time step Effects of several time steps on the simulation results were tested. Several authors have used time step size of 0.001 s for modeling gas particulate flow in fluidized beds [25,54,55]. Fig. 3 shows the effects of different time steps (0.001 s, 0.0005 s, and 0.0001 s) on the simulation results. Time steps of 0.001 s and 0.0005 s led to more stable results and convergence was achieved sooner in compare with time step size of 0.0001 s, for each set of iterations per time steps. Choosing a too small time step size may increase the relative error. In fact, the calculated change of a cell property between two time steps may become smaller than a fixed convergence criterion, resulting in a larger error. In addition, if the time step size is too large, the cell properties between two time steps will not be estimated accurately because of the jump of a cell property between two time steps [56]. The time step of 0.001 s was used as the base case whereas it showed time-step-independency of the results. 4.1.3. Coefficient of restitution Collision of particles plays a major role in dissipative energy of particles. The restitution coefficient of particles describes the variation in kinetic energy of particles due to interparticle collisions. Higher values of restitution coefficient describe higher elasticity of collisions, which result in lower dissipation rates. Higher coefficients of restitution are often used for gas–solid fluidized beds. The effects of different values of 0.85, 0.9, and 0.99 were investigated. The results of evaluations are shown in Fig. 4. It was found that varying coefficient of restitution did not change the results significantly. However, more realistic result was achieved with restitution coefficient of 0.99 as shown in Fig. 4b. Therefore, restitution coefficient of 0.99 was utilized in the evaluation of other parameters. 4.1.4. Convergence criteria Two values of 0.001 and 0.0001 were examined as the convergence criteria for scaled residual components. A convergence criterion of 0.0001 increased computational cost with taking more iterations per each time step. However, minor deviation of the results, in compare with the predictions achieved by using convergence criterion of 0.001, was negligible. Therefore, the value of 0.001 was applied in the model as the sufficient relative error between two successive iterations.

0.9

0.96

0.84

0.78

0.72

0.6

0.66

0.54

0.42

0.48

0.36

0.3

0.24

0.18

0.12

r/R

b Elutriation rate constant (kg/m2s)

0.9

0.96

0.84

0.78

0.72

0.6

0.66

0.48

0.54

0.42

0.3

0.36

0.24

0.18

0.12

0

0.06

r/R

0.06

-0.5 -0.5

0

0

14

Experimental ∆t=0.001 s

12

∆t=0.0005 s ∆t=0.0001 s

10 8 6 4 2 0

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Gas velocity (m/s) Fig. 3. (a) Comparison of particle axial velocities at z = 0.75 m with different time steps. (b) Effect of time step on elutriation rate constant at different gas velocities.

4.1.5. Drag model Different drag models including Syamlal–O’Brien, Gidaspow, and Wen and Yu were applied to evaluate the effect of interphase momentum exchange coefficient between gas and solid phases. A comparison of the modeling results using those drag laws are presented in Fig. 5. The model predictions were almost similar with all of the three drag models. Having considered the fact that Syamlal–O’Brien drag law agreed better with the experimental data, it was used in the optimal CFD model to obtain the most accurate results. Almuttahar and Taghipour [57] compared three drag models including Arastoopour et al. [58], Syamlal–O’Brien, and Gidaspow for a gas–solid riser fluidized bed and concluded that although the results were similar, Syamlal–O’Brien provided a better description of the hydrodynamics of the riser. Pugsley and McKeen [59] compared four drag models including Syamlal– O’Brien, Gidaspow, Ergun [60], and Gibilaro et al. [61] and reported that the models showed significant deviations at low voidage for the Syamlal–O’Brien drag model. 4.2. Optimal model analysis The optimal CFD model was developed according to the investigation of modeling parameters. Elutriation process of limestone particles was studied with the established model. Fig. 6 presents a comparison between estimated elutriation rate constants and experimental data, for various particle sizes at

[()TD$FIG]

[()TD$FIG]

M. Azadi / Journal of Industrial and Engineering Chemistry 17 (2011) 229–236

e=0.99 e=0.9 e=0.85

2.5 Syamlal-O'Brien

1

r/R

b

Experimental e=0.99

10

0.96

0.90

0.84

0.72

0.78

0.60

0.66

0.54

0.42

r/R

12

Elutriation rate constant (kg/m2s)

Elutriation rate constant (kg/m2s)

b

0.48

0.00

0.9

0.84

0.78

0.72

0.6

0.66

0.54

0.48

0.42

0.3

0.36

0.24

0.18

-0.5

0.12

-0.5 0

0

0.06

0

0.36

0.5

0.30

0.5

1.5

0.18

1

Wen & Yu

0.24

1.5

Gidaspow

2

0.06

Particle axial velocity (m/s)

2 Particle axial velocity (m/s)

a

2.5

0.96

a

0.12

234

e=0.9 e=0.85

8

6

4

2

12 Experimental Syamlal-O'Brien

10

Gidaspow Wen & Yu

8

6

4

2

0

0 0.3

0.4

0.5

0.6

0.7

0.8

0.3

0.9

0.4

Gas velocity (m/s)

different fluidization velocities. As shown in Fig. 6, increasing the superficial gas velocity increased the elutriation rate constant regardless of particle size. The effect of limestone particle size on the elutriation rate constant is also presented in Fig. 6. It was observed that increasing the particle size decreased the elutriation rate of particles. As one would expect, elutriation driving force (v0  vt ) increased with increasing the fluidization velocity (up to 0.9 m/s) and decreasing limestone particle size (from 88 mm to 35 mm). Fig. 7 presents the phase distribution of limestone particles with different superficial gas velocities at t = 5 s. It confirmed that the elutriation rate increased at higher superficial gas velocities. Fig. 8 presents the axial profile of limestone particles velocity, predicted with the optimal model. The particle velocity found to be almost axisymmetric. It showed to be the maximum at the center of the fluidized bed. Limestone particle flow pattern formed two kinds of ascending stream at the center, and descending stream at the side walls of the fluidized bed. One of the difficulties in CFD modeling of the elutriation of small particles is that interparticle cohesive forces are typically neglected, and this one makes no exception. As shown in Fig. 6 the model results overestimate the elutriation rate constant with respect to experimental data, which is caused by the effect of neglecting cohesive forces. However, as it can be observed in Fig. 6,

0.6

0.7

0.8

0.9

Gas velocity (m/s) Fig. 5. (a) Comparison of particle axial velocities at z = 0.75 m with different drag laws. (b) Effect of drag law on elutriation rate constant at different gas velocities.

[()TD$FIG] 12

Exp. 35 μm

Elutriation rate constant (kg/m2s)

Fig. 4. (a) Comparison of particle axial velocities at z = 0.75 m with different restitution coefficients. (b) Effect of restitution coefficient on elutriation rate constant at different gas velocities.

0.5

CFD 35 μm

10

Exp. 58 μm CFD 58 μm

8

Exp. 88 μm CFD 88 μm

6

4

2

0 0.3

0.4

0.5

0.6

0.7

0.8

0.9

Gas velocity (m/s) Fig. 6. The optimal model analysis of the effects of fluidization velocity and particle diameter on elutriation rate constant.

[()TD$FIG]

M. Azadi / Journal of Industrial and Engineering Chemistry 17 (2011) 229–236

235

Fig. 7. Effect of superficial gas velocity on the phase distribution of limestone particles.

the overestimation of the results with respect to experimental data was decreased with increasing the particle diameter. As a consequence, the best model prediction was achieved for the particle diameter of 88 mm. It shows that the effect of cohesive forces was less significant for the larger particle diameters. 5. Conclusions The suggested model based on the kinetic theory of granular [()TD$FIG] flow was able to predict the elutriation’s hydrodynamics of a

Particle axial velocity (m/s)

2

1.5

1

binary mixture in a gas–solid fluidized bed. The model showed that the elutriation rate increased with increasing the gas velocity and decreasing the particle diameter (from 88 to 35 mm). Effects of different parameters including restitution coefficient, drag model, time step, and convergence criteria on the simulation results, were studied with the model. The drag model had a minor effect on the results of simulation. However, the Syamlal–O’Brien drag law presented better results with respect to experimental data. Changing the particle–particle restitution coefficient did not affect the results significantly. The presented achievements were in good agreements with other authors’ findings and experimental results which increases the applicability of the model. However, some overestimations of the results were observed for smaller particle diameters which were caused by the cohesive forces. The effect of cohesive forces found to be less significant for larger particle diameters as better model prediction was achieved for larger particle diameters. References

0.5

0

0.9

0.96

0.84

0.78

0.72

0.6

0.66

0.54

0.42

0.48

0.3

0.36

0.18

0.24

0.12

0

0.06

-0.5

r/R Fig. 8. Radial profile of limestone particle axial velocity at z = 0.75 m.

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