Epstein N., Grace J.R. Spouted and Spout-Fluid Beds: Fundamentals and Applications

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Computational ﬂuid dynamic modeling of spouted beds 67

0.9

0.7

0.053 m

0.118 m

z=0.218 m

0.5

Simulations

Experimental

0.3

0.00 0.02 0.04 0.06 0.08

Radial coordinate,r,m

Voidage, α

Figure 4.6. Computed voidage distributions in a spouted bed

compared with the experimental

data of He et al.

at U = 0.59 m/s. (d

= 1.41 mm, ρ

= 2503 kg/m

, ρ

= 1.2 kg/m

, D

0.152 m, D

= 0.019 m, θ = 60

◦

0.9

0.7

0.053 m

0.118 m

z=0.218 m

0.5

Simulations

Experimental

0.3

0.00 0.02 0.04 0.06 0.08

Radial coordinate,r,m

Voidage, α

Figure 4.7. Computed voidage distributions in a spouted bed

compared with the experimental

data of He et al.

at U = 0.70 m/s. (d

= 1.41 mm, ρ

= 2503 kg/m

, ρ

= 1.2 kg/m

, D

0.152 m, D

= 0.019 m, θ = 60

◦

It was shown that different drag models led to signiﬁcant differences in simulations.

Among the drag models tested, the Gidaspow model

gave the best agreement with

experimental observations, both qualitatively and quantitatively. By comparing dif-

ferent drag models, Gryczka et al.

found better agreement with experiments based

on the Schiller and Naumann model.

68 Xiaojun Bao, Wei Du, and Jian Xu

Table 4.2. DEM simulations of spouted beds reported in the literature.

Investigators Model closure Software code Contribution

Kawaguchi

et al.

50,51

Newton’s 3rd law for

coupling between

phases

Author-developed Proposed ﬁrst Eulerian-Lagrangian approach

for spouted beds; obtained typical spouted

bed ﬂow patterns.

Takeuchi

et al.

49,52,53

Author-developed Simulated 3D cylindrical-conical spouted

beds. Proposed new method for treating

boundary conditions.

Zhong et al.

k-ε turbulent model

for gas motion

Author-developed Simulated turbulent motions of the gas and

particles by treating the two phases

separately. Particle motion modeled by

DEM and gas motion by k-ε model.

Zhao et al.

55,56

Low Reynolds k − ε

turbulence model

for ﬂuid phase

Author-developed Simulated ﬂow of particles in 2D spouted

bed with draft plates with a low Reynolds

number k-ε turbulence model for the ﬂuid

phase.

Swasdisevi et al.

Author-developed Simulated aerodynamics of particles and gas

ﬂow in slot-rectangular spouted bed with

draft plates. Calculated U

and pressure

drop agreed well with correlations of

Kalwar et al.

Limtrakul et al.

Author-developed By combining DEM and mass transfer

models, investigated local mass transfer in

gas–solid catalytic spouted bed reactor for

decomposing ozone; results agreed well

with the experimental results of Rovero

et al.

2. Particle–particle collisions have been found to have a signiﬁcant effect on particle

motion in the spout zone and almost no effect in the annular zone.

4,10

3. The computations indicate a major inﬂuence of the frictional stress model and param-

eter selection on modeling dense solids ﬂow.

4,10

4.3 Eulerian-Lagrangian approach

In the Eulerian-Lagrangian approach, the ﬂuid phase is treated as a continuum by solving

the time-averaged Navier-Stokes equations, whereas the dispersed phase is solved by

tracking a large number of individual particles through the computed ﬂow ﬁeld, not

requiring additional closure equations.

Table 4.2 lists previous computations of this

type of model for spouted bed hydrodynamics. The dispersed phase can exchange

momentum, mass, and energy with the ﬂuid phase, and the two phases are coupled by

interphase forces. The DEM approach offers a more natural way to simulate gas–solid

ﬂows, with each individual particle tracked. However, it is much more computationally

demanding, especially as the number of par ticles simulated becomes large.

The framework of the collision model includes hard- and soft-sphere approaches. In

hard-sphere models, trajectories of particles are determined by momentum-conserving

Computational ﬂuid dynamic modeling of spouted beds 69

binary collisions. The interactions between particles are assumed to be pairwise additive

and instantaneous. In the simulation the collisions are processed one by one according

to the order in which they occur. For a not-too-dense phase, hard-sphere models are

considerably faster than soft-sphere models. Hard-sphere models have been used for

spout-ﬂuid beds by Link et al.

47,48

For systems of dense phase or inelastic particles,

soft-sphere models must be used. In these more complex situations, the particles may

interact via short- or long-range forces, and the trajectories are determined by integ rating

the Newtonian equations of motion. Soft-sphere models use a ﬁxed time step; conse-

quently, the particles are allowed to overlap slightly. The contact forces are subsequently

calculated from the deformation history of the contact using a contact force scheme.

Soft-sphere models in the literature differ from each other mainly with respect to the

contact force schemes used. Most current DEM simulations of spouted beds are based

on soft-sphere models.

As pointed out by Takeuchi et al.,

there are two problems in DEM simulation of

spouted beds. One is that it is extremely difﬁcult to establish stable spouting, regardless

of adjustments of related parameters, such as particle diameter, gas velocity, and nozzle-

to-bed size ratio. The other is the fact that two-dimensional simulation of spouted beds is

more difﬁcult to converge for the ﬂuid phase, which makes it difﬁcult to select a proper

turbulence model to describe the central jet.

Kawaguchi et al.

50,51

ﬁrst proposed an Eulerian-Lagrangian approach to simulate

spouted beds, and typical spouted bed ﬂow patterns were obtained. Takeuchi et al.

49,52

further simulated a spouted bed in three dimensions in a cylindrical coordinate system;

the predictions showed typical characteristics of spouted beds, in good agreement with

experimental results. In later work, Takeuchi et al.

proposed a new method for the

treatment of the boundary conditions that satisﬁes both the continuity and momentum-

balance requirements for the gas phase in three-dimensional gas ﬂow along the cone

surface.

To overcome the problem of turbulence modeling, Zhong et al.

simulated the turbu-

lent motion of the gas and particles by treating the two phases separately, with particle

motion modeled by DEM and gas motion by the k − ε turbulent model. Similarly, Zhao

et al.

55,56

simulated the motion of particles in a two-dimensional spouted bed with draft

plates, using a low Reynolds number k − ε turbulence model for the ﬂuid phase. Their

simulation results were in good agreement with experimental data.

The application of spouted beds as chemical reactors is of increasing interest, both

experimentally and for CFD studies. Several authors simulated spouted bed reactors

by DEM models coupled with chemical reactions. This work is also summarized in

Table 4.2.

4.3.1 Numerical approaches

In the Eulerian-Lagrangian approach, the motion of individual particles is predicted by

calculating the contact force on each particle, whereas the gas ﬂow ﬁeld is based on the

continuity and Navier-Stokes equations.

70 Xiaojun Bao, Wei Du, and Jian Xu

4.3.1.1 Discrete approach for particle motions

In the DEM approach, the translational and rotational motions of particle i are given by









n,ij



t,ij



+ m



g +



drag,i

(4.20)

and











t,ij

−



, (4.21)

where





, I

, and



drag,i

are the linear velocity vector, angular velocity vector,

moment of inertia, and drag force, respectively, of particle i, whereas



n,ij



t,ij

, and



are the normal contact force, tangential contact force, and rolling-resistance torque,

respectively, of particle i with neighboring particle j. The normal and tangential contact

forces are given, respectively, by



n,ij

=−



n,ij

+ η









(4.22)

and



t,ij

=−min



t,ij

+ η





,μ





n,ij





, (4.23)

where



n and



s are normal and tangential unit vectors, and δ

and δ

are particle dis-

placements in the normal and tangential directions. Parameters k

and k

are the spring

stiffness of normal force and tangential forces, respectively; η

and η

are the coefﬁcients

of dissipation in the normal and tangential directions.

The drag force exerted on particle i can be described by



f ,i



1 − ε





u −



−∇P



. (4.24)

Here



u is gas velocity, P the gas pressure,



the particle velocity, V

the volume of

each particle (= π d

/6), and β is the drag coefﬁcient predicted by the Ergun equation

for the dense regime and the Wen–Yu equation

for more dilute ﬂow, as given by

Eqs. (4.5) to (4.7).

The relative rotation between contacting particles or between a particle and a wall

in contact produces a rolling resistance because of the elastic hysteresis loss or time-

dependent deformation. Because particles circulate internally in spouted beds, they may

have a great rolling velocity and, therefore, a signiﬁcant rolling resistance. The torque

resistance expression is given

by:

=−μ





)

. (4.25)

Computational ﬂuid dynamic modeling of spouted beds 71

However, there exists a maximum moment:



≤ I





/t. (4.26)

4.3.1.2 Continuum approach for ﬂuid motion

The locally averaged equation of continuity and equations of motion are used to compute

the motion of the gas phase, given, respectively, by

∂

∂t

(α

) +

∂

∂x

(α

) = 0 (4.27)

and

∂(α

)

∂t

∂

∂x

(α

) =

∂

∂x



(μ

+ μ

)



∂u

∂x

∂u

∂x



−α

∂p

∂x

− f

+ ρ

g. (4.28)

Here α

, ρ

, μ

, and f

drag

are the local voidage, gas density, gas viscosity, turbulent

viscosity, and ﬂuid drag force, respectively. The drag force is again obtained from the

Ergun equation

and the Wen–Yu equation,

depending on the voidage, expressed by

Eqs. (4.5) to (4.7).

The solution procedure for the DEM approach differs from that for the TFM approach

in the meshing and solving the pressure and velocity ﬁelds. Unlike TFM, in DEM

computations most researchers – such as Kawaguchi et al.

and Zhong et al.

–have

employed rectangular grids for tapered-wall regions so the grids near the gas entrance

are large enough to contain a sufﬁcient number of particles.

Kawaguchi et al.,

Takeuchi et al.,

and Zhong et al.

solved for the ﬂuid phase

using the SIMPLE method.

Pressure–velocity coupling can be based on the Synthesize

Modes and Correlate (SMAC) method, which has been applied successfully in simulating

other multiphase turbulent ﬂows.

4.3.2 Comparison with experimental results

4.3.2.1 Flow ﬁeld

Figure 4.8 shows the distribution of time-averaged particle velocity reported by

Kawaguchi et al.

for a spouted bed with exactly the same geometry and gas–solid

properties as in the experiments of He et al.

43,44

The three characteristic regions – spout,

fountain, and annulus – can be clearly observed. Figure 4.9 presents the time-average

spout shape and vector ﬁeld of the solids ﬂow predicted by Takeuchi et al.

The spout

radius is taken to correspond to the radial distance at which the time-average axial par-

ticle velocity is zero. The spout diameter varies from 0.01 m at the nozzle exit to 0.02 m

at the bed surface. The predicted spout shape corresponds to one of the possible spout

72 Xiaojun Bao, Wei Du, and Jian Xu

Figure 4.8. Particle velocity vectors predicted by Kawaguchi et al.

for a spouted bed at U/U

1.3. (d

= 3 mm, ρ

= 2500 kg/m

, ρ

= 1.2 kg/m

, D

= 0.152 m, D

= 0.019 m, θ = 60

◦

= 1.46 m/s.)

shapes identiﬁed by Mathur and Epstein.

In the region from the nozzle exit to z =

0.05 m, particles are entrained strongly toward the spout axis with high velocities in the

longitudinal direction, caused by the potential core of gas near the nozzle exit.

Figure 4.10 compares the spout shape computed by Kawaguchi et al.

with the

experimental measurements of He et al.

43,44

Because the sizes of the particles in the

experiments and in the simulation differed, there is some discrepancy. For example,

the diameter of the spout in the cylindrical region obtained experimentally

is smaller

than predicted.

4.3.2.2 Particle velocities in spout

Figure 4.11 shows radial distributions of the vertical component of particle velocity

at eight different heights in the spout computed by Kawaguchi et al.

Experimental

results of He et al.

also appear in the ﬁgure. With increasing height, the particle

velocity decreases along the central axis, but increases at the periphery of the spout.

Qualitatively, the experimental results agree well with the simulation results, except

for z > 0.268 m. The radius of the spout at which the particle velocity becomes zero

did not depend much on z in the experiments, whereas it increased with height in

the CFD simulations. The quantitative difference in the spout diameter between the

simulation and the experimental results arises, at least in part, from different particle

sizes.

Computational ﬂuid dynamic modeling of spouted beds 73

0.2

0.15

spout interface

bed surface

0.1

0.05

0.01 0.03

Radial coordinate,r,m

Height, m

0.05 0.07

Figure 4.9. Time average vector ﬁeld of the solids ﬂow and spout shape in a ﬂat-bottomed

spouted bed predicted by Takeuchi et al.

= 2.4 mm, ρ

= 2650 kg/m

, ρ

= 1.2 kg/m

= 0.14 m, D

= 0.02 m, θ = 180

◦

, U = 1.0 m/s = 1.08U

350

300

250

200

=3 mm

=3 mm)

=1.4 mm)

He et al.

Roy et al.

=6 mm

U/U

=1.3

U/U

=1.3

U/U

=1.2

U/U

=1.2

U/U

=1.1

U/U

=1.1

U/U

=1.3

U/U

=1.3

Vertical coordinate, mm

150

100

350

300

250

200

150

100

0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40

Radial coordinate,r,mm

(a)

(b)

Figure 4.10. Spout shapes: (a) calculated by Kawaguchi et al.

; (b) measured experimentally by

He et al.

and Roy et al.

(ρ

= 2500 kg/m

, ρ

= 1.2 kg/m

, D

= 0.152 m, D

= 0.019 m,

θ = 60

◦

, U

= 1.46 m/s.)

74 Xiaojun Bao, Wei Du, and Jian Xu

(a)

0.318

0.268

0.218

0.168

0.118

0.083

0.052

0.022

0 5 10 15 20 25 30

Radial coordinate,r,mm

Particle velocity, m/s

(b)

0.318

0.268

0.218

0.168

0.118

0.083

0.052

0.022

0 5 10 15 20 25 30

Radial coordinate,r,mm

Figure 4.11. Variation of particle velocity in the spout for U/U

= 1.1 with height from the

bottom of the vessel: (a) from experimental measurements of He et al.

for d

= 1.41 mm, ρ

2503 kg/m

, ρ

= 1.2 kg/m

, D

= 0.152 m, D

= 0.019 m, θ = 60

◦

; (b) predicted by DEM

model of Kawaguchi et al.

for d

= 3 mm, ρ

= 2500 kg/m

, ρ

= 1.2 kg/m

, D

= 0.152 m,

= 0.019 m, θ = 60

◦

, U

= 1.46 m/s.

1.2

1.0

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

30.4

45.6

60.8

76.1

91.2

246810

Draft plate

7.5 mm

Simulation Experimental Z (mm)

0.8

0.6

0.4

0.2

0.0

0204060

Experimental

Simulation

(a)

(b)

80 100 120

Hei

ht,z,mm Radial coordinate,r,mm

Particle velocity, m/s

Figure 4.12. Experimental (PIV) measurements compared with simulated (DEM) proﬁles of

particle vertical velocities in the spout

: (a) vertical proﬁle along spout axis; (b) lateral proﬁle at

various bed heights. (d

= 2.03 mm, ρ

= 2380 kg/m

, ρ

= 1.2 kg/m

, D

= 0.152 m, D

0.009 m, U = 1.58 m/s.)

Zhao et al.

simulated the particle motion in a two-dimensional spouted bed (2DSB)

with draft plates. Figure 4.12 compares experimental and simulated proﬁles of vertical

particle velocities in the spout. DEM simulation is seen to provide good predictions of

the particle vertical velocity proﬁle along the column axis. Par ticle imaging velocimetry

(PIV) measured and DEM-simulated initial particle velocities in the lower part of the

Computational ﬂuid dynamic modeling of spouted beds 75

(a)

(b)

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Radial coordinate,r,mm

Particle velocity, mm/s

Radial coordinate,r,mm

0.318

U/U

= 1.1

0.268

0.218

0.168

0.118

0.083

0.053

0.022

Figure 4.13. Particle velocity proﬁles in the annulus, where z is the height from the bottom of the

vessel: (a) from experimental measurements of He et al.

for d

= 1.41 mm, ρ

= 2503 kg/m

= 1.2 kg/m

, D

= 0.152 m, D

= 0.019 m, θ = 60

◦

; (b) predicted by the DEM model of

Kawaguchi et al.

for d

= 3 mm, ρ

= 2500 kg/m

, ρ

= 1.2 kg/m

, D

= 0.152 m, D

0.019 m, θ = 60

◦

, U

= 1.46 m/s.

spout agree well. This can be attributed to the application of a turbulence model to

describe the jet near the inlet nozzle. The radial proﬁles from both the DEM and PIV

methods indicate that the particle vertical velocities adjacent to the draft plates are lower

than the local average value. As the height coordinate, z, increases from 30.4 to 91.2

mm, the particle vertical velocity increases. The difference between the DEM and PIV

results near the draft plates may result from overestimation of the wall friction.

4.3.2.3 Particle velocities in annulus

Figure 4.13 shows simulated

and experimental

radial distributions of the vertical

component of particle velocity in the annulus. Note that the downward velocity com-

ponents are plotted as positive values in this ﬁgure, with two different ordinate scales.

In the experimental data, the particle velocity proﬁles are nearly ﬂat in the cylindrical

region, but decrease in t he neighborhood of the wall and in the near-boundary region of

the spout. The reduction of the particle downward velocity near the wall is caused by

friction between the particles and the wall. Particle downward velocity increases with

height owing to particle entrainment from the annulus to the spout.

The predicted

particle downward velocity increases with height more than was found experimentally.

This can be attributed to the reduction of the cross-sectional area of the annulus with

increasing height, because, as shown in Figure 4.10, the spout diameter increases with

height.

Radial proﬁles of axial solids velocities in the annulus predicted by Takeuchi et al.

are compared in Figure 4.14 with experimental data of Tsuji et al.,

measured in a

76 Xiaojun Bao, Wei Du, and Jian Xu

0.1

0.08

z = 0.1500 m

0.1000 m

0.0800 m

0.0625 m

0.0375 m

Model Exp.

0.06

0.04

0.02

–0.02

–0.04

–0.06

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Radial coordinate,r,m

Time-average axial velocity, m/s

Figure 4.14. Comparison of radial distributions of particle velocities in the annulus at different

heights above nozzle exit: CFD predictions are from Takeuchi et al.;

experimental results are

from Tsuji et al.

= 2.4 mm, ρ

= 2650 kg/m

, ρ

= 1.2 kg/m

, D

= 0.14 m, D

= 0.02 m,

θ = 180

◦

, U = 1.0 m/s = 1.08U

ﬂat-bottomed vessel of column diameter 0.14 m and nozzle diameter 20 mm. At z =

0.0375 and 0.0625 m, the particles near the nozzle exit are strongly entrained to the core

of the spout. Particles at z = 0.10 and 0.15 m have ﬂat velocity proﬁles in the annulus.

These proﬁles suggest increased downward particle ﬂow at higher levels in the annulus.

The predicted descent velocities are in good agreement with the experimental data.

4.4 Concluding remarks

The characteristic patterns of spouted beds can be well reproduced by both the Eulerian-

Eulerian and the Eulerian-Lagrangian approaches. Parameters such as spout diameter,

minimum spouting velocity and voidage proﬁle all show reasonably good agreement

with experimental data. This indicates that CFD modeling can serve as an important

tool for predicting gas and solid behavior in spouted beds.

The Eulerian-Eulerian approach is usually the ﬁrst choice for simulation because of

its lesser use of computational resources. The successful application of this approach

depends mainly on closure of the momentum equations, as the simulation is sensitive

to the APG, drag coefﬁcient, interparticle coefﬁcient of restitution and solid friction

stresses. To describe the solid-phase stresses, the kinetic theory of granular ﬂow has

been widely adopted.

Although more computational capacity is required, the Eulerian-Lagrangian approach

offers a more physically satisfying way to simulate gas–solid ﬂows, with each individual

particle tracked in the simulation. It can be applied readily for particle tracking – for

example, for residence time and particle circulation studies.

An important challenge in CFD studies of spouted beds is to describe properly the

inherent turbulence for both the solid and gas phases, especially for the spout region.