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

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4 Computational ﬂuid dynamic

modeling of spouted beds

Xiaojun Bao, Wei Du, and Jian Xu

4.1 Introduction

As reviewed in Chapter 3, numerous theoretical and experimental studies have been

carried out in recent decades in an attempt to model the hydrodynamics of spouted beds.

Most of the early models are one-dimensional, with spout and annulus considered sepa-

rately by assuming that some parameters are constant. In addition, these models, though

useful as ﬁrst approximations, are complex, or require parameters to be determined by

experiments.

Thanks to the explosion of computational power, the advance of numerical algorithms,

and deeper understanding of multiphase ﬂow phenomena, computational ﬂuid dynamics

(CFD) modeling has become a powerful tool for understanding dense gas–solid two-

phase ﬂows in the recent past. The main advantage of CFD modeling is that a wide

range of ﬂow proper ties of the gas and solids may be predicted simultaneously without

disturbing the ﬂows.

Currently, there are two main CFD approaches: the Eulerian-Eulerian approach (two-

ﬂuid model, TFM), and the Eulerian-Lagrangian (discrete element method, DEM)

approach. In the f ollowing two sections, the fundamentals and applications of these

two approaches in hydrodynamic modeling of spouted beds are treated separately. In

each section, the main aspects of the CFD approach are introduced brieﬂy, followed

by application to modeling of spouted bed hydrodynamics. Comparison of the CFD

predictions with experimental results is also discussed.

4.2 Eulerian-Eulerian approach

In the Eulerian-Eulerian approach, the ﬂuid and particulate phases are treated mathemat-

ically as interpenetrating continua. Several studies (listed in Table 4.1) have shown that

this approach is capable of predicting gas-solids behavior in spouted beds. Because the

volume of one phase cannot be occupied by the other, the concept of overlapping phases,

each with its own volume fraction, is introduced. Volume fractions of the overlapping

phases are assumed to be continuous functions of space and time, with their sum always

equal to 1. The conservation equations have similar structure for each phase. Owing

Spouted and Spout-Fluid Beds: Fundamentals and Applications, eds. Norman Epstein and John R. Grace.

Published by Cambridge University Press.



Cambridge University Press 2011.

Table 4.1. Eulerian simulations of spouted beds.

Investigators Model closure Software code Contribution

Krzywanski

et al.

Author-

developed

code

Developed a multidimensional model to describe gas

and particle dynamic behavior in a spouted bed.

Lu et al.

3−6

Kinetic theory of

granular ﬂow;

Gidaspow

drag model

with switch

function

K-FIX; M-FIX Viewed spout and annulus as interconnected regions.

Incorporated kinetic-frictional constitutive model:

kinetic theory of granular ﬂow; friction stress was

calculated by combining normal frictional stress

model of Johnson et al.

and modiﬁed frictional shear

viscosity model proposed by Syamlal et al.

;

behavior of agglomerates of nanoparticles in spouted

bed systems was simulated numerically.

Du et al.

9−11

Kinetic theory of

granular ﬂow

FLUENT Found that the descriptions of interfacial forces and

solid stresses play important roles in determining the

hydrodynamics for spouting both coarse and ﬁne

particles.

Wang

et al.

Kinetic theory of

granular ﬂow

FLUENT Found that actual pressure gradient (APG term) in

conical spouted beds signiﬁcantly inﬂuences static

pressure proﬁles.

Shirvanian

et al.

Kinetic theory of

granular ﬂow;

Gidaspow

drag model

FLUENT Developed three-dimensional (3D) simulation model to

describe isothermal liquid–solid two-phase ﬂow in a

rectangular spouted bed; it was able to predict

experimentally observed “choking.”

Wu and

Mujumdar

Kinetic theory of

granular ﬂow;

Gidaspow

drag model

with switch

function

FLUENT Described bubble formation and motion inside a 3D

spout-ﬂuid bed.

Gryczka

et al.

Kinetic theory of

granular ﬂow

FLUENT Compared drag models of Schiller and Naumann,

Wen and Yu,

Syamlal and O’Brien,

Gidaspow

et al.,

Koch and Hill,

van der Hoef et al.,

and

Beetstra et al.

Better agreement with experiments

was obtained by applying Schiller and Naumann

model.

Gryczka

et al.

Kinetic theory of

granular ﬂow

FLUENT Pointed out that an appropriate drag model alone is not

sufﬁcient to ﬁt the simulation results to the

experimental ﬁndings; other contributions such as

particle rotation are also important.

Bettega

et al.

24−25

Kinetic theory of

granular ﬂow

FLUENT Obtained experimental data for a semicylindrical

spouted bed; compared CFD simulations from a 3D

simulation scheme; discussed inﬂuence of ﬂat wall

on solid behavior in semicylindrical vessel. Presented

a numerical scaleup study of spouted beds. Veriﬁed

that the scale-up relationships of He et al.

produced

good numerical results.

Santos

et al.

Kinetic theory of

granular ﬂow

FLUENT Simulated patterns of solids and gas ﬂows in a spouted

bed using 3D Eulerian multiphase model; 3D

predictions showed better accuracy than 2D ones.

Duarte

et al.

Kinetic theory of

granular ﬂow;

Gidaspow

drag model

FLUENT Simulated spouted beds of conical and

conical-cylindrical geometries. Predicted pressure

drops and particle velocities agreed well with

experimental values.

Computational ﬂuid dynamic modeling of spouted beds 59

to the continuum description of the particle phase, two-ﬂuid models require additional

closure laws to describe particle–particle and particle–ﬂuid interactions. The Eulerian-

Eulerian approach is often the ﬁrst choice for simulation because of its lesser use of

computational resources.

The full Eulerian-Eulerian approach includes (1) conservation equations of mass and

momentum for each phase, with an interphase momentum transfer term; (2) closure of

the equations, which requires proper description of interfacial forces, solids stress, and

turbulence of the two phases; and (3) meshing of domain, discretization of equations,

and solution algorithms.

4.2.1 Conser vation equations of mass and momentum

Based on the general Eulerian multiphase model, governing equations of mass and

momentum for spouted beds can be derived by assuming that (1) there is no mass

transfer between spouting gas and bed particles, (2) the bed pressure gradient for stable

spouting is constant, and (3) the densities of both phases are constant.

The partial differential TFM equations for describing particulate and ﬂuid ﬂows in

ﬂuidized beds presented by Gidaspow

are also commonly adopted for spouted beds.

The basic conservation equations are the

volume fraction balance equation (q = g, s):



q=1

= 1 (4.1)

and the

mass conservation equation for phase (q = g, s):

∂α

∂t

+∇·(α

v

) = 0. (4.2)

The momentum conservation equation for phase q (q = g, s)is

∂

∂t

(α

v

) +∇·(α

v

) =−α

∇P +∇·τ



p=1



+ α

(



lift,q



vm,q

), (4.3)

where



is the external body force,



lift,q

the lift force,



vm,q

the virtual mass force, and



the interaction force between the two phases. In most investigations, only drag and

gravity are considered, with the lift force and virtual mass neglected.

4.2.2 Closure of the equations

4.2.2.1 Drag models

The drag force exerted on par ticles in ﬂuid–solid systems is usually represented by the

product of a momentum transfer coefﬁcient, β, and the relative velocity (

v −u) between

60 Xiaojun Bao, Wei Du, and Jian Xu

the two phases:

drag

v −u

f (α

)(v −u) = β(v −u). (4.4)

Ding and Gidaspow

employed the Ergun

equation for dense phase calculation and

the Wen–Yu

equation for dilute phase calculation:

⎧

⎪

⎨

⎪

⎩

Ergun

= 150

+ 1.75

v −u

,α

< 0.8

We n −Yu

v −u

−2.65

,α

≥ 0.8,

(4.5)

where the drag coefﬁcient is expressed by

⎧

⎪

⎨

⎪

⎩

[1 + 0.15(α

)

0.687

], for Re

< 1000

= 0.44, for Re

≥ 1000

(4.6)

and

v −u

. (4.7)

To avoid discontinuity at the boundary (α

= 0.8, α

= 0.2) between these two

equations, Lu et al.

introduced a switching function that gives a rapid transition from

one to the other:

arctan[150 × 1.75(0.2 − α

)]

+ 0.5. (4.8)

Thus, the drag model can be expressed as

β = (1 − ϕ

)β

Ergun

+ ϕ

We n −Yu

. (4.9)

4.2.2.2 Kinetic theory of granular ﬂow

Granular ﬂows can be classiﬁed into two distinct ﬂow r egimes: a rapidly shearing regime,

in which stresses arise because of collisional or translational transfer of momentum, and

a plastic or slowly shearing regime, in which stresses arise because of friction among

particles in contact.

31,32

At high particle concentrations, individual particles interact with

multiple neighbors through sustained contacts. Under such conditions, the normal forces

and associated tangential frictional forces of sliding contacts are the major contributors

to the particle stresses. At low particle concentrations, however, the stresses are caused

mainly by particle–particle collisions. The physical basis for such an assumption may

capture the two extreme limits of granular ﬂow: a rapidly shearing ﬂow regime in

which kinetic contributions dominate, and a quasistatic ﬂow regime in which friction

dominates. With the introduction of the concepts of solid “pressure” and “viscosity,”

Computational ﬂuid dynamic modeling of spouted beds 61

the well-known granular kinetic theory

is well established and is widely employed to

estimate solid stresses.

Closure of the solid phase momentum equation requires a description of the solid phase

stress. Analogous to the thermodynamic temperature for gases, a granular temperature



is introduced as a measure of particle velocity ﬂuctuations:



2

). (4.10)

The granular temperature conservation equation is



∂

∂t

(ρ



) +∇·(ρ

v

)



=(−p

I +τ

):∇v

−∇·(k



∇

) − v

+ φ

(4.11)

where (−p

I +τ

):∇v

is the generation of energy by the solid stress tensor, k



∇

the diffusion of energy (k



is the diffusion coefﬁcient), ∇

is the collisional dissipation

of energy, v

12(1 − e

0,ss

√



3/2

, and φ

=−3β

is the energy exchange

between the ﬂuid and solid phases.

The kinetic solid pressure is given

= α



+ 2ρ

(1 + e

)α

0,ss



, (4.12)

where g

0,ss

is the radial distribution function expressed

0,ss



1 −



s,max





−1

(4.13)

and α

s,max

is the maximum particle volume fraction.

The solid bulk viscosity is given

0,ss

(1 + e

)



, (4.14)

the solid shear viscosity by

0,ss

(1 + e

)



10ρ

√

π

96(1 + e

0,ss



1 +

0,ss

(1 + e

)



(4.15)

and the dissipation ﬂuctuating energy by

= 3



1 − e



0,ss







−∇v



, (4.16)

where e

is the interparticle coefﬁcient of restitution, d

is the particle diameter, and

0,ss

is a radial distribution function.

62 Xiaojun Bao, Wei Du, and Jian Xu

If solid frictional stresses are considered, the solid frictional stress P

is added to the

stress predicted by the kinetic theory of granular ﬂow:

= P

kinetic

+ P

= μ

kinetic

+ μ

(4.17)

The Schaeffer model,

developed for very dense gas–solid systems, is used to predict

the frictional stress P

,giving

= A(α

− α

s,min

)

(4.18)

and

· sin φ





∂u

∂x

−

∂v

∂y





∂v

∂y





∂u

∂x







∂u

∂y

∂v

∂x



, (4.19)

where φ is the internal friction angle of particles. This value can be taken to be 28.5

◦

for glass beads.

A and n are constants.

4.2.3 Meshing, discretization, and solving algorithm

Because most spouted bed columns have conical bases, some researchers

have meshed

the domain with unstructured grids, whereas Lu et al.

employed a body-ﬁtted coordinate

method and reexpressed the equations of continuity and motion for the gas and solid

phases.

Discretization in space produces a system of ordinary differential equations for

unsteady problems and algebraic equations for steady problems. I mplicit or semi-implicit

methods are generally used to integrate the ordinary differential equations, producing a

set of (usually) nonlinear algebraic equations. The most common discretization methods

are the ﬁnite volume method (FVM), ﬁnite element method (FEM), ﬁnite difference

method (FDM), and boundary element method (BEM).

37,38

The FVM is the classical

approach, used most often in commercial software and research codes. Through the

FVM, the governing equations are solved on discrete control volumes.

39,40

As seen in Table 4.1, commercial CFD packages such as FLUENT, CFX, and MFIX,

which solve sets of complex nonlinear mathematical expressions based on the fun-

damental equations of ﬂuid ﬂow, heat, and mass transport, are widely adopted for

computation. The algebraic equations are solved iteratively using complex computer

algorithms embedded within these CFD software packages. The Semi-Implicit Method

for Pressure-Linked Equations (SIMPLE) algorithm, developed by the Spalding group,

is widely used to solve the momentum equations. Details of this algorithm are discussed

in several CFD books, such as that by Patankar.

4.2.4 Comparison with experimental results

Owing to the continuum description of the particle phase, the TFM approach requires

additional closure laws to describe particle–par ticle and particle–wall interactions. In

Computational ﬂuid dynamic modeling of spouted beds 63

the hydrodynamic modeling of spouted beds by the TFM approach, considerable atten-

tion has been paid to closure of the equations, especially to the description of the solid

stress. The Lu group

viewed the spout and annulus as two interconnected regions,

and constitutive equations describing the particulate solids pressure, viscosity, and elas-

ticity moduli were implemented into a hydrodynamic simulation program. They later

incorporated a kinetic-frictional constitutive model for dense assemblies of solids

3,4

that treated the kinetic and frictional stresses of particles additively. The kinetic stress

was modeled using the kinetic theory of granular ﬂow, whereas the friction stress com-

bines the normal frictional stress model proposed by Johnson et al.

and the modiﬁed

frictional shear viscosity model of Syamlal et al.

Interfacial forces other than drag

are less signiﬁcant, and can usually be neglected. Du et al.

9–11

showed that for both

coarse and ﬁne particle spouting, the descriptions of interfacial forces and solid stresses

play important roles in determining the hydrodynamics. Gryczka et al.

15,23

pointed

out that the application of each drag model is limited to a speciﬁc range of particle

Reynolds numbers. The search for an appropriate drag model alone is not sufﬁcient

for an accurate simulation, as other contributions, such as particle rotation, also play

a role.

Simulation results of Wang et al.

showed that the actual pressure gradi-

ent (APG) term in conical spouted beds, introduced as the default gravity term, plus

an empirical axial solid-phase source term, had the greatest inﬂuence on static pres-

sure proﬁles, and that the introduction of this term can improve the CFD simulation

for gas–solid conical spouted beds. On the basis of the foregoing developments, more

and more researchers have begun to use this Eulerian-Eulerian model to investigate

new structures and scaleup of spouted beds (Chapter 17). These works are listed in

Table 4.1.

Although numerical simulation is a useful tool to obtain detailed forecasting of spout-

ing behavior without disturbing the ﬂow, it is important to compare numerical predictions

with corresponding experimental results.

4.2.4.1 Particle velocity ﬂow ﬁelds

Lu et al.

used the Eulerian-Eulerian approach with a kinetic-frictional constitutive model

incorporated to simulate spouted beds. The column geometries and particle properties

matched those studied experimentally by He et al.

43,44

and San Jos

eetal.

to facilitate

comparison with experimental data.

Figure 4.1 shows the distributions of instantaneous particle concentrations and veloc-

ities in a spouted bed for a spouting gas velocity of 12 m/s. The ﬂow patterns in the

spouted bed appear to be qualitatively well reproduced: in the spout zone, the particle

concentration is low and the particle velocity is high, particles move vertically upward

and radially toward the jet axis. Particles are carried by gas to the top of the spout,

forming a fountain from which par ticles cascade downward in the outer region. Particle

concentration in the fountain is higher than in the spout, but lower than in the annulus.

In the annular zone, particles are predicted to move downward and radially inward with

the highest particle concentration and the lowest particle velocity, in agreement with

experimental obser vations.

64 Xiaojun Bao, Wei Du, and Jian Xu

0.9507

0.8520

0.7533

0.6546

0.5559

0.4527

0.3586

0.2599

0.1612

0.0625

t = 6.0 s t = 6.1 s t = 6.2 s t = 6.3 s t = 6.4 s t = 6.5 s

Concentration

Figure 4.1. Instantaneous volumetric concentration and velocity of particles predicted by Lu

et al.

for an inlet gas jet velocity of 12.0 m/s. (d

= 1 mm, ρ

= 1200 kg/m

, ρ

= 1.2 kg/m

= 0.19 m, D

= 0.02 m, θ = 60

◦

4.2.4.2 Particle velocities

Radial proﬁles of time-average vertical particle velocities simulated by Lu et al.

are

shown in Figures 4.2 and 4.3, in comparison with experimental results of He et al.

43,44

measured by optical ﬁber probes. There is reasonable agreement between computational

predictions and experimental results. The vertical particle velocity is seen to decrease

with height at both superﬁcial gas velocities. In both cases, the vertical particle velocity

decreases from the axis in the spout and becomes negative in the annulus.

Figure 4.4 shows simulated

and experimental

distributions of axial particle veloc-

ities in a conical spouted bed for an inlet gas jet velocity of 8.3 m/s. The simulated

results are in good agreement with the experimental ones. The particle velocity reaches

a minimum somewhere in the annular zone. Figure 4.4 indicates that the magnitude of

the axial particle velocity is highest close to the axis, and decreases with distance from

the axis.

Time-averaged horizontal particle velocities predicted by Lu et al.

are plotted in

Figure 4.5. Both the simulated and measured horizontal particle velocities tend to be

inward at low heights in the spout zone, but outward at a higher level.

4.2.4.3 Voidage

Figures 4.6 and 4.7 show simulated

and experimental

proﬁles of time-averaged

voidage for superﬁcial gas velocities of 0.59 and 0.70 m/s, respectively. Comparison

Computational ﬂuid dynamic modeling of spouted beds 65

–1

0.00 0.02 0.04 0.06 0.08

Radial coordinate,r,m

Simulations

Experimental

0.118 m

z=0.218 m

0.053 m

Particle velocity, m/s

Figure 4.2. Comparison between time-mean vertical particle velocities predicted by Lu et al.

with those measured experimentally by 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

◦

Simulations

Experimental

0.053 m

z=0.218 m

–1

0.00 0.02 0.04 0.06 0.08

Radial coordinate,r,m

Particle velocity, m/s

0.118 m

Figure 4.3. Comparison between time-mean vertical particle velocities computed by Lu et al.

and

experimental measurements by He et al.

at U = 0.70 m/s. (d

= 1.41 mm, ρ

= 2503 kg/m

= 1.2 kg/m

, D

= 0.152 m, D

= 0.019 m, θ = 60

◦

of Figures 4.6 and 4.7 indicates that the local voidage increases slightly with increasing

superﬁcial gas velocity throughout the entire spout and annulus zones. The simulation

results show that the local voidage decreases with increasing height and with increasing

distance from the axis. The time-averaged voidage in the annulus zone is slightly higher

than at minimum ﬂuidization.

66 Xiaojun Bao, Wei Du, and Jian Xu

Simulations

Experimental

0.03 m

0.11 m

z=0.17 m

0.00 0.02 0.04 0.06 0.08 0.10

Radial coordinate,r,m

Particle velocity, m/s

–1

Figure 4.4. Computed time-mean vertical particle velocities in a conical spouted bed

compared

with San Jos

eetal.

experimental data for an inlet gas jet velocity of 8.3 m/s. (d

= 1.41 mm,

= 2503 kg/m

, ρ

= 1.2 kg/m

, D

= 0.152 m, D

= 0.019 m, θ = 60

◦

0.3

0.0

–0.3

–0.6

0.00 0.04

Simulations

Experimental

z=0.03 m

z=0.17 m

0.05 m

Radial coordinate,r,m

Particle velocity, m/s

0.08

Figure 4.5. Computed time-mean horizontal particle velocities in a conical spouted bed

compared with San Jos

eetal.

experimental data for an inlet gas jet velocity of 8.3 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

◦

4.2.4.4 Other ﬁndings

1. In CFD modeling of gas–solid two-phase ﬂows, the drag force is the most important

force acting on particles. Thus it has a critical impact on CFD predictions of dense

gas–solid two-phase systems in spouted beds. Du et al.

assessed several widely used

drag models and incorporated some of them into CFD simulations of spouted beds.