Computer simulations of gas–solid flow in spouted beds using kinetic–frictional stress model of granular flow

Chemical Engineering Science 59 (2004) 865 – 878

www.elsevier.com/locate/ces

Computer simulations of gas–solid $ow in spouted beds using kinetic–frictional stress model of granular $ow Lu Huilina , He Yuronga , Liu Wentiea , Jianmin Dingb , Dimitri Gidaspowc;∗ , Jacques Bouillardd a Department

of Power Engineering, Harbin Institute of Technology, Harbin 150001, China Systems Inc., 555 Union Boulevard, Allentown, PA 18109, USA c Department of Chemical and Environmental Engineering, Illinois Institute of Technology, Chicago, IL 60616, USA d INERIS, Parc technologique Alata, Verneuil en Halate, 60550, France b Agere

Received 14 April 2003; received in revised form 22 August 2003; accepted 14 October 2003

Abstract A gas–solid two-$uid $ow model is presented. The kinetic–frictional constitutive model for dense assemblies of solids is incorporated in the simulations of spouted beds. This model treats the kinetic and frictional stresses of particles additively. The kinetic stress is modeled using the kinetic theory of granular $ow, while the friction stress is from the combination of the normal frictional stress model proposed by Johnson et al. (J. Fluid Mech. 210 (1990) 501) and the modi
1. Introduction Spouted beds are suited for the treatments of wood residues including the sawdust, industrial by-products, agro-forest residues by combustion, gasi
Corresponding author. Tel.: +1-312-567-3045; fax: +1-312-567-8874. E-mail addresses: [email protected] (L. Huilin), [email protected] (D. Gidaspow). 0009-2509/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2003.10.018

the use of auxiliary fuels (Khoshnoodi and Weinberg, 1978; Arbib and Levy, 1982; Vuthaluru et al., 2001). The spouted beds also allow for attaining a vigorous gas–solid contact as they operate stably in a wide range of gas $ow rates (Olazar et al., 1992). The spouted bed reactor must ful
866

L. Huilin et al. / Chemical Engineering Science 59 (2004) 865 – 878

zone were predicted by Lefroy and Davidson (1969) using a one-dimensional two-$uid model. They showed that the gas $ow in the annulus zone follows Darcy’s law and the pressure distribution at the interface between spout and annulus zones follows a cosine function. Littman et al. (1985) used the vectorial form of the Ergun equation to predict gas and solids motion in a spouted bed. A modi
2. Model equations The partial di@erential two-$uid model equations for describing particle and $uid $ow in $uidized beds are in Gidaspow (1994). These equations can be numerically solved explicitly in time and with a
(3)

The momentum balance for the gas phase is given by the Navier–Stokes type equation with an interphase momentum transfer term. From the two-dimensional Cartesian coordinate (x and y), the momentum equation of gas phase in the transformed coordinate can be expressed as below:   @ 1 @ @ (g g ’g ) + (g g Ug ’g ) + (g g Vg ’g ) @t J @ @  
 @’g @’g 1 @ g q11 = + q12 J @ J @ @ 
 @’g @’g @ g + q21 + Sg ; (4) + q22 @ J @ @

L. Huilin et al. / Chemical Engineering Science 59 (2004) 865 – 878

867

Table 1 Source terms of gas phase equations in transformed domain Equation

’

Sg

Momentum 

ug

@ @

Momentum 

vg




  

 @ug @ @ug @vg @vg 2 e @Ug @Vg + x + x  x + x + x y )z − + x @ @ @ @ @ 3 J @ @     


 @ug @ @ @ug @vg @vg 2 e @Ug @Vg + + e x + x  x + x + x  y − + x @ @ @ @ @ @ 3 J @ @
 @p @p − g x + x + g g g + gs (Us − Ug ) @ @     


 @ @ug @ 2 e @Ug @ug @vg @Vg @vg + e − y + y  x + y + y  y + y @ @ @ @ @ @ 3 J @ @     


@ug @ @ 2 e @Ug @Vg @ug @vg @vg + + e y + y  x + y + − + y y y )z @ @ @ @ @ 3 J @ @ @
 @p @p @p y + − g y + y + g g g + gs (Vs − Vg ) @ @ @ 

e

where Sg and ’g are respectively the source term of gas phase and the gas phase velocity component given in Table 1. The relations of coordinate transformation are de
q22 = (2x + 2y )J 2 ;

q12 = q21 = (x x + y y )J 2 :

(5)

The dynamic eddy-viscosity of gas phase is modeled by the sub-grid scale (SGS) model. The e@ective viscosity of gas phase may be calculated as (Deardor@, 1971)  (6) g = g; lim + g (0:1)2 Sij : Sij ; where  and Sij are the local
and a plastic or slowly shearing regime, in which stresses arise because of friction among particles in enduring contact. At high particle concentrations, individual particles interact with multiple neighbors through sustained contact. Under such conditions, the normal reaction forces and the associated tangential frictional forces of sliding contacts are the major contribution to the particle stresses. At low particle concentrations, however, the stresses induced mainly from particle–particle collisions. Srivastava and Sundaresan (2003) presented a frictional– kinetic closure for the particle phase stress. This model assumes that the frictional and kinetic stresses are additive. The kinetic stresses are based on the kinetic theory of granular materials. For frictional stress, a model accounting for strain rate $uctuations and slow relaxation of the assembly to the yield surface was proposed. Following Savage (1998), the particulate stress tensor, s , is simply the sum of the kinetic stress tensor k and the frictional stress tensor f , each contribution evaluated as follows: s = k + f :

(8)

The physical basis for such an assumption needs to be proved, however it may capture the two extreme limits of granular $ow: the rapid shear $ow regime where kinetic contributions dominate and the quasi-static $ow regime where friction dominates. Kinetic–frictional theories based on this simple additive treatment have been used to examine a wide variety of $ows such as $ow down inclined chutes and vertical channels (Srinivasa et al., 1997) and $ow in the bubbling $uidized bed (Srivastava and Sundaresan, 2003). The additive theory has been shown to be able to predict the qualitative features of such $ows. In rapid granular $ows, the kinetic energy of mean $ow
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L. Huilin et al. / Chemical Engineering Science 59 (2004) 865 – 878

Table 2 Source terms of solid phase equations in transformed domain Equation

’

Ss

Momentum 

us

@ @

Momentum 

vs




  

 @us @ (s − 2s =3) @Us @us @vs @vs @Vs + x + x  x + x + x y )z + x @ @ @ @ @ J @ @   
 

 @us @ (s − 2s =3) @Us @ @us @vs @vs @Vs + + s x + x  x + x + x y )z + x @ @ @ @ @ @ J @ @

 @p @ps @p @ps + s s g + gs (Ug − Us ) − s x + x + x + x @ @ @ @     


 @ @us @ (s − 2s =3) @Us @Vs @us @vs @vs + s y + + y  x + y + y y )z y @ @ @ J @ @ @ @ @     


 @us @ (s − 2s =3) @Us @ @us @vs @Vs @vs + s + y + y  x + y + + y y y )z @ @ J @ @ @ @ @ @

 @p @ps @p @ps + s s g + gs (Vg − Vs ) − s y + y + y + y @ @ @ @ 

s

is accounted for in the theory by a granular temperature, ", de


  @ 1 @ @ (s s ") + (s s Us ") + (s s Vs ") @t J @ @  
 1 @ ks @" @" q11 = + q12 J @ J @ @ 
 @ ks @" @" + q21 + q22 @ J @ @   (!11 x + !12 y ) @ (Us y − Vs y ) + J @ (x y − y x )   (!11 x + !12 y ) @ (Us y − Vs y ) + J @ (x y − y x )   (!21 x + !22 y ) @ (Us x − Vs x ) + J @ (y x − x y )   (!21 x + !22 y ) @ (Us x − Vs x ) + s ; + J @ (y x − x y )

(9)

where ks is the thermal conductivity coeNcient of solids phase, s the dissipation of $uctuating energy. The dissipation of pseudo-thermal energy due to gas–particle slip is neglected in this study. The coeNcients ! are de
(10)

(11)

The kinetic and the frication contributions to the ps are taken to be the summation of kinetic particle pressure and

L. Huilin et al. / Chemical Engineering Science 59 (2004) 865 – 878

the normal friction stress calculated from a model proposed by Johnson et al. (1990). The particle pressure is calculated as follows: (s − s; min )n ps = [1 + 2g0 s (1 + e)]s s " + F ; (12) (s; max − s )p where s; min is the solid concentration when frictional stresses becomes important, F; n and p are empirical material constants. The values of empirical parameters of s; min ; F; n and p are taken to be 0.5, 0.05, 2.0 and 5.0 for glass beads (Johnson et al., 1990), respectively. The shear viscosity accounts for the tangential forces. The kinetic shear viscosity, kinetic , is from the kinetic theory of granular $ow (Gidaspow, 1994). The frication shear viscosity, friction , is related to the friction normal stress. Syamlal et al. (1993) proposed a frictional viscosity of particles for dense gas–solid system. In present simulation of spouted beds, the modi
s (s; max − s )p



equation. For porosities greater than 0.8, Wen and Yu equation was used (Gidaspow, 1994). Thus the interface momentum transfer coeNcient, gs , can be calculated as follows: gs = ’gs gs |Ergun + (1 − ’gs )gs |Wen & Yu ; gs |Ergun = 150

gs |Wen & Yu = ’gs =

s2 g

g s |Ug − Us | + 1:75 g2 d2 g d

3Cd g s g |Ug − Us | −2:65 g ; 4d

arctan[150 × 1:75(0:2 − s )] + 0:5; *

  24 (1 + 0:15Re0:68 ); Cd = Re  0:44;

− D22 )2 + (D11 )2 + (D22 )2 ] + 14 (D12 + D21 )2

where + is the angle of internal friction. The value of + is taken to be 28:5◦ for glass beads (Johnson et al., 1990). The bulk solids viscosity is as follows:  " 4 2 : (14) s = s s dgo (1 + e) 3 * The dissipation $uctuating energy is (Gidaspow, 1994)


 1 @Us @Vs 4 " 2 2 s = 3(1 − e )s s go " − + ; (15) d * J @ @ where e is the coeNcient of restitution of particles, d the particle diameter, and go the radial distribution function at contact. The radial distribution function, go , can be seen as a measure for the probability of inter-particle contact. The equation of Bagnold (1954) is used in this work:

1=3 −1
s go = 1 − ; (16) s; max where s; max is the maximum particle packing.

Re ¿ 1000;

g 6 0:8; (18) g ¿ 0:8;

(19) (20)

(21)

;

(13)

2.4. Body-;tted transformation functions and boundary conditions Fig. 1 shows a simulated spouted bed with a conical base and the grids distribution. The symmetrical $ow is assumed in the simulations of spouted beds. Hence, all simulations are done in the half of bed in this study. Since the domain has an irregular boundary of the conical base in the physical system, it must be mapped onto a square domain in the computation domain so that computations can be performed on a rectangular transformed
@2 x @2 x @2 x − 2q + q 12 11 @2 @@ @2 = − J 2 [x P(; ) + x Q(; )];

2.3. Interphase momentum exchange In order to couple the momentum transfer between gas and particle phases, a model for the drag force is required. For porosities less than 0.8, the pressure drop due to friction between gas and particles can be described by the Ergun

Re 6 1000;

(17)

where Re is the Reynolds number, Re = d g |Ug − Us |=g .

(s − s; min )n sin + 1 6 [(D11

869

q11

(22)

@2 y @2 y @2 y + q11 2 − 2q12 2 @ @@ @ = − J 2 [(y P(; ) + y Q(; )]:

(23)

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L. Huilin et al. / Chemical Engineering Science 59 (2004) 865 – 878

At the wall, the gas tangential and normal velocities were set to zero. The normal velocity of particles was also set to zero. The following boundary equations apply for the tangential velocity and granular temperature of particles at the wall (Sinclair and Jackson, 1989): 6s s; max vsw = − * s s go J 3"(x2 + y2 )   @vs 2 2 @vs − (y y + x x ) ; (26) × (y + x ) @ @   k" @" @" s "w = − (y2 +x2 ) −(y y +x x ) @ @ 2 2 ew J (x +y ) √ 3* s s vs go " 3=2 + ; (27) 6s; max ew Fig. 1. Array of numerical grids in the half-spouted bed.

The grid control function P(; ) and Q(; ) is given as follows: n  P(; ) = − ai sgn( − i )exp(−ci | − i |)

where ew is the restitution coeNcient of the wall. A modi
i=1

−

m 

3. Simulation results

bj sgn( − j )

j=1

  ×exp −dj ( − j )2 + ( − j )2 ;

(24)

n  ai sgn( − i )exp(−ci | − i |) Q(; ) = − i=1

−

m 

bj sgn( − j )

j=1

  2 2 ×exp −dj ( − j ) + ( − j ) :

(25)

The
For simplicity, a two-dimensional rectangular and symmetrical section of spout beds, instead of cylindrical physical domain, is assumed in the following simulations. Several cases have been modeled in order to investigate the e@ect of operating conditions on the gas–solid $ow pattern in the spouted bed. Fig. 2 shows the distributions of instantaneous concentration and velocity of particles in a spouted bed with an inlet gas jet velocity of 12:0 m=s. The column diameter of the spouted bed and the incline angle of conical base are 190 mm and 60◦ , respectively. The diameter and density of particles are 1:0 mm and 1200 kg=m3 , respectively. The coeNcient of restitution is set to be 0.99. The height of the static bed of particles is 305 mm. The diameter of jet inlet is 20 mm. The formations of spout, fountain and annulus are established, as shown in Fig. 2. As expected, the particle concentration is low and particle velocity is high in the spout, while the particle concentration is high and particle velocity is low in the annular zone. Particles are carried up by gas in the spout, reach to the top of the bed and form a fountain in which the particle concentration is higher than that in the spout, but lower than that in the annular zone. Particles drop down in the bed because of gravity. The particles motion in the spout, annulus and fountain zones forms a circulation of particles in the bed. Fig. 3 shows the instantaneous porosity distribution at three locations respectively in the spout, annulus and fountain with the inlet gas jet velocity of 12:0 m=s. It can be seen that the local porosity $uctuates vigorously with time

L. Huilin et al. / Chemical Engineering Science 59 (2004) 865 – 878

871

Fig. 2. Instantaneous concentration and velocity of particles with an inlet gas jet velocity of 12:0 m=s.

1.0

10 -2

Power spectrum density

ug = 12.0 m/s y = 210 mm, x = 0 mm

Porosity

0.8 y = 540 mm, x = 0 mm

0.6

y = 210 mm, x = 0 mm

10 -6 10 -10 10 -2

y = 540 mm, x = 0 mm

10 -6 10 -10 10 -2

y = 210 mm, x = 30.4 mm

10 -6 ug= 12.0 m/s

10 -10 0.1

y = 210 mm, x = 30.4 mm

0.4

0

2

4

6 Time (s)

1

10

Frequency (Hz) 8

10

Fig. 3. Instantaneous porosity as a function of time at the inlet gas jet velocity of 12:0 m=s.

in the spout and fountain zones while $uctuates weekly in the annular zone. The local porosity $uctuation is mainly due to the motion of particles. Therefore, the local porosity $uctuation re$ects the spatio-temporal pattern of gas–solid $ow in the spout bed. Fig. 4 shows the corresponding power spectral density (PSD) of the porosity $uctuations. It can be seen that the PSD of the local porosity $uctuation exhibits a broad-band character with many spikes over a wide frequency range in the spout and fountain zones compared with that in the annulus. These phenomena imply higher exchange of momentum and energy between gas and particles in the spout and fountain zones than that in the annulus zone. Fig. 5 shows the distributions of time-averaged granular temperature in the spouted bed with the inlet gas jet veloc-

Fig. 4. Power spectra density of porosity at the inlet gas jet velocity of 12:0 m=s.

ities of 7.5 and 12:0 m=s, respectively. Near the inlet, the granular temperature is very low because of low concentration of particles. The granular temperature increases along with the height of the bed in the spout zone because of high particle velocity. The granular temperature of particles is very low in the annulus zone because of high particle concentration and low particle velocity. In the fountain zone the granular temperature is higher than that in the annulus zone, but lower than that in the spout zone. Therefore, the particle motions exhibit di@erent $ow patterns in the spouted bed. 3.1. Simulations of He et al. experiments He et al. (1994a,b) measured particle velocities and solid concentrations in a spouted bed by means of a
872

L. Huilin et al. / Chemical Engineering Science 59 (2004) 865 – 878

vs (m/s)

0.053

5

u su =0.59 m/s Simulations He et al.

3

8

y=0.2

0.11

1

18m

-1 0.02

0.00

0.04

0.06

0.08

Distance from spout axis x (m) Fig. 6. Computed vertical particle velocity in a spouted bed with He et al. (1994b) data at the super
8

vs (m/s)

Simulations He et al.

0.118

Description

Experiment

Computer run

s

g d s; max

Particle density Gas density Particle diameter Maximum solid volume fraction Gas super
2503 kg=m3 1:2 kg=m3 1:41 mm Not reported

Same Same Same 0.593

0:59 m=s; 0:7 m=s Not reported

Same 0.99

0:325 m 0:019 m 0:152 m

Same Same Same

plexiglass column with a inside diameter of 200 mm and a height of 1:4 m, and with a 60◦ conical base. A detailed description of the unit can be found in He et al. (1994a). Gas– solid properties and other related information are listed in Table 3. Initially, the spouted bed was assumed to be at minimum $uidization condition at atmospheric pressure. All simulations are continued for 20 s. The computed time-averaged variables are from the last 15 s. Radial pro
8m .21 y=0

Symbol

Ho Do D

5

2

Table 3 Parameters used for the simulation of He et al. experiments

usu e

0.053

Fig. 5. Time-averaged granular temperature distributions at the inlet gas jet velocities of 7.5 and 12:0 m=s.

u su =0.7 m/s

-1 0.00

0.02

0.04

0.06

0.08

Distance from spout axis x (m) Fig. 7. Computed vertical particle velocity in compared spouted bed compared with He et al. (1994b) data at the super
and goes to zero at the wall. At the gas super
L. Huilin et al. / Chemical Engineering Science 59 (2004) 865 – 878

873

60 u

0.053

0.9

su

=0.59 m/s

u

Solid mass flux (kg/m2s)

Simulations He et al. 0.118

0.7 18m y=0.2

Porosity εg

su

0.5

20

0

-20 y=0.118 m -40

0.3 0.00

0.02

0.04

0.06

= 0.59 m/s

40

y=0.218 m

-60 -0.5

0.08

-0.3

-0.1

0.1

0.3

0.5

Dimensionless distance x/D

Distance from spout axis x (m) Fig. 8. Computed porosity distributions in a spouted bed compared with He et al. (1994a) data at the super
Fig. 10. Computed solid $ux distributions in a spouted bed at the super
0.30 u su =0.7 m/s

0.053

0.9

y = 0.118m

He et al.

Porosity εg

18 m

y=0.2

0.7

Gas fluxes (kg/m2s)

Simulations 0.118

y = 0.218m

0.25 0.20 0.15 0.10 0.05 u

0.00

0.5

su

= 0.59 m/s

-0.05 -0.10 -0.5

0.3 0.00

0.02

0.04

0.06

Distance from spout axis x (m) Fig. 9. Computed porosity distributions in a spouted bed compared with He et al. (1994b) data at the super
annular zone. The solid mass $ux varies from the maximum in the center of the bed, gradually decreases, and reaches a minimum near the walls, then goes to zero at the wall where the particle velocity is set to zero. The calculated total particle mass $ux $owing-up in the spout zone is 26:8 kg=s and is balanced by the mass $ux $owing-down in the annular zone. Fig. 11 shows the gas mass $ux distribution with the gas super
-0.3

-0.1

0.1

0.3

0.5

Dimensionless distance x/D

0.08

Fig. 11. Computed porosity distributions in a spouted bed at the super
Table 4 Parameters used for the simulation of San Jose et al. experiment Symbol Description

s

g d s; max e usp Dc Do Hmf

Particle density Gas density Particle diameter Maximum solid volume fraction Restitution coeNcient of particles Inlet gas jet velocity Column diameter Diameter of air inlet Settled bed height

Experiment kg=m3

Computer run

2420 1:2 kg=m3 3:0 mm Not reported

Same Same Same 0.590

Not reported

0.99

8:3 m=s; 10:0 m=s 0:36 m 0:03 m 0:18 m

Same Same Same Same

bed. The physical properties of the gas and solid particles and the bed dimensions are listed in Table 4. A detailed description of the unit can be found in San Jose et al. (1998).

San Jose et al.

9 0.0 3

Simulations usp =8.3 m/s

vs (m/s)

400

4

300

y=0.07 m y=0.10 m y=0.13 m y=0.14 m y=0.17 m y=0.21 m

200 100 0 α =30o ug=10.0 m/s

-100 -200

H mf = 305 mm Do = 20 mm d= 1.0 mm

0 20 40 60 80 Distance from spout axis x (mm)

400

200

0

-200

(b)

ug =10.0 m/s d=1.0 mm α=60o y=0.06 m y=0.12 m y=0.14 m y=0.17 m y=0.21 m Hmf =305 mm Do =20 mm

0 20 40 60 80 Distance from spout axis x (mm)

1

0.1

(a)

Vertical particle velocity (cm/s)

L. Huilin et al. / Chemical Engineering Science 59 (2004) 865 – 878 Vertical particle velocity (cm/s)

874

Fig. 14. Vertical particle velocity distribution in the spout zone at the inlet gas jet velocity of 10:0 m=s. y=0.1

7m

-1 0.02

0.04

0.06

0.08

0.10

1.00

1.00 ug =10.0 m/s

Distance from spout axis x (m)

d=1.0 mm

Fig. 12. Computed vertical particle velocity in a conical spouted bed compared with San Jose et al. (1998) data at the inlet gas jet velocity of 8:3 m=s.

Porosity (-)

.85

˚

α=30 y=0.04 m y=0.06 m y=0.07 m y=0.12 m

.70 .55 .40

loose-packed

.25

0.3

(a)

0

.85

Porosity (-)

0.00

.55 .40

Hmf=305 mm Do =20 mm

20 40 60 80 Distance from spout axis (mm)

.70

.25

(b)

ug=10.0 m/s d=1.0 mm

˚y=0.04 m

α=60

y=0.06 m y=0.07 m y=0.09 m y=0.12 m

loose-packed H mf =305 mm Do=20 mm

0 20 40 60 80 Distance from spout axis x (mm)

Fig. 15. Porosity distributions in the spouted beds at the inlet gas jet velocity of 10:0 m=s.

y=0.17 m

us (m/s)

0.0 0.05

-0.3

u =8.3 m/s sp Simulations San Jose et al. (1998) y=0.03 m

-0.6 0.00

0.04

0.08

Distance from spout axis x (m) Fig. 13. Computed horizontal particle velocity in a conical spouted bed compared with San Jose et al. (1998) data at the inlet gas jet velocity of 8:3 m=s.

The conical spouted bed was assumed to be initially at minimum $uidization condition. Fig. 12 shows simulated and experimental distributions of axial particle velocities with an inlet gas jet velocity of 8:3 m=s. The simulated results are in good agreement with the experiments. The vertical particle velocities are higher in the spout zone than those in the annular zone at various bed heights. The minimum particle velocity is found to be somewhere between the spout and annular zone. Fig. 12 also indicated that the axial particle velocity is high close to the axis, and decreases away from the axis in the spout bed. The time-averaged horizontal particle velocities are plotted in Fig. 13. It is observed that both simulated and measured horizontal particle velocities increase from zero at the axis to a minimum value corresponding to an intermediate position in the spout zone, and decreases toward to a location

between the spouted and annulus zones, and goes to zero at the height of 0.03 and 0:05 m, respectively. At the bottom of the conical spouted bed, the particles move in a horizontal direction until they enter the spout zone, so particle velocities have a larger horizontal component. For the height of 0:17 m at the surface of the spout, the horizontal component of particle velocity is positive corresponding to trajectories toward the column wall. Comparing with Figs. 12 and 13, it can be seen that the vertical components of the solids $ow are much greater than their horizontal components. 3.3. E=ect of inclined angle of conical base on particle velocity distributions The variation of the vertical component of particle velocity in the spouted beds at several bed heights is shown in Fig. 14 with an inlet gas jet velocity of 10:0 m=s and inclined angles of 30◦ and 60◦ , respectively. The column diameter of spouted bed is 0:36 m. The diameter and density of particles are 1:0 mm and 1200 kg=m3 , respectively. The coeNcient of restitution of particles is 0.99. Due to the geometrical dimension of the conical shape, the particles accelerate as they approach to the bottom. The particle velocity at various heights decreases from its maximum value at the axis to zero at the interface between the spout and annulus zones. The particle velocity is more pronounced near the inlet. The particle velocity decreased with the bed height in the spout zone. The simulated particle concentration distributions at different levels are shown in Fig. 15. The local particle con-

875

0.5

200

u g = 10 m/s

ug=10.0 m/s

100

α =15 α =30o α =45o α =60o α =75o

o

50

Hmf =305 mm Do= 20 mm d = 1.0 mm

0 0

10 20 30 40 Distance from spout axis(mm)

Fig. 16. Spout shape in the spouted bed at the inlet gas jet velocity of 10:0 m=s.

centrations decrease with the bed height. The particle concentration in the annular zone varied at various levels. The simulated results indicated that the common assumptions that the particle concentration in the annular zone is constant and equals to the loose-packing concentration are not accurate. The denser particle concentration in the annular zone is close to the loose-packing concentration. The spout shape, or the diameter of the spout bed, is one of the characteristic properties of spouted beds. Positions at which the vertical component of the particle velocity becomes zero were plotted in Fig. 16. The calculated spout diameter increases monotonously with the height. Numerical results indicated that the diameter of the spout increases with gas velocity. It is observed that as the inclined angle increases from 45◦ to 60◦ the neck was formed at the inlet. The sectional area of conical contour is reduced as the inclined angle increases. The connection point of cylindrical and conical parts deserves special attention as the vertical component of velocity decreases to the surface of the bed. Simulated results shown that the base angle has a greater in$uence on the particle velocity distributions in spouted beds. 3.4. E=ect of the coe>cient of restitution The variation of the granular temperature of particles is shown in Fig. 17 with the inlet gas jet velocity of 10:0 m=s and the inclined angle of 60◦ . The column diameter of spouted bed is 190 mm. The diameter and density of particles are 1:0 mm and 1200 kg=m3 , respectively. In the range of the coeNcients of restitution from 0.90 to 0.99, the granular temperature is high in the spout zone, and gradually decreases toward to annular zone, then becomes zero. Therefore, the particle–particle collisions have a signi
0.4

y = 210 mm

Hmf = 305 mm

α = 60˚

Do = 20 mm d = 1.0 mm

0.3

e = 0.99 e = 0.95 e = 0.90

0.2

0.1

0.0 -0.5

-0.3

-0.1

0.1

0.3

0.5

Dimensionless distance x/D Fig. 17. Granular temperature distribution as a function of coeNcient of restitution at the inlet gas jet velocity of 10:0 m=s.

1.00 u g=10. 0 m /s

.85

Porosity (-)

150

Granular temperature (m/s)2

Vertical distance from inlet (mm)

L. Huilin et al. / Chemical Engineering Science 59 (2004) 865 – 878

α=60˚ y=0.12 m

.70 e = 0.90 e = 0.95

.55

e = 0.99

Hmf =305 mm Do=20 mm d=1.0 mm

.40

.25 0

20

40

60

80

Distance from spout axis (mm) Fig. 18. Porosity distribution as a function of coeNcient of restitution at the inlet gas jet velocity of 10:0 m=s.

3.5. Comparisons of frictional stress models The frictional stress models proposed in the literature are quite empirical. Johnson and Jackson (1987) gave a set of empirical parameters of F; n; p, and the angle of internal friction + in Eqs. (11) and (13). The values they proposed for glass particles for F; n; p, and + are 3:65 × 10−32 , 0.0, 40.0, and 25:0◦ , respectively. In their later paper (Johnson et al., 1990), they suggested a set of parameters of F; n; p, and + for glass beads be 0.05, 2.0, 5.0, and 28:5◦ , respectively. Fig. 19 shows the computed porosities using these two different sets of parameters in the friction stress model as well as without the friction stress model. The computed results are compared with the measured data of He et al. (1994a) in a spouted bed. Without the frictional stress model, the computed porosities in annular zone are higher than the results computed with frictional stress model and experimental data. It is clear that the computed porosities with the parameters proposed by Johnson et al. (1990) are more closed

876

L. Huilin et al. / Chemical Engineering Science 59 (2004) 865 – 878 1.00

0.5

Porosity (-)

.70

Granular temperature (m/s)2

Simulations: Without friction stress Johnson et al. (1987) and Syamlal et al. (1993) Johnson et al. (1990) and Syamlal et al. (1993)

.85

.55

.40

loose-packed α=60 usu=0.59 m/s

˚

He et al. (1994a) z=0.118 m

.25 0

10

20

30

0.3

u g =12.0 m/s Hmf =305 mm y=330 mm

Do=20 mm d=1.0 mm e =0.99

y=210 mm

0.2

0.1

40

Distance from spout axis (mm)

0.0 -0.5

Fig. 19. Comparisons of computed porosity distributions without and with two di@erent frictional stress model parameters at the super
to the experimental results than computed with the parameters from Johnson and Jackson (1987). The computations indicate the important e@ect of friction stress model and parameters selections on modeling dense solids $ow. In dense solids $ow, individual particles interact with multiple neighbors through sustained contact. Under such conditions, the normal reaction forces and the associated tangential frictional forces at these sliding contacts are dominant. Accounting for strain rate $uctuations and slow relaxation of the particle assembly to the yield surface and assuming that the granular material is noncohesive, Srivastava and Sundaresan (2003) gave a frictional model based on the critical state theory of soil mechanics where the granular assembly deforms without any volume change:   √ (s − s min )2 S √ f = F I − ; (28) 2 sin + (s max − s )5 S:S S = 12 [∇ug + (∇ug )T ] − 13 (∇ · ug )I

0.4

(29)

Savage (1998) argued that even in a purely quasi-static $ow there exist $uctuations in the strain rate associated with the formation of shear layers and that these $uctuations reduce the shear stress in the particle assembly. From these arguments, Savage suggested a frictional model:

 √ (s − s min )2 S I − 2 sin +  : f = F (s max − s )5 S : S + "=d2 (30) Eq. (30) becomes Eq. (28) when " = 0. From our simulations shown in Figs. 5 and 17, we see that the granular temperature is close to zero in the annular zone where only the friction are dominant on the particle–particle interactions. Even with a high inlet gas jet velocity of 12:0 m=s, the granular temperature is also close to zero in the dense region of annulus zone, as shown in Fig. 20. Therefore, the di@erence between Eqs. (28) and (30) are negligible when the particles are dense enough. It is necessary to develop a more reliable kinetic and friction stress model for particulate $ow in the near future.

-0.3

-0.1

0.1

0.3

0.5

Dimensionless distance x/D Fig. 20. Pro
4. Conclusions The hydrodynamics of gas–solids $ow in spouted beds was predicted using a hydrodynamic two-$uid computer model in which a kinetic–frictional constitutive model for particle phase stresses is included. This model assumes that the frictional and kinetic stresses are additive. A normal frictional stress model of Johnson et al. (1990) and a modi
L. Huilin et al. / Chemical Engineering Science 59 (2004) 865 – 878

Notation ai bj ci Cd dj d Dc Do e ew g go Hmf I J ps p p(; ) Q(; ) qij Re Ss Sg Sij t umf usp usu u; v U; V x; y

coeNcient coeNcient coeNcient drag coeNcient coeNcient particle diameter, m column diameter, m diameter of air inlet, m coeNcient of restitution of particles coeNcient of restitution of particle–wall collision gravitational force, m=s2 radial distribution function settled bed height, m unit tensor Jacobian determinant solid pressure, N=m2 $uid pressure, N=m2 grid control function grid control function coeNcients of coordinate transformation Reynolds number source term of solid phase source term of gas phase strain-rate tensor, s−1 time, s minimum $uidization velocity, m=s spouting inlet gas velocity, m=s gas super
Greek letters gs s  g s s; max mf " g s ; 

s

g  k f +s +g

$uid–particle friction coeNcient, kg=m3 =s energy dissipation, kg=m3 =s local
’ !ij

877

velocity component of gas or particles, m/s coeNcient

Subscripts g l max min w s

gas laminar $ow maximum packing minimum value wall particle

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