Three-dimensional numerical simulation method for gas–solid injector

sevier.com/locate/powtec

Powder Technology 160

Three-dimensional numerical simulation method for gas–solid injector

Yuanquan Xiong *, Mingyao Zhang, Zhulin Yuan

Key Laboratory on Clean Coal Power Generation and Combustion Technology of Ministry of Education, Thermoenergy Engineering Research Institute,

Southeast University, Nanjing 210096, P. R. China

Received 4 March 2005; received in revised form 15 August 2005; accepted 25 August 2005

Available online 20 October 2005

Abstract

A three-dimensional turbulent gas–solid two-phase flow model for a gas–solid injector is developed in the present study. Time-averaged

conservation equation for mass and momentum and a two-equation k –e closure are used to model the carried fluid phase. The solid phase is

simulated by using a Lagrangian approach. In this model, the drag and lift forces on particles, the multi-body collisions among particles and the

mutual interaction between gas and particles were taken into account. Interparticle interactions and particle–wall collisions are emulated by using

the three-dimensional distinct element method (DEM). A new correlation, b yvpyvp;� yuyvp

;� �

¼ � 2bKð1� sLsLþsd

Þ sLsLþsd

; which represents thetransfer of kinetic energy of the particle motion to kinetic energy of the carrier fluid, is introduced in the additional source term Sd

e of the transport

equation of turbulence kinetic energy, K. The calculated pressure distributions along the axis in the different parts of gas–solid injectors using

pressured pneumatic conveying system under different driving jet velocities, pressures and values of angle of convergent section (a) are found to

be in agreement with the experimental results. The axial mean velocity of particles and the behavior of gas and particles in the gas–solid injector

are calculated, their results reasonably explaining actual phenomenon observed in experiment.

D 2005 Elsevier B.V. All rights reserved.

Keywords: Numerical simulation; Two-phase flow; Gas–solid injector; Pneumatic conveying; Pressure distributions

1. Introduction

With the rapid development of combustion and gasification

techniques under pressures in industry application, the tech-

nology of pneumatic conveying under pressures will be applied

more and more widely. A gas–solid injector, as shown in Fig.

1, is regarded as a key device in the pressured pneumatic

conveying system, whose performance under pressures has

drawn attention. Some researchers [1–3] have investigated

both experimentally and theoretically the Venturi injectors (as

shown in Fig. 2) over the past three decades.

The aerodynamics of two-phase flows in the Venturi injector

was analyzed by Weber [4], Bohnet et al. [5–7], Wagenknecht

[8], Kmiec et al. [9–11] and Wang and Wypych [12]. Weber

developed a one-dimensional theoretical model and carried out

his analysis on the assumption of incompressible flow and

neglecting wall friction. The description of Bohnet and

0032-5910/$ - see front matter D 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.powtec.2005.08.029

* Corresponding author. Tel.: +86 25 8379 4744; fax: +86 25 5771 4489.

E-mail address: [email protected] (Y. Xiong).

Wagenknecht allowed these factors but was based on some

modeling assumptions for analytical integration of one-dimen-

sional continuity and conservation of momentum equations.

They studied static pressure along the injector and velocity of

particles has been made for constant material flow rate, the

assumption that the particles are accelerated in an air jet of

constant mean velocity. Also, they quoted an efficiency of

pressure transformation from experiences and did not incorpo-

rate the effect of the location of the nozzle into their models.

Kmiec continued Bohnet’s work and presented velocity profiles

for mixed particle size. However, his conclusion was that the

over-efficiency of pressure transformation in the gas–solid

injector depended on the expansion of the motive jet only and

that the higher the inlet gas velocity the lower the efficiency of

pressure transformation. Wang established a one-dimensional

mathematical model using the macroscopic mass, momentum

and energy balance for the motive air and air–solid mixture

between the inlet and outlet of the different parts. A lot of

empirical coefficients are involved in the performance modeling

to account for the influence of the friction and variation of flow

area on performance in Wang’s model.

(2005) 180 – 189

www.el

S1

43

1. driving gas nozzle 2. hopper 3. convergent section 4. conveying pipe

2

Fig. 1. Gas–solid injector using pressured pneumatic conveying system.

S1

3

1. driving gas nozzle 2. hopper 3. Venturi nozzle 4. conveying pipe

4

2

Fig. 2. Venturi gas– solid injector used by other researchers.

Y. Xiong et al. / Powder Technology 160 (2005) 180–189 181

Despite the above investigations, the development of a

reliable and reasonable theoretical model has been hindered

by the complexities of gas–solid two-phase flow. In the

present paper, a three-dimensional turbulent gas–solid two-

phase flow model for a gas–solid injector is developed. The

carried fluid phase is modeled by a set of time-averaged

conservation equation for mass and momentum and a two-

equation k–e closure. The solid phase is simulated numeri-

cally based on a Lagrangian approach in which the motion of

each particle is tracked. The drag and lift forces on particles,

the multi-body collisions among particles and the mutual

interaction between gas and particles were taken into account.

The three-dimensional distinct element method is employed

for interparticle interactions and particle–wall collisions. The

pressure distributions along the axis in the different parts of

the gas–solid injector using pressured pneumatic conveying

system is simulated under different driving jet velocities,

pressures and values of angle of convergent section (a) and,also, the axial mean velocity of particles and the behavior of

gas and particles in the gas–solid injector are calculated yet.

The simulation result is compared with the actual phenom-

enon from experiments.

2. Governing equations

2.1. Gas phase transport equations

The gas phase is assumed to be isothermal and incompress-

ible and the particle–fluid interaction was considered to meet

Newton’s third law. The volume fraction of the solid phase is

defined as a =npVp /Vc, where Vp is the volume of a single

particle and np is the number of particles in volume Vc.

A set of conversation equations describing the gas phase, a

two-phase air–solid mixture, may be derived readily by

following the traditional control volume approach. The

instantaneous conversation of mass and momentum [13] can

be expressed as in Cartesian tensor notation,

B uqað ÞBt

þ B

Bxiuqaujuj� �

¼ 0 ð1Þ

B uqaujuj� �Bt

þ B

Bxjuqauiui ujuj� �

¼ � uBp

Bxiþ B

Bxjurij

� �� mi þ uqagi ð2Þ

where qa and u =1�a are the mass density and the volume

fraction of the gas phase, respectively. The instantaneous

gas velocity u is the sum of a mean component u and a

fluctuation component yu, i.e. u =u +yu. p is the fluid

pressure and gi is the ith direction component of gra-

vitational acceleration. mi is the interfacial momentum

transfer term per unit volume. The gas stress tensor rij is

given as,

rij ¼ lBujuj

Bxiþ Buiui

Bxj

�� 2

3dij

Bukuk

Bxk

��

where l is the shear viscosity and dij is the Kroenecker

delta.

Flows with relatively large and massive particles moving at

intermediate particle Reynolds number are considered in the

present study, in which case the important forces on a particle

are the inertia, drag, collision and gravity. The solid to gas mass

density ratio, qp /qa is of the order 103. The particle Reynolds

number, Rep=qavrdp /l, is of the order 103, where dp is the

particle diameter of the order 10�3 m, l is the gas viscosity,

and vr is the relative velocity between the particle and the local

fluid. The ratio of the particle response time to the eddy

turnover time is

spse

¼ 4

3CD

qp

qa

��ue

vr

��dp

le

��

The factor 4 / (3CD) and the ratios ue /vr and dp / le are

approximately of order unity; CD is the drag coefficient and ueand le are the characteristic velocity and length scale of the

large eddies, respectively. As a result, the ratio sp /se is roughlyof the order 103. Consequently, the influence of fluid

turbulence on the particles’ motion would be small. The

fluctuations of flow properties of the solid and the fluid phases

may be assumed to be uncorrelated. The particle volume

fraction, which is a volumetric mean quantity, is assumed to be

independent of the turbulence averaging time-scale of the fluid

phase. After applying the Reynolds time-averaging technique

Y. Xiong et al. / Powder Technology 160 (2005) 180–189182

to Eqs. (1) and (2), the following time-averaged equations for

the fluid phase are obtained [14]

B uqað ÞBt

þ B

Bxiuqauj� �

¼ 0 ð3Þ

B uqauið ÞBt

þ B

Bxjuqauiuj� �

¼ � uBP

Bxiþ B

Bxjusij� �

�Mi þ uqagi ð4Þ

where ui and uj are the ith and jth direction mean velocity

component, respectively. P is the gas pressure and Mi is the

interfacial momentum transfer term. The gas stress tensor sij isgiven as,

sij ¼ l þ ltð Þ Buj

Bxiþ Bui

Bxj

�� 2

3dij

Buk

Bxk

��ð5Þ

where l and lt are the shear viscosity and turbulent viscosity

and dij is the Kroenecker delta. The turbulent viscosity lt in

Eqs. (5) may be expressed as

lt ¼ ClqfK2=e ð6Þ

where Cl is a constant [15].

Neglecting the drift velocity, the interfacial momentum

transfer term, Mi, in Eq. (4) is divided as

Mi ¼ np Fd þ Fl þ Fam þ Fhi þ Fotð Þ ð7Þ

where the forces in the brackets on the right-hand side of Eq.

(7) are the forces acting on a single particle and where np is the

number of particles per unit volume; Fd is the drag force; Fl is

the lift force, which includes the Magnus lift, FlM, and the

Saffman lift, FlS [16,17]; Fam is the added mass force; Fhi is the

history force and Fot is the other force. In the present paper,

Fam, Fhi and Fot are neglected for gas–solid system since the

density ratio qp /qa is very large.

The drag force npFd in Eq. (7) may be expressed as

npFd ¼ b uY� vYp� �

ð8Þ

where b is the drag function and vpY is the mean velocity of the

particles in the volume Vc.

When u >0.8, the Gidaspow [18] correlation is adopted:

b ¼ 3

4dpCDs 1� uð ÞqjvpY� uYju�1:65 ð9Þ

where dp is the particle diameter and the drag coefficient of a

single particle CDs is a function of particle Reynolds number

[19],

CDs ¼24

Re1þ 0:15 Reð Þ0:687

� �; Re < 1000

0:44; Re � 1000

(ð10Þ

with

Reþ qaujvpY� uYjdp=l ð11Þ

When u�0.8, this expression of the drag function b may be

obtained from Ergun equation [20],

b ¼ 1501� uð Þ2lu ndp� �2 þ 1:75

1� uð ÞqajvpY� uYjndp

ð12Þ

where n is the particle spherical coefficient.

Turbulence is a key element in the flow of fluid–particle

mixture. The presence of particles on the turbulence of the

continuous phase is known as turbulence modulation. There

have been numerous studies, both numerical and experimental,

on turbulence modulation. Several sources of turbulence in the

carries phase due to particles have been identified: the stream

line distortion due to the presence of the particles, the wake

generated by particles, the modification of the velocity

gradients in the carrier phase and the associated change in

turbulence generation and the damping of turbulence motion

by the drag force on the particles. However, there still is no

generally accepted model that is applicable to all flow

conditions. In the present paper, the time-averaged equations

for the turbulence kinetic energy, K, and the rate of turbulence

kinetic energy dissipation, e, for the continuous phase, are

written as in Ref. [21]:

B uqaKð ÞBt

þ B

BxjuqaujK� �

¼ B

Bxju l þ lt

rk

��BK

Bxj

��þ uG�uqae þ SKd ð13Þ

B uqaeð ÞBt

þ B

Bxjuqauje� �

¼ B

Bxju l þ lt

rk

��BeBxj

��

þ ueK

C1G� C2qeð Þ þ Sed ð14Þ

with

G ¼ lt

Bui

Bxj

Bui

Bxjþ Buj

Bxi

��

The additional source term SdK in Eq. (13) may be expressed

as in Ref. [22]:

SKd ¼ bjuY� vpYj2þ b yvpyvp

;� yuuyvp;��

ð15Þ

where yu and yvp are the gas phase fluctuating velocity

component and the solid phase fluctuating velocity component,

respectively. The first term on the right-hand side of Eq. (15)

reflects the conversion of mechanical work by the drag force

into turbulence kinetic energy and the second term, the

redistribution term, represents the transfer of kinetic energy

of the particle motion to kinetic energy of the carrier fluid. This

term becomes important when the interparticle collisions are a

source of particle kinetic energy.

In this paper, to compute the additional source term Sdk, a

particle response function Ct =yvp/yu proposed by Gosman et


al. [23], which links the instantaneous velocity fluctuations of

the particle phase to the velocity fluctuations of the gas phase,

is introduced. It is combined with the following correlation

given by Mostafa [24,25]

yuuyuu;

� yuuyvp;

¼ 2K 1� sLsL þ sd

��ð16Þ

where sL=0.35K / e is the carrier phase Lagrangian time scale,

sd ¼4dpqp

3CDsqajuY�vpYj is the particle dynamic relaxation time, qp is

the particle density and dP is the particle diameter.

The turbulent kinetic energy, K, is defined as K ¼ yuuyuu;

=2.Therefore, the second term on the right-hand side of the Eq.

(15) may be deduced as:

b yvpyvp;

� yuudvp;� �

¼ � 2bK 1� sLsL þ sd

��sL

sL þ sdð17Þ

The additional source term Sde in Eq. (14) may be expressed

as in Ref. [26]:

Sed ¼ C3

eKSKd ð18Þ

The constants used in the K –e model are chosen as

Cl = 0.09, C1 = 1.44, C2 = 1.92, C3 = 1.2, rk = 1.0 and

re =1.33. All these constants except C3 have standard,

single-phase flow turbulence values. The constant C3 is

included in the gas–particle interaction term, Sde, of Eq. (14)

and has been determined empirically from turbulent gas–

particle jet flows of Elghobashi and Abou-Arab [27].

2.2. Equations of particle motion

In this study, particles are spheres. A spherical particle has

two types of motion: translational and rotational, the three-

dimensional motion of a particle is described by a set of six

scalar equations. Dependent on Newton’s second law, the

corresponding vectorial equation for the translational and

rotational motion of a particle in turbulent fluid may be written

as

mp

dVYp

dt¼ mpg

Yþ FY

f þ FY

lM þ FY

lS þX

FY

cð19Þ

dxYp

dt¼ Tp

Y

Ið20Þ

where mp is the mass of a particle, Ff

Yis the particle–fluid

interaction force, Fc

Yis the force acting on a particle caused by

collisions, TpYis the total torque and I is the moment of inertia of

the particle.

The particle–fluid interaction force is

Fd ¼1

8qakd

2pCDsjUr

YjUr

Yð21Þ

where the translational relative velocity, Ur

Y, is defined as

Ur

Y¼ u u

Y � VP

Y� �. The drag coefficient CDs is given by

Schiller and Naumann [19]

CDs ¼24

Re1þ 0:15 Reð Þ0687

� �; Re < 1000

0:44; Re � 1000

(

with

Re ¼ qaujVp

Y� u

Y jdp=l

The Magnus lift due to particle rotation is written as in Ref.

[14]

FlM ¼ 0:5qav2r

kd2p4

ClM

xr

Y vrY

jxr

Y jIj vrY j

ð22Þ

where vrY ¼ u

Y � Vp

Yand xr

Y ¼ xf

Y � xp

Yare the relative linear

and angular velocity between the local fluid and the particle,

respectively. The local mean angular velocity of the fluid, xf

Y,

is defined as xY

f ¼ 12l u

Y.

The Magnus lift coefficient ClM is given by Lun and Liu

[14]:

ClM ¼ dpjxr

Y jjvfY j

; RepV1

ClM ¼ dpjxr

Y jjvfY j

0:178þ 0:822Re�0:522p

� �; 1 < Rep

with

Rep ¼qavrdp

l

The rate of angular momentum change of a spherical

particle interacting with a viscous fluid may be expressed as in

Ref. [28]:

mpd2p

10

dYxfp

dt¼

qd5p64

CTjYxfr j

Yxfr ð23Þ

whereYxfr is the relative angular velocity between the local

fluid and the particle, defined asYxfr ¼

Yxf �

Yxfp. The

dimensionless coefficient CT is given by Dennis et al. [29]:

CT ¼ 5:32

Re0:5x

þ 37:2

Rex; Rex < 20

CT ¼ 6:45

Re0:5x

þ 32:1

Rex; 20VRex

with

Rex ¼ qad2p jYxrj= 4lð Þ

The Saffman [16,17] lift due to fluid shearing motion is

FlS ¼ 1:615 ux � Vpx

� �qalð Þ0:5d2pClS

ffiffiffiffiffiffiffiffiffiux

n

�� r

sgnux

n

��;

n ¼ y; zð Þ ð24Þ


where ux and Vpx are the velocities of the fluid and the particle

in the x direction and dux / dn is the shear rate of the mean flow.

ClS is the lift coefficient.

According to Mei [30], the coefficient ClS may be expressed

as

ClS ¼ 1� 0:3314c0:5� �

exp � 0:1Rep� �

þ 0:3314c05;

RepV40� �

ð25Þ

ClS ¼ 0:0524 kRep� �0:5

;

Rep > 40� �

ð26Þ

with

Rep ¼qaj

Yu �Y

V p jdpl

where

c ¼ dux

dn

��dp=2jYvr j; n ¼ y; zð Þ:

The particle–particle collision forceYF c is based on DEM

[31], which can be divided into normal and tangential

components:YFcnij and

YFctij ,

Fcnij

Y¼ � kndnij � gnV

Y

rij

Ynij

� �Ynij ð27Þ

Fctij

Y¼ � kt

Ydtij � gt

YV tij ð28Þ

with

YV tij ¼

YV rij �

YV rij I

Ynij

� �Ynij þ

dp

2

Yxi �

Yxj

� �Ynij

where dnij and dtij are the normal and shear relative

displacement increments, respectively; kn and kt are the normal

and shear spring constants, respectively;YV rij is the relative

velocity of particle i to particle j andYV tij is the tangential

velocity at the contact point;Yxi and

Yxj are the angular velocity

of particle i and particle j, respectively;Ynij is the unit normal

vector; and gn and gt are the normal and shear damping

coefficients, respectively, which can be calculated by the

following equations [32]:

gn ¼ � 2lne

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimpkn

k2 þ ln2e

s

gt ¼ � 2lne

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimpkt

k2 þ ln2e

s

where the restitution coefficient, e =0.9 for mono-size [33,34].

If, however, the following relation is satisfied

jYFctij j > lf

YFcnij

where lf is the coefficient of friction, then sliding is taken to

occur and the tangential force is given as:

YFctij ¼ � lf j

YFctij j

Ytij ð29Þ

where tij is the unit vector in the tangential direction defined byYtij ¼ Y

V tij=jYV tij j.

For the treatment of particle–wall collision, the computation

of particle–wall collision is similar to that of particle–particle,

except for substituting the particle j for wall and assuming that

particle j is not moving.

The tangential component of collision force acting on a

particle,YFctij , causes particle rotation. This rotation motion can

be expressed as:

dYxcp

dt¼ rp

PYFctij

Ið30Þ

where rp is the particle radius andYxcp is the angular

acceleration caused by the contact force.

The new velocities and position of the particle after the time

step Dt are calculated explicitly:

YV p ¼

YV p0 þ

YVYV pDt ð31Þ

Yxp ¼

Yxp0 þ

YxYxpDt ð32Þ

YSp ¼

YSp0 þ

YV pDt ð33Þ

where subscript 0 denotes the initial value.

3. Numerical procedure and computation conditions

The governing equations for gas phase and solid phase are

discretized on a non-staggered mesh. The equations for the gas

phase were solved using the SIMPLE method [35] for a

collocated grid. The particle equations of motion are integrated

using a second-order finite-difference algorithm in a small time

step Dt. The smaller the time step, the more stable the

calculation. However the time step should be as large as

possible to save computation time. So it is very important in

this simulation to find the optimal value of the time step. In this

present simulation, cases based on various time steps were

examined. It was found that the calculation became unstable

when the particles traverse more than 20% of the particle

diameter in a time step. For the particles traversing no more

than 20% of the particle diameter in a time step, the simulation

results are in well agreement with our experimental results

[36]. Therefore, in the present calculation, a time step adopted

must satisfy that the particles traverse no more than 20% of the

particle diameter.

At the wall, no-slip boundary conditions for the gas phase

and the free slip wall approximation for particles are used. At

the gas inlet, ux and uy =uz =0 are assumed. At the particles

inlet, it is assumed that the particle velocity is vx =vy=vz =0

and the particle density profile is distributed randomly and the

particles are input continuously into the calculating domain.

Table 1

Computing conditions

Number of cells at X axes 50

Number of cells at Y axes 5

Number of cells at Z axes 36

Air density, qa (kg/m3) 1.4¨5.0

Gas viscosity, l (m2/s) 18.2410�6

Gas velocity of driving jet, U0 (m/s) 140¨220

Position of driving jet, S (mm) 130

Convergent section angle of an injector, a (-) 10¨35

Mass flow rate of solid material (kg/h) 893

diameter of particles, dp (mm) 2.5

Particle density, qp (kg/m3) 2480

Maximal particle number, N 15000

Restitution coefficient, e 0.9

Friction coefficient of particle–particle, lpf 0.25

Friction coefficient of particle–wall, lwf 0.3

Spring constant, k 1000

Time step of solid phase (s) 110�6

0 110 220 330 440 550

49500

50000

50500

51000

51500

52000

52500

S=130 mm, α=150, GP=893kg/h,

U0=210m/s, Pb=50kPa Calculated resultsS

tatic

Pre

ssur

e p

(Pa)

Position X (mm)

Experimental results

Fig. 4. Experimental and calculated results on static pressure distributions in an

injector.

305000 Pb=300 kPa,α=150,Gs=893 kg/h


The exit boundary conditions for the gas phase were treated

into the local one-way. The initial value of the turbulent kinetic

energy, K, and the turbulent kinetic energy dissipation rate, e,for the gas phase are assumed to be given as:

K ¼ 0:5¨1:5%u202

e ¼ clqK20

lt

where lt is given by qu0delt

¼ 100¨1000, de is an equivalent

diameter of the driving gas inlet.

Computation is carried out for 5 s. A single-phase turbulent

fluid flow field is initially computed for the volume fraction of

the gas phase u =1 in an injector.

The computing conditions are listed in Table 1. Fig. 3 shows

gas–solid injector configuration and its coordinates used for

the experiments and calculation. The calculations of gas flow

and particle motion are 3-dimensional. A square driving gas

nozzle is of size 810 mm, the hopper is of size 10040 mm

and height 70 mm, the inlet of the convergent section of

injector is of size 40100 mm, and the outlet of the

convergent section of injector and the conveying pipe are of

size 4020 mm. The computational domain is of size 50040

1. driving jet; 2. hopper tube; 3. convergent section

of injector; 4. conveying pipe

x

zy

α

S

solid

air3

4

2

1

Fig. 3. Schematic diagram of an injector using the present model.

mm and height 180 mm and the cell size for calculation is

108 mm and height 10 mm.

4. Numerical results and discussions

Fig. 4 shows the calculated and experimental static pressure

distributions in the injector. It can be seen that the calculated

results are in reasonably good agreement with the experimental

results. This agreement shows the reliability of this model in

the present investigation.

4.1. Static pressure distributions in the injector

In Fig. 5, a substantial rise in static pressure occurs in the

inlet region of the convergent section of injector since later on

the static pressure falls gradually due to friction and acceler-

ation losses that exceed the pressure recovery. Ultimately, in

the outlet region of the convergent section of the injector there

is a rapid drop in static pressure. However, it is remarkable that

0 100 200 300 400 500

300000

301000

302000

303000

304000

Sta

tic P

ress

ure

p (P

a)

Position X (mm)

U0=143.2 m/s U0=177.3 m/s U0=210 m/s

Fig. 5. Calculated static pressure distribution in injector at different gas flow

rates.

0 100 200 300 400 500

50000

51000

52000

53000

54000

55000

Sta

tic P

ress

ure

p (P

a)

Positon X (mm)

Pb=50kPa, U0=210m/s, GP=893kg/h

α=150

α=250

α=350

Fig. 6. Static pressure distribution in the injector affected by the convergent

section angle under.

0 100 200 300 400 500 600 700 800 900

-2

0

2

4

6

8

10

12

14

16

X A

xial

Mea

n V

eloc

ity o

f Par

ticle

s (m

/s)

Position X (mm)

Pb=350kPa,U0=177m/s

α=10o

α=15o

α=25o

α=35o Fitting curve

Fig. 8. X-axial mean velocity distribution of particles in the injector affected by

the convergence.


the transformation of pressure energy is related to the gas flow

rates (shown in Fig. 5). At low gas flow rates the transforma-

tion of kinetic into static pressure energy occurs only in the

inlet region of the convergent section of injector. However, at

high driving jet velocities the pressure recovery arises in a

wider region.

Figs. 6 and 7 illustrate the calculated static pressure

distributions at different convergent section angle a. It is

observed that the static pressure increases with the value of

angle of convergent section in the injector. It can be also found

from Figs. 6 and 7 that the length of the pressure recovery

enlarges with the value of angle of convergent section and at

higher pressure (i.e. 300 kPa) the length of gradual drop in

pressure decreases with the value of angle of convergent

section of injector (Fig. 7).

4.2. Gas–solid flows in the injector

Fig. 8 shows the calculated results of X-axial mean velocity

distributions of particles along the injector axis. It can be seen

that the X-axial mean speed of particles in the gas–solid injector

increases due to the high-velocity driving jet initially as the X-

axial distance is increased and that later is retarded because of

0 100 200 300 400 500

300000

302000

304000

306000

Sta

tic P

ress

ure

p (k

Pa)

Position X (mm)

Pb=300 kPa, U0=177.3m/s,GP=893kg/h

α=15o

α=25o

α=35o

Fig. 7. Static pressure distribution in the injector affected by the convergent

section angle under.

interparticle interactions and particle–wall collisions and fric-

tions in the convergent section of injector. Ultimately, in the

outlet region of the convergent section of the injector, the

particles are accelerated by conveying gas once more to their

final velocity in the adjoining conveying pipe. For relatively

large and massive particles in a confined-flows situation, the

particle motion is considerably influenced by the wall collision

force [37]. The volume and length of the convergent section of

gas–solid injector becomes rapidly smaller as a increases, where

the interparticle interactions and particle–wall collisions and

frictions are enhanced. When a�15, the particle motion in the

convergent section of injector is acutely retarded and many

particles move against the flow due to rebounding off the wall,

which results in the minimum value of the particle mean velocity

to occur, as shown in Fig. 8. However, owing to a rapid

weakening of the influences of particle–wall collisions and

frictions at a =10-, a continuous increase for the particle velocityin the injector is observed from Fig. 8. The calculated results in

Fig. 8 effectively explain the reason why the mass flow rate of

material conveyed decreases when a is increased in experiments

[36,38]. It can be indicated that the convergent section angle,a, isabout 7- to 10- and helps in achieving the goal of optimum

particle velocity in present injector.

Figs. 9 and 10 show the calculated results of gas–solid flow

in injector between 0 and 5 s. In Fig. 9, it can be found that most

particles from hopper tube are accelerated by the driving gas jet

between the driving nozzle and the beginning of the convergent

section and then collided with the bottom wall of the entrance of

the convergent section. Subsequently, depending on the effects

of the collision force, the particle–fluid interaction force and the

fluid lift force, the particles entrancing the convergent section

move ahead. In this case, the transportation becomes steady after

3 s around from the initial state. As seen in Fig. 9, due to the

effects of the gravity of solid particles and the circumfluence of

gas, the solid particles are distinctly accumulated near the

t=0 s t=0.126 s

t=3 s

t=0.198 s t=0.342 s

t=1 s

t=4 s t=5 s

Fig. 9. Behavior of particles in injector (U0=210 m/s, Gs=893 kg/h, Pb=50 kPa, a =15-).


bottom of the left-hand region of the injector; however, those are

few near the top of the entrance of the convergent section, which

describes well the actual phenomenon observed in experiment.

Near the beginning of conveying pipe, the impacts and friction

of gas and particles against the walls of injector and the multi-

body collisions among particles are largely enhanced due to

increase of the solid volume fraction and the particle velocity,

which results in the pressures rapidly dropping along the flow. It

corresponds to the calculated results shown in Figs. 5–7 and the

experimental results of Xiong and Zhang [38].

Fig. 10 shows that the values of expansion angles of the

driving gas jet along the flow increase and the range and velocity

of the gas jet are retarded when the solid particles enter the gas jet

region (at t =0.198 s) owing to the strong interactions between

gas and particles, which are in agreement with the experimental

results of Bohnet and Wagenknecht [5] and Kmiec and

Leschonski [11]. In this case (a constant feeding rate is 893

kg/h), the effect on the gas field by the solid particles maintains

fixedness on the whole and the flow patterns of the driving gas

jet are approximately unaltered after 3 s. It can be observed from

Fig. 10 that there is a relatively steady driving gas jet in the axial

center of the gas–solid injector to ensure the particles to be

transported into the conveying pipe.

Furthermore, as shown in Fig. 10, the circumfluence of gas

becomes steady near the bottom of the left-hand region of the

injector after 3 s (shown in Fig. 10f–h), which illuminates the

number of accumulated particles attained that is approximately

constant, as shown in Fig. 9f–h. However, this phenomenon

increases the energy losses of the driving gas and also is

harmful to normal transportation for easily cohesive material

conveyed due to existing moisture of conveying gas. So, this

paper proposes that the inside configuration near the bottom of

the left-hand region of a gas–solid injector should be designed

into a streamline inclined plane in order to ensure the conveyed

materials to be immediately transported to the conveying pipe,

which not only reduces the energy losses but also gets rid of

adhering of conveyed material together on account of the solid

accumulation.

5. Conclusions

(1) Numerical simulation method of the solid phase field

based on the Lagrangian description and the gas phase

field based on the Eulerian description is first used to

research the behavior of gas and particles in gas–solid

injectors under pressure, in which not only the interac-

20 m/s t=0 s 20 m/s t=0.126 s

20 m/s 20 m/s

20 m/s 20 m/s t=3 s

20 m/s 20 m/s

t=0.198 s t=0.342 s

t=1 s

t=4 s t=5 s

Fig. 10. Behavior of gas in injector (U0=210 m/s, Gs=893 kg/h, Pb=50 kPa, a =15-).


tions between gas and particles are taken into account

but also the particle/particle collisions. The particle/

wall collisions are emulated by DEM. In the present

model, a new correlation that represents the transfer of

kinetic energy of the particle motion to kinetic energy

of the carrier fluid is introduced in the additional

source term Sde of the transport equation of turbulence

kinetic energy, K.

(2) The calculated results on static pressure distributions in

the present gas–solid injector are in reasonably good

agreement with the experimental results. The static

pressure distributions depend on driving jet velocities,

pressures and values of angle of convergent section, a.(3) The X-axial mean speed of particles in gas–solid injector

increases initially as the axial distance is increased and

later is retarded. Ultimately, in the outlet region of the

convergent section of the injector, the particles are

accelerated once more to their final velocity in the

adjoining conveying pipe.

(4) Increasing a can enhance the influences of particle–wall

collisions and frictions in gas–solid injector, which

results in the X-axial mean velocity of particles decreas-

ing as a whole. These calculated results effectively

explain the reason why the mass flow rate of material

conveyed decreases when a is increased in experiments.

It can be observed that a is no more than 10-, whichhelps in achieving the goal of optimum particle velocity

in the present injector.

(5) Based on analyzing the calculated results of the behavior

of gas and particles in this gas–solid injectors under

pressure, this paper proposes that the inside configuration

near the bottom of the left-hand region of gas–solid

injector should be designed into a streamline inclined

plane in order to ensure the conveyed materials to be

immediately transported to the conveying pipe.

Acknowledgements

This work is financially supported by a research grant from

the National Basic Research Program of China under contracts

No. 2004CB217702-03 and No. G199902210535. The authors

also expressed sincere gratitude to the respected professors

Prof. Y. Tsuji, for kindly presenting us some of their valuable

papers in year 2003, and Prof. M. Horio, for constructive


advice, specially during his visitation period in our laboratory,

which contributed to our researches.

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Three-dimensional numerical simulation method for gas–solid injector

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