Numerical Simulations of the Equations of Particle Motion in the Gas Flow

doi:10.4236/ojapps.2013.31B1005

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Open Journal of Applied Sciences, 2013, 3, 21-26

doi:10.4236/ojapps.2013.31B1005 Published Online April 2013 (http://www.scirp.org/journal/ojapps)

Numerical Simulations of the Equations of

Particle Motion in the Gas Flow

Kelong Zheng1*, Liuxiang Zhang2, Haiyan Chen2

1School of Science, Southwest University of Science and Technology, Mianyang, China

2School of Environment and Resource, Southwest University of Science and Technology, Mianyang, China

Email: [email protected]

Received 2013

ABSTRACT

Under the assumption of considering the gravity and without gravity, two different acceleration models to describe par-

ticle’ motion in the gas flow are formulated, respectively. The corresponding numerical simulations of these models do

not only show the trend of the velocity o f the particle in different density and particle diameter sizes, but also the rela-

tionship between the maximum particle velocity and its diameter size.

Keywords: Numerical Simulation; Particle Motion; Acceleration; Jet Grinding

1. Introduction

The principle of jet grinding is that solids particles are

accelerated by high-speed gas flow and fragmented due

to multiple particle-particle collisions in interacting gas-

particle jet. The development of jet grinding technology

mainly includes the development of basic theory of jet

grinding and its device, and the former is the foundation

of the latter. According to the principle of jet grinding,

the grind energy of particles comes from high-speed gas.

Therefore, the analysis on the particle acceleration pro-

gress is a key point in the design of jet grinding device.

Whether the particles in the gas flow can be effectively

accelerated and collided at its maximum velocity is the

important condition to improve the efficiency of jet

grinding. Researches of particles’ acceleration in nozzles

have been reported in some literatures [1-3], but there

have not any reports about particles’ acceleration in flu-

idized bed. If the accelerated distance is too far, when the

particle is accelerated to the maximum velocity, it will be

affected by the gas flow and solid s, an d slow down ; if the

accelerated distance is too close, particle can not be ac-

celerated enough. So, to determine the optimal injection

distance is very important to improve the efficiency of jet

grinding. Reference [4,5] pointed out that it was a pref-

erable choice that the accelerated distance of nozzles was

as 10 - 20 times as the dimensionless distance, but the

result is too wide. On the other hand, although many pa-

pers on the analysis of particle motion do not consider

the influence of gravity, for tough particle, the gravity, as

well as the angle of inclination of the nozzle, has great

influence on the efficiency of the fluidized bed.

To investigate the particle motion in the high-speed gas

flow better, this paper establishes some different mathe-

matical models to describe particle acceleration with and

without considering gravity, respectively. Meanwhile,

numerical simulations show the trend of the velocity of

the particle under different conditions and the relation-

ship between the maximum particle velocity and its di-

ameter size.

2. Particle Acceleration Model without

Gravity

2.1. Mathematical Model

Without considering gravity, the equation of a single

particle motion [6] is as follows

0.75 (() )

sd s

duC ut u

dt d





(1)

where

u is the velocity of particle, is the veloc-

ity of gas, ()ut



is the gas density,



is the particle den-

sity,

d is the particle diameter size and is the

particle drag force coefficient. d

Equation (1) indicates the velocity of particle

u is a

function in t. By transformation, it can be rewritten to an

equation about th e particle velocity and the jet distance,

0.75 (() ),

sde

duC d

dx d





(2)

Where

is the dimensionless distance which equals to

the ratio of the distance of jet stream section to the nozzle

*Corresponding author.

K. L. ZHENG ET AL.

outlet and the diameter of the nozzle outlet, ()ux is the

velocity of gas in the dimensionless distance and is

the diameter size of the nozzle outlet. e

Depending on the injecting high-speed gas flow, most

particles can be accelerated in axis velocity area, but the

existing problem is that some particles are accelerated at

the edges of area (see Figure 1). Thus, except for the

axis velocity of gas, we should also consider the mean

velocity of quality of gas. Only in this way, can we sci-

entifically evaluate the acceleration effect.

Assume that the axis velocity of g as changes in accor-

dance with the jet.

In initial part, 4.4x

(), ( ),u xuAxisvelocity (3)

() 1 1.525.28

(



ux axax )

ean velocity ofquality

(4)

where is the coefficient of turbulence.

0.076a

In main part,

6.3

() ( ),

1.93



u xAxisvelocity

x (5)

()

1.93

(



ux x).

ean velocity ofquality

(6)

2.2. Influence of the Axis Velocity on Particle

Acceleration

Using air as the working medium, and by FLUENT soft-

ware we get the testing data of gas flow field in the 30

times dimensionless distance, that is, array [,]

y, wh er e

is the dimensionless distance, is the parameter

group including gas v e locity, ga s density an d viscosity .

Next, letmm, and

16

d2650



kg/m3 and

7950



kg/m3, respectively.

Figure 1. The velocity vectors of gas in fluidized bed jet

grinding.

We also take e= 50 µm, 150 µm, 250 µm, 500 µm

and 750 µm, respectively, and employ equation (3) and

(5) to compute the velocity of gas. After discretization of

equation (2) and by Newton iteration, we get the rela-

tionship between the particle acceleration and the dimen-

sionless distance (see Figure 2 and Figure 3).

It is clearly observed from Figure 2 and Figure 3 that,

when the curve of particle acceleration crosses the curve

of the velocity of gas, particle reaches its maximum ve-

locity, and the corresponding dimensionless distance is

the optimal distance for nozzle jet.

In general, the smaller the particle diameter size is, the

easier it is to be accelerated. When the particle density is

low, the smaller the particle diameter size is, the more

obvious the acceleration of velocity is; but if the density

is high, the speed change will not be obvious no matter

the size of the particle is big or small.

05 10 15 202530354045 50

200

400

600

800

1000

1200

Velocity of gas

50m

150m

250m

500m

750m

Figure 2. The velocity of particle and the axis velocity of gas

with s





2650 kg/m3.

05 10 15 20 25 30 35 40 4550

200

400

600

800

1000

1200

V elocit y of gas



150



250



500



750



Figure 3. The velocity of particle and the axis velocity of gas

with s





7950 kg/m3.

K. L. ZHENG ET AL. 23

For 2650



kg/m3, when particle diameter is

= 50 µm, the 0 - 10 times dimensionless distance is the

particle’s rapidly accelerating phase, the 10 - 20 times

dimensionless distance is its slow rise and slow reduction

phase, and the 20 - 30 times dimensionless distance is

particle velocity’s sharp decline phase. At 14 times di-

mensionless distance, particle reaches its maximum ve-

locity of 378 m/s.

At the same time, Figure 4 also shows the relationship

between the particle maximum velocity and the particle

diameter size with particle density



= 5300 kg/m3.

2.3. Influence of the Mean Velocity of Quality on

Particle Acceleration

Parameters ,,



and d are taken as in sec-

tion 2.2. Here we employ equation (4) and (6) to compute

the velocity of gas. By the same computation method, we

get the relationship between the particle acceleration and

the dimensionless distance (see Figure 5 and Figure 6).

From the above figures, we can see that the quality mean

velocity of gas flow decays faster than the axis velocity

of gas. In 0 - 25 times dimensionless distance, the dif-

ference between the quality mean velocity and axis ve-

locity increases along with the increase of dimensionless

distance, because the gas flow is more and more diver-

gent with the increase of the dimensionless distance.

In 25 - 30 times, the situation conversed. The main

reason is that when gas collision happens in the grinding

center, the anti-shock wave will be formed, and then the

axis velocity of gas will rapidly decay.

The trend of the particle velocity gets something simi-

lar to the abovementioned result, but the concrete nu-

merical result is slightly different. For 2650



kg/m3,

when particle diameter is ds = 50 µm, the 0 - 8 times di-

mensionless distance is the particle’s rapidly accelerating

0100 200 300400 500600 700800

100

150

200

250

300

Figure 4. The relationship between the particle maximum

velocity and the particle diameter size with particle den-

sity s



= 5300 kg/m3.

05 10 15 20 25 30 35 40 45 50

100

200

300

400

500

600

700

800

900

1000

Velocity of gas



150



250



500



750



Figure 5. The velocity of particle and the mean velocity of

quality of gas with s



= 2650 kg/m3.

051015 202530 3540 4550

100

200

300

400

500

600

700

800

900

1000

Velocity of gas



150



250



500



750



Figure 6. The velocity of particle and the mean velocity of

quality of gas with s



= 7950 kg/m3.

phase, the 8 - 17 times dimensionless distance is its slow

rise and slow reduction phase, and the 17 - 30 times di-

mensionless distance is particle velocity’s sharp decline

phase. At 10 times dimensionless distance, particle

reaches its maximum velocity of 223 m/s.

Similarly, Figure 7 also shows the changing relation-

ship between the particle maximum velocity and the par-

ticle diameter size with particle density



= 5300 kg/m3.

3. Particle Acceleration Model with Gravity

3.1. Mathematical Model

With considering the gravity of the particle, equation (2)

can be revis ed as follows,

0.75 (() ),

sxd e

sx sx

duC d

dx d



(7)

K. L. ZHENG ET AL.

,

where

(8)

xsy

particle, resp

mean the horizontal and vertical velocity

of the ectively. The particle velocity

u can

be obtaine d by

,

sx sy

uuu (9)

Because the direction of movement is no longer hori-

zontal, the corresponding angle of

zle outlet can appro ximately represent as:

inclination of the noz-

arctan /



(10)

where ,

mean the horizontal and vertical displace-

ment oarticle at its maximumf the p speed, respectively.

3.2. Influence of the Axis Velocity on Particle

Acceleration

Here we only take



= 2650 kg/m3 to discuss the in

ence of the axis velocity. Other parameters and axis

flu-

ve-

locity are taken as in section 2.2. After discretization of

that

equation (7) and (8) Newton iteration, we also get the

relationship between the particle acceleration and the

dimensionless distance with gravity (see Figure 8).

It can be perceived from the Figure 8 that, the change

of particle velocity does not have much impact, with

considering the gravity. The main reason for this is

e particle density is not high, and particle size is small,

which makes the effect of gravity on particle acceleration

process not so obvious. However, gravity will produce

the vertical direction of the displacement. To achieve

better crushing impact, we must adjust nozzle angle ac-

cording to the influence of gravity. For example, Figure

9 shows Particle trajectory with



= 2650 kg/m3 and

d = 750 µm.

By equation (10), we can get

0.0011

l m, 0.5680 0.1



l m,

0100 200 300400 500 600 700800

100

120

140

160

180

Figure 7. The relationship between the particle maximum

velocity and the particle diameter size with particle den-

sity s



0510 15 20 253035 40 45 50

200

400

600

800

1000

1200

Velocity of gas



150



250



500



750



Figure 8. The velocity of particle and the axis velocity of gas

with s



= 2650 kg/m3 under considering gravity.

00.1 0.2 0.30.4 0.5 0.6 0.70.8

0x 10

-3

0. 5

1. 5

Figure 9. Particle trajectory with s



= 2650 kg/m3 and

= 750 µm.

e may correspondingly increase. To verify this

ct, we also present the result in Figure 10 which shows

If high-density and coarse particle for parameters are

taken, angl

farticle trajectory with



= 7950 kg/m3 and

d = 1000

µm.

We get (Figure 11)

0.0068



lm, 0.7101



lm, 0.5489



3.3. Influence of the Mean Velocity of Quality on

Particle Acceleration

corre ulations are given as

In this section, we also investigate the trend of velocity

of the particle by the quality mean velocity of gas. The

sponding results of these sim

= 5300 kg/m3. follows, The reason is similar to the statement in section

K. L. ZHENG ET AL. 25

2.3, and we do not explain it here repeatedly.

From Figure 12, w e h ave

0.0014

lm, 0.4840

lm, 0.16



Accordingly, (Figure 13)

0.0109

lm, m, .

0.6667

l0.9



From the numerical simulation in section 3.2 and 3.3,

it can be seen that the influence of ity in th

of particle acceleration is not obvious. While aimed at

acceleration of the particle

grinding. On the one hand, based on

on model, the relationships of particle

grave process

gh-density and coarse particle, the influence of gravity

needs to be considered.

4. Conclusions

This paper mainly focuses on

in the process of jet

a basic particle moti

velocity with gas velocity, particle density and particle

diameter size are obtained through numerical simulations.

00.1 0.2 0.30.4 0.5 0.6 0.7 0.8

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Figure10. Particle trajectory with s



= 7950 kg/m3 =

1000 µm.

05 10 15 20 2530 3540 4550

100

200

300

400

500

600

700

800

900

1000

Vel oci ty of gas



150



250



500



750



Figure 11. The velocity of particle and the mean velocity of

quality of gas with



00.1 0.2 0.30.4 0.5 0.6 0.70.8

0x 10

-3

0.5

1.5

Figure 12. Particle trajectory with s



= 2650 kg/m3 and

= 750 µm.

00.1 0.2 0.30.4 0.5 0.6 0.70.8

= 2650 kg/m3 under considering grav-

ity.

0. 002

0. 004

0. 006

0. 008

0. 01

0. 012

0. 014

0. 016

0. 018

0. 02

Figure 13. Particle trajectory with s



= 7950 kg/m3 and

= 1000 µm.

Onnd, the mathematical model of particle

acceleration r the influence of gravity is established

nd its numerical simulation is also carried out. Theo-

Re-

University of Science and

nd the National Science and

the other ha

unde

retical analysis will have great guidance to the improve-

ment of the fluidized bed jet grinding technology.

5. Acknowledgements

This work is supported by the Doctoral Program

search Fund of Southwest

Technology (No.11zx7129) a

Technology Support Program (No. 2011BAA04B04).

K. L. ZHENG ET AL.

34, No. 1, 1983, pp. 81-86.

doi:10.1016/00

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