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
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
2
0.75 (() )
sd s
ss
duC ut u
dt d


 (1)
where
s
u is the velocity of particle, is the veloc-
ity of gas, ()ut

is the gas density,
s

is the particle den-
sity,
s
d is the particle diameter size and is the
particle drag force coefficient. d
C
Equation (1) indicates the velocity of particle
s
u is a
function in t. By transformation, it can be rewritten to an
equation about th e particle velocity and the jet distance,
2
0.75 (() ),
sde
ss
ss
duC d
uu
dx d



xu
(2)
Where
x
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.
22
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
d
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)
22
() 1 1.525.28
(

u
ux axax )
M
ean velocity ofquality
(4)
where is the coefficient of turbulence.
0.076a
In main part,
6.3
() ( ),
1.93

u
u xAxisvelocity
x (5)
3
()
1.93
(

u
ux x).
M
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 [,]
x
y, wh er e
x
is the dimensionless distance, is the parameter
group including gas v e locity, ga s density an d viscosity .
y
Next, letmm, and
16
e
d2650
s

kg/m3 and
7950
s

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).
d
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
0
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
0
200
400
600
800
1000
1200
V elocit y of gas
50

m
150

m
250

m
500

m
750

m
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
s

kg/m3, when particle diameter is
s
d
= 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
s

= 5300 kg/m3.
2.3. Influence of the Mean Velocity of Quality on
Particle Acceleration
Parameters ,,
s
se
dd

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).
C
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
s

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
0
50
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
0
100
200
300
400
500
600
700
800
900
1000
Velocity of gas
50

m
150

m
250

m
500

m
750

m
Figure 5. The velocity of particle and the mean velocity of
quality of gas with s

= 2650 kg/m3.
051015 202530 3540 4550
0
100
200
300
400
500
600
700
800
900
1000
Velocity of gas
50

m
150

m
250

m
500

m
750

m
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
s

= 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,
2
0.75 (() ),
sxd e
sx sx
ss
duC d
uu
dx d


xu (7)

K. L. ZHENG ET AL.
24
,
sy
du
g
dt
where
(8)
,
s
xsy
uu
particle, resp
mean the horizontal and vertical velocity
of the ectively. The particle velocity
s
u can
be obtaine d by
22
,
s
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 /
y
x
ll

(10)
where ,
x
y
ll
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
s

= 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
by
that
th
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
s

= 2650 kg/m3 and
s
d = 750 µm.
By equation (10), we can get
0.0011
y
l m, 0.5680 0.1  .
x
l m,
0100 200 300400 500 600 700800
20
40
60
80
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
0
200
400
600
800
1000
1200
Velocity of gas
50

m
150  m
250

m
500

m
750

m
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
1. 5
2
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
pa
s
d
If high-density and coarse particle for parameters are
taken, angl
farticle trajectory with
s

= 7950 kg/m3 and
s
d = 1000
µm.
We get (Figure 11)
0.0068

y
lm, 0.7101

x
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
y
lm, 0.4840
x
lm, 0.16  .
Accordingly, (Figure 13)
0.0109
y
lm, m, .
0.6667
x
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
hi
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
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.
s
d
05 10 15 20 2530 3540 4550
0
100
200
300
400
500
600
700
800
900
1000
Vel oci ty of gas
50  m
150  m
250

m
500

m
750

m
Figure 11. The velocity of particle and the mean velocity of
quality of gas with
s

00.1 0.2 0.30.4 0.5 0.6 0.70.8
0x 10
-3
0.5
1
1.5
2
Figure 12. Particle trajectory with s

= 2650 kg/m3 and
= 750 µm.
s
d
00.1 0.2 0.30.4 0.5 0.6 0.70.8
0
= 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
s
d
the other ha
unde
a
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.
26
ol
34, No. 1, 1983, pp. 81-86.
doi:10.1016/00
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