3D turbulent flow modeling in the separation column of a

Powder Technology 235 (2013) 82–90
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Powder Technology
journal homepage: www.elsevier.com/locate/powtec
3D turbulent flow modeling in the separation column of a circumfluent cyclone
Baocheng Shi a, Jinjia Wei a,⁎, Pingzhong Chen b
a
b
State Key Laboratory of Multiphase Flow in Power Engineering, Xi'an Jiaotong University, Xi'an 710049, China
School of Petroleum Engineering, Yangtze University, WuHan430100, China
a r t i c l e
i n f o
Article history:
Received 16 November 2011
Received in revised form 22 September 2012
Accepted 6 October 2012
Available online 13 October 2012
Keywords:
Column coordinate system
Separation column
3D turbulent flow field
Separation efficiency
a b s t r a c t
A detailed study of the internal flow field for the separation column in a circumfluent cyclone separator has
been of great significance in understanding its separation mechanism and thus improving its efficiency. The
turbulent flow viscidity theory and average velocity field were used, based upon the Reynolds and continuity
equations for the column geometry and coordinate system. An approximate analysis of the velocity and pressure
distribution of the flow field inside the separating column was carried out, for which basic equations for threedimensional velocity, pressure gradient and distribution of static pressure were given. Results obtained from experiments and prediction for tangential velocity and static pressure distribution are in very good agreement according
to a specific set of experiments. More specifically, the tangential velocity of the inside separation column is characterized by its centrally symmetric distribution pattern, and the axial velocity is in basically axial symmetric distribution with the exception of the outlet vicinity. The value near the axial center reaches its maximum, which decreases
gradually along the radial direction, and the value near the wall approaches zero. The value for static pressure near
the axial center is the lowest one. When the length of the radius increases, the intensity of the pressure will also
increase. Predictions showed that the axial line of the vortex core fishtails along the geometrical axial center
when the gas in the separation column rotates at a high speed and results in an unstable flow field which reduces
the separation efficiency.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
The presence of the ultrafine sand particle from unconsolidated
siltstone sources in the gas production is a problem, which is difficult to
be solved. Tiny sand in natural gas pipeline brings about much potential
safety risk, such as the decrease of the treating capacity of equipment
caused by the wearing and blockage of the pipeline and equipment,
which makes the control operation out of order, and more gas loss caused
by the process flow unclosed. Therefore, many researchers began to study
the features of the internal flow field inside separators to improve
separation efficiency [1–8]. They found that ordinary cyclones provide
high efficiency in capturing dusts larger than 10 μm, but low efficiency
in capturing dusts less than 2 μm, which limits their application scope.
To solve the problem in capturing dusts less than 2 μm in the used
cyclones, a great deal of experiments and theoretical studies on cyclones
were carried out. Bloor and Ingham [9] obtained an exact solution for
internal flow field in the cyclone with the assumptions of no viscosity
and axisymmetry involved. Jia and Zhang [10], and Boysan [11] and
Xu [12] analyzed the internal flow field of cyclone separator, and
the result showed that the tangential velocity agrees well with the
experiment result. However, because the viscous term of Reynolds
equation was ignored and the mathematical model was too simplified,
⁎ Corresponding author.
E-mail address: [email protected] (J. Wei).
0032-5910/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.powtec.2012.10.009
the other velocities only approximate to the experiment results. Basing
on multiphase fluid dynamics principle, Zhou and Soo [13,14] solved
the gas flow field of cyclone separator and obtained the polynomial
expressions of axial velocity, tangential velocity and pressure drop.
Ogawa [15] derived a combined vortex model based on the energy
equation, and the theoretical calculation results agree well with the
measured values. Wang et al. [16] derived the tangential velocity formulas
for cyclone both with and without reducing pressure drop stick (Repds)
based on the theory of viscous fluid mechanics. The calculation results
of both formulas have a good agreement with the experiments. Chen
and Shi [17] established a set of simplified gas motion equations for the
internal flow in a cyclone separator by adopting an algebraic eddy
viscosity model. By using the assumed axial or radial velocity profile, a
semi-empirical three-dimensional velocity solution to the simplified
equations for the separation space was deduced. Wang [18] systematically analyzed the cyclone gas flow field of a circumfluent cyclone separator
based on the Navier–Stokes equation and the continuity equation of ideal
flow under columnar coordinates. The Adomian decomposition-inverse
operator method was used to solve the problem in flow field. The basic
representations for radial velocity, tangential velocity and axial velocity of the ideal flow were derived. Dewil et al. [19] and Chan
et al. [20] did a great work on the direct measurement of particle velocity and particle occupancy in a circulating fluidized bed (CFB)
cyclones operating at high solid loadings, and particle trajectories
within the cyclone were obtained through positron emission particle
tracking (PEPT).
B. Shi et al. / Powder Technology 235 (2013) 82–90
Nomenclature
Notation
ur
uθ
uz
p
r
z
u′ r
u′ θ
u′ z
R
Vin
uθ0
uz0
U
V
W
P
V0
W0
P0
Re
k1, k2
c
Radial velocity: m/s
Tangential velocity: m/s
Axial velocity: m/s
Pressure: Pa
Radial coordinate: m
Axial coordinate: m
Radial fluctuating velocity: m/s
Tangential fluctuating velocity: m/s
Axial fluctuating velocity: m/s
Inside radius of the separating column: m
Inlet velocity of primary cyclone: m/s
Recirculation inlet tangential velocity: m/s
Axial velocity near the inlet of the separating column: m/s
Dimensionless radial velocity
Dimensionless tangential velocity
Dimensionless axial velocity
Dimensionless pressure
Dimensionless tangential velocity of calculation section
Dimensionless axial velocity of calculation section
Dimensionless static pressure of calculation section
Reynolds number
Dimensionless axial parameters
Dimensionless tangential parameters
Greek letters
ρ
Fluid density: kg/m 3
χ
Dimensionless radial coordinate
ξ
Dimensionless axial coordinate
ε
Dimensionless turbulent viscosity
υ
Kinematic viscosity: m 2/s
Turbulent viscosity: m 2/s
υt
η
Separation efficiency of circumfluent cyclone separator
Subscripts
r
Radial
θ
Tangential
z
Axial
In general, although many valuable results have been achieved
during large aggregation studies, some shortcomings still exist. For
example, the calculation models are either too simple [9–12], such as
the assumptions of no viscosity and ideal flow, of which the precision
of the calculation results cannot meet the requirements of engineering
practices; or too complex [13–17], the results depend on the theoretical
research of the specialist, the professional software and the high performance computer, so it is difficult to be carried out in engineering. Moreover, fewer researches are focused on effectively removing submicron
dusts contained in the dusty gas. In recent years, a new type of cyclone,
the circumfluent cyclone separator was developed in China which is
able to remove submicron dusts contained in the dusty gas successfully
[21,22], as shown in Fig. 1. However, only a preliminary study was
carried out on the circumfluent cyclone separator. The flow field
of the circumfluent cyclone system differs greatly from that of the
common cyclone. Firstly, both the upper and lower swirling flows along
the axis are generated in the same rotational direction in the common
cyclone, whereas only the upper swirling flow is generated inside the
separation column for the circumfluent cyclone system. Secondly,
there is only a tangential gas entrance in the common cyclone, which
is the original power for gas rotational movement; however, there are
83
two entrances in the circumfluent cyclone system, namely the exhaust
of the primary cyclone and the reflux inlet of the second cyclone. In addition, both the exhaust and the reflux outlets are located in the top of
the separation column. Hoffmann and Qian [23,24] carried out experiments and theoretical studies on the 3D turbulent flow field inside the
separation column (Fig. 2) of a circumfluent cyclone separator, the
mathematical model of the three-dimensional velocity of the separation
column of the circumfluent cyclone separator was established, and the
analytical solution for the three-dimensional velocity distribution was
given. Unfortunately, there are two problems existing in it, one is the
unreasonable theoretical hypothesis of the fully developed flow pattern,
which causes the momentum term in Navier–Stokes equation to be
neglected. In fact, it is the viscosity term but the momentum term
might be neglected, because the Reynolds number of the flow field inside
the separation column is 10 5 ~ 106 (gas max rotation speed inside the
cyclone separator will reach ten to hundred meters per second. Gas
density is 1.2 kg/m3, gas viscosity is generally 1 ~ 2 × 10−5 Pa s) [25].
The other reason is that the effect of that reflux gas on the flow field inside
the separation column was neglected, which causes the deviation of the
calculated results from experimental data.
The circumfluent cyclone separator has excellent performance.
Therefore, it is extensively applied in various industries and engineering
fields. The separation column is one of its key separation components, of
which the efficiency plays a decisive role in that of the whole separator.
Consequently, a detailed study of the internal flow field in the separation column has been of great significance in understanding its separation mechanism and thus improving its efficiency. The turbulent flow
viscidity theory and average velocity field were used, based upon the
Reynolds and continuity equations for the column geometry and coordinate system. An approximate analysis of the velocity and pressure distribution of the flow field inside the separating column was carried out,
for which the basic equations of three-dimensional velocity, pressure
gradient and distribution of static pressure were given.
2. Mathematical models
The assumptions of the gas flow in the separation column of the
circumfluent cyclone separator (Fig. 1) are as follows: (1) symmetric
along the axis, (2) incompressible viscous and (3) steady.
Based on the above assumptions, Reynolds and continuity equations
in the cylindrical coordinate system can be expressed as follows:
∂ur ∂uz ur
þ
þ ¼0
r
∂r
∂z
ð1Þ
ur
2
∂ur u2θ
1 ∂p
∂ ur 1 ∂ur ur
1 ∂ru′ 2r 1 ′ 2
−
þ
þv
− ¼−
−
þ uθ
r
ρ ∂r
r ∂r
r
∂r
∂r 2 r ∂r r 2
ð2Þ
ur
2
∂ur ur u2θ
∂ uθ 1 ∂uθ uθ
∂u′ 2r u′ θ 2 ′ ′
¼v
þ
þ
− 2 −
− u θu θ
2
r
r ∂r r
r
∂r
∂r
∂r
ð3Þ
ur
∂ur
∂u
∂2 uz 1 ∂uz
þ
þuz θ ¼ v
r ∂r
∂r
∂r
∂r2
!
−
∂u′ r u′ z 1 ′ ′
− u ru r
r
∂r
ð4Þ
Wherein uθ, ur, uz, p are the mean values of velocity and pressure,
and u′ θ ; u′ r ; u′ z are the velocity fluctuations. It can be seen that Reynolds
equation is very similar to the Navier–Stokes equation except for the
addition of some turbulence terms caused by Reynolds stress. By
using the average velocity field model proposed by Boussinesq
[26,27], the Reynolds stress can be deduced as the function of mean velocity gradient,
−u′ i u′ j ¼ vt
∂ui ∂uj
þ
∂xj ∂xi
!
ði; j ¼ r; zÞ
ð5Þ
84
B. Shi et al. / Powder Technology 235 (2013) 82–90
Fig. 1. Structure of circumfluent circulation cyclone separator.
Thereupon, Reynolds stress equation is
∂u
2 ∂ur ur
−u′ r u′ r ¼ vt 2 r −
þ
r
∂xr 3 ∂xr
∂u
2 ∂uθ uθ
þ
−u′ θ u′ θ ¼ vt 2 θ −
r
∂xr 3 ∂xr
−u′ r u′ θ ¼ vt r
In a strongly swirling turbulent flow with Re ≥ 10 5, the Reynolds
stress is much greater than gas viscous stress, so the viscous shear
stress terms can be neglected. Therefore, for the column geometry
and coordinate system, the boundary conditions and Eqs. (1)–(4)
can be simplified as follows
−
r ¼ 0 : uθ ¼ 0; r ¼ R ; z ¼ 0 : uθ ¼ uθ0 ; r ¼ 0; z ¼ 0 : uz ¼ uz0 ; r
−
¼ R ; z ¼ 0 : uz ¼ 0
∂ uθ ∂u
; −u′ r u′ z ¼ vt r r
∂r r
∂r
ð6Þ
∂ur ∂uz ur
þ
þ ¼0
r
∂r
∂z
ð7Þ
2
2
∂u
u
1 ∂p
∂ ur 1 ∂ur ur
þ
ur r − θ ¼ −
þv
− 2
r
ρ ∂r
r ∂r
∂r
∂r 2
r
ur
∂ur ur u2θ
∂2 uθ 1 ∂uθ uθ
¼v
þ
þ
− 2
r
r ∂r
∂r
∂r 2
r
ur
∂ur
∂u
∂2 uz 1 ∂uz
þ
þ uz θ ¼ v
r ∂r
∂r
∂r
∂r 2
!
ð8Þ
!
ð9Þ
!
ð10Þ
The coordinate system is shown in Fig. 3. For the convenience of
analysis, Eqs. (7)–(10) and boundary conditions Eq. (6) should be in
a dimensionless form. Here, we define, χ ¼ Rr , ξ ¼ Rz , U ¼ Vuinr , V ¼ Vuinθ ,
W ¼ Vuinz , P ¼ ρVp2 , ε ¼ RVvtin , so the result is:
in
−
χ ¼ 0 : V ¼ 0; χ ¼ 1 ; ξ ¼ 0 : V ¼ V 0 ; χ ¼ 0; ξ ¼ 0 : W ¼ W 0 ; χ
−
¼ 1 ; ξ ¼ 0 : W ¼ 0:
ð11Þ
Hence, the normalized form of Eqs. (7)–(10) can be rewritten as
∂U ∂W U
þ
þ ¼0
∂χ
∂ξ χ
U
Fig. 2. Structure of separation column.
∂U V 2
∂P
∂2 U 1 ∂U
U
¼−
þ
−
þε
−
χ
∂χ
∂χ
∂χ 2 χ ∂χ χ 2
ð12Þ
!
ð13Þ
B. Shi et al. / Powder Technology 235 (2013) 82–90
85
2.1. Tangential velocity
According to the above assumptions, the tangential velocity V is only
related to radial coordinate χ, and Eq. (18) is converted into the following
ordinary differential equation:
idV 1 k1 h
d2 V 1 dV
V
2
exp −cχ −1
þ
þ V ¼ε
−
χ
dχ χ
dχ 2 χ dχ χ 2
!
ð20Þ
Furthermore,
ε
i dðχV Þ
d 1 dðχV Þ
k h
2
− 12 exp −cχ −1
¼0
dχ χ dχ
dχ
χ
ð21Þ
By dividing each side of Eq. (21) with χ, we can obtain the follow
equation
ε
(a) Velocity
(b) Static pressure field
Fig. 3. The coordinate systems and the distribution of measurement holes. (a) Velocity
and (b) static pressure field.
U
U
∂U UV 2
∂2 U 1 ∂V
V
¼ε
þ
þ
−
χ
∂χ
∂χ 2 χ ∂χ χ 2
∂U
∂W
∂2 W 1 ∂W
þ
þW
¼ε
∂χ
∂χ
∂χ 2 χ ∂χ
ð14Þ
!
ð15Þ
Obviously, this is a nonlinear partial differential equation which is
difficult to be solved directly. Usually, in order to solve this equation,
U or W is assumed as one distribution function, before being
substituted into the corresponding equation to convert the nonlinear
partial differential equations into ordinary differential equations. In
accordance with [28], U is assumed to be an exponential distribution
function, before being substituted into the corresponding equation
and being solved by a continuity equation. Because the value of U is
less than the values of V or W by a dozen times, the former has little
influence on the flow field [29]. Hence, U is substituted by an approximate solution of radial velocity in an ideal state [30], and is given as
follows:
U¼
i
k1 h
2
exp −cχ −1
χ
ð16Þ
∂W
2
¼ 2ck1 exp −cχ
∂ξ
ð17Þ
By substituting Eqs. (16) and (17) into Eqs. (14) and (15), we can
obtain the dimensionless tangential and axial momentum equations
which are shown as follows:
2
i ∂V 1 k1 h
∂ V 1 ∂V
V
2
exp −cχ −1
þ
þ V ¼ε
−
χ
∂χ χ
∂χ 2 χ ∂χ χ 2
!
i ∂W
k1 h
∂2 W 1 ∂W
2
2
exp −cχ −1
þ
þ 2ck1 exp −cχ W ¼ ε
χ
∂χ
∂χ 2 χ ∂χ
ð22Þ
If we define that x ¼ 12 χ 2 ; y ¼ χV, then the Eq. (22) can be
transformed as follows:
ε
!
i 1 dðχV Þ
1 d 1 dðχV Þ
k h
2
− 12 exp −cχ −1
¼0
χ dχ χ dϕ
χ dχ
χ
d 2 y k1
dy
− ½ expð−2cxÞ−1
¼0
dx
dx2 2x
ð23Þ
This is a second order ordinary differential equation with variable
coefficients. ε distribution plays a decisive role in the properties of
characteristic equation. For ε = 0, Eq. (23) can be converted into the
following first order ordinary differential equation
k1
dy
½ expð−2cxÞ−1
¼0
2x
dx
ð24Þ
Obviously, the solution of the above equation is:
y ¼ ηV ¼ Const
ð25Þ
The tangential velocity values in external rotating areas under inviscid fluid model are in good agreement with experiment data
[31,32]. Hence, Eq. (25) can be used as an approximate solution for
the tangential velocity in external rotating area.
Furthermore, the following equation can be obtained by truncating appropriately Taylor series expansion of Eq. (23),
k1
½ expð−2cxÞ−1≈k1 c
2x
ð26Þ
Wherein c is a constant obtained through the experiment.
By substituting Eq. (26) into Eq. (24), we obtain the following
equation:
d2 y ck1 dy
þ
¼0
ε dx
dx2
ð27Þ
The solution of the above equation is:
ð18Þ
!
ð19Þ
The main inertia term as well as Reynolds stress term of turbulence
characteristics are preserved in the result worked out in this way.
ck
y ¼ A0 þ A1 exp − 1 x
χ
ð28Þ
If we define that x ¼ 12 χ 2 ; y ¼ χV, the tangential velocity can be
obtained through the following equation
V¼
1
ck
2
A0 þ A1 exp − 1 χ
2ε
χ
ð29Þ
86
B. Shi et al. / Powder Technology 235 (2013) 82–90
When boundary conditions of Eq. (10) are considered, the following
equation can be obtained:
V0
V
; A1 ¼
0 A0 ¼
ck1
1
1− exp − 2ε
exp − ck
2ε −1
ð30Þ
h
i
ck1 2
V 0 1− exp − 2ε χ
h
i
V¼
χ
1
1− exp − ck
2ε
ð31Þ
2.2. Axial velocity
We can note from the above equation that the axial velocity is
equal to an approximate linear function of axial position. Therefore,
by using variable separation method, we can obtain the following
equation:
W ¼ g ðξÞf ðχ Þ
ð32Þ
By substituting Eqs. (10) and (32) into Eq. (27), we can obtain the
following equation:
ð33Þ
ð34Þ
Obviously, it is difficult to solve the above equation since it contains a regular singular point. The complete solution can be found by
both variable separation and series methods near the regular singular
point,
f 1 ðχ Þ ¼
∞
X
∞
X
n
ð35Þ
n
dn χ þ mf 1 ðχ Þ lnχ
n
ð36Þ
0
n
dn χ ≈d0 y1 þ a0 y2
ð38Þ
n¼0
3 2 1 1 2
4
y2 ¼ − aχ −
a −ca χ
4
16 2
1
1 3
1
19 2 1 4
2
6
2 2
8
þ
7ca þ a χ −
3c a þ
ca þ a χ
144
4
1024
9
9
If we define that a ¼ b1 ¼ ckε1 and m = 1, Eq. (35) is given by,
f 1 ðχ Þ ¼ a0 y1
ð39Þ
f 2 ðχ Þ ¼ a0 y1 lnχ þ d0 y1 þ a0 y2
ð40Þ
We note that Eq. (32) is a linear function contains f1 and f2.
f ðχ Þ ¼ A1 f 1 ðχ Þ þ A2 f 2 ðχ Þ
ð41Þ
Substituting Eqs. (37) and (38) into Eq. (39), we obtain
f ðχ Þ ¼ A1 a0 y1 þ A2 ða0 y1 lnχ þ d0 y1 þ a0 y2 Þ
ð42Þ
Obviously, a singularity (χ = 0) is contained in the term of ln χ,
hence boundary conditions cannot be used directly. If
ð43Þ
Wherein χ stands for a micro amount.
Substituting boundary conditions Eqs. (10) and (43) into Eq. (42),
we obtain the following:
W 0 y21 þ y11 da0
y11
W0
0
; A2 ¼ −
A1 ¼ y21 −y11 lnη a0
a0 y21 −y11 ln η0
2
2 2
3
1
1
ca þ 256
c a þ 13 ca
Where y11 ¼ 1 þ 12 a− 18 ca− 48
ð44Þ
3
1 1 2
y21 ¼ − a−
a −ca
4 16 2
1
1 3
1
19 2 1 4
2
2 2
7ca þ a −
3c a þ
ca þ a
þ
144
4
1024
9
9
2
n an þ b1 ðn−2Þan−2 þ b2 an−4 þ b3 an−2 ¼ 0
g ðεÞ ¼ k1 ε þ k2
ð37Þ
The coefficient of each item is solved according to recursion
relation. If we substitute the above eight terms, then, a1 = a3 =
1
½ð2b1 þ b3 Þa2 þ b2 a0 ; a6 ¼
a5 =a7 =0; a2 ¼ − 14 b3 a0 ; a4 ¼ − 16
1
1
− 36
½ð4b1 þ b3 Þa4 þ b2 a2 ; a8 ¼ − 64
½ð6b1 þ b3 Þa6 þ b2 a4 Similarly, d1 = d3 = d5 = d7 = 0; d2 ¼ − 14 ð4a2 þ b1 a0 þ b3 d0 Þ; d4 ¼
1
þ b3 Þd2 þ b2 d0 þ 8a4 þ b1 b2 ;
d6 ¼ − 36
½ð4b1 þ b3 Þd4 þ
1
½ð6b1 þ b3 Þd6 þ b2 d4 þ 16a8 þ b4 b2 b2 d2 þ 12a6 þ b1 b4 ; d8 ¼ − 64
ð45Þ
Substituting Eq. (45) into Eq. (27), we can obtain the axial velocity
solution as
W ¼ ðk1 ε þ k2 Þf ðχ Þ
Wherein m, an, dn are the respectively undetermined constant and
coefficients.
1
½ð2b1
− 16
n¼0
∞
X
Considering
an χ
0
f 2 ðχ Þ ¼
n
an χ ≈a0 y1 ;
χ ¼ χ 0 ;ε ¼ 0; W ¼ W 0
!
Furthermore, the following equation can be obtained by truncating
appropriately Taylor series expansion of Eq. (33)
ck
d2 f
2ck1 1 df
2
1
−cχ
χ
þ
þ
−1
þ
¼0
ε
ε
χ dχ
dχ 2
∞
X
2 2 1 3 8
1
1
Where y1 ¼ 1 þ 12 aχ 2 − 18 caχ 4 − 48
ca2 χ 6 þ 256
c a þ 3 ca χ
By substituting Eq. (30) into Eq. (29), the approximate solution of
the tangential velocity inside the separation column can be obtained
as follows:
i df
k1 h
d2 f
1 df
2
2
exp −cχ
þ
þ 2ck1 exp −cχ f ¼ ε
χ
dχ
dχ 2 χ dχ
Furthermore,
ð46Þ
2.3. Static pressure distribution
Substituting tangential velocity Eq. (30) and radical velocity Eq. (16)
into radical momentum Eq. (13) yields the following expression for the
static pressure drop:
!
∂P
1 2
∂2 U 1 ∂U
U
∂U
þ
¼ V þε
− 2 −U
2
χ ∂χ χ
∂χ χ
∂χ
∂χ
,
i2
V2
k c 2 2
k c 2
k2 h
2
1− exp − 1 χ
¼ 03 1− exp − 1 χ
þ 13 1− exp −cχ
2ε
2ε
χ
χ
2ck2
h
i
2
2
2
2
1
1− exp −cχ
exp −cχ
−4k1 εc exp −cχ −
χ
ð47Þ
B. Shi et al. / Powder Technology 235 (2013) 82–90
Evidently, radial pressure drop distribution is a complex exponential
function, where ε ≪ 1 and k1 ≪ 1, and the right three terms can be
neglected. Therefore, we have
,
2
∂P V 0
k c 2 2
k c 2
1− exp − 1 χ
¼ 3 1− exp − 1 χ
2ε
2ε
∂χ χ
,
V 20
k1 c 2 2
1 2
1− exp − aχ
χ
¼ 3 1− exp −
2ε
2
χ
ð48Þ
Radial static pressure distribution can be derived from integration
of Eq. (42)
∂P
þ C0
∂χ
Kp
1
1
1 2
2
1− 2 þ aψ −1; a −ψ −1; aχ þ aψ −1; aχ −aψð−1; aÞ
¼ P0 þ
2
2
2
χ
P¼∫
ð49Þ
−1aχ 2 −2
where −1 12 at −2 e−t dt, ψ −1; 12 aχ 2 ¼ ∫−12 t e−t dt, ψ(−1, aχ2) =
aχ2 −2 −t
∫−1
t e dt,
a
a) = ∫−1
t −2e −tdt
ψ(−1,
In Eq. (47), the effects of different turbulent viscosities on pressure
grade distribution are embodied through function ψ.
3. Introduction to the experiment
In Sebei Gas Field, the particle size distribution of sand from gas
fields is mostly 1 ~ 5 μm. As the circumfluent cyclone separator has
excellent performance to remove submicron dusts from gas fields
successfully, the test could be carried out. The experimental facilities
consisted of a dedusting system, an air supply system, a dust generating
system, a dust collecting system, a data acquisition system and a
computer data processing system. The dedusting system was the
most important part as shown in Fig. 1. Amount of reflux gas was
about 10–15%, and the input of dirty gas included nature gas and
ultrafine sand particles. Dirty gas flew into primary cyclone from the
tangential entrance with the help of fans, and dusts were separated
from gas flow by the centrifugal force which was caused by the rotation
of dusty gas in the core separation column, and the clean gas flew out
from exhaust in the top of the separation column. However, the gas
flow containing bigger dusts concentration rotated upward along the
wall surface, and flew into the second cyclone from tangential outlet
for secondary separation of dusts. Then, the clean gas flew out from
the exhaust of the second cyclone and back to gas entrance through a
circulating pipeline, which formed a loop. This test result shows [33]
that the cyclone system's maximum collection efficiency will be up to
99% when the sand size distribution is 2 ~ 5 μm, which will greatly
improve the separating efficiency of the cyclone separator to remove
submicron dusts.
Dimensions of the dedusting system were: the diameter of primary
cyclone was 250 mm with a height of 900 mm; the diameter of
secondary cyclone was 150 mm with a height of 350 mm; the diameter
of separation column was 120 mm with a height of 1000 mm; the
exhaust-tube diameter on the top of the separation column was
60 mm; the tangential-outlet diameter on the top of the separation
column was 40 mm; and the entrance diameter at the bottom end
of the separation column was 40 mm. Air supply system consisted
of a fan, an orifice flowmeter, a U-type differential pressure meter, pipelines and butterfly valves. The flow rate of the fan was 800 ~ 1800 m3/h
with a pressure of 8000 Pa, and the power was 5 kW. Flow rate was
controlled with a valve. Dust generating system consists of a compressor, a buffer tank, a generator and pipelines. Dust concentration was
controlled with air flow rate of the compressor.
87
Velocity field was obtained by IFA. There were 300 hot wire (film)
anemometer in the separation column. Fig. 3(a) shows the distribution
of velocity field measurement holes. 12 acquisition sections were set
along the axial direction of separation column, and measurement
holes were arranged in each section at 0°, 90°, 180° and 270°. 10 acquisition points were set along radial positions with r = 3, 6, 12, 18, 25, 31,
38, 45, 50 and 55 mm respectively. The axial distance of each acquisition section was 50 mm.
Static pressure field was obtained by U-type tube pressure difference meter. Fig. 3(b) shows the distribution of static pressure field measurement holes. 6 acquisition sections were set along the axial direction
of separation column, and the measurement holes were arranged in
each section at 0° and 180°. 12 acquisition points were set along radial
positions with r = 0, 3, 6, 9, 12, 19, 25, 31, 37, 44, 50 and 57 mm, respectively. The axial distance of each acquisition section was 100 mm.
Separation efficiency was obtained by filtration method which was
in accordance with the guidelines of methods for testing the performance of air filtration of JB/T9747 internal-combustion engine.
η¼
1−
ΔM
100%
M
ð50Þ
Where η is separation efficiency; M is the amount of talcum powder;
and ΔM is the net increases of air filtration.
4. Numerical calculations and analysis
For the purpose of comparison, mathematical analysis was carried
out under the same experimental conditions [32]. The parameters
were: R = 0.05 m, r0 = 0.02 m, L = 0.6 m, Vin = 30 m/s, qi = 0.1 m 3/s,
ρ = 1.22 kg/m 3, V0 = 0.55, W0 = 0.61, i = 0.3, P0 = 1.92. The value of
vt was 400 times gas viscosity [34], so in this study, ε, the value of dimensionless turbulent viscosity, was 0.003. The parameters, k1 and c,
were obtained by experiment or derived by Eqs. (51) and (52) in [8].
Results obtained by both methods were consistent, and the parameters
of this paper, k1 and c, were conducted under the same experimental
conditions of [8]. Substituting the values of these parameters into corresponding equations, the dimensionless radial velocity U, the dimensionless tangential velocity V, the dimensionless axial velocity W, the radial
∂P
and the radial static pressure P distribution were
pressure drop ∂χ
obtained.
k1 ¼
"
#
1
Q0
W
2 − 0
2c
l 2π 1− exp cβ
cQ i ð1 þ iÞ ¼ π½1− expð−cÞ
ð51Þ
ð52Þ
4.1. Comparison experiments of computational results
The distinction between experimental results and predicted results
is shown in Figs. 4–7. The experimental values are indicated with solid
experimental values), and the predicted values with
symbols (
solid lines ( predicted values).
Distribution of the theoretical and experimental results of tangential
velocity (z =250 mm and z=300 mm) and axial velocity is respectively
compared in Figs. 4–6. Considering the effect of turbulent viscosity, the
experimental results agreed well with the calculated results, but for the
tangential velocity distribution, the deviation increased in central part.
The difference was mainly due to the existence of forced vortex. The
flow in axial center was extremely complex, three-dimensional, unsteady and irregular with rotation, and the vortices had different sizes
and shapes in flow field. In this case, average velocity field hypotheses
might not be suitable, but the calculation may become much more complex if using other models to calculate the flow fields. On the other hand,
although the deviation increased the predicted results of tangential
88
B. Shi et al. / Powder Technology 235 (2013) 82–90
Tangetial velocity V
1.6
experimental values
predicted values
1.4
1.2
1
0.8
0.6
0.4
Axial velocity W
1.8
0.2
0
-1
-0.8 -0.6 -0.4 -0.2
0
0.2
0.4
0.6
0.8
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1
experimental values
predicted values
-1
-0.8 -0.6 -0.4 -0.2
0
0.2
0.4
0.6
0.8
1
χ=r/R
χ=r/R
Fig. 6. Axial velocity distribution (z = 300 mm).
Fig. 4. Tangential velocity distribution (z = 250 mm).
velocity in central part (r/R=0~0.1), it was not the important separation
parameter because the separation quality mainly depended on the outer
tangential velocity and axial velocity [35–37]. So the deviation among
the predicted results of tangential velocity in central part has little influence on separation efficiency. Yang, Luo and Li [38–40] analyzed the factors of separation efficiency of a cyclone separator, and regarded the
distribution of internal flow field inside the cyclone separator as the
key influence factor. The internal flow field is divided into three regions:
(I) r/R=0~0.1; (II) r/R=0.1~0.5; and (III) r/R=0.5~ 1.0. The distribution of tangential velocity in Region (I) has little influence on separation
efficiency. The tangential velocities should be high enough in Regions II or
III to move particles to the wall, so they have a great effect on separation
efficiency. The axial velocity is much smaller than the tangential velocity,
but it is very important to improve the separation efficiency of a cyclone
separator. Therefore, although the model fails to accurately predict the
details of cyclone separators which are not important, the key influence
factors are all reasonably considered in the model. In this paper, we derived simpler formulas by considering key factors comparing with other
models. And the predicted results of the tangential velocity(II and III)
and axial velocity agreed well with experimental values. The mathematical model and computation equations are proved reasonable, and very
easy and convenient to be carried out in engineering. In the middle area
of separation column, the tangential velocity is characterized by the
centrally symmetric distribution pattern. However, the maximum
tangential velocity in various sections has different radial positions.
It may be caused by the vortex core fishtailing along the geometrical
axial center when the gas in separation column rotates spirally at a
high speed. The instability of the vortex core leads to a low-pressure
fluctuation, which results in dust back-mixing and poor separation
efficiency.
As shown in Fig. 6, the theoretical prediction of the variation of
axial velocity agrees well with experimental data. The theoretical
data are less than the experiments, which shows that the turbulent
viscidity has a certain influence on axial velocity. The axial velocity
basically has axial symmetric distribution except for the vicinity of
the outlet, and the value near axial center reaches the maximum.
When the radius increases, the axial velocity tends to decrease gradually,
and the values will be less than the tangential velocity and tend to be
zero near the wall. This is because, on one hand, the axial gas velocity
gradually increases due to contraction of the exhaust pipes; but on the
other hand, part of the gas flow out from tangential direction caused
by tangential reflux at the upper end of the separation column.
Fig. 7 shows the radial static pressure distribution (z = 300 mm).
The calculated results are in good agreement with experimental data.
Near the axial center, a low-pressure center is formed in the column.
When the radius increases, the pressure will also increase. This is because that the gas moves upward spirally with the maximum velocity
near the axis, which transforms partial static pressure into dynamic
pressure in center area. On the other hand, the centrifugal force caused
by high speed rotation of the gas flow is helpful to generate a lower or
sometimes negative pressure in the center.
4.2. Influence of turbulence viscosity coefficient
The radial distribution of tangential velocity, axial velocity, pressure
drop and static pressure is shown in Figs. 8–11, respectively.
The tangential velocity decreases rapidly with the increase of dimensionless viscosity ε in the vicinity of axis. In most regions near the
boundary, the tangential velocity hardly changes with dimensionless
viscosity coefficient. As dimensionless viscosity coefficient increases,
the radial position of the largest tangential velocity also increases as
shown in Fig. 8. It means that the interface of internal and external
rotating zone moves ceaselessly outward with increasing dimensionless viscosity coefficient. It can also be inferred that the turbulent flow
of separation column will become a rigid rotating flow field when the
dimensionless viscosity tends to infinity.
As shown in Fig. 9, although the model taking into account of the
turbulent viscosity makes axial velocity results closer to the measured
values, the dimensionless viscosity has no significant influence on
axial velocity. That is to say, changes in axial velocity are insensitive
to dimensionless viscosity.
2.5
1.6
experimental values
predicted values
1.4
1.2
1
0.8
0.6
0.4
1.5
1
0.5
predicted values
experimental values
0
-0.5
0.2
0
-1
2
Static pessure P
Tangential velocity V
1.8
-0.8 -0.6 -0.4 -0.2
0
0.2
0.4
0.6
0.8
χ=r/R
Fig. 5. Tangential velocity distribution (Z = 300 mm).
1
-1
-1
-0.8 -0.6 -0.4 -0.2
0
0.2
0.4
0.6
χ=r/R
Fig. 7. Static pressure distribution (z = 300 mm).
0.8
1
B. Shi et al. / Powder Technology 235 (2013) 82–90
89
25
1.6
ε=0.02
ε=0.04
ε=0.06
1.2
1
Pressure gradient
Tangential velocity V
1.4
0.8
0.6
0.4
20
ε=0.02
ε=0.04
ε=0.06
15
10
5
0.2
0
-1
-0.8 -0.6 -0.4 -0.2
0
0.2
0.4
0.6
0.8
0
-1
1
-0.8 -0.6 -0.4 -0.2
χ=r/R
Fig. 8. Tangential velocity distribution with different turbulent viscosity (z = 300 mm).
As shown in Fig. 10, the changes in radial pressure gradient and
radial tangential velocity are similar with the changes in dimensionless
viscosity. With increasing dimensionless viscosity coefficient, pressure
gradient decreases rapidly at the interface of internal and external
parts of the cyclone, and changes more slowly along the radial direction.
There is little variation near the wall.
As shown in Fig. 11, the static pressure increases with increasing
viscosity, and the increasing rate of the static pressure is largest at the
axis and becomes smaller with increasing radius. The viscosity has no
effect on static pressure near the boundary.
4.3. Effects of inlet velocity and dust concentration on separation efficiency
The circumfluent cyclone separator has excellent performance to
remove submicron dusts contained in dust effectively. Experimental
study on relevant influencing factors of separation efficiency of the
circumfluent cyclone separator has been carried out. In the experiment,
the particle size distribution of talcum powder is that the talcum
powder with a size distribution of 10 μm ~ 2 μm accounts for 20% of
the total and that with a size distribution of less than 2 μm accounts
for 80%. Effects of inlet velocity and dust concentration on separation
efficiency of the circumfluent cyclone separator are shown in Figs. 12
and 13, respectively. When the inlet velocity V1 is in the range of
25 ~ 40 m/s, the separation efficiency will change between 97% and
99.9%, which can meet the requirements as is shown in Fig. 12.
Dust concentration and inlet velocity have some influence on separation
efficiency of the circumfluent cyclone separator, and the best separation
results could be obtained when the inlet velocity are in the range of
30~40 m/s, as is shown in Fig. 13. Higher efficiency of circumfluent
cyclone separator is caused by increase of fine dust separation efficiency.
The separation column is the key separation element which determines
the fine dust separation efficiency. Inside the separation column, the
0.6
0.4
0.2
0
-1
-0.8 -0.6 -0.4 -0.2
0
0.2
0.4
0.6
0.8
1
χ=r/R
Fig. 9. Axial velocity distribution with different turbulent viscosity (z = 300 mm).
0.6
0.8
1
5. Conclusions
(1) In the column geometry and coordinate system, based upon the
Reynolds and continuity equations, the turbulent flow viscidity
theory and average velocity field were used to approximately analyze the velocity and pressure distribution of the flow field inside the separating column, by which the basic equations for
three-dimensional velocity, the pressure gradient and the static
pressure were obtained.
(2) Results obtained from experiments and prediction for tangential
velocity and static pressure distribution are in very good agreement according to a specific set of experiments. More specifically,
the tangential velocity of the inside separation column is characterized by its centrally symmetric distribution pattern, and the
axial velocity is in basically axial symmetric distribution with the
exception of the outlet vicinity. The value near the axial center
reaches its maximum, which decreases gradually along the radial
direction, and the value near the wall approaches zero. The value
for static pressure near the axial center is the lowest one. When
the length of the radius increases, the intensity of the pressure
will also increase. Predictions showed that the axial line of the
vortex core fishtails along the geometrical axial center when the
gas in the separation column rotates at a high speed, and results
in an unstable flow field which reduces the separation efficiency.
(3) Dust concentration and inlet velocity have some influence on the
separation efficiency of circumfluent cyclone system. When the
inlet velocity V1 is in the range of 25~40 m/s, the separation efficiency will change between 97% and 99.9%, so the efficiency can
meet the requirements.
Static pressure P
Axial velocity W
ε=0.02
ε=0.04
ε=0.06
1
0.4
unitary upward swirl flow, low fluid velocity gradient and low turbulence intensity can improve the fine dust separation efficiency.
1.4
0.8
0.2
Fig. 10. Pressure gradient distribution with different turbulent viscosity (z = 300 mm).
1.6
1.2
0
χ=r/R
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
ε=0.02
ε=0.04
ε=0.06
-1
-0.8 -0.6 -0.4 -0.2
0
0.2
0.4
0.6
0.8
1
χ=r/R
Fig. 11. Static pressure distribution with different turbulent viscosity (z = 300 mm).
90
B. Shi et al. / Powder Technology 235 (2013) 82–90
separation efficiency η
100
99
98
97
96
95
94
10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44
Inlet velocity (m/s)
Fig. 12. Effects of inlet velocity on separation efficiency (c1 = 20 g/m3).
separation efficiency η
100
99
98
v1=20m/s
v1=40m/s
97
96
95
6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54
dust concentration C1(g/cm3)
Fig. 13. Effects of dust concentration on separation efficiency.
Acknowledgements
We gratefully acknowledge the financial support from the Fundamental Research Funds for the Central Universities and the project of
National Natural Science Foundation of China (NO. 51121092).
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