Phenomenological Model of Cyclone Separator
Kuo-Tai Hsieh (謝國台), King-Jang Yang (楊錦章) and Chin-Hung Ko (柯慶宏)
Department of Applied Mathematics, Chung Hua University
No. 707, Sec. 2, Wufu Rd., Hsinchu, 30012, Taiwan, R.O.C.
Tel: 03-518-6389, Fax: 03-518-6435
Email: kahsieh@chu.edu.tw, kingjang@chu.edu.tw, lhocvye@yahoo.com.tw
Abstract
The development of a mathematical model of cyclone separator based on physics of fluid flow provides
details of gas flow and particle motion. The model is validated by comparing the predicted grade-efficiency with
experimental data reported in the literature. A new particle tracking procedure proposed in this model accurately
predicts the short-circuiting of coarse particles
Keywords: cyclone separator, mathematical model, size classification, short-circuiting flow
1. Introduction
upon which the current cyclone design is almost
The cyclone separator is extensively used to
entirely founded. This usually rests on a number of
collect particles from gases in various industrial
expressions to obtain an overall pressure drop, and
processes. The reasons for its popularity lie in the
more importantly a characteristic grade-efficiency
simplicity
low
curve as a function of design and operating variables.
maintenance and operating costs, and the small
Some popular models include the Lapple (1951),
physical size of the unit. In a typical cyclone
Barth (1956), Leith-Licht (1972), Dietz (1982), and
operation, particle laden gas entering the cyclone
Iozia (1990) models [1-5]. Strauss [6] as well as
from a tangential inlet causes a swirling motion
Buonicore and Davis [7] gave reviews of cyclone
within the device that generates centrifugal force. In
design methods in current use. Over the years, the
this centrifugal force field, solid particles move
Navier-Stokes equations gradually made its way into
towards the wall and are swept downwards to the
model formulation with a number of assumptions.
dust receiver where they are collected. As the gas
Boysan et al. [8] proposed a mathematical model
spirals down towards the apex, the direction of flow
which provided all details of gas flow and particle
is reversed since there is no outlet in the dust receiver;
motion in a cyclone separator. They modeled the
after which the gas spirals upwards in a central
highly swirling flow by means of an algebraic
column of fluid and exits through a vortex finder.
Reynolds stresses approach. The time-averaged
of
its
design
and
operation,
A considerable amount of experimental data
Navier-Stokes
equations
were
solved
by
a
exists on the behavior of flows in cyclones, obtained
semi-implicit scheme developed by Patankar [9] with
for the most part in the 1930s and 1940s before the
the assumption of axisymmetry. Grade-efficiency
availability of laser-Doppler anemometry. Although
curves were obtained by a stochastic particle-tracking
the reliability of this data can not be warranted, it has
technique and showed fair agreement with the
formed the basis of many semi-empirical correlations
experimental data determined by Stairmand [10].
However, the three-dimensional inlet stream was
solid phases within the cyclone. A full multiphase
improperly
two-dimensional
approach will not be pursued, as it would be difficult
formulation. Consequently, the short-circuiting of
to keep computation time reasonable. In view of the
coarse particles was greatly underestimated. By
fact that the dispersed phase is dilute and the particle
adopting the    turbulence model, Yoshida et al.
slip velocities are likely to be small, alternatively, we
[11] solved the three-dimensional Navier-Stokes
assume that particle-fluid momentum coupling is
equations numerically. They indicated that flow field
absent and thus break the modeling work into two
of the cyclone changes with circumferential angle,
parts. First, the gas-phase fluid-flow problem is
and that upward and downward velocity components
solved to predict gas-phase velocities; then, the
merge strongly near the entrance of the dust receiver.
particle motion with respect to fluid is computed by
They also obtained fair agreement between calculated
balancing all forces acting on the particle itself. We
and experimental particle separation efficiency.
postulate that if we can calculate particle slip
However, the turbulence closures based on the
velocities, then particle trajectories of each particle
assumption of isotropy have been found to be
size, from inlet to outlet can be predicted. This in turn
inapplicable in the case of highly swirling flows.
yields the grade-efficiency curve of the cyclone.
modeled
in
a
Besides, they did not validate the computed flow
2. Modeling of the Gas Phase
field.
The swirling fluid flow is a classic fluid
mechanics problem which can be analyzed with the
Navier-Stokes equations. The difficulties, however,
are that the current system comprises multiple phases
− gas, and solid particles (each particle size
represents an additional phase), and that the swirling
flow is turbulent. By introducing a new particle
tracking procedure, the objective of this study is to
model cyclone performance with the Navier-Stokes
equations. The unique aspect of modeling in this
manner is that in a numerical solution, the geometry
of a cyclone is implicitly accounted for. Hence, a
validated model is useful for the design of cyclones
modeling
through a single-inlet will quickly distributes itself
around the cylindrical section; axisymmetry is
naturally assumed for this modeling work. The
assumption of axisymmetry implies that the inlet
flow has to be distributed in some manner to obtain
appropriate
initial
conditions
for
this
two-dimensional model. It is therefore decided to
model the inlet flow through a full 360° inlet ring
[12], which allows the same amount of fluid to enter
the cyclone.
Although it is possible to numerically solve the
governing
Navier-Stokes
equations
written
in
primitive variables, many successful solutions have
in specific applications.
Complete
It is believed that particle laden gas entering
work
of
cyclone
performance involves predicting gas phase velocities,
slip velocities of particles with respect to the gas
phase, and ultimately the grade-efficiency curve of
the device. The problem becomes complex, because
utilized the vorticity-stream function approach. For
two-dimensional incompressible turbulent flow, with
constant properties and
adding to this complexity is the presence of gas and
the
dimensionless modeled transport equations are as
follows [12].
the governing fluid-flow equations are non-linear,
simultaneous partial differential equations, and
no body forces,
Vorticity:
  1   2  u  w



 t r3  z
r
z

viscosity/diffusivity concept, which relates turbulent
(1)
1    1     





Re   r 2 r  r r 2  z 2 
2
2
For boundary-layer flow developing over rigid
boundaries, the mixing-length hypothesis combines a
Stream function:
good mixture of accuracy and simplicity. Therefore, a
 2 1   2


  r
 r2 r  r  z2
(2)
 v v
w
 
r
 r r
t   m  2 

 u u  w



t
r
r
z
(3)
1    1   




Re   r 2 r  r  z 2 
2
modified Prandtl mixing-length model [12] will be
used in this modeling work which reads:
Angular spin velocity:

transport terms to gradients of mean-flow quantities.
2



(5)
It should be noted that variables with an overbar
represent their respective dimensional quantities. The
proposed turbulence model takes into account the
and
effect of the presence of solid particles on fluid
1 
1 

 w, 
 u,
 v,
r r
r z
r
(4)
turbulence by assuming a particulate density that
depends on solid concentration. A turbulent viscosity
where Re is Reynolds number defined as RcU 0  . It
that in turn depends on the particulate density and
should be noted that the normalization constant used
viscosity is also assumed. Equation 5 involves a
here is based on the advective time scale Rc U 0 ,
single unknown parameter, the mixing-length  ,
where Rc is the cyclone radius and U 0 is the mean
which is a characteristic length scale of turbulent
inlet velocity.
motion and whose distribution over the flow field has
Although the correlation between turbulence and
to be prescribed with the aid of empirical information.
Reynolds number has not been established for
We consider that the mixing length varies both
vortical flow, it is generally believed that turbulent
radially and axially in a typical cyclone operation,
conditions exist within the body of the cyclone,
and propose the following correlation:
especially since the inlet Reynolds number is as high
as 105 to 106 for most practical applications.
1 4
R   r 
  0 . 2 R0c  z   
 Rc   Rc 
1 2
(6)
Reviews on models available for calculating turbulent
Where Rz is the radius of the cyclone at the
stresses are abundant in the literature [13-14].
horizontal level under consideration.
Use
of the    turbulence model in its standard form is
The boundaries where steady-state conditions
unrealistic for the current system, since anisotropy is
must be specified are the inlet, the outlet, and the
imposed on the flow by high swirling stresses [15].
solid wall. The inlet conditions are known beforehand.
The algebraic stress model proposed by Boysan et al.
Therefore, a mass inflow of gas and a concentration
[8] is, on the other hand, too complicated and time
of solids are preset at the beginning of computation.
consuming. In addition, it does not take into account
Conditions at the outlet are not known, and
the effect of presence of solid particles on the
zero-gradient conditions are applied. Values of
properties of fluids. The Prandtl mixing-length
angular spin velocity at the wall are approximated by
hypothesis is one of the first turbulence models
a logarithmic wall function. Boundary values of
proposed, and interestingly is still among the most
vorticity are more difficult to obtain and are
widely
extremely
used
models.
It
employs
the
eddy
important
quantities.
The
vorticity
transport equation for   t determines how  is
advected and diffused. Total  is conserved at
36.7mm
interior points, but at the no-slip boundary  is
produced.
It is in fact the diffusion and subsequent
advection of vorticity, which is produced at the wall
that drives the problem. If  w  1 is expanded by the
225mm
75mm
Taylor series around the wall value  w and the
no-slip condition is used, we get:
r w   2
Where
 w 1   w
n2
 O (n)
(7)
100mm
124mm
165mm
n is the dimensionless distance from
( w1) to w, normal to the wall.
The numerical problem on hand is a set of
coupled parabolic and elliptic partial differential
equations. Equations 1 and 3 are parabolic, and they
pose an initial-value problem. These equations are
405mm
solved using the Hopscotch method, where a solution
is stepped out from some initial conditions. Equation
2 is elliptic which poses a boundary-value problem,
and is solved by the successive over-relaxation (SOR)
75mm
20mm
method. A rectangular mesh system has been adopted
where small radial mesh size would be favored due to
high velocity gradient in the radial direction. A small
axial mesh size should also be used in the vortex
Fig. 1 Cyclone separator
finder region to account for short-circuiting flow;
while in the conical part, it has been set in such a way
Figure 2 shows the predicted tangential velocity
that the node points fall exactly on the conical wall.
profile, flow field, and streamlines. When modeling
The iterative scheme starts from the establishment of
highly swirling flows, it has been found that the
initial values of  ,  ,  , u and w everywhere at t
correct prediction of the tangential velocity is
= 0, and converges to the correct solution after
essential to adequately reproduce the rest of the flow
performing a number of iterations.
field. The largest tangential velocity appears in the
inlet zone. A forced-free vortex is well predicted by
3. Predicted Gas-Phase Flow Patterns
The dimensions of a 225-mm cyclone with apex
cone used in this study are shown in Fig. 1. We adopt
the same design parameters and operating conditions
as those of Bohnet’s experiments [16]. Specifically,
the gas flow rate to the cyclone was 0.05 m 3 sec and
the mean inlet velocity was 13.6 m sec .
the model. The peak tangential velocity at each
horizontal level decreases as the gas spirals
downwards. Due to the presence of an apex cone, the
tangential velocity drops rapidly as the gas enters the
dust receiver. The short-circuiting flow in the upper
part as well as the flow reversal are observed in the
flow field. Non-swirling velocity components are
very small in the dust receiver and thus drastically
tracked from the inlet to either the dust receiver or the
reduce the re-entrainment of fine particles. The
outlet. Depending on the size of each particle, there is
streamlines show the expected upward and downward
lag in the gas velocity. Particle slip velocities are
moving streams. It is estimated that about 6.3% of the
determined by a dynamic force balance on the
inlet gas stream short-circuits to the vortex finder,
particle itself. For a particle of diameter d p and
and 9.0% of the total gas stream flows through the
density  p , by balancing the centrifugal force
dust receiver.
against the radial drag force and then solving for the
Tangential Velocity
30
24
18
12
6
0
m/sec
Flow Field
radial slip velocity u s , we obtain
Streamlines ( ψ ×10 )
6
0 mm
4   p  m
us  
3   g
30
60
197
90
density, and
150
757
180
12
(8)
Where  m is the particulate density,  g is the gas
500
120
 v 2 dp

 r CD
CD
is the drag coefficient [15].
Similarly, by balancing the gravity force against the
210
axial drag force and solving for the axial slip velocity
1375
240
ws , we have
270
300
4   p  m
ws  
3   g
2712
330
360
 dp
 g
 CD
12
(9)
4140
390
5. Predicted Particle Trajectories and
Grade-Efficiency
420
5549
450
As shown in Fig. 3, the gas stream translates
480
510
from a vertical plane containing line
6857
AB to
540
another vertical plane including line AB after entering
570
the cyclone through a tangential inlet. In other words,
600
7450
630
the fluid flows in a non-swirling fashion. Since there
7760
660
are no significant forces acting on the particles
7957
690
contained in the gas stream, they move along
Fig. 2 Predicted tangential velocity profile, flow field,
and streamlines
swirling motion of the gas stream will not take place
until it passes plane ABCD, where the linear
4. Modeling of the Particulate Phase
The
general
performance
of
cyclones
is
influenced both by design variables such as cyclone
dimensions, and also by operating variables such as
particulate
concentration
streamlines of the gas phase in the inlet zone. The
and
gas
flow
rate.
Grade-efficiency curves are commonly used to
evaluate cyclone performance. To determine this
curve, trajectories of particles for each size must be
momentum of the gas stream starts to distribute into
the tangential, axial, and radial directions. We
therefore propose a new particle tracking procedure
that starts from plane ABCD instead of a peripheral
line segment AD.
Incidentally, we divide plane
ABCD into 100  49 entry points where the particle
tracking procedure begins.
For a tangential inlet, we
postulate that particles of any size are equally likely
to be present at any of the aforementioned entry
points.
For an involute inlet, however, particles will
be unevenly distributed among these entry points
because centrifugal force already affects particle
motion as they enter the cyclone.
r
This in turn
r
z
z
reduces the short-circuiting of coarse particles to a
certain extent. Starting from each of the entry points,
dp = 0.8 μm
dp = 1.7 μm
dp = 2.0 μm
dp = 3.0 μm
the particle tracking procedure begins. Particle
displacement for a small time interval is calculated.
dp = 0.8 μm
dp = 1.7 μm
dp = 2.0 μm
dp = 3.0 μm
The computational routine is repeated until the
particle is collected or exits the cyclone through the
vortex finder.
A
A'
B'
B
49
A
B
100
D
C
(a)
(b)
Fig. 4 Computed particle trajectories
at which it enters the cyclone. As seen in Fig. 4a,
after entering the cyclone from a peripheral entry
point 10 mm below the roof, all particles move
downwards along the cylindrical section and then
into the conical part. Strong inward flow in the
conical section brings fine particles toward the axis of
symmetry. These fine particles reverse flow direction
after passing their respective equilibrium orbits, and
move toward the vortex finder along with the central
upward flow. As for the coarse particles, the large
Fig. 3 Entry points of particle trajectories
centrifugal force pushes them toward the conical wall
despite the strong inward flow. Then they join the
The essence of the cyclone’s separation effect is
clearly demonstrated in Fig. 4, where trajectories of
four different size particles are shown. The trajectory
of a particle depends on both its size and the location
boundary-layer flow along the conical wall into the
dust receiver, where they are collected. Particle
trajectories that start from a peripheral entry point 80
mm blow the roof display a similar pattern as seen in
Fig. 4b. Although in both cases, 0.8  m particles
short-circuit to the vortex finder should they start
escape and 3.0  m particles are collected; the
from entry point B.
outcome of intermediate size (1.7 and 2.0  m )
For a certain particle size in the inlet stream, the
particles is quite different. While in one case these
particle trajectories are tracked from each of
particles escape as shown in Fig. 4a; they are
the 100  49 entry points. The number of particle
collected in the other case as seen in Fig. 4b.
trajectories terminating at the dust receiver divided
A
A'
B'
B'
B
by the total number of entry points denotes the
A
A'
collection efficiency of that particle size. Repeating
B
this procedure for all particle sizes in the inlet stream
yields a grade-efficiency curve. It should be noted
that, based on Bohnet’s experimental conditions,
r
B
A
particles are assumed to move through dilute
r
z
z
A
particulate where particle/particle interactions are
B
negligible.
dp = 3.5 μm
dp = 3.5 μm
100
80
60
This Model
Experimental (Bohnet)
Barth Model
Lapple Model
40
20
Dietz Model
0
2
4
PARTICLE DIAMETER
6
8
10
( MICRONS )
Fig. 6 Predicted and experimental grade-efficiency
(a)
(b)
Fig. 5 Short-circuiting of coarse particles
Grade-efficiency curve predicted by this model
is compared with Bohnet’s experimental results in
Fig. 6. This model will not only predict the cut size
The short-circuiting of coarse particles to the
accurately but also the entire grade-efficiency curve
vortex finder is clearly seen in Fig. 5. For
with precision. The new particle tracking procedure
3.5  m particles entering the cyclone from any entry
proposed in this model seems to work extremely well,
point 20 mm below the roof, they are all collected as
since the short-circuiting of coarse particles is closely
shown in Fig. 5a. For particles of the same size
predicted. It is believed that deficiency of the
entering the cyclone from entry points 80 mm below
hypothetical ring inlet in the model formulation has
the roof, however, the outcome could be different
been remedied by this new particle tracking
depending on the respective location of entry points.
procedure. Grade-efficiency curves predicted by a
As can be seen in Fig. 5b, while particles are
number of semi-empirical models reported in the
collected if they start from entry point A; they
literature are also shown in Fig. 6. Considerable
discrepancies are observed both in cut size and
general trend of the separation curve. Specifically, the
collection efficiency of coarse particles is greatly
underestimated by these models.
[6] Strauss, W., “Industrial Gas Cleaning,” Pergamon,
1966, Ch. 6.
[7] Buonicore, A.J. and Davis, W.T., “Air Pollution
Manual,” Van Nostrand Reinhold, 1992, Ch. 3, pp.
71-78.
6. Conclusions
[8] Boysan, F., Ayers, W.H. and Swithenbank, J., “A
Based on the physics of fluid flow, a theoretical
approach to the modeling of cyclone performance has
been developed. Close agreements between predicted
and
measured
grade-efficiency
characteristics
indicate that the basic construction of the model is
sound
and
well
justified.
The
fundamental
mathematical modeling technique is capable of
providing detailed information about the gas and
particle dynamics within the device. In addition, this
information allows for a better understanding of the
short-circuiting mechanism, so that a rigorous
approach to cyclone separator design can be
undertaken.
Fundamental Mathematical Modeling Approach
to
Cyclone
Design,”
Trans.
Institution
of
Chemical Engineers, Vol. 60, 1982, pp. 222-230.
[9] Patankar, S.V., “Numerical Heat Transfer and
Fluid Flow,” Hemisphere, 1980.
[10] Stairmand, C.J., “The Design and Performance
of Cyclone Separators,” Trans. Institution of
Chemical Engineers, Vol. 29, 1951, pp. 356-383.
[11] Yoshida, H., Saeki, T., Hashimoto, K. and
Fujioka, T., “Size Classification of Submicron
Powder by Air Cyclone and Three-Dimensional
Analysis,” Journal of Chemical Engineering of
Japan, Vol. 24, 1991, pp. 640-647.
[12] Hsieh, K.T. and Rajamani, R.K., “Mathematical
References
[1] Lapple, C. E., “Process Uses Many Collector
Model of the Hydrocyclone Based on Physics of
Types,” Chemical Engineering, Vol. 58, 1951, pp.
Fluid Flow,” AIChE Journal, Vol. 37, No. 5,1991,
144-151.
pp. 735-746.
[2] Barth, W., “Design and Layout of the Cyclone
[13] Rodi, W., “Examples of Turbulence Model for
Separator on the Basis of New Investigations,”
Incompressible Flows,” AIAA Journal, Vol. 20,
Brennstoff-Warme-Kraft, Vol. 8, No. 4, 1956, pp.
No. 7, 1982, pp. 872-879.
[14] Sloan, D.G., Smith, P.J. and Smoot, L.D.,
1-9.
[3] Leith, D. and Licht, W., “The Collection
“Modeling of Swirl in Turbulent Flow Systems,”
Efficiency of Cyclone Type Collectors ─ A New
Progress of Energy Combustion Science, Vol. 12,
Theoretical Approach,” AIChE Symposium series,
1986, pp.163-250.
[15] Pericleous, K.A., Rhodes, N. and Cutting G.W.,
Vol. 68, No. 126, 1972, pp. 196-206.
[4] Dietz, P.W., “Electrostatically Enhanced Cyclone
Separators,”
Powder
Technology,
Vol.
19,
1982, pp. 221-226.
[5] Iozia, D.L. and Leith, D., “The Logistic Function
and Cyclone Fractional Efficiency,” Aerosol
“A Mathematical Model for Predicting the Flow
Field in a Hydrocyclone Classifier,” Proceedings
of International Conference on Hydrocyclones,
1984, pp. 27.
[16] Bohnet, M., “Cyclone Separators for Fine
Science and Technology, Vol. 12, 1990, pp.
Particles and Difficult Operating Conditions,”
598-606.
KONA, No. 12, 1994, pp. 69-76.