1 MODELING THE GAS AND PARTICLE FLOW INSIDE CYCLONE SEPARATORS Cristóbal Cortés * and Antonia Gil Center of Research of Energy Resources and Consumptions (CIRCE) Department of Mechanical Engineering University of Zaragoza Maria de Luna 3, 50018 Zaragoza, Spain Abstract This paper reviews the models developed for the flow field inside inverse-flow cyclone separators. In a first part, traditional algebraic models and their foundations are summarized in a unified manner, including the formulae for tangential velocity and pressure drop. The immediate application to the prediction of collection efficiency is also reviewed. The approach is the classical, treating first the dilute limit (clean-gas correlations), and afterwards correcting for “mass loading” effects. Although all these methods have had a remarkable success, more advanced ideas are needed to model cyclones. This is put forward by exploring the work done on the so-called “natu- ral” length of the cyclone, that has led to the discovery of instability and secondary flows. The re- sort to CFD in this case is difficult, however, due to the very nature of the flow structure. A closing section on the subject reviews past and recent CFD simulations of cyclones, both single- and two- phase, steady and unsteady, aiming at delineating the state-of-the-art, present limitations and per- spectives of this field of research. Keywords: CFD, cyclone, gas-solid flow, swirling flow Contents 1. Introduction. 2. Basics of cyclone separators. 3. Flow field and pressure drop. 3.1. Velocity distribution inside cyclones. 3.2. Models of the velocity distribution. 3.3. Pressure field in cyclones. 3.4. More on cyclone velocity patterns. * Corresponding author. Tel.: +34 976 762034; fax: +34 976 732078; e-mail address: [email protected]
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1
MODELING THE GAS AND PARTICLE FLOW INSIDE CYCLONE SEPARATORS
Cristóbal Cortés* and Antonia Gil Center of Research of Energy Resources and Consumptions (CIRCE)
Department of Mechanical Engineering University of Zaragoza
Maria de Luna 3, 50018 Zaragoza, Spain
Abstract This paper reviews the models developed for the flow field inside inverse-flow cyclone
separators. In a first part, traditional algebraic models and their foundations are summarized in a
unified manner, including the formulae for tangential velocity and pressure drop. The immediate
application to the prediction of collection efficiency is also reviewed. The approach is the classical,
treating first the dilute limit (clean-gas correlations), and afterwards correcting for “mass loading”
effects. Although all these methods have had a remarkable success, more advanced ideas are
needed to model cyclones. This is put forward by exploring the work done on the so-called “natu-
ral” length of the cyclone, that has led to the discovery of instability and secondary flows. The re-
sort to CFD in this case is difficult, however, due to the very nature of the flow structure. A closing
section on the subject reviews past and recent CFD simulations of cyclones, both single- and two-
phase, steady and unsteady, aiming at delineating the state-of-the-art, present limitations and per-
3.1. Velocity distribution inside cyclones. 3.2. Models of the velocity distribution. 3.3. Pressure field in cyclones. 3.4. More on cyclone velocity patterns.
6. Computational fluid dynamics applied to cyclones.
6.1. CFD studies on single-phase cyclone flow. 6.2. Numerical computation of two-phase flow in cyclones.
7. Conclusions.
Acknowledgements.
References.
Nomenclature.
a inlet section height
Ai inlet area
AS inner cyclone friction surface area
b inlet section width
B discharge duct diameter
Ccr critical load
Csi inlet solids loading
dp generic particle size
D diameter
Dc cyclone diameter
De vortex finder diameter
DLn diameter of the cyclone cone at the vortex end position
f(x) particle size distribution
fg parameter of Alexander pressure drop model, Eq. (39)
3
Frc inlet Froude number, c2ic gDvFr =
Fre vortex finder Froude number, e2ee gDvFr =
g acceleration of gravity
H total cyclone height
h height of cylindrical section of cyclone
h* height of the separation surface in the models of Barth [29]
K empirical constant [20]
L generic interparticle spacing in two-phase flow
Ln cyclone vortex length
m slope of the fractional efficiency curve, Eq. (52)
p Pressure
r Radius
ra radius of maximum tangential velocity, in particular as defined in [20]
re radius of the vortex finder
rc cyclone radius
Rec cyclone Reynolds number, gigcc vDRe μρ=
rt transition radius [20]
S height of the vortex finder
St cyclone Strouhal number, Eq. (65)
Sw swirl number, Eqs. (63), (64)
T Temperature
V& volumetric gas flow rate
vb mean axial velocity at cyclone body, 24 cb DVv π&=
vi inlet gas velocity
vr radial gas velocity
4
vt tangential gas velocity
vt max maximum tangential velocity [20]
vte tangential gas velocity at the inner vortex [29]
vtw velocity in the vicinity of the wall
vtw* wall velocity just after inlet [17]
vz axial gas velocity
vz0 mean axial velocity, ( )220 / tcz rrVv −= π&
vze Axial gas velocity at vortex finder
x particle size in cyclones
x50 cut size of the cyclone
xm particle mass median diameter
Greek χ coefficient from Meissner and Loffler [17]
Λ parameter [20]
α coefficient in Barth theory [29]
αp particle volume fraction
εstr porosity of the strands, Eq. (49)
η(x) fractional collection efficiency
η total collection efficiency
λ friction factor
μg gas dynamic viscosity
ρb bulk density
ρg gas density
ρs solids density
ξc cyclone pressure coefficient, 22
1 )/( igcc vp ρξ Δ=
5
cΦ angle of the cyclone cone
Γ constant, Eq. (30)
Ω angular velocity, Eq. (29)
Subscripts b body of the cyclone
c cyclone
g gas
i inlet
r radial component
s solids
t tangential component
w wall
z axial component
Abbreviations CFBC Circulating Fluidized Bed Combustion
CFD Computational Fluid Dynamics
DNS Direct Numerical Simulation
FCC Fluid Catalytic Cracking
LDA Laser-Doppler Anemometry
LES Large Eddy Simulation
LRR Launder, Reece and Rodi implementation of a differential RSTM, also known as the
“basic” implementation. Variant: LRRG, after the modification by Gibson and
Launder [83]
PFBC Pressurized Fluidized Bed Combustion
PSD Particle Size Distribution
6
PSI-Cell Particle-source-in cell methods of calculation of two-way coupled two-phase flow
PVC Precessing Vortex Core
RANS Reynolds Averaged Navier-Stokes Equations
RNG Re-Normalization Group theory
RSTM Reynolds Stresses Transport Model
SGS Subgrid Scale Model
SSG Speziale, Sarkar and Gatski implementation of a differential RSTM [83]
TRANS Transient Reynolds Averaged Navier-Stokes Equations; equivalent to URANS
URANS Unsteady Reynolds Averaged Navier-Stokes Equations; equivalent to TRANS
1. Introduction Cyclone separators have been a decisive factor in the development of coal combustion
technologies. Among diverse possibilities for hot gas cleaning, these devices have demonstrated
the most favorable balance of separation efficiency and cost of investment, operation and mainte-
nance. Able to handle any combination of gas pressure, temperature and very high solids loading,
their performance is tolerable as compared with more efficient separation equipment (i.e., ceramic
filters), being at once much more simple, robust and reliable. Presently, cyclones are a key compo-
nent in most advanced coal utilization concepts, such as pressurized and circulating fluidized bed
combustion (PFBC and CFBC). In PFBC, cyclones are essential to maintain the integrity of the gas
turbine, and thus the advantages of the concept itself [1], [2]. In CFBC, the scaling-up of the
equipment to sizes compared to conventional coal firing is being developed partly based on new
designs of integrated, compact cyclonic separators [3].
As a consequence, there is still a great need of a sound knowledge of the principles of op-
eration. Since cyclones were developed in the last decades of 19th century, extensive experimental
work has been done in order to explain their flow characteristics, as a fundamental step to under-
stand pressure drop and separation efficiency. At the same time, and based on the data gathered,
theoretical models have been advanced to predict the basic features of the flow field, mostly on
7
semi-empirical grounds. This generic modeling strategy is still in use; in fact, many formulae and
methods derived in the past remain very useful indeed for design purposes nowadays. Neverthe-
less, as in many other fields of study, advances in experimental and computational methods have
brought to light many additional details and subtleties of the question.
In the case of cyclone separators, some of them have turned out to be of a rather fundamen-
tal nature, and at the same time, of paramount importance from the point of view of applications.
Unsteadiness and asymmetry are for example two features not considered in classical cyclone the-
ory that may affect the velocity distribution to a great extent, thus changing the model of the sepa-
ration mechanism. In close relationship, our picture of the end of the separation vortex has been
evolving until very recently. Latest research is revealing that in the vicinity of this region, the flow
can be unstable and the vortex can attach to the solid wall, rotating on it. Consequently, the predic-
tion of the total length of the vortex (the so-called “natural length” of the cyclone) is subjected to
great uncertainty, as the great discrepancy between different calculation methods found in the
literature already attested. This parameter is essential for an optimal design, specially in those
units equipped with a dipleg,† and thus constitutes a current challenge for new cyclone models.
The modern study of cyclone separators has taken advantage of several sophisticated ex-
perimental and numerical techniques. Amongst the former, Particle Image Velocimetry (PIV) and
Laser Doppler Anemometry (LDA) are prominent, even though the need of seeding the gas with
small particles, in turn too prone to separate, poses some intrinsic difficulties. These are absent in
intrusive methods (whose chief example is the miniature X-hot-wire anemometer), that in ex-
change may suffer from inaccuracy when dealing with unsteady flow, and from probe deteriora-
tion and damage due to the aggressive environment in two-phase flow. On the other hand, as in
many other fields, Computational Fluid Dynamics (CFD) currently emerges as an efficient alterna-
tive to traditional, approximate analytical models of the time-averaged flow.
† A prolonged tube used to accommodate the flow of solids in some applications, notably FCC and PFBC.
8
A validated numerical technique is of course extremely powerful for analyzing geometry
and operating conditions and pursuing an optimal design. However, the flow inside cyclone sepa-
rators again entails special difficulties in this respect, so that the use of CFD is not as widespread as
it can be expected. The high anisotropy of the turbulent field in such a confined, strongly swirling
flow demands the adoption of modeling expedients that are quite expensive in terms of computing
resources, such as Reynolds Stresses Transport Models (RSTM) and higher order discretization.
Furthermore, the very unsteady nature of the flow implies that any valid CFD scheme should be
transient as well, which strictly leads to extremely costly techniques of turbulence modeling, i.e.,
Large Eddy or Direct Numerical Simulation (LES or DNS). Simpler schemes, such as the tradi-
tional steady-state models, or even time-dependent versions of them, are not sound from a funda-
mental standpoint, and should be scrutinized in depth.
In this paper, we undertake a review of the most relevant semi-empirical models proposed
for the time-averaged flow in cyclones, as found in the literature. These are in turn connected with
the results of the latest CFD simulations, both steady state and transient. Predicted flow field, cy-
clone natural length, pressure drop and collection efficiency are discussed and compared with ex-
perimental data and between different models and calculations. The paper discusses all the pecu-
liar features mentioned above, analyses the relative performance of the models developed
throughout the years, and attempts to outline general recommendations and future perspectives.
2. Basics of cyclone separators. The basic principle of cyclone separators is the forcing of the particle-laden gas into a vor-
tex, where inertia and gravitational forces effect particle separation. Among existing cyclones and
diverse cyclonic equipment, there is a basic model that at once has been used by industry exhaus-
tively: the inverse flow cyclone, Fig. 1.
In this device, the fluid enters tangentially into the cylindrical chamber with a high rota-
tional component. The flow descends rotating near the wall, until a certain axial location where the
9
axial velocity component reverses itself, thus making the flow to ascend. This is referred to as the
vortex end position. The ascension proceeds near the cyclone axis and, since the flow rotation con-
tinues, a double vortex structure is formed, as indicated in the figure. The inner vortex finally leads
the flow to exit through a central duct, called the vortex finder. The vortex finder protrudes within
the cyclone body, which serves both to shield the inner vortex from the high inlet velocity and to
stabilize it. It is also worth to mention that the inversion leading to this peculiar flow structure is
apparently originated by the pressure field inside the cyclone, and not directly influenced by the
conical shape or the geometrical length.
The textbook explanation of the separation is that the solid particles, denser than the gas,
are subjected to a high centrifugal force, which directs them to the walls, where they collide, lose
momentum and became disengaged from the flow. The solids thus separated descend sliding on
the conical wall and are collected or extracted at the lower part. This mechanism obviously suffers
from a number of imperfections: small particles that follow the gas, particles that rebound and are
re-entrained, and direct re-entrainment or by-pass at the lowest conical section, near the inversion
zone, at the vortex finder lip, and all along the inner/outer vortex boundary. All this factors add
up to the variables that affect the centrifugal force (mainly geometry and inlet velocity), to make
the collection efficiency of cyclones highly variable.
Actually, the centrifugal force (or any other parameter of the gas dynamics) can only ex-
plain the separation of solids in dilute flows. It is well known that above a certain solids loading,
particles are collected as soon as they enter the device, forming dust strands that descend helicoid-
ally along the walls. The centrifugal force only acts upon the remaining dust, usually a small frac-
tion of the total [4] [5]. As we will see later, this effect is explained essentially by inertia and inter-
action between particles.
There are several systems to insert the flow into the cyclone with a high tangential velocity
component; some inlet designs are shown in Fig. 2. Most frequent are the tangential and the scroll
configurations. The scroll inlet is usually designed to wrap around up to 180º of the cylindrical
10
cross-section. Higher arcs are not used, because they are ineffective and unnecessarily increase
pressure drop [7], although shorter ones are not unusual. Depending on their use and particular
properties of the dusts, many different cyclone designs have been developed throughout the years.
As an example, Table 1 and Fig. 3 show sixteen different designs of the same inlet area (0.01 m2)
performing the same duty at a given inlet velocity, as compiled in the monograph by Hoffmann
and Stein [8].
From an engineering point of view, cyclone performance is measured by collection effi-
ciency (the fraction of solids separated) and pressure drop. These two parameters are the direct
outcome of the flow developed inside the device, in turn described by the velocity, solids concen-
tration and pressure fields. Given the wide range of solids loadings that cyclones are apt to handle,
the flow is generically biphasic; interaction between particles and two-way coupling can only be
neglected for low concentrations of solids. In spite of this, traditional cyclone models proceeded
from clean-gas velocity measurements, through explanation and correlation of the observed pro-
files, to arrive at relatively simple formulations of measured efficiency and pressure drop. We will
follow roughly this historical path, signaling here and there the effects of a high solids concentra-
tion and how these are taken into account in traditional cyclone modeling. A final section on CFD
calculations will give a more integrated view, dealing with work done and perspectives on clean-
gas and two-phase flow simulations—mostly the latter in the latter case.
3. Flow field and pressure drop 3.1. Velocity distribution inside cyclones. The first studies of the features of cyclone inner flow were undertaken in 1930-1950 [9]–[10],
promptly revealing their extraordinary complexity, that initiated an enormous wealth of experi-
mental and theoretical work on the subject. Figures 4 and 5 show classical measurements and ex-
planations, attempting to describe the three velocity components inside a cyclone separator.
The time-averaged flow is made up mainly of a vortex, thus dominated by tangential veloc-
ity and strong shear in the radial direction. As a first and simple approximation, the profile can be
11
described as a Rankine vortex, a combined free and forced vortex. The tangential velocity distribu-
tion vt in the radial direction is assumed to obey a law of the form:
Crv nt = (1)
where C is a constant, r the radius and the exponent n depends on r. This variation is sketched in
Fig. 6. In the inner region near the cyclone axis, n is close to –1 (forced vortex), whereas n ap-
proaches 1 (free vortex) near the wall. The forced vortex only encompasses a region fairly close to
the centerline, so that the point at which the velocity attains a maximum is well inside the radius
covered by the vortex finder, as seen in Fig. 5. Actually, the inner rotation is somewhat different
from that of a solid, and shear in the outer region is too high for the influence of viscosity to van-
ish. In practice, a “forced vortex” exponent close to –1 is observed for the inner part of the profile,
but the outer part (excluding the sudden decrease very close to the wall) is better correlated by a
exponent n in the range 0.4 - 0.8.
A further common assumption is no axial variation of vt, acknowledging the fact that it is
indeed fairly low, at least within the main separation space. As we will see, many algebraic models
have relied on the adequate correlation of n, but this approach cannot take into account two impor-
tant parameters, namely, the wall friction and, for dense flows, the concentration of solids. Both
directly influence the strength of the vortex, and thus the exponent n changes with them, in a
manner that cannot be ascertained but by blind empiricism. Clearly, more elaborated models of the
velocity field are needed.
Also of importance are the other two velocity components. The axial velocity is responsible,
more than gravity, for the transport of particles to the collection device [10]. A simple quantitative
model is suggested by the double vortex structure: radially-constant values for the outer vor-
tex/downward flow and the inner vortex/upward flow zones. Both values should be zero at the
axial position of the vortex end. At the vortex finder, the inner value is given by the volumetric
flow rate and the cross-sectional area; its axial variation can be then adjusted by a simple (linear)
function, and the outer value deduced from continuity. This requires however some rather drastic
12
assumptions about the entire flow structure, Fig. 7: a vortex end position dictated by geometry or
pre-established in some other, simple form, and a hypothetical cylinder having the vortex finder
diameter as the locus of flow inversion.
Obviously, actual profiles are not flat, but exhibit maxima. Descending flow always has a
maximum close to the wall, as can be expected, but axial velocity at the inner vortex is either re-
ported as an inverted V or W-shaped profile, i.e., with a maximum or a dip at the symmetry axis,
as can be observed in Figs. 4 and 5. The W-pattern exhibits a maximum roughly at the radial posi-
tion of the vortex finder; sometimes the drop in momentum is so severe as to cause backflow [8].
This curious behavior is frequently observed in experimental measurements and CFD simulations;
we will return later to its explanation and effects. An implication is that the locus of flow inversion
cannot be simply a cylinder, for obvious reasons of continuity.
The radial velocity is important in some models of particle collection; evidently, it always
will be a factor when analyzing by-pass and losses of efficiency. Frequently, it is assumed of much
lesser magnitude than the other components, but this is only true concerning the outer vortex. The
radial velocity grows steeply towards the vortex core, aimed inwards, specially in the vicinity of
the vortex finder [7], [18]. An average, perhaps characteristic, value can be derived [4], [17], [29] by
assuming that the gas flows evenly through the imaginary cylinder CS seen in Fig. 7, which needs
the approximations previously mentioned.
Finally, in cyclones equipped with diplegs, several flow regimes may develop within them,
depending on the procedures for extracting the solids. In PFBC applications, a deep penetration of
the swirl inside the dipleg has been experimentally measured, [14]–[16] caused by the (small) frac-
tion of gas used for particle transport. This is mainly absent in FCC cyclones, where the gravity-
assisted, intermittent flow of solids gives way to zones of dense-phase transport [13].
13
3.2. Models of the velocity distribution. Table 2 summarizes the most relevant, algebraic models of cyclone flow, as taken from the
literature. A scheme of the key hypotheses and main formulae is given; the complete details can be
looked up at the original references. As we will see, cyclone models started from crude considera-
tions on vortex flow, and evolved to incorporate more classes of phenomena and more sophisti-
cated principia. Most models deal only with the tangential component; axial and radial velocities
are usually handled through the simplified expedients mentioned above.
Alexander [9], Eqs. (2)–(4). This is a purely empirical model that addresses two separate
questions. Firstly, it correlates the ratio of the tangential velocity in the vicinity of the wall vtw to
the (given) mean inlet velocity vi, considering it as a purely geometrical parameter, which is rea-
sonable for the high Reynolds numbers usually found in practice. Secondly, a correlation is given
for the exponent n that characterizes the radial profile of the tangential velocity in the outer vortex.
This is made to depend on the cyclone diameter and also on absolute temperature, since the ex-
perimental census comprised cyclones treating hot gases, whose tangential velocity additionally
changes due to the change of viscosity. Compared to more modern models and measurements, the
value of n is normally underpredicted; on the other hand, variation of the wall friction, having a
significant effect on the flow field, is not easily handled in this manner.
Barth [29], Eqs. (5) – (8). This is a simple and still useful model, by which friction was first
introduced in cyclone modeling. As we shall see later, the velocity profiles of Barth were immedi-
ately applied as a first construct to predict collection efficiency.
Similarly to Alexander’s, this model considers as a geometric constant the ratio α between
average angular momentum of the gas at the inlet and that of the gas rotating inside the cylindrical
body of the cyclone, close to the wall. This constant is obtained for several entrance geometries,
being unity for scroll, 360º inlets and less than unity for tangential inlets, where a considerable ac-
celeration obtains.
14
The tangential velocity at the wall vtw is then related to the tangential velocity at the control
surface CS of Fig. 7, which approximately represents the swirl intensity of the inner vortex. The
method is an angular momentum balance that assumes another imaginary surface of diameter
ec DD ⋅ and height h* where all frictional losses are concentrated; the analysis leads to Eq. (6).
Losses are represented by a lumped wall friction coefficient λg, empirically adjusted. The height h*
can be naturally identified as the length of the vortex; Barth made a purely geometrical interpreta-
tion of this parameter, via Eqs. (8).
Muschelknautz [4]. Muschelknautz and co-workers have worked upon Barth’s ideas to de-
velop empirical models that combine in admirable measures both simplicity and realism. Concern-
ing the velocity distribution, the table summarizes one of their earliest models. The essential con-
cepts of the coefficient α and the friction surface are maintained and perfected. Data on α is corre-
lated by means of analytical formulae, Eqs. (9) and (10), to replace the original graphs. In addition,
the value of the gas friction coefficient λg is readjusted.
But beyond that, the model is modified to give a quantitative prediction of the effect of the
concentration of solids in the regime of dense flow. By means of streaks of particles directly sepa-
rated at the entrance, this is to augment wall friction and thus weaken the vortex intensity.
Muschelknautz’s original expressions for the increase of λg with the inlet solids loading Csi are
given in the table as Eqs. (11). As we will see later, this discovery opened the way to modeling
mass loading effects, both in pressure drop and separation efficiency.
Meisnner & Loffler [17],[18]. Similar to Barth, they derived an empirical expression for the
geometric relationship between the tangential velocity at the cyclone wall *twv and the inlet veloc-
ity iv , Eq. (12), and a momentum balance to take wall friction into account, Eq. (13). Only that two
values of the tangential velocity are considered: *twv just at the inlet slot where acceleration occurs
and the “developed” value vtw at the cyclone body. As indicated by the formulae, the momentum
balance is different from Barth’s; it only refers to the reduction from *twv to vtw as a result of flow
15
along the cyclone wall. Only slot or tangential inlets are handled by this model; scrolls are explic-
itly excluded.
To calculate the tangential velocity at CS, an angular momentum balance is applied to a
hollow cylinder of differential thickness between r and r + dr, which leads to Eq. (16) for the tan-
gential velocity at any radius vt(r). The velocity at CS is simply found by substituting r = re. Ideas
similar to those discussed at the beginning, are adopted for the radial, Eq. (18), and axial velocity at
the outer vortex, Eq. (19). Note in the formulae that they refer to geometries more restrictive than
those considered by Barth.
The second angular momentum balance implies the use of two additional friction factors,
different from λg: those corresponding to the upper and lower metal surfaces of the cyclone that
bound the control volume. Although in latter studies the same value was given to the three coeffi-
cients, it is by no means clear that their physical significance be equivalent. For this reason, the
model is only strictly valid for dilute flows; the inclusion of mass loading effects is more difficult
than with Barth or Muschelknautz.
Reydon & Gauvin [19]. Both theoretical and experimental flow studies were carried out at
different operating conditions and for different geometric parameters, in an effort to obtain more
general expressions for vt. The results are divided in two regions corresponding to the outer and
inner vortex, Eqs. (20) and (21), and the coefficients were adjusted with experimental data by linear
regression.
Ogawa [20], [21]. This author developed the most complex algebraic construct to date, based
on theoretical considerations. The wall tangential velocity vtw, that intermediate variable used in
previous work, is absent here. A classical outer “free vortex” region is deduced, Eq. (22), with con-
stants K0 and n that depend on Reynolds number and geometry. A law for the “forced vortex” re-
gion is also deduced, Eq. (23), introducing a maximum tangential velocity vt max at a radial position
ra from which solid body rotation prevails in the inner vortex. These results are based on consid-
16
erations on axial swirl stability, as can be consulted in [22]. Axial and radial velocities at the two
flow regions are also derived, Eqs. (25)-(27).
Finally, the strategy proposed in [23] can serve to integrate recent models [17],[19],[20] in a
supposedly coherent manner:
1. Use Ogawa’s Eq. (23) to determine the radial evolution of vt in the inner vortex.
2. The radius of maximum tangential velocity ra, needed to obtain the constant Λ, is ob-
tained by intersection of Eqs. (21) and (16), i.e., the “forced vortex” of Reydon & Gauvin
and the “free vortex” of Meissner & Löffler.
3. The “free vortex” region is described by Eq. (16), and the transition radius rt between
this region and the “forced vortex” can be obtained as the intersection with the Ogawa
curve, Eq. (23).
Figure 8, taken from [23], shows the results and a comparison of different models and
measurements. Interestingly, the framework of the most recent and complex models is apparently
inviscid, although friction is of course introduced by the adoption of correlating exponents and
functions for the inner and outer vortexes. However, this hardly can take into account the effect of
a high concentration of solids [26]; in this sense, older approaches, such as Muschelknautz’s, are
perhaps more practical.
3.3. Pressure field in cyclones. Some discussion of pressure distribution in swirl flows is in order here, since some fluid
mechanics effects are very special, leading to ideas that contradict the usual intuition drawn from
unidirectional, swirl-free flows. These have caused more than one confusion in cyclone literature.
Let us begin with the equilibrium between centrifugal force and radial pressure gradient:
drdp
rvt =
2
ρ (28)
17
It is worth to remember that this equation is exact for idealized, axially symmetric one-
dimensional flows, being directly derived from the momentum equation in the radial direction. In
other words, it is equally valid for ideal or viscous, for laminar or turbulent, steady flows.
If we assume that the inner zone is a pure, forced vortex rrvt Ω=)( , being const.=Ω the
angular velocity, Eq. (28) is easily integrated to obtain
220 2
1)( rprp Ω+= ρ (29)
where p0 = p(0) is the pressure at the cyclone axis. In a similar fashion, for an outer zone that obeys
a pure, free vortex law rrvt /)( Γ= with const.=Γ , the integral results in
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−Γ+= 22
2 1121
rrprp
aa ρ (30)
where ra is an arbitrary radius and pa = p (ra). Simply matching both velocity and pressure laws at
r= ra, we get
22
2
21
aoa
a
rpp
r
Ω+=
Ω=Γ
ρ (31)
and using Eq. (31) on Eq. (30), the outer pressure is
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−Ω+= 2
222 2
21
rrrprp a
ao ρ (32)
Fig. 9 shows an example of the radial pressure distribution given by Eq. (29) for r < ra and
Eq. (32) for r > ra, and normalized to a unitary maximum pressure difference. The implication is
clear: due to the centrifugal force, the effect of a swirl is to decrease pressure towards the axis of
rotation. In real cyclones, it has been estimated that the difference may be as high as to rise pres-
sure 30 % above the cross-sectional average [8].
3.4 More on cyclone velocity patterns.
The consequences of this fact are diverse. Firstly, it somewhat serves to complete the pic-
ture of inner cyclone velocity patterns. Due to the cross-sectional pressure gradient, any rotated
18
flow develops secondary components that are forced to evolve in the axial and radial directions.
As we have already mentioned, these secondary flows can be of relevance, since they directly con-
tribute to by-pass or leakage of particles to the inner vortex, thus leading to losses of separation
efficiency. A generic sketch of secondary flow patterns in cyclones is given in [8]. A pattern re-
cently verified by numerical calculations, e.g. in [12], [104], is the so-called “lip leakage”: a strong
radial component of velocity, inwardly directed just under the rim of the vortex finder. There is
possibly much more to say about secondary flow structures; for instance, Ref. [105] documents
(also numerically) four different classes of them along the cyclone. However, the question is possi-
bly much harder to rationalize.
Other feature that can be explained now is the existence of W-shaped profiles of axial veloc-
ity. They result from the attenuation of swirl by the walls of the vortex finder, which flattens a
pressure distribution like that in Fig. 9. In this manner, the pressure gradient is positive down-
stream and near the centerline, so that a dip in the profile of axial momentum obtains [11], [96].
The effect is more pronounced with larger vortex finder diameters. Inside and nearby the vortex
finder area, the velocity normally becomes negative, i.e., a region of backflow develops; the phe-
nomenon is conceptually the same as the recirculation “bubble” of swirl tubes [108] and burners
[111], an can be classified as a vortex breakdown of type 0 [52]. Some results that further document
the flow in this region of a cyclone are the numerical calculations of [12] and [107] and the visuali-
zation described in [27] .
In many cases, the dip in velocity persists inside the inner separation space, well below the
vortex finder. This can be seen for instance in the data of [96], [104], [109], [112] and in older and
newer numerical simulations [90]-[92], [107]. According to [11], this reflects the high sensitivity of
the swirling flow to the conditions in the vortex finder: If swirl attenuation is also provided at the
other end (the dust extraction end) by means of a prolonged tube, inner W-shaped profiles are
suppressed due to the same influence coming from the opposite direction. Numerical results of
[12] for a cyclone equipped with a dipleg only exhibit V-shaped profiles, thus contrasting with
19
those of [107] with direct dust discharge, and confirming the explanation of Ref. [11]. Although a
W-shaped profile with backflow may appear beneficial for the separation of particles, it is not;
separation is actually more efficient with V-shaped profiles due to the fact that the ascending flow
region is narrower and thus subjected to higher swirl.
Finally, the radial pressure gradient makes the vortex flow inherently unstable. Separation
of solids can be upset as a result if the flow near the wall encounters otherwise normal deviations
from an ideal surface: weld seams, measuring probes, riveted unions and the like. This makes cy-
clone performance difficult to predict, even by sophisticated calculations. In fact, as we will see
later, instability of the double-vortex structure influences all the cyclone flow features, being the
key to completely explain its operation.
3.5. Pressure drop in cyclones. Logically, vortex motion and its associated radial pressure gradient cannot be ignored
when considering pressure drop in cyclones. As if the flow were unidirectional, pressure drop for
clean, smooth-wall cyclones can be split in three contributions: (1) losses at the inlet; (2) fluid fric-
tion in the double vortex within the separation space; and (3) losses in the vortex finder and exit
duct. Among them, the first is usually of minor importance and the last is the largest. However,
the underlying mechanism is not the usual one, as the following experimental trend clearly points
out: cyclone pressure drop decreases with increased wall friction coefficient, concentration of solids
or length of the apparatus.
The explanation of these perplexing effects is common and lies in the fact that it is not
merely the normal or “static” fluid pressure p what is being lost, decreased or dissipated. As a cor-
rectly written mechanical energy balance may put forward, viscous dissipation results in a de-
crease of the quantity ,221 vp ρ+ which we usually call “total” pressure. Here v is the modulus of the
velocity, but the axial component is of like magnitude at cyclone inlet and outlet, as a result of con-
20
tinuity and similar cross-sectional areas. Accordingly, one can think of a total pressure inside the
cyclone made up of the static pressure plus the kinetic energy of the vortex, .221
tvp ρ+
Pressure losses in cyclones are dominated by the viscous dissipation of this quantity in the
vortex finder. Such a dissipation is roughly proportional to the absolute magnitude of ,2tv so that
any influence that tends to increase the strength of the vortex increases the losses and vice versa.
For instance, an increase of the wall friction coefficient results in increased losses in the separation
space, as it would be in the absence of rotation. However, at the same time it also brings about a
decrease in the magnitude of vt, which in turn leads to decreased losses in the vortex finder. Since
vt is higher in the inner vortex, and the relevant variable is actually vt squared, the second effect is
always the largest by far, so that, surprisingly (or not so by now), increased wall friction decreases
pressure drop. This has been documented by many studies; see a good example in [26]. Exactly the
same argument explains the effect of solid loading and body length. For instance, the experimental
studies of [32] and [33] verified that lower tangential velocities do result when dust concentration
is increased, which was attributed to increased wall friction due to separated solids covering the
wall.
A second subtlety concerns the very meaning of pressure drop measurements. As we have
defined it, cyclone total pressure is equivalent to stagnation pressure, or, in other words, the pres-
sure that would be measured by reversibly stopping the fluid that swirls at a velocity vt in an ideal
flow rectifier. However, this is irrelevant here. Pressure drop in cyclones is measured by the usual
means of static pressure taps on the wall of inlet and outlet ducts. Assuming good instrumentation
practices, such a pressure probe senses perpendicular force on the wall, be it under a swirl compo-
nent of velocity or not. Therefore, the magnitude detected is the static pressure in any case. The
fact that there is a centrifugal force has nothing to do with stagnation whatsoever; otherwise, the
equilibrium expressed by Eq. (28) will be counted twice, so to speak.
The question is then that measurements at cyclone outlet (and thus pressure drop data) are
in principle difficult to interpret. For normal cyclone arrangements, it happens that a tangential
21
velocity component persists at the exit duct connected to the vortex finder. Therefore, according to
our arguments, what is really measured is a static pressure at the wall that is higher than the cross-
sectional average that should make up an overall balance. On the other hand, the “dynamic” com-
ponent 221
tvρ of the “total” pressure is neither taken into account.
If (and only if) we define “pressure drop” or “pressure losses” as the viscous dissipation of
total pressure, which is of course what makes more sense, both effects are opposite: Detecting only
static pressure tends to increase apparent losses, whereas the fact that the measurement is higher
than the average tends to decrease them. Strictly speaking, there is no reason why these two ten-
dencies should exactly compensate, so that the question remains open and we can still expect a
high uncertainty, say perhaps of ± 20 %, when interpreting pressure drop reports.
Fortunately, things are this once much more friendly to the researcher or engineer. For in-
stance, if we assume solid body rotation inside the vortex finder, Eq. (28) applies for 0 ≤ r ≤ re. Us-
ing it, it is easy to show that both static pressure at the wall )r(p e and the cross-sectional average of
the total pressure 220
2221)( rprrp Ω+=Ω+ ρρ attain a common value, .22
21
0 erp Ω+ ρ These are of
course very idealized velocity and pressure profiles, but it has been shown that the same sort of
compensation takes place approximately with real vortex finder flows [26].
Summarizing, for simple exit arrangements, the usual (static) pressure measurement gives
an adequate account of cyclone pressure losses if the latter are understood as total, dissipative vis-
cous losses, not just simply the input/output difference in static values. Cyclone designers and
users should be aware however than dissipation of the swirl will certainly continue downstream of
the exit duct, up to the point that almost all the “dynamic” pressure is lost, with no reversible re-
covery of any (static) pressure. Therefore, total system loss is higher than the figure reported by
cyclone studies. Conversely, if the cyclone discharges to the atmosphere, and no exit pressure
measurement is done, the excess pressure at inlet directly indicates total losses, since the sudden
dissipation of swirl at the outlet is added up by this procedure. But then the pressure drop figure is
not comparable to that obtained by using pressure taps at the exit duct.
22
On the other hand, the use of flow rectifiers really diminishes pressure losses by means of a
(partial) recovery of the dynamic component as a static pressure, and certainly makes the meas-
urement downstream unequivocal. The idea is very old, as it is the dispute about the pressure re-
covery attainable and the side effects on vortex dynamics and separation efficiency. The recent
work reported in [27] reviews the question and contributes to its enlightenment. According to it, a
rectifier located downstream of the vortex breakdown inside the vortex finder has little effect on
cyclone performance but a limited potential, since the swirl has been largely dissipated already. To
attain reductions of 30-50 %, as reported in early work, the device should be located somewhat
protruding from the vortex finder, where the tangential velocity is still high. But then, logically,
there is also a significant, deleterious effect on vortex stability and separation.
3.6. Calculating the pressure drop. Dimensional analysis helps to identify relevant variables and organize empirical or theory-
based formulae for pressure drop in cyclones. Taking for instance the approach of [15], we get, as a
rather complete list of variables:
),,,,(21 2 gscisc
ig
cc ReCFrgeometryf
vp
ρρρ
ξ =Δ
≡ (33)
Dimensionless pressure drop cξ (also called the Euler number, Eu) is customarily defined
with reference to the inlet gas velocity, although there are other possibilities. As in most equipment
operating under turbulent flow, the dependence on Reynolds number is only relevant up to a cer-
tain value, and usually negligible for the values found in practice [24]. Other two parameters, the
Froude number c2ic gD/vFr = and the solid to gas density ratio ρs /ρg, can be also eliminated, on
the basis that their variation for a certain class of cyclone designs and operating conditions is usu-
ally small; it can also be added that their influence is nil in the dilute flow limit.
This leaves us with the sole influences of cyclone geometry and solids loading; this is a pos-
sible explanation of the structure of most pressure drop models for cyclones. In many of them, the
23
two effects are introduced as independent, multiplicative factors to compose an overall Euler
number:
sgc ξξξ = (34)
where ξg represent the limit of dilute flow, i.e., the loss that would occur in the absence of particles
and thus it only can be a function of geometry. ξs is a correction factor that accounts for the pres-
ence of a high concentration of solids. Of course, the latter is an ad hoc correction, so that the fac-
torization expressed by Eq. (34) is devoid of any fundamental significance.
The most widely used correlations for the “clean” pressure loss coefficient ξg are summa-
rized in Table 3, Eqs. (35)–(44). Some of them are empirical, such as the simple formulae of [28]
and [30], but also the more complex formulation of Barth [29]. Actually, the formula for the loss in
the cyclone body, Eq. (39), is theoretical, based on the friction surface concept, but, according to the
data, it gives figures considerably lower than the main contribution: the loss in the vortex finder
calculated by Eq. (40), which is purely empirical. Muschelknautz & Kambrock method [54] pat-
terns the general scheme of Barth.
According to our analysis, some methods just assume that ξg is only a function of geometry,
which in fact constitutes an elementary scaling rule, for smooth-walled cyclones operating at high
Reynolds numbers and low solid loadings. However, Alexander [9] already took into account
variations of gas viscosity with temperature, by using in his formulae, Eq. (36) and (37), the expo-
nent n, Eq. (4). The models of Barth [29] and Muschelknautz & Kambrock [54] use the velocities vtw
and vte given by Eq. (5)-(6), which amounts to introduce a explicit friction coefficient λg for cyclone
walls. This is included in “geometry” or already assumed of typical value in the dimensionless
Eq. (33), but in this way, variations of wall roughness can be handled.
Table 4 compares data from the experimental rig of [31] with the predictions of formulae in
Table 3. The experimental value of ξg is an actual “clean” pressure drop, i.e., measured without
solids loading. All the correlations perform rather modestly, although Muschelnautz’s is clearly
the best. However, the experimental cyclone of [31] was a model of a PFBC unit equipped with a
24
long dipleg. Allowing the vortex end to penetrate the latter (which is actually an observed fact of
the experiment), we were able to predict ξg with total accuracy by Muschelknautz method. The
comparison is fair for the rest of formulae because such a modification is not possible with them,
whereas tangential velocities of Eq. (42)-(43), and thus ξg, depend on vortex end position through
Barth’s vortex length, Eq. (8), Fig. 7.
The factor ξs ≤ 1 estimates the reduction in pressure drop due to a high solids loading.
There is considerable uncertainty in cyclone literature as to what should be understood by “high”
in this context. The right answer lies possibly within the range of 25-50 g/kg, considering only
effects in observed pressure drop. Equations (45) to (49) in Table 5 summarize the most cited mod-
els for ξs. Most of the studies have taken the simplified approach of assuming ξs only a function of
the inlet solids loading Csi — the alternate variable Csiρg (kg/m3) being also very popular, if not
dimensionally coherent. This amounts to our reasoning that Froude and density ratio numbers do
not vary typically much, plus the additional assumption that the multiplicative factor ξs is univer-
sal, not dependent on geometry, or that the formula is restricted to geometrically similar appara-
tuses of certain design.
On the other hand, if we recall Eq. (11) in Table 2, it is clear that Muschelknautz method [4]
can go a step beyond, being able to account for the effect through an augmented friction coeffi-
cient, which is much more sound from a physical standpoint. The table shows the way of introduc-
ing this within the framework of Eq. (34). The method also considers geometry and Froude and
density ratio numbers, possibly being the most complete available (and not proprietary of cyclone
manufacturers).
Figure 10 evaluates the different methods by comparing their predictions with data from
the PFBC cyclone of [31]. The value of ξs is the truly correction factor defined in Eq. (34), calculated
from measured pressure drops with and without solids loading. Calculations following
Muschelknautz need the clean pressure drop, which is calculated as above, taking into account
25
vortex penetration in dipleg. These results offer again a good agreement with measured data, al-
though the recommendations of Baskakov et al. perform remarkably well, if not better.
The studies that led to the correlations of Table 5 and Fig. 10 also revealed another signifi-
cant fact. For low-to-medium inlet solid loadings, the pressure coefficient ξs deceases monotoni-
cally with solids concentration, in accordance with the physical explanation of the effect. Neverthe-
less, for very high solid loadings (such as in CFBC) the opposite has been detected. Baskakov et al.
[36] found a minimum located in the ξs–Csi curve at 200 g/kg. This most probably represents the
very limits of the effect, i.e., the point at which no more friction due to particle strands can de-
crease further the tangential velocities, so that the “normal” effect of an increased friction begins to
dominate. Chen et al. [37] obtained similar results for various cyclone geometries, although the
minimum was located at higher inlet concentrations, in the range 400-700 g/kg. Similar trends
have been found for CFBC cyclones in other experimental studies [38]-[40].
4. Collection efficiency All the ideas on the mechanics of cyclone flow we have discussed thus far are only a part of
the way to explain how the basic purpose of this kind of equipment is effected: separation of solids
from dust-laden gas streams. Let’s give here a brief account of the question. Collection of particles
inside a cyclone is naturally a result of the forces acting on them, whose resultant drives them to
cyclone walls. Literature always lists centrifugal, drag and gravitational forces, but in addition,
there might be others, not entirely understood and often neglected, such as particle-particle and
particle-wall interaction, that surely influence the collection process.
The collection ability of a cyclone is measured by its collection efficiency η, defined as the
fraction of the inlet flow rate of solids separated in the cyclone. Since a cyclone usually collects
particles possessing a wide range of sizes, it is common to work also with different efficiencies,
each defined for a particular and narrow interval of particle sizes. Imagining indefinitely small
intervals, we get a continuous function η(x) that can be thought of as the fractional or grade-
26
efficiency of the cyclone for particles of size x. Reference [8] gives a mathematical definition of η(x)
in relationship with η that is very enlightening. Here we limit ourselves to the reverse relation-
ship, most obvious: if f(x) is the particle size distribution (PSD) at cyclone inlet,
∫∞
=0
)()( dxxxf ηη (50)
The grade-efficiency curve can be conceived too as the true measure of the cyclone effect,
since by its own definition, it depends only on cyclone characteristics, but not on inlet PSD. On the
contrary, we can see in Eq. (50) that the total efficiency depends on both, so that it is not only a
characteristic of the apparatus.
It is also pretty obvious that very large particles will be always separated, whereas very fine
material will always escape. If fluid and particulate flow were always laminar and ordered, there
will be an abrupt cut at some intermediate x at which particles would cease to be separated to es-
cape or vice versa. (Some slight dispersion will appear however due to differences in the position
of the particle at the inlet section.) This is not the case; for many reasons, a fixed particle size is
separated with a probability greater than cero and less than unity. As a consequence, η(x) has the
generic shape of a Sigma function between the limits η(0) = 0 and η(∞) → 1. Then, a simple charac-
terization is made up of the so-called cut size x50, the size which is separated half the time,
η(x50) = 0.5, and the slope of the grade-efficiency curve at that point. Fig. 11 is a scheme of the rele-
vant definitions concerning cyclone collection efficiency.
4.1 Models of collection efficiency As with pressure drop, cyclone efficiency was first modeled in the dilute limit, and after-
wards corrected for high concentrations of dust, the so-called “mass loading effects.” For the first
class of models, it has been tradition to build theoretical constructs from very idealized arrange-
ments of particle forces and velocities, far removed from the chaos associated with turbulent fluc-
27
tuations and dispersion, and the natural instability of the vortex structures. In spite of this, some
models have been surprisingly successful.
Most popular hypothesis are 1) gravitational field negligible compared to centrifugal
forces, 2) gas density negligible vs. particle density, 3) particles are spherical, of low size and the
relative velocity is small enough for Stokes law to apply, and 4) relative velocity is purely radial.
Under these hypotheses, the following equation of motion of a spherical particle rotating at an ar-
bitrary radial position applies:
rt
sr
s xur
vxdt
dux πμρπρπ 366
233
−= (51)
where ur is radial velocity of the particle relative to the gas, x is particle diameter, ρs is particle den-
sity and vt is tangential velocity of both particles and gas. The first right-side term is the centrifugal
force and the second the drag. For the particle sizes and Reynolds numbers encountered in indus-
trial cyclones, the Basset and displaced-mass terms are clearly negligible. However, turbulent dif-
fusion is flagrantly ignored.
Classical models for calculating cyclone efficiency result from integrating Eq. (51) under a
manifold of flow situations and hypotheses. Afterwards, some notion (also simplified) must be
imposed to decide if a given particle is collected or not. The calculated parameter is often x50; the
complete curve can be then adjusted by some other means, frequently of empirical nature. For in-
stance, a very popular curve-fit has the form
m
i
i
xx
⎟⎟⎠
⎞⎜⎜⎝
⎛+
=501
1η (52)
for discrete sizes xi and fractional efficiencies ηi. The exponent m is the slope of the curve, to be
adjusted to the data.
Algebraic efficiency models pertain to either of two broad classes. In principle, the idea
could be to perform the integral of Eq. (51) from the inlet, and see whether the particle in question
has time to reach the wall. A model that does exactly this is called a “time-of-flight” model. The
28
idea needs however a complete battery of simplifications, especially if a more or less closed, alge-
braic method of prediction is pursued. For this reason, such drastic hypotheses have been used
that the concept of a Lagrangian track of particles is almost lost. The most successful has been the
so-called notion of “equilibrium orbit” firstly put forth by Barth.
Most popular formulae for cyclone efficiency calculation are summarized in Table 6.
Lapple [41]. It is similar to the method for sizing sedimentation chambers, the simplest ver-
sion of the time-of-flight ideas. A radial initial position is taken at the middle of the entrance duct
(b/2), a uniform particle distribution across the inlet section is assumed and a constant tangential
velocity is used, all of this in order to ease the integration of Eq. (51). This serves to calculate the
time for a particle of given size to reach the wall. The particle whose time equals the gas residence
time is considered to be separated with 50 % efficiency, and thus its size is x50.
Barth [29]. Barth introduced the notion, reasonable but arbitrary, that a particle subjected to
radial force equilibrium just at the CS surface (Fig. 7) will be separated half the time, since under
ideal conditions it would remain there forever. Therefore, equating Eq. (51) to zero is all it is
needed to calculate x50. The result is included in Table 6 as Eq. (54). Its resemblance with the time-
of-flight formula given by Eq. (53) is at least surprising. However, differences are significant. One
of them is the fact that Barth used his own tangential velocity at the surface CS. This makes the
method sensible to an important parameter: the diameter of the vortex finder De , a geometrical
feature that greatly influences cyclone efficiency. In contrast, only the inlet velocity is used in most
time-of-flight models, such as Lapple’s; they lack accordingly this important capability.
In this respect, it has been shown recently [104], both by measurement and numerical calcu-
lation, that the reasonable “Barthian” cylinder CS doesn’t really exist, in the sense that the width of
the ascending flow tube seems to be not influenced by De, but only by the cyclone diameter Dc, at
least for some kind of apparatuses. The effect of De on collection efficiency would result from the
change of local flow patterns just under the vortex finder.
29
Leith & Licht [42]. This is a more elaborated time-of-flight model that permits to determine
the entire grade-efficiency curve. To this end, the authors considered a continuous flux of dust
instead of single particles. Equation (51) is thus not used, but the continuous advection of material
balanced by centrifugal forces and inertia. Other important assumptions are negligible gas radial
velocity (as in most time-of-flight models) and a constant concentration of particles along the ra-
dius. Assuming this being a result of turbulent diffusion served to overcome the limitation we
commented above. Results are expressed as a function of the cyclone “natural length”, a parameter
that was assumed geometric by Barth. In section 5.1 we will return to the importance of this fact.
Although very popular, the model of Leith & Licht was incorrect in its derivation, as put
forward in [44], due to a basic flaw in the definition of average residence time for a continuous
flow system. Correcting the mistake led to a simpler formula, but not to a proper model of the frac-
tional efficiency curve.
More sophisticated models can be found in the literature, such as Dietz’s [43], where the
cyclone is separated in three regions, assuming perfect mixing of solids in the radial direction. This
model is hybrid, using ideas both from time-of-flight and equilibrium-orbit paradigms. It had the
same basic flaw as Leith & Licht’s, but could be corrected satisfactorily, with a lesser shift of the
predicted grade efficiency curve [44]. As can be imagined, Muschelknautz work [4] served also to
improve Barth’s efficiency, by including the effects of wall friction and solids loading through the
tangential velocity calculation (more on this in the next section). Other models available are even
more complex in nature, although in essence, most reduce to modifications of Barth´s equilibrium
orbit, a seminal idea. For instance, the extension to cyclone zones different from the main separa-
tion space is a recurrent idea, as in [45]. Finally, Mothes & Löffler [18] is a hybrid model that added
a finite particle dispersion coefficient in an effort to improve the prediction.
30
4.2 Mass loading effects. All these models of collection efficiency usually perform reasonably well for low solids
loadings, less than 5-10 g/m3 at cyclone inlet. However, many industrial units handle higher loads,
that can ever reach 10 kg/kg, e.g. in FCC, PFBC or CFBC applications. Under these conditions, the
gas cannot be treated as if the particles were absent, and the latter cannot be modeled as a single
particle, roughly following the gas and without interactions with other particles. In other words,
the simplicity of a dilute flow should be abandoned and more involved ideas should be applied.
This has been accomplished only to a partial success. Experimental facts are, again, perhaps
perplexing: collection efficiency η increases as the concentration of solids at inlet Csi is increased. A
part of the perplexity is explained because the increase in η proceeds at a lower pace than that of
Csi, so that the absolute amount of solids emitted – lost– always increases with Csi. Another part
can be accepted if we consider that the physics of a concentrated, two-phase flow actually favors
particle separation
This is the point adopted by the main approach to model mass loading effects: the concept
of a “critical load”, firstly proposed by Muschelknautz in his early studies [4]. Tracing an analo-
gous with sedimentation effects in pneumatic transport, Muschelknautz reasoned that a form of
sedimentation of particles occurs under the centrifugal force, just at the cyclone entrance. The tur-
bulent energy of the gas stream only supports a limited weight of solids, a critical load Ccr, settling
the rest unclassified. Therefore, if solid inlet concentration Csi is increased above Ccr, cyclone effi-
ciency increases regardless of particle size.
In this manner, collection efficiency is calculated differently for the regimes of dilute
(Csi ≤ Ccr) and dense (Csi > Ccr) two-phase flow:
crsi CC ≤= 0ηη (57a)
crsisi
cr
si
cr CCCC
CC
>+⎟⎟⎠
⎞⎜⎜⎝
⎛−= 01 ηη (57b)
31
In Eq. (57b), the first term accounts for the efficiency of the solids separated by “sedimenta-
tion” at cyclone inlet, whereas the second stands for the collection efficiency η0 of the remaining
load in the separation space. In principle, η0 is equal to the collection efficiency of the cyclone at
low solids loadings, and this is assumed in Eq. (57a). However, as we have seen, the solids sepa-
rated at the entrance have a definite influence on cyclone flow, decreasing tangential velocity
trough increased wall friction. This obviously influences collection efficiency and thus can make
the η0 in the second formula different from the η0 in the first one. As we have seen too, some mod-
els of flow and efficiency in cyclones can handle this difference.
Other than this, the question reduces itself to estimate the value of the critical load Ccr. The
original formula of Muschelknautz [4] was derived from reasoning on his velocity distributions:
twtemsc
e
eccr
vvxDD
DDC
212 ρ
μλ
⎟⎟⎠
⎞⎜⎜⎝
⎛−
= (58)
where xm is the particle mass-median diameter and the other terms are derived from flow field
calculations.
However, more recent experimentation [5] has led to different expressions, not directly
based on sedimentation studies:
( ) 1.010025.0 4.050 <= sisim
cr CCxxC (59a)
( ) 1.010025.0 15.050 >= sisim
cr CCxxC (59b)
Reference [5], based on the work of Muschelknautz [4] and Barth [29], also developed a
method to correct the dilute efficiency η0 for the effect of a decreased vortex intensity, that can
reach up to 60 % for values of Csi = 1 [51]. But the main novelty of this work was the hypothesis
that some classification also occurred at the inlet, changing the PSD of the particles finally sus-
pended in the vortex. By considering a mean centrifugal acceleration, a settling velocity was calcu-
lated at the entrance duct, from which an inlet cut-size was determined.
32
This has been signaled as inconsistent with an observed uniform increase of the grade-
efficiency curve that would not warrant any kind of sharp classification due to the cyclone effect.
In any case, fractional efficiency and its variations with load usually exhibit lifts and hooks that are
not easily reconciled with physical evidence. An example from [31] is shown in Fig. 12. The fact
that efficiency seems to be higher for smaller particles leads us to a second mass-loading effect that
has been far less explored.
It consists in the possibility that the temporary adhesion of small particles to larger ones, or
the swept of the former by the latter inside the vortex, could be another significant factor for an
increased efficiency at high solids loadings [18], [46], [47]. In fact, this is almost the only way a
minimum in grade efficiency can be explained, once particle attrition is accounted for. The ag-
glomeration effect seems to have been detected for hot gas cleaning applications in the
Grimethorpe PFBC cyclones [49]. However, up to now, particle interaction effects have not been
neither confirmed nor modeled in this context, neither as a modification of the η(x) curve, nor to
the total η value.
Finally, to give an idea of the performance of present efficiency models when used in the
high loading regime, Fig. 13 compares data from [31] with the calculations according to Trefz &
Muschelknautz [5]. Agreement is much better than that obtained with simpler, older efficiency
models, limited to low concentrations, which is well explained by the separation of the critical load
at inlet. Moreover, agreement is even better if the dilute efficiency is calculated taking into account
an extended vortex length, as we did with the pressure drop.
5. Special phenomena associated with the flow field in cyclones. 5.1. Natural turning length. As we have seen, in a reverse-flow cyclone, the outer vortex weakens and changes its direc-
tion at a certain axial distance Ln from the vortex finder.‡ This magnitude is usually called the
‡ Customarily measured from its lower rim, as in Fig. 7.
33
“turning length”, “natural length” or “vortex length” of the cyclone, and the axial position is re-
ferred to as “the end of the vortex”. This point can be imagined as an effective end of the appara-
tus, since almost all the gas has leaked entirely out to the inner vortex at this position. Therefore,
both pressure losses and particle separation are mainly determined by events occurring above;
what is left below is an induced, secondary vortex that cannot contribute much, only to re-
entrainment, in poorly-designed extraction systems.
As a consequence, cyclone designs with a natural length some measure greater than the
physical length are advisable [47], [48], since a lack or performance or an oversized unit results
otherwise. The question is then what determines this parameter. In his influential theories, Barth
assumed that h*, the effective cyclone length, Fig. 7, was a mere function of geometry, i.e., that it
changed exclusively with cyclone design. Although many useful results were derived in part from
this hypothesis (as we have seen), it is not generally true. We have learned from experience that
the natural length of cyclones is influenced by dynamic factors. Moreover, the vortex end can be
itself a dynamic and complicated phenomenon.
It is thus not strange that its true nature wasn’t fully explained until very recently. Initially
it was assumed to be an axisymmetric flow structure caused by the axial and radial pressure gra-
dients, perhaps related to the phenomenon of vortex breakdown as observed in once-through
swirling flows [52]. However, this is possibly true only at low velocities and high wall friction coef-
ficients, i.e., at low swirl [53]. Under realistic conditions, the double vortex bends and attaches it-
self to the lateral wall, and, superposed to the vortex swirl, the bend itself rotates at frequencies in
the order of several tens of Hz. Thus, the cyclone vortex doesn’t end inside the fluid, but on its
boundary, and it is not, definitely, an axisymmetric and stationary structure.
The history of how this was discovered deserves some attention; a swirling flow that de-
scends, bends on a wall, reverses its stream direction and continuously changes the axial plane
where everything happens is certainly not easy to detect. In fact, modern studies, both experimen-
34
tal and numerical, normally miss the detail if not looking specifically at it [67], [107], and papers
continue to be published still seemingly unaware of the true nature of the vortex end [106].
Observations of a bended vortex attached to the wall were already made by Muschelknautz
in the 1970s [54]; by the end of the 1990s, the fact seemed almost to pertain to the traditional
knowledge on cyclones, see for instance the “regimes” explained in [53]. A definite evidence was
[139] Qian, F., Huanga, Z., Chena, G., and Zhang M. , Numerical study of the separation
characteristics in a cyclone of different inlet particle concentrations, Comput. Chem.
Eng (2006), doi:10.1016/j.compchemeng.2006.09.012.
76
Figure captions
Fig. 1. (a) Qualitative drawing of the principle of operation and flow patterns in cyclones. (b) Main parts and dimensions of an inverse-flow cyclone: (1) cyclone body, (2) conical part, (3) inlet duct, (4) exit duct, (5) vortex finder [14], [12]................................................................................ 77 Fig. 2. Main inlet arrangements: (a) tangential, (b) scroll, (c) helicoidal, (d) axial [6]. ........................... 78 Fig. 3. Scale drawings of the cyclone designs of Table 1 [8]. .................................................................... 78 Fig. 4. Measured profile of velocity components in a reverse-flow cyclone: (a) tangential, (b) radial, (c) axial [10]. ........................................................................................................................................ 79 Fig. 5. Sketch of cyclone velocity profiles: 1- radial, 2- axial, 3- tangential [18]. ..................................... 80 Fig. 6. Sketch of the tangential velocity profile in cyclones [12]................................................................ 80 Fig. 7. Imaginary cylinder of Barth theories [29]........................................................................................ 81 Fig. 8. Comparison of tangential velocity profiles predicted with different flow models. Adapted from [23]. .......................................................................................................................................... 81 Fig. 9. Radial distribution of static pressure in a Rankine vortex with ra/rc = 0.6 [12]. ......................... 82 Fig. 10. Cyclone pressure coefficient vs. solids concentration at inlet. Comparison between correlations and measured data [31]............................................................................................................. 82 Fig. 11. Typical fractional efficiency curve................................................................................................... 83 Fig. 12. Experimental collection efficiency as a function of solids inlet concentration [31]................... 83 Fig. 13. Comparison of total collection efficiency obtained experimentally and calculated. (a) Model of Trefz & Muschelknautz [5], (b) same model, accounting for the length of the vortex [31]. .................................................................................................................................................................... 84 Fig. 14. (a) Cyclone operation with Ln > H. (b) Operation with Ln < H and the vortex-end attached to the lateral wall . Adapted from [53].......................................................................................... 84 Fig. 15. Comparison of experimental and predicted values of the pressure drop coefficient, as a function of inlet solids loading. Lines are power fits to the data [31]............................................... 85 Fig. 16. Visualization of the precessing vortex core (PVC) at the exit duct of a hydrocyclone [73]. .................................................................................................................................................................... 85 Fig. 17. Sketch of tangential velocity profiles as affected by the phenomenon of the PVC. Adapted from [74] ........................................................................................................................................... 86 Fig. 18. Comparison of tangential velocity profiles [96]............................................................................. 86 Fig. 19. Comparison of tangential velocity profiles [98]............................................................................. 87 Fig. 20. Comparison between predictions of tangential velocity by algebraic models and CFD computation. [12]............................................................................................................................................. 88 Fig. 21. Unsteady field of axial velocity obtained by a URANS simulation of cyclone flow [12], signaling the existence of a PVC. .......................................................................................................... 89 Fig. 22. Instantaneous locus of the surface of zero axial velocity inside the ensemble of a cyclone + dipleg, colored according to the modulus of velocity [12]....................................................... 90 Fig. 23. Regimes of dispersed two-phase flow as a function of the particle volume fraction/ interparticle spacing. Adapted from [117]. .................................................................................................. 91 Fig. 24. Snapshots of particle concentration (scale not given) at five instants of time inside a Stairmand cyclone separator, as calculated by LES + Lagrangian tracking [107]. ................................. 91
77
(a)
a
bDe
S
h
HDc
B
(5)
(1)
(2)
(4)
(3)
(b)
Fig. 1. (a) Qualitative drawing of the principle of operation and flow patterns in cyclones. (b) Main parts and dimensions of an inverse-flow cyclone: (1) cyclone body, (2) conical part,
Fig. 6. Sketch of the tangential velocity profile in cyclones [12].
81
Fig. 7. Imaginary cylinder of Barth theories [29].
Fig. 8. Comparison of tangential velocity profiles predicted with different flow models.
Adapted from [23].
82
Fig. 9. Radial distribution of static pressure in a Rankine vortex with ra/rc = 0.6 [12].
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 50 100 150 200 250
Csi , g solids/kg gas
Cyc
lone
pre
ssur
e co
efic
ient
, ξs
Briggs (1949)Smolik (1975)Baskakov et al. (1990)Muschelknautz (1972)PFBC cyclone
[1]
[29][30]
[31]
Measured data [26]
Fig. 10. Cyclone pressure coefficient vs. solids concentration at inlet. Comparison between
correlations and measured data [31].
[4]
[31]
[36] [35] [34]
83
0
10
20
30
40
50
60
70
80
90
100
1 10 100 1000Particle size , μm
Frac
tiona
l eff
icie
ncy
%
x50
Fig. 11. Typical fractional efficiency curve.
80
85
90
95
100
105
1 10 100
71 g/kg
114 g/kg
169 g/kg
Particle size , μm
Frac
tiona
l eff
icie
ncy,
%
Fig. 12. Experimental collection efficiency as a function of solids inlet concentration [31].
84
94
95
96
97
98
99
100
0 50 100 150 200 250
PFBC cyclonePredicted
Inlet solid concentration, g solids/kg gas
Col
d m
odel
ove
rall
effic
ienc
y %
separation at inlet
inner separation
(a)
94
95
96
97
98
99
100
0 50 100 150 200 250
PFBC cyclonePredicted
Inlet solid concentration, g solids/kg gas
Col
d m
odel
ove
rall
effic
ienc
y %
separation at inlet
inner separation
(b)
Fig. 13. Comparison of total collection efficiency obtained experimentally and calculated. (a) Model of Trefz & Muschelknautz [5], (b) same model, accounting for the length of the
vortex [31].
Fig. 14. (a) Cyclone operation with Ln > H. (b) Operation with Ln < H and the vortex-end at-
tached to the lateral wall . Adapted from [53].
85
0.5
0.55
0.6
0.65
0.7
0.750.8
0.85
0.9
0.95
1
0 50 100 150 200 250
Csi , g solids/kg gas
Cyc
lone
pre
ssur
e co
efic
ient
, ξs
Cold model
Muschelknautz (1972)
Muschelknautz (1972) including vortex penetration in dipleg
[1]
Measured data [26]
[1] ,
Fig. 15. Comparison of experimental and predicted values of the pressure drop coefficient,
as a function of inlet solids loading. Lines are power fits to the data [31].
Fig. 16. Visualization of the precessing vortex core (PVC) at the exit duct of a hydrocyclone
[73].
Measured data [31]
[4]
[4]
86
Fig. 17. Sketch of tangential velocity profiles as affected by the phenomenon of the PVC.
Adapted from [74]
Fig. 18. Comparison of tangential velocity profiles [96].
87
Fig. 19. Comparison of tangential velocity profiles [98].
88
Fig. 20. Comparison between predictions of tangential velocity by algebraic models and
CFD computation. [12]
89
Fig. 21. Unsteady field of axial velocity obtained by a URANS simulation of cyclone flow
[12], signaling the existence of a PVC.
t = 0.50 t = 0.75
t = 0 t = 0.25
t = 1
90
Fig. 22. Instantaneous locus of the surface of zero axial velocity inside the ensemble of a cy-
clone + dipleg, colored according to the modulus of velocity [12].
91
Fig. 23. Regimes of dispersed two-phase flow as a function of the particle volume fraction/ interparticle spacing. Adapted from [117].
Fig. 24. Snapshots of particle concentration (scale not given) at five instants of time inside a Stairmand cyclone separator, as calculated by LES + Lagrangian tracking [107].
Volume fraction αp
Interparticle spacing L/dp
One-way coupling
Dilute-dispersed two-phase flow
Two-way coupling
Dense-dispersed two-phase flow
100 10 1
Four-way coupling
10-8 10-6 10-4 0.01 0.1
92
Table 1 Relevant geometric dimensions of several standard cyclone designs [8].