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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: tdyfqdb@unizar.es

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3.5. Pressure drop in cyclones. 3.6. Calculating the pressure drop.

4. Collection efficiency.

4.1 Models of collection efficiency. 4.2 Mass loading effects.

5. Special phenomena associated with the flow field in cyclones.

5.1. Natural turning length. 5.2. Precessing vortex core.

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)

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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

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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

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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

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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.

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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

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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

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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

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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

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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].

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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.

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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

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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-

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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)

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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

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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

blatantly absent however. Hoffmann and Stein [8] firstly reported unpublished visualization ex-

periments with stroboscopic lighting that fully supported the fact. Finally, the same research group

was able to duplicate and document the experiment, publishing solid evidence [56]. They further

characterized the phenomenon in terms of frequencies and parametric dependencies; more re-

cently, they have made consistent observations by visualization with a neutrally buoyant tracer

and time-resolved wall pressure measurements [110].

The explanation agreed very reasonably with a vast ensemble of indirect evidence obtained

throughout the years. For instance, the erosion ring at the lower part of the conical section, fre-

quently encountered in cyclones operating with hard powders [57]-[59], is simply a witness of the

circular movement of the vortex end on the wall. So is the ring made of residual dust frequently

observed in experimental units with transparent walls, e.g., in [16] – see also the interpretation of

[56]. Also, the decrease in cyclone efficiency with a shortened vortex, that cannot be completely

explained by the mere reduction of Ln [25]. And finally the drop in static wall pressure and tangen-

tial velocity measurements observed by many, e.g., [12], [16].

In the numerical arena there is clear evidence too. Pioneers are Montavon et al [101], who

reported “interaction with the wall” apparently unaware of the experimental work done on the

question. In reference [12], CFD data was used to “visualize”, in striking resemblance to the

stroboscopic experiments, the attachment of the vortex to the wall. (We will return to this later).

This picture cannot be complete without explaining why the cyclone vortex behaves in this

way. The reason is that the whole vortex is inherently an unstable structure, due to the radial pres-

sure gradient developed by itself, as already mentioned in Sect. 3.4. As a consequence, it oscillates

35

radially at any axial location, which is known as the phenomenon of the Precessing Vortex Core

(PVC), common to any swirl flow of practical interest. In cyclone separators, the vortex bends, at-

taches to the wall and rotates on it because this is simply the way in which the PVC ends in that

case. The implication is that unsteadiness of cyclone flows is not limited to the position of the vor-

tex end, but is rather a feature of the whole pattern [55].

5.1.1. Relevant parameters influencing natural turning length.

Predicting natural length of cyclones is a complex endeavor, and there is no complete

agreement about the influential parameters. A review of the possibilities most explored in the lit-

erature is given below.

Inlet Reynolds number. There was no documented evidence of the influence of this variable

until the last decade, what amounts to say that the standard, “Barthian” picture of a stable flow

only determined by geometry has been the dominant view until recent times. A review of cyclone

research [53] found, after analyzing an exhaustive data base, a strong dependency of the natural

length Lm on the inlet Reynolds number. In another exhaustive review [61], two distinct types of

cyclone operation were identified, as depicted in Fig. 14:

• Operation with Ln ≥ H, Fig. 14a. In this case, the natural vortex length is equal or longer

than the physical length, and the vortex end reaches the bottom of the cyclone, or even

the dust collection or extraction hardware. This is the most advisable mode of opera-

tion, since it leads to a high collection efficiency. It corresponds to cyclones with rela-

tively high inlet Reynolds numbers (approximately ranging from 2×103 to 2×104) and

H/Dc ratios within 2-10.

• Operation with Ln < H, Fig 14 b. It entails poor collection performance, because the vor-

tex end attaches to the cyclone wall, disturbing the solids strands that are already sepa-

rated and decreasing efficiency through instability and re-entrainment. This mode of

36

operation corresponds to low inlet Reynolds numbers (<103) or very tall cyclones

(H/Dc> 15).

These “regimes” have been recently observed by flow visualization [110]. Reynolds num-

bers in practice are usually high enough to allow high values of cyclone height, so that the majority

of cyclones should normally operate in the range of good performance (Ln ≥ H). There is however

not total agreement on the quantitative effect. For instance, the influence detected in [62] is signifi-

cant but not as strong as that found in [61].

Geometric parameters. Aside from the ratio H/Dc itself, the most relevant geometric parame-

ters are the following:

• Inlet area. Alexander found [9] that Ln decreased proportionally to the inlet area Ai, but

the opposite trend has been also reported [63]. More exhaustive research seems to sup-

port classical Alexander’s findings [61].

• Length and diameter of the vortex finder. There is experimental evidence [25], [62] that the

length of the vortex finder S does not influence the position of the end of the vortex.§

On the contrary, the vortex finder diameter De has been identified as a definite influ-

ence. The trend most frequently observed is an increased natural length with an in-

creased De [8], [9], [62]. However, again, the opposite has been reported too [63].

• Discharge zone design. In industrial cyclones, vortex stabilization devices are frequently

used to attach the vortex at the lower end of the cyclone, thus making the natural length

equal to the geometrical length and stabilizing long cyclones. These are conical-shaped

pieces, with an upward-oriented apex and an axis aligned with the cyclone axis.

In the opposite situation of a short cyclone, the vortex end is located in the dust hopper

and induces there an unusual swirl, which may jeopardize the collection efficiency

§ Which would invalidate the traditional origin for measuring Ln, making more meaningful the roof of the

cyclone for this role [25].

37

through re-entrainment [110]. A swirl tube or dipleg can be then desirable, to accom-

modate the vortex end and avoid this effect.

In relation with this, also the use of gas extraction flows for solids evacuation is known

to increase the natural length of the cyclone and exert a strong stabilization effect on the

vortex [31].

Wall roughness and solids loading. In laboratory tests with gas cyclones operating with smoke,

it was found that the increase in wall roughness caused by condensate covering the wall decreases

Ln making the vortex shorter [25]; the same effect was consistently found with higher solids load-

ings in standard operation. The same observation has been reported also in experiences with PFBC

cyclones [16],[31]. Recalling the influence of the Reynolds number, the general implication seems

to be that any parameter that attenuates the swirl intensity leads to a shortening of the natural

length.

Instability and apparent randomness of the vortex end position. The deleterious effect of a short

vortex is not only due to the reduced value of Ln and the movement of the vortex end structure.

Actually, reduction of the natural length is accompanied by instability, in the sense that a short

vortex losses space- and time- coherence to a great extent. For instance, the investigation in [25]

found that the vortex-end position (as visualized by smoke) was much more variable for larger

vortex finder diameters, i.e., for shorter natural lengths. Also in [25], and in [16] among others, the

vortex end position (visualized by a ring of dust) was reported to move up and vary continuously

and apparently at random under some operating conditions. The ensuing highly variable flow

patterns obviously entail a reduced efficiency and an increased pressure drop, and a high uncer-

tainty in any measurement or prediction. The reverse trend is also true, i.e., a long vortex is more

coherent and stable, which favors the separation effect and reduces losses and uncertainty.

38

5.1.2 Calculation of Ln.

Up to now, adequate calculation methods of the cyclone natural length have not been de-

vised. None of the available proposals (e.g., [9], [63], [65], [66]) takes into account all the observa-

tions made in relationship with this phenomenon. Moreover, most of them still consider Ln as a

geometric parameter, and, when compared among themselves, they deliver diverging results and

also trends, as it can be easily verified with the following expressions:

Alexander [9]: 312

3.2 ⎟⎟⎠

⎞⎜⎜⎝

⎛=

abD

DD

DL c

c

e

c

n (60)

Zenz [65], Bryant [66]: 21

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛=

e

icn A

ADL (61)

Zhongli [63]: 361.0225.2

4.2−−

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

i

c

c

e

c

n

AD

DD

DL

(62)

Considering the characteristics of the vortex end we have just mentioned, it is not strange

that these formulae perform very badly. (Alexander’s geometries and conditions were also too

different from modern designs). Although no correlation effort has been reported incorporating

dynamic parameters, there has been some success in predicting Ln by numerical calculation. Velilla

[12] used CFD results to calculate a natural length, defining it as the locus where isobaric surfaces

bended to the wall. The figure thus obtained was in very reasonable agreement with experimental

observations of the dust ring. Table 7 shows those results; the (completely deviant) predictions of

the above formulae are also shown for illustration.

Nevertheless, these experiences were for clean gas and the vortex end stabilized inside the

dipleg. More general conditions, i.e., a short and unstable vortex provoked or not by a high solid

loading, have not been considered to date by this kind of methods. There are however favorable

indications in the literature; for instance the numerical simulations of [101], and somewhat also

those of [106], apparently reproduced very realistically a highly chaotic PVC for non-interacting

two-phase flows.

39

5.1.3. Influence of cyclone natural length on collection efficiency and pressure drop.

The influence of the natural length is clearly shown when this parameter is included in

some efficiency models of advanced capabilities, as we have demonstrated in Fig. 13 [16], [31] us-

ing the calculation methods of Muschelknaut, [4], [5]. However, there the vortex end was stable

and located well inside a dipleg, so that the sole effect of a larger separation space manifests, as it

should be for a properly operated cyclone.

In other circumstances, if a short vortex is present, we get a regime similar to Fig. 14b and

efficiency falls. As an example, the study in [62] demonstrated that efficiency increases with geo-

metric length: a 20 % increase was detected when using different models of cyclones with H/Dc

equal to 4.65 and 5.65 instead of 2.65 and 3.15, respectively. However, when H/Dc was higher than

5.65, collection efficiency slightly decreased, which was interpreted as the effect of the vortex end

entering the separation space and disturbing the flow there.

Only two approaches in the classical cyclone literature account for the effect of the vertical

dimension of the cyclone on the total pressure drop. Barth [29] included the length H–S in his cal-

culation, Eq. (39), and in tangential velocity models, Eq. (6). In [49], tangential velocity and pres-

sure drop are related to the length of the separation zone by considering the total inner friction

surface of cyclone walls AS.

As in the case of efficiency, if the measured vortex length is included in classical models, for

instance, those of Muschelknautz, Table 3, the predicted values fairly agree with experimental

data. This has been already shown in Table 4; figure 15 further shows the question under variable

solids loading. The explanation lies in the identification of Ln as an effective length. Then, as ex-

plained in Sect. 3.5, a longer vortex increases friction, causing lower tangential velocities and thus a

reduced pressure drop.

Again, all this refers to stable vortex ends; it can be expected that the higher dissipation re-

sulting from a highly variable flow will increase pressure drop.

40

5.2. Precessing vortex core (PVC). The precessing vortex core (PVC) is a flow phenomenon originally observed in once-

through swirling flows, such as those in swirl tubes, burners and combustors. In general, the swirl

intensity is characterized by the swirl number Sw [68]:

DG

GSx

tw

2= (63)

where Gt is the axial flux of angular momentum, Gx is the axial flux of axial momentum, and D is a

characteristic diameter. In most occasions, the exact calculation of Sw is too involved. But using

reasonable approximations, in the constant-density case, Eq. (63) can be shown to reduce to a geo-

metric factor. The appropriate expression for a cyclone separator is, for instance [69]:

i

cew A

DDS4π

= (64)

Swirling flow patterns change dramatically with Sw. For swirl numbers about 0.6 and in-

creasing Reynolds numbers, a recirculation “bubble” is formed. It is initially axisymmetric, but if

the swirl is further increased, it develops a fundamental instability and starts to rotate axially. At

the same time, the zone close to the centerline of the flow also becomes unstable downstream, loses

symmetry and starts to precesse (locally) at a frequency depending on geometry [67]. The back-

flow zone also oscillates axially; it is finally shaped by the downstream oscillation itself. In this

way, a coherent, quasi-periodic flow structure is formed, which we known as the “Precessing Vor-

tex Core” (PVC). The term precession is somewhat misleading; the whole structure resembles

more an spiraling or helical movement.

In once-trough swirling flows, all these features are also characteristic of what it is called a

vortex breakdown of type 0. For an unconfined flow, the recirculation is caused by the down-

stream attenuation of the swirl in the open volume; the dip in the profile of radial pressure ac-

counts for the instability [52]. As we have seen, cyclone separators develop similar structures for

similar reasons, with the backflow “bubble” located inside the vortex finder. The peculiarity of a

41

reverse-flow disposition is that the PVC extends in both directions: down to the separation space

encompassing the double-vortex and the vortex end structure, and also downstream of the vortex

finder, inside the connecting tube. As an example, Fig. 16 illustrates flow visualization at the exit

cross-section of a laboratory hydrocyclone model [73]; the displacement of the vortex core from the

axis is clearly noticed. Cyclone separators operating at practical values of swirl and Reynolds

numbers always exhibit a PVC as a quasi-periodic feature of the flow. Its properties depend how-

ever on flow parameters. The coherence is high or low under the conditions for a stable or unstable

vortex end, as explained in the preceding section, respectively. Frequency of the movement is also

variable, as explained in Sect. 5.2.2 below.

In swirl burners and combustors, where mixing is desired, a PVC is indeed highly benefi-

cial [68]. On the contrary, given the objective of cyclone separators, the phenomenon can be detri-

mental in them. With independence of this, the cyclone PVC has been found to influence aerody-

namic noise [60]. Under some conditions, it may lead to deterioration of the equipment through

coupling with the natural frequency of large apparatuses, and even to damage of some elements

[70]. In any case, the PVC phenomenon has associated a significant energy expense, which means

that it increases pressure losses [71].

Other peculiarity of the PVC in cyclones is that the backflow zone, being confined by the

vortex finder, adopts the spiraling shape and movement of the PVC [12]. A common feature with

once-through PVCs is the formation of coherent eddies in the axial-radial plane downstream of the

backflow zone, that detach in an alternating pattern (synchronized with the PVC movement), and

migrate following a helicoidal trajectory to the exit section of the cyclone, being quickly dissipated

some diameters downstream [12], [60].

5.2.1. Influence of the PVC in the flow field.

By studying tangential velocity profiles in a cross-sectional plane at cyclone exit, the motion

of the vortex core has been exhaustively described, rotating around the geometric center of the

42

apparatus [74]. Figure 17 represents a sketch of the flow at this position. The precession movement

displaces the vortex core from the geometrical center of the device. This produces a “negative”

tangential velocity near the axis and a maximum of “positive” velocity near the wall ; both can be

explained by an angular momentum balance through the plane formed by the precession and the

geometric centers. The flow is “pressed” to the wall by the displacement of the vortex core, forcing

it to pass through a restricted area, and causing an increase of the tangential velocity [60].

The vortex nucleus describes a closely circular path around the geometrical center of the

device. A stationary observer located at a point on the xx- line detects tangential velocity fluctua-

tions, which are roughly sinusoidal. Thus, a sinusoidal wave can be measured, such as that re-

ported by [73] in a cold-flow model of a swirl combustor. The statistical distribution of the tangen-

tial velocity is obviously bimodal at the points situated within the regions of change of the flow

direction. This behavior has been used to detect the existence of the PVC phenomenon [75], [76].

Some numerical simulations and experimental campaigns have dwelt on the effect of the

PVC on velocity fluctuations [77]. The outcome is, briefly, that a maximum of the RMS value of

axial and tangential velocity fluctuations is induced by the PVC in the central part of the cyclone.

This influence roughly accounts for 80 % of the total fluctuation, the rest being due to “unstruc-

tured” turbulence, whose contribution is of the same magnitude as in elementary flows such as

boundary layers.

As for pressure patterns, the PVC appearance implies that a irregular and twisting zone of

low pressures (the displaced inner vortex zone) must be detected. No measurements confirm this

up to now, but numerical simulations show it clearly [12], [77].

43

5.2.2. Influence of operating conditions on PVC.

The frequencies of oscillation of the PVC in cyclones don’t differ much from those of the

swirl of the gas itself. Customarily, for a frequency f (Hz), a cyclone Strouhal number is defined as

i

c

vfDSt = (65)

Several studies consistently report a value of St of about 0.5-0.6 for PVC and vortex end

structures in cyclones [56], [72], [77], almost independent of the Reynolds number. Other refer-

ences agree in reporting an approximately linear increase of frequency with Rec [73], [78]. A de-

crease has been found with increased swirl number [60], which can be interpreted, in agreement

with previous observations, as an stabilization effect. Consistently, a slight decrease of frequency

along the length of the vortex has been reported [56].

On the other hand, many efforts have been devoted to control PVC by cyclone design as a

means to favor cyclone efficiency. Under a fixed inlet Reynolds number, the behavior of the PVC at

the lower part of the cyclone is a mere question of geometry. But things can be changed at the

other end [74]. When a 90º-bend is attached to the vortex finder, it has been found that the PVC is

distorted and damped. By adding a straight section 6.5 diameters long, the intensity of the PVC is

further decreased, reducing tangential velocity and centering the vortex. Finally, if a 60º-bend is

added, the precession movement completely disappears. Research has also focused at mitigating

PVC amplitude and frequency in the vortex finder by installing centering devices and guide vanes

[60], [67].

6. Computational Fluid Dynamics applied to cyclones. Up to this point, we have seen what traditional, semi-empirical and algebraic, models of

complex flow phenomena can offer to the practical task of calculating cyclones. Let’s see what the

modern computational artillery can bring in.

Flow in the inner space of a cyclone separator possesses several characteristics that makes

its numerical simulation difficult. Obviously, intense swirl and shear and confined and unstable

44

flow structures won’t be easy to duplicate. But, to be more concrete, the main issue here is rather

the necessity of a model of turbulence that simultaneously accounts for

1. High curvature of the average streamlines.

2. High swirl intensity and radial shear.

3. Adverse pressure gradients and recirculation zones.

These three features make turbulence inside cyclones highly anisotropic, which means that

most models based on first order turbulence closure, including two-equation models (i.e., the

k−ε model and its variations), are very weak at capturing the real flow.

As we will see, several attempts were made to overcome this limitation. The outcome has

been that at least second order closure, i.e., a Reynolds Stresses Transport Model (RSTM) is needed

to capture anisotropy and achieve a realistic simulation of cyclone flow [83], [84]. Both algebraic

and differential RSTMs have been employed. The algebraic RSTMs made a great difference with

respect to simpler models, but it became clear very early that a differential RSTM was always pref-

erable – in case the extra cost of the calculation were affordable [79]–[81]. With the computer capa-

bilities of nowadays, even the difference between a basic and an advanced differential RSTM can

be of relevance in this case [83].

There are also purely numerical issues in the hydrodynamic simulation of cyclones. Specifi-

cally, the discretization of advection terms cannot be the usual, first-order scheme, since this brings

along excessive numerical diffusion (for any kind of discretization geometry of reasonable size)

and prevents an adequate simulation of flows possessing the features mentioned above. Along

with an adequate turbulence model, higher accuracy discretization schemes, at least second order,

are needed.

At a first sight, the hypothesis of an axisymmetric swirling flow can be advantageous. It is,

of course, in terms of computational time and cost. However, such a computation cannot duplicate

the features derived from the entrance, inherently asymmetric, and, most importantly, fully 3-D

45

effects, such as secondary eddies and flows, and the PVC. This is especially important at the lower

part of the cyclone and in the vortex finder.

These phenomena arise because of a transient behavior: cyclone flow turns asymmetric

mostly because it turns unsteady. Accordingly, although much work has been done on steady CFD

simulation of cyclones, one must face the fact that a realistic flow field can only be duplicated by

solving a time-dependent, transient problem. In other words, it is almost a certitude that early CFD

modelization of cyclones produced steady-state solutions only because the limited spatial resolu-

tion available at the time, which resulted in artificial damping of instabilities. In modern times, it

has been recognized that the problem really demands an adequately space- and time- resolved

numerical calculation.

Computational cost grows even further; the fact that the unsteadiness is quasi-periodic is

hardly of any help. To accomplish the task, there are however several degrees of complexity:

1. Simulation of the Unsteady or Transient Reynolds-Averaged Navier-Stokes equations

(TRANS or URANS).

2. Large Eddy Simulation (LES).

3. Direct Numerical Simulation (DNS).

The first consist in simply retaining the transient term in the RANS equations, thus solving

a time-dependent problem instead of a stationary one. Turbulence closure can be accomplished by

first or second order methods; roughly speaking, the observations above still hold, so that a RSTM

is preferable for swirling flows, although this can be arguable (see later). In the second family of

methods, filtered, although exact, conservation equations are solved, so that large scales (“eddies”)

are exactly duplicated. Filtered-out details are represented by means of a standard turbulence

model; this is called a Subgrid Scale Model (SGS). Most commonly, a isotropic eddy viscosity

method suffices as a SGS. Finally, DNS is the numerical solution of all the spatial and temporal

scales of the time-dependent Navier-Stokes equations, thus attempting to duplicate all the detail of

the turbulent flow.

46

This ladder of complexity obviously implies a gradation of realism and computational cost.

Direct simulation of turbulence is presently feasible only at limited sizes or for model problems,

and low Reynolds numbers. Flows in industrial equipment are, and will continue to be, out of the

scope of DNS. LES is relatively affordable to duplicate great structures in external flows, but its

computational cost grows as the wall is approached, since the “large” scales gets progressively

smaller there. LES for full-size equipment is presently possible, but still costly. URANS methods

are comparatively far less expensive; in fact, most commercial CFD solvers usually handle a tran-

sient, taking as convergence criterion the cancellation of time derivatives. Only the extra memory

and the size and number of time steps needed to attain fully developed periodicities make the dif-

ference between solving RANS or URANS.

In an attempt to take advantage of the LES concept at a lower cost, there are also methods

that can be qualified as a mixture of LES and RANS. This is somewhat in exchange for a minor loss

of accuracy, although some RANS abilities, i.e., the treatment of the boundary layer, can be very

valuable in a LES context. Very Large Eddy Simulation (VLES) consists in an advanced URANS

scheme for the SGS, allowing less space resolution for the resolved scales. Detached Eddy Simula-

tion (DES) is essentially a RANS solver that changes to subgrid mode when the computed turbu-

lent length scale exceeds the grid side; or put in other way, it is a LES that switches to RANS near

the wall. The economy of all these methods is still under test; on the other hand, and contrary to

pure URANS or pure LES, they are not standard, easily-implemented methods.

As for the realism of the simulation, the ladder goes naturally the other way. There is a ba-

sic problem with URANS: it assumes two distant ranges of time scales, those pertaining to the tur-

bulent fluctuations and to the unsteady structures of the main flow. The question of exactly which

flows meet such a condition is as diffuse as the condition itself, so that it has always been a source

of confusion; for instance, literature on the simulation of combustion inside the cylinders of IC re-

ciprocating engines is a good example. In the case of cyclones, experimental observations and nu-

merical calculations readily showed that there is not a clear-cut of frequencies in this kind of flow.

47

Therefore, URANS methods lack a sound theoretical basis [87], and the preferred method —DNS

discarded— should be LES [86].

However, although the superiority of LES has been established against URANS, the specific

models used in the later were limited to two-equation models. The simulations were able in any

case to show up periodicities and coherent structures but, as can be expected, these were some-

what schematic, not completely duplicating intermediate details of the flow. In contrast, when

powerful, differential RSTMs are brought into play, differences between LES and URANS dimin-

ish, and it seems that the second is outperforming its very theoretical basis [88].

Presently, URANS with advanced RSTM appears as an attractive option, offering realism at

a much lower cost than LES [85]. This seems to be true not only for turbulent flows with large co-

herent structures [89], but also for more intricate cases such as flow in cyclone separators. This is,

however, still a point for research and discussion; for instance, there have been URANS simula-

tions of swirling flows in which a k-ε model was deemed more adequate than second-order closure

at capturing the instabilities [82].

In any case, it should not be forgotten that the fast growing capabilities of modern com-

puters will surely favor complexity, which in this case is almost a pledge of LES or intermediate

methods.

6.1. CFD studies on single-phase cyclone flow. The first CFD simulation of cyclone flow was performed about twenty years ago [90]-[92],

using the finite-element method. It was the first to detect that the standard k−ε turbulence model

was not able to accurately simulate this kind of flows. Using a mixed algebraic-differential, sta-

tionary RSTM, the authors found instead good agreement between experimental data and simula-

tion of overall pressure drop (both in absolute value and tendencies), and they could correctly du-

plicate the radial velocity profile. At the same time, advanced k−ε schemes, that took partial ac-

48

count of the turbulence anisotropy, were employed for simulating hydrocyclone flow [93]-[95],

arriving at acceptable results.

In one of the first comparative studies of several turbulence models, the relative perform-

ance of the k−ε model, the differential RSTM known as the LRRG model [83], and a variation of

the k−ε model based on ReNormalization Group (RNG) theory was evaluated [96]. Simulations

were also compared with velocity measurements carried out by means of Laser-Doppler Anem-

ometry (LDA). Tests were performed with three different vortex finder diameters, which produced

three different swirl numbers. For all runs, the k−ε model predicted solid-body rotation, a logical

result given its foundations, albeit contrary to experiment. The RNG k−ε model showed great im-

provement over this, but as expected, the Reynolds stresses model exhibited the best behavior.

Figure 18 pictures the results for tangential velocity. The same tendency was found when compar-

ing the same models when simulating an exit device called postcyclone [97].

Among all RSTMs available, it is still not clear which is the most suitable for a specific ap-

plication. For instance, another comparative study was made [98] including two differential RSTM

implementations, those known as LRR and SSG (see nomenclature), as well as the standard k−ε

and a k−ε modified to account for the streamline curvature (k−ε cc). Although the pressure distri-

bution showed acceptable agreement for all models, the flow field was very sensitive, and the

typical Rankine profile could only be obtained by means of RSTMs. Figure 19 gives a clear illustra-

tion. Only small differences are observed between LRR and SSG models, especially at the lower

part of the cyclone. At this position, the SSG model produces more acceptable results, whereas the

LRR method underpredicts tangential velocity near the cyclone center.

The high computational costs associated to Reynolds stresses models justified great interest

in alternate methods that account for the turbulence anisotropy without excessive computational

effort. Aside from the RNG k−ε and the k−ε cc, other hybrid models based on two-equation ideas,

such as that of [99] have been used to obtain more acceptable results

49

In later years, the need of using unsteady solvers and second-order discretization was es-

tablished; some key papers on this issue are Refs. [100]-[102]. For instance, in [101] a URANS

scheme was used to detect that the PVC structure develops thorough the cyclone length, is de-

stroyed at a certain height, and finally stabilizes itself in the form of a quasi-periodic phenomenon.

The study reported in [102] used a stationary RSTM simulation as an initial condition to trigger a

LES, to finally capture the PVC phenomenon with great realism. It was also established that the

flow was dominated by the greater scales, since the influence of the specific SGS used in the LES

solver was minor.

To provide some illustrations to the simulation of PVC phenomena, lets quote recent re-

search on a pilot PFBC cyclone, equipped with a long dipleg [12]. Single-phase gas flow was simu-

lated by three methods: k−ε, RNG k−ε and the SSG implementation of a RSTM. Predictions were

compared with hot-film anemometry and pressure measurements taken at the dipleg. The study

discovered that, under normal operating conditions, the vortex enters well within the dipleg, so

that the flow possess there many characteristics of the flow in a cyclone body [15], [16]. In order to

simulate it, second order discretization and a differential RSTM were needed. When the latter was

run, however, it did not converge to a stationary solution. With an adequate time step and retain-

ing an adequate number of time iterations, quasi-periodicities were discovered and the simulation

transformed itself in a URANS implementation.

Results showed a reasonable agreement in pressure drop and tangential velocity up to the

lower part of the dipleg. Velocity data suffered however from the fact that experimental measure-

ments were taken as pure averages, i.e., without acknowledging the fact that large oscillations will

show up at fixed positions as a consequence of the PVC. Figure 20 shows the simulated, phase-

averaged, tangential velocity at the cylindrical part of the cyclone (it does not vary much axially,

even at the conical part), compared with several algebraic models. The scheme resulting from the

combination Meissner-Reydon-Ogawa shows the best agreement.

50

Special phenomena associated with cyclone flow were also detected for the PFBC cyclone

dipleg, both experimentally and numerically. The position of the vortex end was detected by pres-

sure measurements (see also Ref. [16]) and by the numerical simulation. The adopted URANS

scheme seemed to capture the PVC in a similar fashion to other unsteady calculations [101]-[104],

some of them being also URANS models but including LES as well. Figure 21 shows the evolution

of the vortex core in a cross-sectional plane of the dipleg. It describes a nearly circular path around

the geometrical center, rotating in the same direction as the flow. To complete the picture, Fig. 22

shows the 3-D surface of zero axial velocity inside the cyclone and dipleg, which separates the

upward and downward vortex flows. The surface is colored in proportion to the velocity modulus.

In very apparent agreement with the stroboscopic experiments of Hoffmann [56], the vortex end,

identified as a 90o twist of the structure, just separating the zones of high and low velocity, is seen

to adhere to the wall of the dipleg and rotate on it. Observations of a ring of dust in the same ex-

perimental rig, coherent with pressure measurements, confirm this [12], [16]. On the other hand,

the “precession” is seen in this way more as a helicoidal movement than just a simple wavering of

the double vortex structure.

The latest CFD studies [103]-[104] continue to show the capacity of CFD models for predict-

ing interesting aspects of centrifugal separators. Further work using RSTMs is reported in [105],

[106]. LES simulation of swirling and cyclone flows is presently becoming a new standard, based

as already noted on increased computer capabilities. For example,. Derksen & Akker at Delft Uni-

versity [77] were among the first to simulate the PVC phenomenon by this method, but only at-

tempting to model small scale cyclones at a moderate inlet Reynolds number of 14 000. Subsequent

investigations with the same numerical method have been able to cope with industrial equipment

at Rec = 280 000 [107]. The capabilities and sensibilities of the particular technique (LES with stan-

dard SGS and a lattice-Boltzmann discretization) in simulating confined turbulent vortex flows

have been further reported in [108].

51

It is not easy to forecast the future of this line of research. On the one hand, LES methods

seem to offer a completely realistic simulation barring some minor details. However, the relative

competence of URANS with an advanced RSTM has not been established yet, and it should be,

given the very relevant cost factor involved. On the other hand, only relatively stable and ordered

flows have been simulated. LES and URANS methods must be tested for conditions of a highly

incoherent and variable PVC, to ascertain their usefulness under such conditions. Obviously, this

might demand also empirical input. In any case, a full perspective must take into account two-

phase flow; we address the question in Sect. 6.2 below.

6.1.1 Non-intrusive velocity measurements in cyclone separators.

Finally, it is worth noting that modern CFD studies have relied largely on advanced, non-

intrusive measurements of velocity patterns inside the cyclone, as a source of detailed validation

data. Although we cannot intend here a full account of the question, a brief review is in order.

Early measurements of velocity profiles inside cyclones were undertaken by means of Pitot tubes–

see for instance [28], which quickly began to compete with hot-wire (or hot-film) sensors. Both

kinds of instruments have been in use until recent times, the second having the advantages of a

smaller size and the possibility of very good time resolution. In both, however, determination of

the direction of velocity can be an issue and there is of course the essential problem of an intrusive

measurement.

The sensor head and its support may indeed disturb the flow and introduce a lingering and

mostly unknown uncertainty in the magnitude and profile of the measured velocity. This fact has

been long recognized by cyclone experimenters, as attested by reports of induced asymmetry [55],

attenuation of the swirl [104] and inconsistent readings near the centerline [12]. In the context of a

highly unstable and easily disturbed flow, the necessity of non-intrusive measurements soon be-

came evident.

52

As we have already mentioned, these have been accomplished mostly by laser doppler

anemometry, starting with the classical investigations at Delft [96] and continuing with many

other works, most notably [98], [104], [108]. LDA techniques are used in cyclones with apparent

success, not only for the determination of time-averaged profiles, but also resolving turbulent

quantities and detecting periodic fluctuations (PVC). The only drawback is that the tracer particles

may be segregated inside the volume, leaving a void in the immediate vicinity of the centerline

where the time resolution can be limited [96] or even no measurement is possible at all [104].

Recent developments in Particle Image Velocimetry (PIV) have qualified this technique as a

competent alternative to LDA, with the added capability of a quantitative appreciation of global

aspects of the flow pattern, instead of single-point velocity values. (The problem of tracer segrega-

tion is however pretty much the same.) Actually, the reports of its use in swirling flows date also

from the end of the 1990s. In particular, the group at Cardiff University has been using both LDA

and PIV in cyclonic combustors and separators to measure velocity and detect coherent structures

[67], [71], [110]. Their method for resolving the PVC consists in phase-locking the PIV measuring

system with the aid of fast pressure sensors. More recent references related to this technique in

cyclones are [112] and [113], having been the first studies to introduce stereoscopic PIV in this con-

text.

A perennial field in cyclone experimentation is by far and large flow visualization. This has

been accomplished by diverse methods, ranging from the simple observation of the separated par-

ticle streaks on Perspex model walls to the use of streamers and tracers. A couple of recent and

representative examples are the already-mentioned works on PVC characterization by strobo-

scopic lighting [56] and by neutrally–buoyant tracers combined with pressure measurements of

high time-resolution [109].

53

6.2. Numerical computation of two-phase flow in cyclones.

Overwhelmed by the complexity of the flow of gas in cyclones, we have perhaps forgotten

that it is the combined flow of gas and particles what really matters in the operation of such an

apparatus. Compared to single-phase, gas flow simulations, simulations of two-phase flows, and

its immediate results concerning cut sizes and grade-efficiency, are relatively scarce in the litera-

ture about cyclones. Let’s give a brief account of the state-of-the-art.

Dispersed two-phase flows can be classified according to the importance of the interaction

mechanisms [114], [117]. Figure 23 shows an elementary and practical sketch. Generally, two dif-

ferent regimes are distinguished, depending on the existence of mutual, significant interaction be-

tween particles: dilute and dense two-phase flow. The approximate borderline is a volume fraction

αp ≈ 10–3, which translates generically as an interparticle spacing L/dp ≈ 8. Within the dilute regime,

all the influence of particles on the gas can be neglected for αp < 10–6 (L/dp > 80), which is known as

“one-way coupling”; for higher volume fractions, it needs to be accounted for (“two way cou-

pling”). Aside from obvious issues of volume and continuity, the generic flow feature most af-

fected is turbulence. In our context, the effect is normally an attenuation, but an enhancement may

also result under some circumstances. In the dense regime (αp > 10–3, L/dp < 8), interparticle inter-

actions become of importance, both physical collisions and indirect influence through the nearby

flow field. The collisions can lead to coalescence and break-up, which must be considered too. This

regime is called frequently “four-way coupling”.

Within this framework, numerical models have developed in two parallel paths according

to the manner in which the dispersed phase is treated, either by a Lagrangian or by an equivalent

Eulerian formulation. In the Lagrangian model, particle trajectories are obtained by integrating the

particle equation of motion. In the Eulerian approach, conservation equations are written for the

mass, momentum and energy of the particle cloud as a continuum, and integrated to predict vol-

ume-averaged properties thorough the field.

54

The suitable choice is often problem-specific. Generally speaking, Lagrangian methods are

more suited to dilute flows, free of numerical diffusion, less influenced by other errors and more

stable for large particle velocity gradients. Also, the treatment of realistic poly-dispersed particle

systems is straightforward. However, for highly loaded flows, limitations related to computer

storage, calculation times and convergence arise [115]. In those cases, the Eulerian approach be-

comes more adequate. It seems also the right framework for modeling particles’ mutual interaction

[116], although Lagrangian methods have been also applied [117], [118].

6.2.1 Eulerian-Lagrangian models of cyclone operation.

For dilute, one-way coupled flows, a Lagrangian tracking of particles is adequate, which

consists in integrating the equation of motion of a particle given by the general form of Eq. (51). In

the context of cyclone CFD, gas velocities are not longer the result of an algebraic model but the

computed 3D velocity field; the calculation is equivalent to a very refined time-of-flight model. A

Lagrangian tracking is normally decoupled from the gas flow calculation; it can be undertaken as a

post-process computation by a CFD solver, i.e., as a calculation using previous gas velocity results

from a clean, gas-only run.

Equation (51) neglects however a basic fact: only particles of large size will behave as exclu-

sively influenced by the time-averaged gas flow. Very small particles will tend to fluctuate follow-

ing turbulent fluctuations of the gas velocity, and there will be a complete range of intermediate

behaviors between these two extremes. This phenomenon is known as turbulent diffusion. There

are several approaches to simulate it by including additional terms in a time-averaged equation

such as Eq. (51). The most popular nowadays amount to assume a prefixed probability distribution

of velocity, solve the equation of motion for discrete velocities (and particle sizes) and make an

average of the forces. These are referred to as “stochastic” models of particle turbulent diffusion

[119]–[121].

55

Lagrangian calculations following these principles have been profuse in cyclone literature

[90], [96], [105], [122]–[124]. Computational simulations frequently obtained larger cut-sizes than

measured for different flow conditions, which is clearly a signal of the complexity of the flow and

the approximate nature of the models. To the intricacy of the single-phase gaseous flow, it is in-

deed very easy to add-up difficulties arising from the dispersed phase: For instance, with inde-

pendence of inlet solids loading, the accumulation of particles in specific regions (i.e., cyclone

walls as a result of the collection process) leads to high local values of particle concentration, which

violate the basic assumption of a simple Lagrangian tracking, and are not easily modeled, even

considering their effect as a boundary condition.

In recent years, as we have seen, LES is increasingly being applied to simulate time-

resolved gas flow in cyclones. Of course, the resulting velocity field has also been used to calculate

particle flow by means of a Lagrangian tracking [107], [125]. As an example of such a modern cal-

culation, Fig. 24 shows snapshots of particle concentration at five instants of time inside a high-

efficiency, lightly-loaded cyclone [107]. The leftmost drawing represents the time at which the

feeding of particles is stopped; subsequent evolution is shown at equally-spaced intervals to the

right. As expected, particles tend to accumulate at the bottom bin (note also the spiral streaks so

familiar in Perspex experiments), but there is also a persistent, high local concentration at the roof

of the cyclone. The fate of these particles is still undecided, which means that the collection process

can be quite a lengthy one.

This fact leads to an excessively costly method to predict cyclone performance; in this re-

spect, it should be noted that the enormous amount of time-dependent data generated by LES pre-

cludes post-processing and requires at once repeated tracking of a large amount of identical parti-

cles. (The method is neither free of the necessity of a stochastic model of particle turbulent diffu-

sion due to the unresolved part of the velocity.) Accordingly, several alternatives are proposed,

based on average, frozen and periodic LES-velocity fields. Although no definitive conclusions are

drawn, the periodic approximation seems to be the best, but also the more expensive, logically. In

56

any case, calculated efficiencies compare much better with experiment than former, classical La-

grangian calculations [107].

For interacting flows, the two-way transfer of momentum between particulate and fluid

phases is modeled by particle–source–in Cell (PSI-Cell) methods, originally developed in [126].

The flow field is calculated first without particle-phase source terms until a converged solution is

achieved. Then, a large number of “parcels” (i.e. discrete particles representing large groups with

the same properties) are tracked through the flow field, typically a minimum of 10 000-20 000.

Source terms are thus obtained for a second Eulerian calculation of the gas flow. The procedure is

repeated iteratively until convergence; to avoid strong fluctuations, under-relaxation is frequently

used [127]. Special models of turbulence accounting for the effect of particles are needed, at least

for steady RANS– see a good review in [128]. The main drawback of this type of methods is obvi-

ously the high computational effort, if large numbers of particles are needed to represent the dis-

persed phase.

This seems to be the case of cyclone separators; in contrast with simple Lagrangian track-

ings, the use of PSI-Cells methods is still at its very infancy. Actually, we can only signal a relevant

reference at the time of writing (December 2006): [129], which is a logical continuation of the work

of the same authors we have just reviewed.

Curiously enough, the intermediate step of a stationary two-phase calculation seems to be

absent, and the work deals directly with LES-resolved, time-dependent flow. In any case, the con-

tribution is worth a good look. In it, the previously-reported LES+Lagrangian tracking [107] is

modified to assemble a PSI-Cell calculation, attempting to simulate mass-loading effects for mod-

erate, but still high, values of Csi = 0.05–0.1, that fall into the regime of dilute flow but involve

forcibly two-way coupling. To this end, a deemed-modest number of ten million parcels is tracked,

assumed representative of a much larger number of real particles. The simulation uses a standard

SGS, with no proviso for the particles effect on unresolved scales, which is considered, also provi-

sionally, adequate.

57

One outcome of the investigation is that such a computational load is indeed enormous and

converged results for gas and particle fields (in the quasi-periodic sense) cannot be reported yet, so

that conclusions are only tentative. However, while we wait for the end of the calculation, the

main thesis of the paper is very interesting.

This is that the dependence of separation efficiency on inlet solid loading is not really

caused by a singular separation process as postulated classically by Muschelknautz, but the result

of a competition between two effects of a high particle concentration: attenuation of swirl, which

lowers efficiency due to a lowered centrifugal force, and attenuation of turbulence, which aug-

ments efficiency through a decreased turbulent diffusion of particles. The second effect having

received very little attention in the traditional cyclone literature, this is of course quite interesting.

If definitive results support the hypothesis (as the provisional ones seem to do), Muschelknautz’s

views will become, at least for moderate solids loadings, another very useful, but purely imagi-

nary, theory on cyclone operation.

6.2.2 Eulerian-Eulerian models of cyclone operation.

The Eulerian–Eulerian approach treats both the solid particles and the fluid as interpene-

trating continua, each governed by conservation laws whose general structure is known albeit spe-

cific terms that must be modeled. These are those representing the interaction between phases, i.e.,

the result of the forces the gas imposes on the particles (average draft and turbulent diffusion, act-

ing on a continuum) on the one side, and the effect in the gas of the presence of solid particles on

the other [128]. A generic inconvenient of Eulerian models is that they require multiple levels of

modeling (e.g., boundary conditions for the dispersed phase, mutual interaction between particles,

interactions with the wall). On the other hand, complex phenomena such as agglomeration and

break–up can be taken into account by using a population balance model [130].

Eulerian–Eulerian schemes have been used in many practical multiphase flow simulations,

for instance, and typically, in fluidized-bed applications [131], where they are specially suited.

58

However, Eulerian models cannot treat different particle sizes directly, but considering every size

class as a distinct phase, which is evidently a serious drawback for practical problems. As a conse-

quence, simplified Eulerian multiphase approaches have been developed, mostly based on an al-

gebraic velocity slip between phases. This is adequate under conditions of a strong coupling, with

particle relaxation times much shorter that the characteristic time-scale of the flow, which is fre-

quently reasonable. At once, it represents a significant computational saving, allowing the sole

integration of the mixture equations, with the only additions of the volume fractions and slip rela-

tionships [132].

This kind of approaches is referred to as Mixture Models in some commercial CFD codes.

Although specifically signaled as apt for modeling cyclone separators, reports in the literature are

very scarce. Simplified Eulerian-Eulerian models have been used recently in the related field of

hydro- and dense-medium cyclone separators [133]–[137], with promising, but not completely sat-

isfactory, results. Aside for that, as for gas cyclones, these authors only know of two investigations.

In [138], a mixed k-ε/algebraic RSTM, stationary gas flow model is used along with a simple Eule-

rian model of a diluted, monodispersed particulate phase. The results, clearly exploratory, predict

several effects reasonably, most notably the reduction of swirl leading to a reduction of pressure

drop.

The very recent work reported in [139] deals with a simulation of two-phase cyclone flow

in the dense regime, with solids loadings up to 2 kg/m3N. The Fluent’s implementation of the

multiphase algebraic slip scheme is used (details on the gas flow model are not given), with no

account of interparticle effects. Results on collection efficiency, as compared with dedicated ex-

periments, are not conclusive or even deviant. However, interestingly, the attenuation of both

swirl and turbulence with mass loading is clearly noticed.

59

6. Concluding remarks and perspectives. Cyclones are robust devices, widely used in the chemical and process industries, as well as

for heavy-duty hot gas-cleaning service in several applications of combustion of solid particles.

Much ingenuity has been devoted to the prediction of their performance parameters, collection

efficiency and pressure drop, which has required a previous knowledge of their basic flow pat-

terns.

The classical view of a steady flow, that can be calculated as a clean gas, single-phase flow

corrected by mass loading effects has been relatively successful. Many algebraic models and semi-

empirical formulas have been developed throughout the years; among them, the more complex

models of Muschelknautz [4] for pressure drop, and Trefz and Muschelknautz [5] for collection

efficiency give most accurate results as compared to experimental data.

However, the complexity of the flow in cyclones is due to instability, as modern experimen-

tal and numerical techniques have demonstrated. Specifically, the double-vortex structure that

makes up the basic flow is essentially unstable, and develops a phenomenon of quasi-periodic os-

cillations known as a “precessing vortex core”. This instability can be a threat to the cyclone objec-

tive of solids’ separation, but also affects a basic flow feature, formerly thought of as stationary,

called the cyclone natural length. This is in turn a major influence in cyclone velocity patterns,

pressure drop and collection efficiency.

As a result, it can be said that the flow in cyclones is not yet completely understood. Given

the kind of phenomena involved, classic, algebraic models are positively not well suited to a

proper account; modern progress should evidently resort to CFD calculations. However, the very

nature of the problem makes the task difficult. Unusual high precision in the numerical discretiza-

tion and unsteady simulation methods (LES or URANS) are needed to possibly capture the whole

complexity of the unsteady flow of gas inside a cyclone. The simplest of these methods is computa-

tionally very costly, much more than the usual steady-state simulation of industrial equipment.

Perhaps this has prevented up to now a closed account of several related design aspects, such as a

60

precise methodology to reasonably estimate the cyclone natural length, and thus the advisable

geometric length, the circumstances for and the effects of a loss of coherence of the vortex and the

ensuing chaotic flow patterns, the effect of swirl-stabilization devices,…. More systematic research

is obviously needed in this direction.

On the other hand, more work should be devoted to ascertain if, and under what circum-

stances, a URANS solver with advanced turbulence models (differential RSTMs) can supplant a

more rigorous, but more costly, LES or its variants.

Finally, two-phase flow simulation of cyclones is certainly at its very beginning and de-

mands several advances. On the one hand, factors like interparticle phenomena and conditions at

the wall have not received yet the attention they surely deserve. But most importantly, modern

two-phase flow simulations of cyclones have turned out to be too costly, due to the necessity of

reproducing unsteadiness of the gas flow and combine it with the simulation of a poly-disperse

particulate system. Even for dilute flow, even for one-way coupling, present computer capabilities

are insufficient, at best making the CFD calculation a costly expedient.

Simplified schemes are obviously needed. To the possibilities explored in the literature of

using “abridged” LES data to this end, we may add the necessity of directly testing URANS meth-

ods, not yet realized. A development of the PSI-Cell techniques is the sense of coping more effi-

ciently with time-variable problems seems also, obviously, very convenient. As for the calculation

of highly-loaded cyclones, the experience to date is so meager that we can only hope for simplified,

algebraic-slip models that can compete in simulating dilute flows and then can be applied to the

dense regime.

In any case, the incipient experience with two-phase flow simulations seems to be in the

verge of another revolution, in the sense that classical, well-established theories on cyclone particle

flow might be compromised. This would be of course stimulating, and surely a great step forward

in our understanding of cyclone operation.

61

Acknowledgements Authors are grateful to Dr. José Velilla, now on General Electric (USA), who spent several

years at CIRCE facilities (experimental and numerical) untangling cyclone flow: Many thanks. A

part of our research on cyclones was financed by the Escatrón pressurized fluidized bed combus-

tion power station (contracts No. 40.702/94, 40.815/95, 40.590/96, 41.090/97, 41.114/97 and

40.700/98), then owned by Endesa Generación SA (Spain). Authors also wish to thank Prof. Alex

C. Hofmann for his useful comments and suggestions.

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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,

(3) inlet duct, (4) exit duct, (5) vortex finder [14], [12].

78

Fig. 2. Main inlet arrangements: (a) tangential, (b) scroll, (c) helicoidal, (d) axial [6].

Fig. 3. Scale drawings of the cyclone designs of Table 1 [8].

79

(a) (b)

(c)

Fig. 4. Measured profile of velocity components in a reverse-flow cyclone: (a) tangential, (b) radial, (c) axial [10].

80

Fig. 5. Sketch of cyclone velocity profiles: 1- radial, 2- axial, 3- tangential [18].

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].

Name Dc De S H h A b B

Muschelknautz E 680 170 311 934 173 173 58 228

Muschelknautz D 357 119 318 863 262 187 54 195

Storch4 260 117 176 1616 909 260 38 91

Storch3 192 107 200 821 462 167 60 92

Storch 2 225 108 239 1097 464 188 53 84

Storch 1 365 123 142 1943 548 100 100 64

Tengbergen C 337 112 145 930 187 100 100 112

Tengbergen B 210 112 224 604 324 179 56 112

Tengbergen A 277 112 157 647 180 135 74 202

TSN-11 348 136 242 959 219 184 54 154

TSN-15 266 158 350 1124 589 166 60 119

Stairmand High Efficiency 316 158 158 1265 474 158 63 119

Stairmand High Flow 190 141 165 755 283 141 71 71

VanTongeren AC 325 100 325 1231 436 149 67 130

Vibco 286 111 124 720 228 111 90 66

Lapple GP 283 141 177 1131 566 141 71 71

93

Table 2. Algebraic models of the velocity field in cyclones.

Model Equation Remarks

Alexander [9] n

ctwt r

rvv ⎟⎠⎞

⎜⎝⎛= (2)

n

ec

i

i

tw

DDA

vv

⎟⎟⎠

⎞⎜⎜⎝

⎛= 15.2 (3)

( )3.0

14.0

28367.011 ⎟

⎠⎞

⎜⎝⎛−−=

TDn c (4)

Experiments with scroll and tangential inlets.

Air and combustion gases, up to 1100 ºC.

T in K and Dc in m.

Barth [29] ctw

ii

Dvrv2

=α (5)

Vvrhrrvv

twc

ectwet

&πλ*

1 +

⋅= (6)

( ) 2bDr ci += (7) BDSHh e ≤−=* (8a)

( )( ) BDShBD

DDhHh ec

ec >−+−

−−=* (8b)

Scroll inlets 1≈α

For tangential inlets, values of α are given by graphs

λ = λg ≈ 0.02

Muschelknautz [4]

cDb2.11 −=α for tangential inlets (9) abriπλα 31 += for scroll inlets** (10)

( )( ) 131

121

>+=

<+=

sisig

sisig

CC

CC

λλ

λλ (11a-b)

Eq. (9) is valid for 8.19.0 2 <π< irab

Csi in kg solids/kg gas.

005.0≈gλ for high Reynolds numbers

Meissner & Loffler [17]

1*889.0204.0

−

⎥⎦

⎤⎢⎣

⎡+

−=

ci

twr

bv

v (12)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+= 5.025.01

5.0**

*b

twzg

zgb

twv

vhhv

v λ

λ (13)

24 cb DVv π&= (14) ( )[ ]

c

c

cz r

hrbarah +⎥⎦

⎤⎢⎣⎡ −

−−= 1

21cos2*

ππ (15)

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+

=

cc

twt

rr

rr

vv11 χ

(16)

( )⎥⎦⎤

⎢⎣

⎡Φ

+=c

kx

b

twvv

sinλλχ (17)

Radial velocity

( ) ( )( )SHrVrvrv

eercr −

==π2

0& (18a-b)

Axial velocity ( )

( )( )SHrrzHVv

ecz −−

−= 220 π

& (19)

χ accounts for the angular mo-mentum exchange between wall and gas

0075.00065.0,, ÷≈kxg λλλ for smooth walls at room tem-peratures

Reydon & Gau-vin [19]

Free-vortex 72.079.14 −= rvv it (20)

Forced vortex [ ]rvrvv iit

17.0153.0exp35.1

−

−= (21)

r in cm

** This equation doesn’t always give consistent values. [4]

94

Ogawa [20],[21] Free vortex ( )n

cin

t brvKrv −= 0 (22) Forced vortex

( )rKrvt Λ−= 1 (23)

a

maxt

rv2K = (24a)

nnrt +

+=Λ

21 (24b)

5.0=Λar (24c) VDK e&65.25.66 −= (24d)

Mean axial velocity

( )220tc

z rrVv

−=

π

& (25)

Radial and axial velocities in the free vortex

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

rr

rr

hr

vv c

c

c

z

r*

0 2

*0

1hz

vv

z

z −= (26 a-b)

Radial and axial velocities in the forced-vortex

rhrrr

vv

t

tc

z

r*2

22

0 2−

−= ⎟⎠⎞

⎜⎝⎛ −

−= 1*2

22

0 hz

rrr

vv

t

tc

z

z (27 a-b)

K0, n = f (Re, geometry)

95

Table 3. Algebraic models of the “clean” pressure drop gξ in cyclones.

Model Equation Remarks

Shepherd & Lapple [28] 2

16e

g Dab

=ξ (35) Tangential inlet; ambient air condi-tions

Alexander [9] ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ −

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

n

e

cg

n

e

c

ecg D

Dfn

nDD

DDab

221162.4ξ (36)

( )

( ) ( )⎥⎦⎤

⎢⎣

⎡ +⎟⎠⎞

⎜⎝⎛ −

−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

−⎟⎟⎠

⎞⎜⎜⎝

⎛ −−

=

nn

n

g

nn

nn

nnf

22

2

25.11122.0

1324

118.0

(37)

( )3.0

14.0

28367.011 ⎟

⎠⎞

⎜⎝⎛−−=

TDn c (4)

Experiments with scroll and tangential inlets.

Air and combustion gases, up to 1100 ºC

Barth [29] ( ) ( )ebe

g Dab ξξ

πλλξ +⎟⎟

⎠

⎞⎜⎜⎝

⎛==

2

2 4 (38)

loss in the cyclone body

( )⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−

=2

2

5.0

1ze

te

ete

zec

eb v

v

DSH

vvD

D

λ

ξ (39)

loss in the vortex finder

234

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

ze

te

ze

tee v

vvvKξ (40)

3.41 < K < 4.4

Muschelknautz & Kambrock [54]

( ) ( )eb

2

2e

g4D

abξ+ξ⎟

⎟⎠

⎞⎜⎜⎝

⎛

π=λ=λξ (41)

( ) 5.1tetw

gSb vv

2V9.0A ρ

λ=ξ&

(42)

2

ze

te34

ze

tee v

vvv32 ⎟

⎠

⎞⎜⎝

⎛+⎟⎠

⎞⎜⎝

⎛+=ξ (43)

Tangential and scroll inlets.

Flow field based on Barth´s model [29].

Ambient P,T condi-tions

λ=λg≈0.006.

AS is the total inner area of cyclone con-tributing to friction

Casal et al. [30] 33.23.112

2 +⎟⎟⎠

⎞⎜⎜⎝

⎛=

eg D

abξ (44) Comparative study of six correlations

96

Table 4. Comparison between measurement of ξg and predictions [31]. ξg Discrepancy

PFBC cyclone, measured [31] 11.6 -

Shepherd & Lapple [28], Eq. (35) 15.7 35.3 %

Alexander [9], Eq. (36) 13.6 17.2 %

Muschelknautz, [4] [54] , Eq. (41) 14.1 21.6 %

—, including vortex penetration in dipleg [31] 11.7 1 %

Casal et al. [30], Eq. (44) 14.2 22.4 %

Table 5. Formulae for the coefficient of solid loading effects sξ [31].

Reference Equation Remarks

Briggs [34] ( ) 5.00086.011

gsis C ρ

ξ+

= (45) Air at ambient conditions.

Multivane cyclone of 0.23 m

diameter.

Solids: rock dust 0-44 μm, Csi

range: 0.17-192 g/kg air.

Smolik [35] ( ) 6.002.01 gsis C ρξ −= (46) Experimental data from sev-

eral sources

Baskakov et al. [36] isis

s CC

67.01.311

7.0 ++

=ξ (47) Hot gases at ambient pressure

and 150-250ºC.

Helical inlet cyclone of 0.204 m

diameter.

Solids: alumina xm=40 μm., Csi

up to 0.7 kg/kg gas.

Muschelknautz [4] ( )g

ss ξ

λλξξ

== (48)

( )

8/55.0

125.0

−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+

+=

e

c

strs

gesi

gs

DDFrC

ερρ

η

λλ (49)

See table 3 for ξg.

97

Table 6. Formulae for the cyclone collection efficiency. Reference Equation

Lapple [41] cis

g

Nvb

xπρ

μ2

950 =

( )a

hHhNc2−+

= (53a-b)

Barth [29] 2tws

reeg50 v

vD9x

ρμ

= (54)

Leith & Licht. [42],

corrected by Clift

et al [44]

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛μρ

−−=η resc

i

g

s tDxv 22

9exp1 (55)

VVtres &= (56)

Table 7. Calculations of the natural vortex length [12] . Ln/Dc Remarks

Experiment 9.51 Stable vortex end on the dipleg wall.

CFD 9.85 +3.5 % off from experiment

Alexander (1949) 1.56 Predicts vortex end on cyclone cone.

Bryant et al. (1983) 2.24 Ibid.

Zhongli et al. (1991) 14.62 Vortex end out of the dipleg