MODELING THE GAS AND PARTICLE FLOW INSIDE CYCLONE SEPARATORS Cristbal Corts* 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 natural length of the cyclone, that has led to the discovery of instability and secondary flows. The resort 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 twophase, steady and unsteady, aiming at delineating the state-of-the-art, present limitations and perspectives 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 1

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 Ai AS b B Ccr Csi dp D Dc De DLn f(x) fg inlet section height inlet area inner cyclone friction surface area inlet section width discharge duct diameter critical load inlet solids loading generic particle size diameter cyclone diameter vortex finder diameter diameter of the cyclone cone at the vortex end position particle size distribution parameter of Alexander pressure drop model, Eq. (39)

Frc Fre g H h h* K L Ln m p r ra re rc Rec rt S St Sw T

inlet Froude number, Frc = v i2 gDc

2 vortex finder Froude number, Fre = v e gDe

acceleration of gravity total cyclone height height of cylindrical section of cyclone height of the separation surface in the models of Barth [29] empirical constant [20] generic interparticle spacing in two-phase flow cyclone vortex length slope of the fractional efficiency curve, Eq. (52) Pressure Radius radius of maximum tangential velocity, in particular as defined in [20] radius of the vortex finder cyclone radius cyclone Reynolds number, Re c = Dc g v i g transition radius [20] height of the vortex finder cyclone Strouhal number, Eq. (65) swirl number, Eqs. (63), (64) Temperature volumetric gas flow rate
& mean axial velocity at cyclone body, v b = 4 V Dc2

& V
vb vi vr

inlet gas velocity radial gas velocity

vt

tangential gas velocity

vt max maximum tangential velocity [20] vte vtw vtw* vz vz0 tangential gas velocity at the inner vortex [29] velocity in the vicinity of the wall wall velocity just after inlet [17] axial gas velocity

& mean axial velocity, v z 0 = V / rc2 rt2 Axial gas velocity at vortex finder particle size in cyclones cut size of the cyclone particle mass median diameter

vze x x50 xm

Greek

coefficient from Meissner and Loffler [17] parameter [20] coefficient in Barth theory [29] particle volume fraction porosity of the strands, Eq. (49)

p str

(x) fractional collection efficiency g b g s c

total collection efficiency friction factor gas dynamic viscosity bulk density gas density solids density cyclone pressure coefficient, c = pc /( 1 2 ) g vi2

angle of the cyclone cone constant, Eq. (30) angular velocity, Eq. (29)

Subscripts b c g i r s t body of the cyclone cyclone gas inlet radial component solids tangential component

w wall z axial component

Abbreviations CFBC Circulating Fluidized Bed Combustion CFD DNS FCC LDA LES LRR Computational Fluid Dynamics Direct Numerical Simulation Fluid Catalytic Cracking Laser-Doppler Anemometry Large Eddy Simulation 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
5

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 SSG Subgrid Scale Model 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 maintenance. 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 component 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 operation. 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 understand 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
6

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. Nevertheless, 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 fundamental 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 theory that may affect the velocity distribution to a great extent, thus changing the model of the separation 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 prediction 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 experimental 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 exchange may suffer from inaccuracy when dealing with unsteady flow, and from probe deterioration 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 alternative 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.

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 separators 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 traditional steady-state models, or even time-dependent versions of them, are not sound from a fundamental 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, cyclone natural length, pressure drop and collection efficiency are discussed and compared with experimental data and between different models and calculations. The paper discusses all the peculiar 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 vortex, 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 exhaustively: the inverse flow cyclone, Fig. 1. In this device, the fluid enters tangentially into the cylindrical chamber with a high rotational component. The flow descends rotating near the wall, until a certain axial location where the

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 continues, 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 explain 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 helicoidally along the walls. The centrifugal force only acts upon the remaining dust, usually a small fraction of the total [4] [5]. As we will see later, this effect is explained essentially by inertia and interaction 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 180 of the cylindrical

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 efficiency (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 concentration 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 profiles, 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 concentration 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 cleangas and two-phase flow simulationsmostly 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 experimental and theoretical work on the subject. Figures 4 and 5 show classical measurements and explanations, 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 velocity and strong shear in the radial direction. As a first and simple approximation, the profile can be
10

described as a Rankine vortex, a combined free and forced vortex. The tangential velocity distribution vt in the radial direction is assumed to obey a law of the form:

vt r n = C

(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 approaches 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 vanish. 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 important 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 vortex/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
11

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 reported 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 position 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) fraction of gas used for particle transport. This is mainly absent in FCC cyclones, where the gravityassisted, intermittent flow of solids gives way to zones of dense-phase transport [13].

12

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 considerations on vortex flow, and evolved to incorporate more classes of phenomena and more sophisticated 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 reasonable 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 experimental 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 immediately applied as a first construct to predict collection efficiency. Similarly to Alexanders, this model considers as a geometric constant the ratio 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, 360 inlets and less than unity for tangential inlets, where a considerable acceleration obtains.

13

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
Dc De and height h* where all frictional losses are concentrated; the analysis leads to Eq. (6).

Losses are represented by a lumped wall friction coefficient g, empirically adjusted. The height h* can be naturally identified as the length of the vortex; Barth made a purely geometrical interpretation of this parameter, via Eqs. (8).
Muschelknautz [4]. Muschelknautz and co-workers have worked upon Barths ideas to de-

velop empirical models that combine in admirable measures both simplicity and realism. Concerning the velocity distribution, the table summarizes one of their earliest models. The essential concepts of the coefficient and the friction surface are maintained and perfected. Data on is correlated by means of analytical formulae, Eqs. (9) and (10), to replace the original graphs. In addition, the value of the gas friction coefficient g 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 separated at the entrance, this is to augment wall friction and thus weaken the vortex intensity. Muschelknautzs original expressions for the increase of g 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 v* and the inlet veloctw ity vi , Eq. (12), and a momentum balance to take wall friction into account, Eq. (13). Only that two values of the tangential velocity are considered: v* just at the inlet slot where acceleration occurs tw and the developed value vtw at the cyclone body. As indicated by the formulae, the momentum balance is different from Barths; it only refers to the reduction from v* to vtw as a result of flow tw

14

along the cyclone wall. Only slot or tangential inlets are handled by this model; scrolls are explicitly 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 tangential 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 g: 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 coefficients, 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 constants K0 and n that depend on Reynolds number and geometry. A law for the forced vortex region 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-

15

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 Ogawas 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 , is obtained by intersection of Eqs. (21) and (16), i.e., the forced vortex of Reydon & Gauvin and the free vortex of Meissner & Lffler. 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 Muschelknautzs, 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:

vt2 dp = r dr

(28)

16

It is worth to remember that this equation is exact for idealized, axially symmetric onedimensional 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 vt (r ) = r , being = const. the angular velocity, Eq. (28) is easily integrated to obtain

p(r ) = p0 +

1 2 r 2 2

(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 vt (r ) = /r with = const. , the integral results in p(r ) = p a + 1 2 1 1 2 2 r 2 a r

(30)

where ra is an arbitrary radius and pa = p (ra). Simply matching both velocity and pressure laws at r= ra, we get
= ra2 1 p a = p o + 2 ra2 2 and using Eq. (31) on Eq. (30), the outer pressure is p(r ) = p o + r2 1 2 ra2 2 a2 2 r

(31)

(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 pressure 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 picture of inner cyclone velocity patterns. Due to the cross-sectional pressure gradient, any rotated
17

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 contribute 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 recently 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 possibly much harder to rationalize. Other feature that can be explained now is the existence of W-shaped profiles of axial velocity. 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 downstream 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 phenomenon 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 visualization 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

18

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 cyclone 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 friction 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 correctly written mechanical energy balance may put forward, viscous dissipation results in a de1 crease of the quantity p + 2 v 2 , 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-

19

tinuity and similar cross-sectional areas. Accordingly, one can think of a total pressure inside the
1 cyclone made up of the static pressure plus the kinetic energy of the vortex, p + 2 vt2 .

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 vt2 , 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 pressure 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 component 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
20

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 crosssectional average that should make up an overall balance. On the other hand, the dynamic component
1 2

vt2 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 tendencies 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 instance, if we assume solid body rotation inside the vortex finder, Eq. (28) applies for 0 r re. Using it, it is easy to show that both static pressure at the wall p( re ) and the cross-sectional average of
1 1 the total pressure p(r ) + 2 2 r 2 = p0 + 2 r 2 attain a common value, p0 + 2 2 re2 . 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 viscous 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 recovery 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.
21

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 measurement downstream unequivocal. The idea is very old, as it is the dispute about the pressure recovery 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 theorybased formulae for pressure drop in cyclones. Taking for instance the approach of [15], we get, as a rather complete list of variables:

pc = f ( geometry , Frc , C si , Rec , s g ) 1 2 g vi2

(33)

Dimensionless pressure drop c (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 certain value, and usually negligible for the values found in practice [24]. Other two parameters, the Froude number Frc = v i2 / gDc and the solid to gas density ratio s /g, can be also eliminated, on the basis that their variation for a certain class of cyclone designs and operating conditions is usually 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 possible explanation of the structure of most pressure drop models for cyclones. In many of them, the

22

two effects are introduced as independent, multiplicative factors to compose an overall Euler number:

c = g s

(34)

where g 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. s is a correction factor that accounts for the presence of a high concentration of solids. Of course, the latter is an ad hoc correction, so that the factorization expressed by Eq. (34) is devoid of any fundamental significance. The most widely used correlations for the clean pressure loss coefficient g are summarized 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] patterns the general scheme of Barth. According to our analysis, some methods just assume that g 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 exponent 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 g 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 g is an actual clean pressure drop, i.e., measured without solids loading. All the correlations perform rather modestly, although Muschelnautzs is clearly the best. However, the experimental cyclone of [31] was a model of a PFBC unit equipped with a
23

long dipleg. Allowing the vortex end to penetrate the latter (which is actually an observed fact of the experiment), we were able to predict g 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 g, depend on vortex end position through Barths vortex length, Eq. (8), Fig. 7. The factor s 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 models for s. Most of the studies have taken the simplified approach of assuming s only a function of the inlet solids loading Csi the alternate variable Csig (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 s is universal, not dependent on geometry, or that the formula is restricted to geometrically similar apparatuses 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 coefficient, which is much more sound from a physical standpoint. The table shows the way of introducing 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 s 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

24

vortex penetration in dipleg. These results offer again a good agreement with measured data, although 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 significant fact. For low-to-medium inlet solid loadings, the pressure coefficient s deceases monotonically with solids concentration, in accordance with the physical explanation of the effect. Nevertheless, for very high solid loadings (such as in CFBC) the opposite has been detected. Baskakov et al. [36] found a minimum located in the sCsi 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 decrease 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. Lets 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 , 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 (x) that can be thought of as the fractional or grade25

efficiency of the cyclone for particles of size x. Reference [8] gives a mathematical definition of (x) in relationship with that is very enlightening. Here we limit ourselves to the reverse relationship, most obvious: if f(x) is the particle size distribution (PSD) at cyclone inlet,

= f ( x ) ( x )dx
0

(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 escape 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, (x) has the generic shape of a Sigma function between the limits (0) = 0 and () 1. Then, a simple characterization is made up of the so-called cut size x50, the size which is separated half the time,

(x50) = 0.5, and the slope of the grade-efficiency curve at that point. Fig. 11 is a scheme of the relevant 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 afterwards 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 arrangements of particle forces and velocities, far removed from the chaos associated with turbulent fluc-

26

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 arbitrary radial position applies:

x 3
6

dur x 3 v2 = s t 3xur dt 6 r

(51)

where ur is radial velocity of the particle relative to the gas, x is particle diameter, s is particle density 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 industrial cyclones, the Basset and displaced-mass terms are clearly negligible. However, turbulent diffusion 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 instance, a very popular curve-fit has the form

i =

1 x 1 + 50 x i
m

(52)

for discrete sizes xi and fractional efficiencies i. 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
27

idea needs however a complete battery of simplifications, especially if a more or less closed, algebraic 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 version 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 timeof-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 Lapples; they lack accordingly this important capability. In this respect, it has been shown recently [104], both by measurement and numerical calculation, that the reasonable Barthian cylinder CS doesnt 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.

28

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 radius. 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 fractional efficiency curve. More sophisticated models can be found in the literature, such as Dietzs [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 & Lichts, 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 Barths 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 Barths equilibrium orbit, a seminal idea. For instance, the extension to cyclone zones different from the main separation space is a recurrent idea, as in [45]. Finally, Mothes & Lffler [18] is a hybrid model that added a finite particle dispersion coefficient in an effort to improve the prediction.

29

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 increases as the concentration of solids at inlet Csi is increased. A part of the perplexity is explained because the increase in 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 analogous 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 turbulent 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 efficiency 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:

= 0
= 1
C cr C si C cr + C 0 si

C si C cr
C si > C cr

(57a) (57b)

30

In Eq. (57b), the first term accounts for the efficiency of the solids separated by sedimentation at cyclone inlet, whereas the second stands for the collection efficiency 0 of the remaining load in the separation space. In principle, 0 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 separated 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 0 in the second formula different from the 0 in the first one. As we have seen too, some models 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:

C cr =

Dc De
D 2 2 1 e s x m vte vtw Dc

(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:

C cr = 0.025 C cr = 0.025

x 50 (10C si )0.4 xm x 50 (10C si )0.15 xm

C si < 0.1 C si > 0.1

(59a) (59b)

Reference [5], based on the work of Muschelknautz [4] and Barth [29], also developed a method to correct the dilute efficiency 0 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 suspended in the vortex. By considering a mean centrifugal acceleration, a settling velocity was calculated at the entrance duct, from which an inlet cut-size was determined.

31

This has been signaled as inconsistent with an observed uniform increase of the gradeefficiency 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 agglomeration 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 (x) curve, nor to the total 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 direction 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. 32

turning length, natural length or vortex length of the cyclone, and the axial position is referred to as the end of the vortex. This point can be imagined as an effective end of the apparatus, 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 reentrainment, 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 wasnt fully explained until very recently. Initially it was assumed to be an axisymmetric flow structure caused by the axial and radial pressure gradients, 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 coefficients, i.e., at low swirl [53]. Under realistic conditions, the double vortex bends and attaches itself 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 doesnt 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 descends, 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-

33

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 experiments 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 recently, 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, frequently 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 tangential 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 pressure gradient developed by itself, as already mentioned in Sect. 3.4. As a consequence, it oscillates

34

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, attaches 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 vortex 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 literature 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 operation, since it leads to a high collection efficiency. It corresponds to cyclones with relatively high inlet Reynolds numbers (approximately ranging from 2103 to 2104) and H/Dc ratios within 2-10. Operation with Ln < H, Fig 14 b. It entails poor collection performance, because the vortex end attaches to the cyclone wall, disturbing the solids strands that are already separated and decreasing efficiency through instability and re-entrainment. This mode of

35

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 numbers 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 significant but not as strong as that found in [61]. Geometric parameters. Aside from the ratio H/Dc itself, the most relevant geometric parameters 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 support classical Alexanders 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 influence. The trend most frequently observed is an increased natural length with an increased 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].

36

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

37

5.1.2 Calculation of Ln. Up to now, adequate calculation methods of the cyclone natural length have not been devised. None of the available proposals (e.g., [9], [63], [65], [66]) takes into account all the observations 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:
13

Alexander [9]:

Ln D D 2 = 2 .3 e c Dc Dc ab
A Ln = 2 Dc i A e
12

(60)

Zenz [65], Bryant [66]:

(61)
0.361

Zhongli [63]:

D Ln = 2 .4 e D Dc c

2.25

Dc 2 A i

(62)

Considering the characteristics of the vortex end we have just mentioned, it is not strange that these formulae perform very badly. (Alexanders 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.

38

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] using 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 geometric 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 HS in his calculation, Eq. (39), and in tangential velocity models, Eq. (6). In [49], tangential velocity and pressure 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 explained 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 resulting from a highly variable flow will increase pressure drop.

39

5.2. Precessing vortex core (PVC).

The precessing vortex core (PVC) is a flow phenomenon originally observed in oncethrough swirling flows, such as those in swirl tubes, burners and combustors. In general, the swirl intensity is characterized by the swirl number Sw [68]: Sw = 2 Gt Gx D
(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 geometric factor. The appropriate expression for a cyclone separator is, for instance [69]: Sw =

De Dc
4 Ai

(64)

Swirling flow patterns change dramatically with Sw. For swirl numbers about 0.6 and increasing 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 backflow 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 Vortex 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 downstream attenuation of the swirl in the open volume; the dip in the profile of radial pressure accounts 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
40

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 however 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 beneficial [68]. On the contrary, given the objective of cyclone separators, the phenomenon can be detrimental in them. With independence of this, the cyclone PVC has been found to influence aerodynamic 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

41

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 fluctuations, which are roughly sinusoidal. Thus, a sinusoidal wave can be measured, such as that reported by [73] in a cold-flow model of a swirl combustor. The statistical distribution of the tangential 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 unstructured 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].

42

5.2.2. Influence of operating conditions on PVC. The frequencies of oscillation of the PVC in cyclones dont differ much from those of the swirl of the gas itself. Customarily, for a frequency f (Hz), a cyclone Strouhal number is defined as
St = fDc vi

(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 references agree in reporting an approximately linear increase of frequency with Rec [73], [78]. A decrease 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 90-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 60-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. Lets 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
43

flow structures wont 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 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 preferable in case the extra cost of the calculation were affordable [79][81]. With the computer capabilities 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. Specifically, 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

44

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 resolution 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.

45

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 transient, 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 difference 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 Simulation (DES) is essentially a RANS solver that changes to subgrid mode when the computed turbulent 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 basic problem with URANS: it assumes two distant ranges of time scales, those pertaining to the turbulent 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 reciprocating engines is a good example. In the case of cyclones, experimental observations and numerical calculations readily showed that there is not a clear-cut of frequencies in this kind of flow.

46

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 somewhat schematic, not completely duplicating intermediate details of the flow. In contrast, when powerful, differential RSTMs are brought into play, differences between LES and URANS diminish, 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 coherent 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 simulations of swirling flows in which a k- 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 computers 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 turbulence model was not able to accurately simulate this kind of flows. Using a mixed algebraic-differential, stationary RSTM, the authors found instead good agreement between experimental data and simulation of overall pressure drop (both in absolute value and tendencies), and they could correctly duplicate the radial velocity profile. At the same time, advanced k schemes, that took partial ac-

47

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 performance of the k model, the differential RSTM known as the LRRG model [83], and a variation of the k model based on ReNormalization Group (RNG) theory was evaluated [96]. Simulations were also compared with velocity measurements carried out by means of Laser-Doppler Anemometry (LDA). Tests were performed with three different vortex finder diameters, which produced three different swirl numbers. For all runs, the k model predicted solid-body rotation, a logical result given its foundations, albeit contrary to experiment. The RNG k model showed great improvement 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 comparing 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 application. 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 and a k modified to account for the streamline curvature (k cc). Although the pressure distribution 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 illustration. 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 and the k cc, other hybrid models based on two-equation ideas, such as that of [99] have been used to obtain more acceptable results

48

In later years, the need of using unsteady solvers and second-order discretization was established; 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 destroyed 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 research on a pilot PFBC cyclone, equipped with a long dipleg [12]. Single-phase gas flow was simulated by three methods: k, RNG k 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 retaining 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 measurements 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, phaseaveraged, 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.

49

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 pressure 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 experimental 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 predicting 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 University [77] were among the first to simulate the PVC phenomenon by this method, but only attempting 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 standard SGS and a lattice-Boltzmann discretization) in simulating confined turbulent vortex flows have been further reported in [108].

50

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 twophase 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, nonintrusive 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 became evident.

51

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 segregation 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 context. 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 particle 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 stroboscopic lighting [56] and by neutrallybuoyant tracers combined with pressure measurements of high time-resolution [109].

52

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 literature about cyclones. Lets 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 different regimes are distinguished, depending on the existence of mutual, significant interaction between particles: dilute and dense two-phase flow. The approximate borderline is a volume fraction

p 103, 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 p < 106 (L/dp > 80), which is known as one-way coupling; for higher volume fractions, it needs to be accounted for (two way coupling). Aside from obvious issues of volume and continuity, the generic flow feature most affected is turbulence. In our context, the effect is normally an attenuation, but an enhancement may also result under some circumstances. In the dense regime (p > 103, L/dp < 8), interparticle interactions 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 volume-averaged properties thorough the field.

53

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 becomes 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 exclusively influenced by the time-averaged gas flow. Very small particles will tend to fluctuate following 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].

54

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 indeed very easy to add-up difficulties arising from the dispersed phase: For instance, with independence 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 timeresolved 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 calculation, Fig. 24 shows snapshots of particle concentration at five instants of time inside a highefficiency, 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 respect, it should be noted that the enormous amount of time-dependent data generated by LES precludes post-processing and requires at once repeated tracking of a large amount of identical particles. (The method is neither free of the necessity of a stochastic model of particle turbulent diffusion 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

55

any case, calculated efficiencies compare much better with experiment than former, classical Lagrangian calculations [107]. For interacting flows, the two-way transfer of momentum between particulate and fluid phases is modeled by particlesourcein 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 obviously the high computational effort, if large numbers of particles are needed to represent the dispersed phase. This seems to be the case of cyclone separators; in contrast with simple Lagrangian trackings, 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 contribution 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 moderate, but still high, values of Csi = 0.050.1, 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 provisionally, adequate.

56

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 augments 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), Muschelknautzs views will become, at least for moderate solids loadings, another very useful, but purely imaginary, theory on cyclone operation.

6.2.2 Eulerian-Eulerian models of cyclone operation.

The EulerianEulerian approach treats both the solid particles and the fluid as interpenetrating continua, each governed by conservation laws whose general structure is known albeit specific 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, acting 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 breakup can be taken into account by using a population balance model [130]. EulerianEulerian schemes have been used in many practical multiphase flow simulations, for instance, and typically, in fluidized-bed applications [131], where they are specially suited.

57

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 consequence, simplified Eulerian multiphase approaches have been developed, mostly based on an algebraic 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 frequently 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 relationships [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 satisfactory, results. Aside for that, as for gas cyclones, these authors only know of two investigations. In [138], a mixed k-/algebraic RSTM, stationary gas flow model is used along with a simple Eulerian 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 Fluents 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 experiments, are not conclusive or even deviant. However, interestingly, the attenuation of both swirl and turbulence with mass loading is clearly noticed.

58

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 patterns. 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 semiempirical 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 experimental 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 oscillations known as a precessing vortex core. This instability can be a threat to the cyclone objective 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 discretization 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 computationally 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

59

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 circumstances, 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 demands 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 methods, not yet realized. A development of the PSI-Cell techniques is the sense of coping more efficiently 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.

60

Acknowledgements

Authors are grateful to Dr. Jos 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 Escatrn pressurized fluidized bed combustion 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 Generacin SA (Spain). Authors also wish to thank Prof. Alex C. Hofmann for his useful comments and suggestions.

References

[1]

lvarez, M., Anthony, E. J. (Eds.), Pressurized Fluidized Bed Combustion. Blackie Academic & Professional, 1995.

[2]

Robertson, A., Fan, Z., Goldstein, H., Horazak, D., Newby, R., Bose A.C., Second generation PFB plant with super critical pressure, Proc. Int. Pittsburgh Coal Conf., 2005.

[3]

Venalainen, I., Psik, R., 460 MWe supercritical CFB boiler design for Lagisza power plant, Proc. Power-Gen Europe, 2004.

[4]

Muschelknautz, E., Die Berechnung von Zyklonabscheidern fr Gase, Chem-Ing-Tech 1972; 44:63-71.

[5]

Trefz, M., Muschelknautz, E., Extended cyclone theory for gas flows with high solids concentrations, Chem. Eng. Tech. 1993; 16: 153-160.

[6]

Dirgo, J., Leith, D., Design of cyclone separators, in Encyclopedia of Fluid Mechanics, Vol.4Gas-solid Flows, Ed. NP Cheremisinoff, Gulf Publishing Company, Houston, 1986.

[7]

Reznik, V.A., Matsnev, V.V., Comparing the characteristics of the elements in batteries of cyclones, Thermal Engineering 1971; 18:34-39.

61

[8]

Hoffmann, A.C., Stein, L.E., Gas Cyclones and Swirl Tubes. Principles, Design and Operation, Springer-Verlag, Berlin, 2002.

[9]

Alexander, R. McK., Fundamentals of cyclone design and operation, Proc. Australas. Inst. Min. Metall. 1949; 152:203-228.

[10]

Ter Linden, A.J., Investigations into cyclone dust collectors, Proc. Inst. Mech. Eng. 1949; 160: 233-251.

[11]

Hoffmann, A.C., de Groot, M., Hospers, A., The effect of the dust collection system on the flowpattern and separation efficiency of a gas cyclone, Can. J. Chem. Eng. 1996; 74:464-470.

[12]

Velilla, J., Study of the flow at a PFBC cyclone dipleg. PhD Thesis, University of Zaragoza, 2005.

[13]

Wang, J., Bouma, J.H., Dries, H., An experimental study of cyclone dipleg flow in fluidized catalytic cracking, Powder Technol. 2000; 112: 221-228.

[14]

Gil, A., Experimental cold flow model of a PFBC primary cyclone. Ph. D. Thesis, University of Zaragoza, 2000.

[15]

Gil, A., Romeo, L.M., Corts, C., Cold flow model of a PFBC cyclone, Powder Technol. 2001; 117: 207-220.

[16]

Gil, A., Corts, C. Romeo, L. M. and Velilla, J., Gas-particle flow inside cyclone diplegs with pneumatic extraction, Powder Technol. 2002; 128: 78-91.

[17]

Meissner P. and Lffler F., Zur Berechnung des Strmungsfeldes im Zyklonabscheider, Chem-Ing Tech 1978; 50: 471.

[18]

Mothes, H., Lffler, F., Motion and deposition of particles in cyclones, Ger. Chem. Eng. 1985; 8:223-233.

62

[19]

Reydon R.F., Gauvin W.H., Theoretical and experimental studies in combined vortex flow, Can. J. Chem. Eng. 1981; 59: 14-23.

[20]

Ogawa, A., Separation of particles from air and gases, I and II, CRC Press, Florida, 1984.

[21]

Ogawa, A., Mechanical separation process and low patterns of cyclone dust collectors, Appl. Mech. Rev. 1997; 50:97-129.

[22]

Comolet R., Bonnin J. Stabilit dun vortex, in Mcanique Experimentale des Fluides, Masson, Paris; 1986, pp. 417-423.

[23]

Kirch R., Der Einfluss der Turbulenz auf die Partikelbewegung im Gaszyclon, in VDIFortschritts-Berichte Reihe 7: Strmungstechnik, 145, VDI-Verlag, Dsseldorf, 1988

[24]

Morweiser, M., Bohnet, M., Influence of temperature and pressure on separation efficiency and pressure drop of aerocyclones, in High Temperature Gas Cleaning, Ed. E. Schmidt, Institut fr Mechanische Verfahrenstechnik und Mechanik, 1996, pp. 26-39.

[25]

Hoffmann, A.C., De Jonge, R., Arends, H. and Hanrats, C., Evidence of the natural vortex length and its effect on the separation efficiency of gas cyclones, Filtr. & Sep. 1995; 32: 799-804.

[26]

Hoffmann A.C.; Van Santen A.; Allen R.W.K.; Clift R., Effect of geometry and solid loading on the performance of gas cyclones. Powd. Technol. 1992; 70:8391.

[27]

Hoffmann, A.C., Peng, W., Dries, H., Regelink, M., Foo, K.-K., Effect of pressure recovery vanes on the performance of a swirl tube, with emphasis on the flow pattern and separation efficiency, Energ. Fuel. 2006; 20:1691-1697.

[28]

Shepherd, C.B., Lapple, C.E., Flow pattern and pressure drop in cyclone dust collectors, Ind. Eng. Chem. 1939; 31:972-984.

63

[29]

Barth, W., Berechnung und Auslegung von Zyklonabscheidern auf Grund neuerer Untersuchungen, Brennst.-Waerme-Kraft 1956; 8: 1-9.

[30]

Casal, J., Martnez-Benet, J.M., Valencia, Clculo y diseo de ciclones, Ing. Quim. 1989; 2:115-124.

[31]

Gil, A., Romeo, L.M., Corts, C., Effect of the solid loading on a PFBC cyclone with pneumatic extraction of solids, Chem. Eng. Technol. 2002; 25:407-415.

[32]

Yuu, S., Jotake, T., Tomita, Y., Yoshida, K., The reduction of pressure drop due to dust loading in a conventional cyclone, Chem. Eng. Sci. 1978; 33:1573-1580.

[33]

Parida, A., Chand, P., Turbulent swirl with gas-solid flow in a cyclone, Chem. Eng. Sci. 1980;35: 949-954.

[34]

Briggs, L.W., Effect of dust concentration on cyclone performance, Trans. Am. Inst. Chem. Eng. 1946; 42: 511-526.

[35]

Smolik, J. Air pollution abatement, Part I, Scriptum No. 401-2099, Technical University of Prague, 1975.

[36]

Baskakov, A.P., Dolgov, V.N., Goldovin, Yu. M., Aerodynamics and heat transfer in cyclones with particle-laden gas flow, Exp. Therm. Fluid Sci. 1990; 3:597-602.

[37]

Chen, J., Luo, X., & Shi, M., Experimental research on the pressure drops of PV-type cyclone separators at various cyclone concentrations, Proc. Int. Symp. Multiph. Flow, 1997, pp. 332-334.

[38]

Lewnard, J.J., et al., Effect of design and operating parameters on cyclone performance for Circulating Fluidized Bed Risers, in Ed. Avidan, A.A., Circulating Fluidized Bed Technology IV. AIChE, New York, 1994, pp. 525531.

64

[39]

Ushiki, K., et al., Experimental study on the performance of very small cyclones for the recycle of ultra-fine powders, in Ed. Avidan, A.A., Circulating Fluidized Bed Technology IV. AIChE, New York, 1994, pp. 504510.

[40]

Cankurt, N.T., Yerushalmi, J., Some observations on the performance of cyclones in a fast-fluidized bed system. Journal of Powder & Bulk Solids Technology 1978; 3: 2935.

[41]

Lapple, C.E., Gravity and centrifugal separation, Industrial Hygiene Quart., 1950; 11:40-47.

[42]

Leith, D., Licht, W., The collection efficiency of cyclone type particle collectors- a new theoretical approach, AIChE Symp. Ser. 1972;68:196-206.

[43] [44]

Dietz, P.W., Collection efficiency of cyclone separators, AIChE J. 1981; 27: 888-892. Clift, R. ,Ghadiri, M., Hoffmann, A.C., A critique of two models for cyclone performance, AIChE J., 1991; 37: 285-289.

[45] [46]

Lorentz, T., Heigasentstaubung mit Zyklonen, VDI-Verlag, Dusseldorf, 1994. Abrahamson, J., Martin, C.G., Wong, K.K., The physical mechanisms of dust collection in a cyclone, Trans. Inst. Chem. Eng. 1978; 56:168-177.

[47]

Abrahamson, R.W.K Allen, The efficiency of conventional return-flow cyclones at high temperatures, AIChE Symp. Ser. 1986;99 :31-43.

[48]

Stairmand C.J. The design and performance of cyclone separators. Trans. Inst. Chem. Eng. 1951; 29: 357-383.

[49]

Wheeldon, J.M. & Burnard, G.K., Performance of cyclones in the off-gas path of a pressurized fluidized bed combustor, Proceedings of the Filtration Society, Filtr. & Sep. 1987; 3 :178-187.

65

[50]

Muschelknautz, E., Greif, V., Cyclones and other gas-solids separators, in Eds. Grace, J.R., Avidan, A.A., Knowlton, T.M., Circulating Fluidized Beds, Blackie, London, 1997, pp. 61816213.

[51]

B. Reinhardt, A. Cordonnier and P. Florent, Use of a isokinetic sampling probe. Results in a cyclone, Powder Technol. 1999; 101: 8190.

[52]

Lucca-Negro, O., ODoherty, T., Vortex breakdown: a review, Prog. Energ. Combust. 2001; 27: 431-481.

[53]

Lidn, G. and Gudmundsson, A., Semi-empirical modeling to generalize the dependence of cyclone collection efficiency on operating conditions and cyclone design, J. Aerosol Sci. 1997; 28: 853874.

[54]

Muschelknautz, E., Kambrock, W., Aerodynamische Beiwerte des Zyclonabscheiders aufgrund neuer und verbesserter Messungen , Chem-Ing-Tech 1970; 42: 247-255.

[55]

Smith, J.L., An analysis of the vortex flow in the cyclone separator, J. Basic Eng. 1962; 609-618.

[56]

Peng, W. , Hoffmann, A.C., Dries, H.W.A., Regelink, M.A., Stein, L.E., Experimental study of the vortex-end in centrifugal separators: The nature of the vortex end, Chem. Eng. Sci. 2005; 60: 6919-6928.

[57]

Caplan, K.J. Source control by centrifugal force and gravity in Air Pollution 1968; 3:359407.

[58]

Geiger, F.J.A, Dries, H.W.A., Bakker, E., Erosion, fouling and cracking of FCC cyclone systems resolved, VDI-Berichte 1511, VDI-Verlag, Dsseldorf, 1999.

[59]

MacAuley, R. Dries, H., FCC cyclone systems: a vital element for FCC profitability, in European Refining Technology Conference, ERTC, Rome 2000.

66

[60]

Griffiths, A.J. Yazdabadi, P.A., Syred, N. Alternate eddy shedding set up by the nonaxisymmetric recirculation zone at the exhaust of a cyclone dust separator, J. Fluids Eng. 1998; 120: 193-199.

[61]

Bttner, H., Dimensionless representation of particle separation characteristics of cyclones, J. Aerosol Sci. 1999; 30:1291-1302.

[62]

Hoffmann A.C., Groot M., Peng W., Dries H.W.A., Kater, J., Advantages and risks in increasing cyclone separator length, AICHE J. 2001; 47: 2452-2460.

[63]

Zhongli J., Xiaolin, W., Minxian, S., Experimental research on the Natural Turning Length in Cyclones, Proc. Filtech Europa 91, Karlsruhe, The Netherlands, 1991, pp. 583-89.

[64]

Ito S., Ogawa K. and Kuroda C., Turbulent Swirling Flow in a Circular Pipe, J. Chem. Eng. Jpn. 1980;13: 6-10.

[65]

Zenz, F.A. , Cyclone separators, in Manual on Disposal of Refinery Wastes, Volume on Atmospheric Emissions, American Petroleum Institute Publication 961, Chapter 11, 1975

[66]

Bryant, H.S., Sylverman, R.W., Zenz, F.A., How dust in gas is affects cyclone pressure drop, Hydro. Proc. 1983; 62: 87-90.

[67]

ODoherty, T., Griffiths A.J., Syred, N., Bowen, P., Fick, W., Experimental analysis of rotating instabilities in swirling and cyclonic flows, Dev. Chem. Eng. Mineral Proces. 1999; 7: 245-267.

[68]

Syred, N., Ber, J.M., Combustion in swirling flows: A review, Combust. Flame 1974; 23:143-201.

[69]

Gupta, A. K., Lilley, D. G., Syred, N., Swirl Flows, Abacus Press, Tunbridge Wells, UK, 1984.

67

[70]

Williams J.M., Investigations into the Precessing Vortex Core Phenomenon within Dust Sperating Ciclones, Cardiff School of Engineering, Cardif University of Wales, Internal Report no. 1772, 1993.

[71]

Fick, W., Griffiths, A.J., ODoherty, T., Visualization of the precession vortex core in an unconfined swirling flow, Opt. Diagn. Eng. 1997; 2:19-31.

[72]

Solero, G, Cogher, A., Experimental fluid dynamics characterization of a cyclone chamber, Exp. Therm. Fluid Sci. 2002; 27:87-96.

[73]

Styles, A. C. , Syred, N., Najim, S.A., A study of modulatable cyclone combustors using gaseous fuel, J. Inst. Energ. 1979; 159 160-8.

[74]

Yadzabadi, P.A., Griffiths, A.J., Syred, N., Axial and tangential velocity components at the exhaust end of highly complex cyclonic flows, in Developments in laser techniques and applications to fluid mechanics, Springer, 1996, pp. 67-82.

[75]

Claypole T.C., Evans P., Hodge J., and Syred N., The influence of the PVC on Velocity measurements in Swirling Flows, Proceedings of the 3rd International Symposium on Application of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 1986.

[76]

Gorton-Hlgerth A., Messung und Berechnung der Geschwindigkeitsfelder und Partikelbahn im Gaszyclon, VDI-Fortschritt-Berichte, Reihe 7, Nr. 357, 1999.

[77]

Derksen, J.J., Akker, V.D., Simulation of vortex core precession in a reverse flow cyclone, AICHE J. 2000;46: 1317-1330.

[78]

Griffiths, A.J. Yazdabadi, P., Syred, N., The use of centre bodies and de-swirl vanes in the exhaust of cyclone dust separators, Proc. Instn. Mech. Engrs. 1996; 210: 193-203.

[79]

Wilkes, N.S., Clarke, D.S., Turbulent flow predictions using algebraic stress models, Techn. Report, AERE-R 12694, Harwell Laboratory, 1987.

68

[80]

Clarke, D.S., Wilkes, N.S., The calculation of turbulent flows in complex geometries using a differential stress model, Techn. Report, AERE-R 13428, Harwell Laboratory, 1989.

[81]

Hogg, S., Leschziner, M.A., Computational of highly swirling confined flow with a Reynolds stress turbulence model, AIAA J. 1989; 27: 57-63.

[82]

Guo B., Langrish, T. A.G., Fletcher, D. F., Simulation of Turbulent Swirl Flow in an Axisymmetric Sudden Expansion, AIAA J. 2001; 39:96-102.

[83]

Hankalic, K., Closure Models for Incompressible Turbulent Flows (Lecture Series), in Beeck, J. P. A. J., Benocci, C. (Eds.), Introduction to Turbulence Modeling, Von Karman Institute for Fluid Dynamics, 2004, 1-75 (or http://www.vki.ac.be/educat/lectser/2004/turbulence2004/hanjalic.pdf)

[84]

Jakirlic, S., Hanjalic, K., Modeling Rotating and Swirling Turbulent Flows: A Perpetual Challenge, AIAA J. 2002; 40:1984-1996.

[85]

Speziale, S. G., Turbulence Modeling for Time-Dependent RANS and VLES: A Review, AIAA J. 1998; 36:173-184.

[86]

Prasad, R., Bakker, A. Cyclone Modeling using LES. Fluent Users Group Meeting, Danvers, Massachusetts, 1999.

[87]

Ferziger, J.H., The physics and simulation of the turbulence, in VKI Lecture Series, Introduction to turbulence modeling, von Karman Institute for Fluid Dynamics, Springer Verlag, 2002.

[88]

Laurence D., Applications of Reynolds Averaged Navier Stokes Equations to Industrial Flows, in Introduction to Turbulence Modeling, VKI von Karman Institute for Fluid Dynamics, Bruxelles, 2002.

69

[89]

Spalart, P.R., Strategies for turbulence modeling and simulations, Int. J. of Heat and Fluid Flow 2000;21: 252-263.

[90]

Boysan, F., Ayers, W.H. and Swithenbank, J. A fundamental mathematical modeling approach to cyclone design. Trans. Inst. Chem. Eng. 1982; 60: 222-230.

[91]

Boysan, F., Ewan, B. C. R, Swithenbank, J. Ayers, W. H. Experimental and theoretical studies of cyclone separator aerodynamics, Inst. Chem. Eng. Symp. Ser. 1983; 69:305319.

[92]

Boysan, F., Swithenbank, J., Ayers, W. H., Mathematical modelling of gas-particle flows in cyclone separators, in Encyclopedia of Fluid Mech., Vol. 4, Chap. 42, Gulf Publishing Co., 1986.

[93]

Dyakowsky, W., Modelling turbulent flow within a small-diameter hydrocyclone, Chem. Eng. Sci. 1993; 48: 1143-1152.

[94]

Griffiths, W.D., and Boysan, F., Computational Fluid Dynamics (CFD) and Empirical Modelling of the Performance of a Number of Cyclone Samplers, J. Aerosol Sci. 1996; 27: 281-304.

[95]

Averous, J., Fuentes, R., Advances in the numerical simulation of hydrocyclone classification, Can. Metall. Quaterly 1997; 36:309-314.

[96]

Hoekstra, A.J. , Derksen, J.J., Van Den Akker, H.E.A. , An experimental and numerical study of turbulent swirling flow in gas cyclones, Chem. Eng. Sci. 1999, 54: 2055-2065.

[97]

Jo, Y., Tien, C. Ray, M.B., Development of a post cyclone to improve the efficiency of reverse flow cyclones, Powd. Technol. 2000, 113, 97-108.

[98]

Grotjans H., Application of higher order turbulence models to cyclone flows, VDIBerichte 1511, VDI-Verlag Dsseldorf, 1999, pp. 175-182.

70

[99]

Meier, H.F., Mori, M., Anisotropic behavior of the Reynolds Stress in Gas and GasSolid flows in cyclones, Powd. Technol. 1999; 101, 108-19.

[100]

Witt, P.J., Mittoni, L.J., Validation of a CFD model for predicting gas flow in a cyclone, CHEMECA 99, Newcastle, Australia ,1999.

[101]

Montavon, C.A., Grotjans, H., Hamill I.S., Philips H.W., Jones, I.P., Mathematical modelling and experimental validation of flow in a cyclone, BHR Conference on Cyclone technologies, Warwick, 2000.

[102]

Slack, M.D., Prasad, R.O., Bakker,A., Boysan F., Advances in Cyclone Modelling Using Unstructured Grids, Trasn. I. Chem. 2000, 78, 1098-104.

[103]

Peng, W., Boot, P., Udding, A., Hoffmann, A.C., Dries, H.W.A., Ekker, A., Kater, J., Determining the best modelling assumptions for cyclones and swirl tubes by CFD and LDA, International Congress for Particle Technology, 2001, p. 1-8.

[104]

Peng, W., Hoffmann A. C., Boot, P.J.A.J., Udding, A., Dries, H.W.A., Ekker, A., Kater, J., Flow pattern in reverse-flow centrifugal separators, Powd. Technol. 2002; 127: 212222.

[105]

Wang, B., Xu, D. L., Chu, K. W., Yu, A.B., Numerical study of gas-solid flow in a cyclone separator, Appl. Math. Model. 2006;30:1326-1342.

[106]

Qian, F., Zhang, J., Zhang, M., Effects of the prolonged vertical tube on the separation performance of a cyclone, J. Hazar. Mater. 2006;B136:822-829.

[107]

Derksen, J. J., Separation Performance Predictions of a Stairmand High-Efficiency Cyclone, AIChE J. 2003; 49:1359-1371.

[108]

Derksen, J. J., Simulations of confined turbulent vortex flow, Comp. & Fluids 2005; 34:301-318.

71

[109]

Peng, W., Boot, P.J.A.J., Hoffmann A. C., Dries, H.W.A., Kater, J., Ekker, A., Flow in the inlet region in tangential inlet cyclones, Ind. Eng. Chem. Res. 2001; 40:5649-5655

[110]

Peng, W., Hoffmann, A. C., Dries, H. W. A., Regelink, M., Foo, K.-K., Neutrally buoyant tracer in gas cleaning equipment: a case study. Meas. Sci. Technol. 2005, 16:24052414

[111]

Syred, N., A review of oscillation mechanisms and the role of the precessing vortex core (PVC) in swirl combustion systems, Prog. Energ. Combust. Sci. 2006; 32:93-161.

[112]

Liu, Z., Zheng, Y., Jia, L., Jiao, J., Zhang, Q., Stereoscopic PIV studies on the swirling flow structure in a gas cyclone, Chem. Eng. Sci., 2006; 61:4252-4261.

[113]

Liu, Z., Jiao, J., Zheng, Y., Zhang, Q., Investigation of turbulence characteristics in a gas cyclone by stereoscopic PIV, AIChE J. 2006; 52:4150-4160.

[114]

Elghobashi, S.E., On predicting particle-laden turbulent flows, Appl. Sci. Res. 1994; 52: 309-329.

[115]

Chiesa, M., Mathisen, V., Melheim, J. A., Halvorsen, B. M. Numerical simulation of particulate flow by the Eulerian-Lagrangian and the Eulerian-Eulerian approach with application to fluidised bed, Comput. Chem. Eng. 2005, 29(2): 291-304..

[116]

Loth, E., Numerical Approaches for motion of dispersed particles, droplets and bubbles, Prog. Energ. Combust. Sci. 2000; 26:161-223.

[117]

Sommerfeld, M. 2000. Theoretical and Experimental Modelling of Particulate Flows, Lecture Series 2000-06, von Karman Institute for Fluid Dynamics, 2000.

[118]

Ho, C.A., Sommerfeld, M., Modelling of micro-particle agglomeration in turbulent flow, Chem. Eng. Sci. 2002; 57: 30733084.

[119]

Gosman, A.D., Ioannides, E., Aspects of computer simulation of liquid-fuelled combustors. AIAA Paper 81-0323, 1981
72

[120]

Berlemont, A., Desjonqueres, P., Gouesbet, G., Particle Lagrangian simulation in turbulent flows, Int. J. Multiph. Flow 1990, 16, 1934.

[121]

Sommerfeld, M., Kohnen, G., Rger, M., Some open questions and inconsistencies of Lagrangian particle dispersion models. Proceedings, 9th Symposium on Turbulent Shear Flows, Kyoto, Japan, Paper 151, 1993.

[122]

Ma, L., Ingham, D.B., Wen, X., Numerical modelling of the fluid and particle penetration through small sampling cyclones, J. Aerosol Sci. 2000; 31: 10971119.

[123]

Frank, Th. Wassen, E. Yu, Q., Lagrangian prediction of disperse gas particle flow in cyclone separators, Third International Conference on Multiphase Flow, ICMF 98, Lyon, France, 1998.

[124]

Frank, Th. Application of Eulerian-Lagrangian prediction of gas particle flows to cyclone separators, in Theoretical and Experimental Modelling of particulate flow, von Karman Institute for Fluid Dynamics, Lecture Series 2000-06.

[125]

Narasimha, M. Brennan, M.S. Holtham P.N. & Napier-Munn, T.J., A comprehensive CFD model of dense medium cyclone performance, Miner. Eng. 2006,

doi:10.1016/j.mineng.2006.10.004. [126] Crowe, C.T., Sharma, M.P. & Stock, D.E.: The Particle-source-in-cell (PSI-cell) model for gas-droplet flows. J. Fluids Eng. 1977; 99: 325-332. [127] Kohnen, G., Rger, M. & Sommerfeld, M., Convergence behaviour for numerical calculations by the Euler/Lagrange method for strongly coupled phases. Numerical Methods in Multiphase Flows, ASME Summer Meeting, Lake Tahoe, U.S.A., FED-Vol. 185, 191-202, 1994. [128] Crowe, C.T., Troutt, T.R., Chung, J.N., Numerical models for two-phase turbulent flows, Annu. Rev. Fluid. Mech. 1996; 28:1143.

73

[129]

Derksen, J.J., Sundaresan, S., van den Akker, H.E.A., Simulation of mass-loading effects in gassolid cyclone separators, Powder Technol. 2006; 163: 59-68.

[130]

Ibsen, C.H., Helland, E., Hjertager, B.H., Solberg, T., Tadrist, L., Occelli, R., Comparison of multifluid and discrete particle modelling in numerical predictions of gas particle flow in circulating fluidised beds, Powder Technol. 2004; 149: 2941.

[131]

Enwald, H., Peirano, E., & Almstedt, A.E., Eulerian two- phase flow theory applied to fluidization, Int. J. Multiphase Flow 1996; 22: 21-66.

[132]

Manninen, M. Taivassalo, V. & Kallio, S., On the mixture model for multiphase flow, VTT Publications 288, Espoo, Finland, 1996.

[133]

Narasimha, M., Sripriya, R. and Banerjee, P. K., CFD modelling of hydrocyclone prediction of cut size. Int. J. Miner. Proces. 2005; 75: 53-68.

[134]

Brennan, M.S., Multiphase CFD simulations of dense medium and classifying hydrocyclones, Proceedings of the Third International Conference on CFD in the Minerals and Process Industries, CSIRO Melbourne Australia, 2003, pp. 5963.

[135]

Narasimha, M. Brennan M.S.& Holtham, P.N., Numerical simulation of magnetite segregation in a dense medium cyclone, Miner. Eng. 2006; 19 (10): 1034-1047.

[136]

Brennan, M.S., Narasimha, M., & Holtham, Multiphase modelling of hydrocyclones prediction of cut-size, Miner. Eng. 2006, doi:10.1016/j.mineng.2006.10.010

[137]

Grady, S. A., Wesson, G. D., Abdullah, M., & Kalu, E. E. (2003). Prediction of 10-mm hydrocyclone separation efficiency using computational fluid dynamics. Filtr. & Sep. 2003; 40: 4146.

[138]

Meier, H.F, Mori, M., Gas-solid flow in cyclones: The Eulerian-Eulerian approach. Comput. Chem. Eng. 1998; 22 (Supplement 1): S641-2644.

74

[139]

Qian, F., Huanga, Z., Chena, G., and Zhang M. , Numerical study of the separation characteristics in a cyclone of different inlet particle concentrations, Comput. Chem. Eng (2006), doi:10.1016/j.compchemeng.2006.09.012.

75

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

76

(a)

De b a
(3) (5)

S h

(1)

Dc H
(2)

(4)

(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].

77

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].

78

(a)

(b)

(c) 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].

Fig. 6. Sketch of the tangential velocity profile in cyclones [12].

80

Fig. 7. Imaginary cylinder of Barth theories [29].

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].

1 s Cyclone pressure coeficient, 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 50 100 150 200 250

[29] Briggs (1949) [34] [30] Smolik (1975) [35] [31] Baskakov et al. [36] (1990) Muschelknautz[4] (1972) [1] PFBC cyclone [31] Measured data [26]

C si , g solids/kg gas

Fig. 10. Cyclone pressure coefficient vs. solids concentration at inlet. Comparison between correlations and measured data [31].

82

100 90 Fractional efficiency % 80 70 60 50 40 30 20 10 0 1 10 Particle siz e , m 100 1000

x50

Fig. 11. Typical fractional efficiency curve.

105

Fractional efficiency, %

100

95 71 g/kg 90 114 g/kg 85 169 g/kg

80 1 10 Particle size , m 100

Fig. 12. Experimental collection efficiency as a function of solids inlet concentration [31].

83

100 99 98 97 96 95 94 0 50 100 150 Inle t solid conce ntration, g solids/kg gas 200 250 PFBC cyclone Predicted

Cold model overall efficiency %

inner separation

separation at inlet

(a)

100 99 98 97 96 95 94 0 50 100 150 Inle t solid conce ntration, g solids/kg gas 200 250 PFBC cyclone Predicted

Cold model overall efficiency %

inner separation

separation at inlet

(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 attached to the lateral wall . Adapted from [53].
84

1 s Cyclone pressure coeficient, 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0 50 100 150 200 250 Muschelknautz (1972) including vortex penetration in dipleg [4] , [1] Cold model Measured data [26] Measured data [31] [1] Muschelknautz [4] (1972)

C si , g solids/kg gas

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].

85

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].

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

t=0

t = 0.25

t = 0.50

t = 0.75

t=1

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

Interparticle spacing L/dp 100 10 Dilute-dispersed two-phase flow Dense-dispersed two-phase flow

One-way coupling

Two-way coupling

Four-way coupling

10-8

10-6

10-4

0.01

0.1

Volume fraction p

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].

91

Table 1 Relevant geometric dimensions of several standard cyclone designs [8].

Name Muschelknautz E Muschelknautz D Storch4 Storch3 Storch 2 Storch 1 Tengbergen C Tengbergen B Tengbergen A TSN-11 TSN-15 Stairmand High Efficiency Stairmand High Flow VanTongeren AC Vibco Lapple GP Dc 680 357 260 192 225 365 337 210 277 348 266 316 190 325 286 283 De 170 119 117 107 108 123 112 112 112 136 158 158 141 100 111 141 S 311 318 176 200 239 142 145 224 157 242 350 158 165 325 124 177 H 934 863 1616 821 1097 1943 930 604 647 959 1124 1265 755 1231 720 1131 h 173 262 909 462 464 548 187 324 180 219 589 474 283 436 228 566 A 173 187 260 167 188 100 100 179 135 184 166 158 141 149 111 141 b 58 54 38 60 53 100 100 56 74 54 60 63 71 67 90 71 B 228 195 91 92 84 64 112 112 202 154 119 119 71 130 66 71

92

Table 2. Algebraic models of the velocity field in cyclones. Model Alexander [9] Equation
r vt = vtw c r
n

Remarks (2)

n

Ai vtw = 2.15 D D vi c e

Experiments with scroll and tangential inlets. Air and combustion gases, up to 1100 C. T in K and Dc in m. Scroll inlets 1 For tangential inlets, values of
are given by graphs

(3)

T n = 1 1 0.67 Dc 0.14 283

0.3

(4) (5) (6) (7) (8a)

De > B

Barth [29]

2 v i ri v tw Dc

vt e =

ri = (Dc + b ) 2
*

vtw rc re h * r v 1 + c & tw V

= g 0.02

h = H S De B (H h )(Dc De ) * h = +hS Dc B

(8b) (9) (10) (11a-b) Eq. (9) is valid for

0.9 < ab ri2 < 1.8

Muschelknautz [4]

= 1 1.2 b Dc for tangential inlets

= 1 + 3 ri

= g (1 + 2 C si ) = g (1 + 3 C si )

ab for scroll
C si < 1 C si > 1
1

inlets**

Csi in kg solids/kg gas.

g 0.005 for high Reynolds numbers

(12)

Meissner & Loffler [17]

* vtw 0.204b = + 0.889 vi rc

vtw 1 = * vb g hz

* * 0.25 + g h z vtw vb

0.5

0.5

(13) (14) (15) (16)

accounts for the angular momentum exchange between wall and gas
for smooth walls at room temperatures

g , x , k 0.0065 0.0075

2 & vb = 4 V Dc * hz =

a 2 a cos[(b rc ) 1] h 1 + rc 2 rc
vtw r r 1 + 1 rc rc

vt =

k vtw x + vb sin ( c )

(17)

Radial velocity
vr (rc ) = 0 vr (re ) =

& V 2re (H S )

(18a-b)

Axial velocity
vz0 =

rc2

& V (H z ) re2 (H S )

(19) r in cm (20)
0.17

Reydon & Gau- Free-vortex vin [19] vt = 14.79 vi r 0.72 Forced vortex
vt = 1.35 vi r exp 0.153 vi

(21)

**

This equation doesnt always give consistent values. [4]

93

Ogawa [20],[21] Free vortex

vt r n = K 0 vi (rc b )n

K0, n = f (Re, geometry) (22) (23) (24a) (24b)

Forced vortex
vt = Kr (1 r ) 2v K = t max ra 1+n r t = 2+n ra = 0.5
K = 66 .5D e

2.65

& V

(24c) (24d) (25)

Mean axial velocity

vz0 =

rc2

& V rt2

)
vz z =1 * vz0 h

Radial and axial velocities in the free vortex

vr r r r = c c v z 0 2 h * rc r vr r2 r2 = c 2 t* r vz0 2 rt h

(26 a-b)

Radial and axial velocities in the forced-vortex

v z rc2 rt2 z = 1 vz0 rt2 h *

(27 a-b)

94

Table 3. Algebraic models of the clean pressure drop g in cyclones. Model Shepherd & Lapple [28] Alexander [9] Equation
g =
16 ab 2 De
ab Dc Dc De De
2n

Remarks (35)
1 n D 1 +f c n g De
2n

Tangential inlet; ambient air conditions Experiments with scroll and tangential inlets. Air and combustion gases, up to 1100 C

g = 4.62

(36)

1 4 2 2n f g = 0. 8 n(1 n ) 3

1 n n

(37)

1n 2n + 0. 2 2 2 n 1 + 1. 5 2 n

( )
0.3

T n = 1 1 0.67 Dc 0.14 283

(4) (38) 3.41 < K < 4.4

Barth [29]

= g = 2 ( b + e ) D 4 e

ab

loss in the cyclone body

2 v D 1 te b = e 2 Dc v (H S ) v ze ze v 0.5D e te

(39)

loss in the vortex finder

e = K te v ze
v
43

v + te v ze
2

(40) (41) Tangential and scroll inlets. Flow field based on Barths model [29]. Ambient P,T conditions

Muschelknautz & Kambrock [54]

ab ( = g ) = 2 D e 4 ( b + e )

b =

AS g (v tw v te )1.5 & 0.9V 2

43

(42) (43)

v e = 2 + 3 te v ze

v + te v ze

=g0.006.
AS is the total inner area of cyclone contributing to friction

Casal et al. [30]

g = 11.3

ab 2 De

+ 2.33

(44)

Comparative study of six correlations

95

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

g
PFBC cyclone, measured [31] Shepherd & Lapple [28], Eq. (35) Alexander [9], Eq. (36) Muschelknautz, [4] [54] , Eq. (41) , including vortex penetration in dipleg [31] Casal et al. [30], Eq. (44) 11.6 15.7 13.6 14.1 11.7 14.2

Discrepancy 35.3 % 17.2 % 21.6 % 1% 22.4 %

Table 5. Formulae for the coefficient of solid loading effects Reference Briggs [34] Equation
s =
1 1 + 0.0086 C si g

[31]. Remarks

0.5

(45)

Air at ambient conditions. Multivane cyclone of 0.23 m diameter. Solids: rock dust 0-44 m, Csi range: 0.17-192 g/kg air.

Smolik [35]

s = 1 0.02 C si g

0.6

(46)

Experimental data from several sources

Baskakov et al. [36]

s =

1 + 0.67C si 0 1 + 3.1C si.7

(47)

Hot gases at ambient pressure and 150-250C. Helical inlet cyclone of 0.204 m diameter. Solids: alumina xm=40 m., Csi up to 0.7 kg/kg gas.

Muschelknautz [4]

s =

( = s ) g

(48)

See table 3 for g.

s = g +
+ 0.25 C si Fre0.5 Dc s (1 str ) De

5 /8

(49)

96

Table 6. Formulae for the cyclone collection efficiency. Reference Lapple [41] Equation
x 50
=

9 g b 2s v i N c

Nc =

h + (H h ) 2 a
9 g De v re
2 s vtw

(53a-b)

Barth [29]

x 50 =

(54)

Leith & Licht. [42], corrected by Clift et al [44]

2 xvi 2 = 1 exp s tres 9 g Dc

(55)

V t res = & V

(56)

Table 7. Calculations of the natural vortex length [12] . Ln/Dc Experiment CFD Alexander (1949) Bryant et al. (1983) Zhongli et al. (1991) 9.51 9.85 1.56 2.24 14.62 Remarks Stable vortex end on the dipleg wall. +3.5 % off from experiment Predicts vortex end on cyclone cone. Ibid. Vortex end out of the dipleg

97