Volume 204, Numbers 3-4, 203-216

Acta Mech 204, 203–216 (2009)DOI 10.1007/s00707-008-0086-7

Stefan Pirker · Damir Kahrimanovic · Georg Aichinger

Modeling mass loading effects in industrial cyclonesby a combined Eulerian–Lagrangian approach

Received: 30 April 2008 / Revised: 4 June 2008 / Published online: 13 August 2008© Springer-Verlag 2008

Abstract Cyclone separation is studied by means of numerical simulations. While the gas flow is modeledby a modified Reynolds stress (RS) model, the behavior of the particles is pictured by a combined Eulerian–Lagrangian approach. A mono-disperse Eulerian particle phase is utilized to account for inter-particle collisions,while the effects of fractional separation and particle-wall collisions are considered by poly-disperse Lagrangianparticles. The above particle models interact in two ways. On the one hand, the Lagrangian particles determinethe local mean diameter of the substitute Eulerian particle class. On the other hand, especially in regions of highparticle concentration, the Eulerian particle phase exerts an additional collisional force onto the Lagrangianparticle trajectories. An industrial cyclone is chosen as a test case and the numerical results are evaluated withrespect to pressure drop as well as to global and fractional separation efficiency. In this context the influenceof the cyclone’s mass loading and wall roughness is highlighted. Simulations indicate that the separationefficiency improves with increasing mass loading until an excess loading is reached while at the same timethe pressure drop is reduced. Furthermore, it can be shown that rough walls lead to a reduction of separationefficiency while simultaneously the pressure drop decreases. The simulations results are compared with bothan analytic theory of Muschelknautz [Die Berechnung von Zykonabscheidern für Gase. Chem Ing Techn 44,(1+2):63–71, 1972] as well as with real plant measurements.

List of Symbols

B mass loading (.)CD,(L) drag, (lift) coefficient (.)d particle diameter (m)e restitution coefficient (.)E wall function constant (=9.793)

f specific force vector (m/s2)g radial distribution function (.)g gravitational acceleration vector (m/s2)

k specific turbulent kinetic energy (m2/s2)

K + roughness height (m)Kqp momentum exchange coefficient (kg/m3s)

S. Pirker (B) · D. KahrimanovicJohannes Kepler University, Linz, AustriaE-mail: [email protected]

G. AichingerSiemens VAI Metals Technologies, Linz, Austria

204 S. Pirker et al.

p pressure (Pa)m, m mass, mass flow rate (kg, kg/s)r radial coordinate (m)Re dimensionless Reynolds number (=uL ρ/m)t time (s)u∗ dimensionless velocity (.)u velocity vector (m/s)x Cartesian coordinates (m)y∗ dimensionless wall distance (.)

Greek symbols

α volume fraction (.)ε turbulent dissipation rate (m2/s3)κ von Karman constant (=0.4187)µ dynamic viscosity (kg/ms)ρ density (kg/m3)φ specularity coefficient (.)ϕ angle, angular coordinate (.)τ stress tensor, wall stress vector (kg/ms2)

� granular temperature (m2/s2)

Subscripts

g continuous gas phasep discrete Lagrangian particless continuous Eulerian particle phase

1 Introduction

Cyclones for separating solid particles out of a gas flow have been in use for over a century. Their behaviorhas been comprehensively investigated and many empirical and analytical theories for predicting the cyclone’sperformance have emerged (a recent overview is given in [2]). In the field of numerical simulations, cycloneshave been initiating applications that triggered the early development of commercial Computational FluidDynamics (CFD) tools [3]. Nevertheless, if a more detailed look on the separation process (e.g., for highlyladen cyclones) is required, efficient simulation models are still on need.

From a technological point of view, there are three main criteria describing the performance of a cycloneseparator. First, the global separation efficiency expressed by the mass flow rate of separated particles dividedby the incoming particle mass flow rate is of major interest. Next, the exhaust particle spectrum can befocused: what minimum particle diameter can be separated by a certain type of cyclone? Finally, the costs ofoperation expressed by the cyclone’s pressure loss should be considered. Beside these main criteria additionaltechnological requests might involve minimizing the wear of the cyclone’s lining or reducing the attrition ofthe processed particles.

Addressing the above technological design criteria by means of CFD simulations represents a tough chal-lenge for a particle flow model. On the one hand, the model should be detailed enough to consider interactionsbetween the particles and the most likely turbulent gas flow field as well as inter-particle and particle-wallcollisions. On the other hand, the model should remain affordable with respect to the computational costs.Especially in highly laden cyclones this second requirement quickly constitutes the limiting factor for detailedparticle simulations.

In the last decades an impressive portfolio of particle flow models has emerged. In principle these particlemodels can be organized into two fundamental modeling concepts—Lagrangian and Eulerian models. Whilewithin the first approach the trajectories of distinct particles, or representative parcels of particles are traced in aLagrangian frame of reference, the second model smears out individual particles and considers the particles asan additional particle phase instead. Both approaches have their pros and cons. Within the Lagrangian model,physically relevant mechanisms like particle rotation and Magnus force, the interaction between particles andthe fluid’s turbulence or the detailed particle rebound mechanism at rough walls can be readily modeled.Nevertheless, in this context the modeling of inter-particle collisions is often restricted to statistical methods

Modeling mass loading effects in industrial cyclones by a combined Eulerian–Lagrangian approach 205

or overlapping soft sphere models. In case of dense particle flow situations (e.g., in the wall region of highlyladen cyclone separators) even those simplified methods might fail due to the immense computational costs.In the latter case of dense particle flows in many cases Eulerian particle models have proven to sufficientlyrepresent the global particle flow situation (e.g., Gidaspow [4]). By applying mathematical approaches knownfrom the kinetic theory of nonuniform gases (Chapman and Cowling [5]) the effect of inter-particle collisionsis expressed by global quantities like a granular viscosity, a granular temperature or a granular pressure. Never-theless, within the Eulerian approach modeling individual particle properties like different particle diametersor densities requires the definition of several Eulerian particle phases and thus leads to a tremendous increasein computational costs. Furthermore, at least in its standard formulation the Eulerian particle model lacks thepossibility to account for detailed particle rebounds at walls or the effect of particle rotation.

With these introducing remarks a combined Eulerian–Lagrangian model seems to be obvious. The basic ideaof a combined model is to account for the locally dominating inter-particle collisions by means of a Eulerianapproach while the detailed behavior of distinct particles in dilute regions is captured by the Lagrangianmodel. Zwinger [6] has realized this coupling methodology by a domain decomposition method. Thereby,he introduced an infinitely thin re-suspension layer (RSL) in between the dense and the dilute region of anavalanche. At this RSL he defined the exchange of particles by applying Reynolds analogy between momentumand particle diffusion. Pirker and Kahrimanovic [7] subsequently adopted his method for the simulation ofhighly laden cyclone separators. In another coupling approach Popoff and Braun [8] introduced an additionalgranular pressure force for the Lagrangian particles. This granular pressure force is derived from the localgradient of the granular pressure that stems from a Eulerian particle model.

In this paper, another candidate for a combined Eulerian–Lagrangian particle model is presented. In thismodel a collisional force is defined for the Lagrangian particles while a locally varying particle diameteris suggested for the Eulerian particle phase. After a description of the mathematical model in Sect. 2 themodel is applied to an industrial cyclone separator in Sect. 3. Thereby, emphasis is laid on the effect of massloading because with increased mass loading inter-particle collisions inside the particle strand at the wall startdetermining the global separation efficiency. Finally, a conclusion and an outlook complete this paper.

2 Modeling

The following section starts with a short description of the gas flow modeling. Thereby, a single-phase flow isconsidered first before the influence of particles is accounted for later on. Next, the basic modeling concepts forthe Lagrangian and Eulerian particle models are outlined before finally the coupling mechanisms are presented.

2.1 Gas flow model

The single-phase gas flow is based on the incompressible Navier–Stokes’s (NS) equations

∂ρg

∂t+ ∇ · (

ρgug) = 0,

∂(ρgug

)

∂t+ ∇ · (

ρgugug) = −∇ p + ∇ · τ g + ρgg

(1)

with

τ g = µg(∇ug + ∇uTg ) − τ g,RS (2)

denoting the gas-phase stress tensor that explicitly accounts for the Reynolds stresses

τg,RS,i, j = ρgu′i u

′j . (3)

Due to the dominating swirl inside the cyclone separator an anisotropic turbulence field establishes. Tho-roughly resolving this turbulence pattern is mandatory in order to properly picture the mean flow’s vortexstructure that is responsible for the cyclone’s inner separation efficiency. The Reynolds Stress (RS) model hasproven to be an efficient method for modeling anisotropic turbulence [9]. In this model a transport equation isdefined for each Reynolds stress component,

∂(ρgu′

i u′j

)

∂t+ ∇ ·

(ρgugu′

i u′j

)= Di, j + Pi, j + φi, j − 2

3δi jρgε. (4)


Table 1 Geometric extensions, properties and selected boundary conditions

H 13.8 m b 1.33 m ρg 1.39 kg/m3

HC 5.8 m S 1.2 m µg 1.8 10−5kg/ms

HD 6 m D 1.6 m ρp(s) 3615 kg/m3

Hv f 1.33 m Dv f 0.76 m vin,g(p) 17 m/sa 0.35 m DD 0.46 m K + 5 mm (rough walls)

The terms on the right side of above equations describe laminar and turbulent diffusion, stress production,linear pressure strain interactions and dissipation, respectively. Details on the modeling of these terms can befound in, e.g., [10]. In this place it should be sufficient to note that all of these terms are functions of the meangas-phase velocity gradients. Note that the dissipation rate ε in the last term is modeled as a scalar quantitywhich is expressed by a further transport equation,

∂(ρgε

)

∂t+ ∇ · (

ρgugε) = Dε + Cε1

2Pi,i

ε

k− 2

3Cε2

ε2

k. (5)

The three terms on the right side of Eq. (5) describe the effective diffusion, production and destruction ofthe dissipation rate with the specific kinetic energy k being defined by

k = 1

2u′

i u′i . (6)

At the inlet of the cyclone separator a constant gas velocity is set (the boundary conditions are summarizedin Table 1). The gas is assumed to enter the cyclone homogeneously in a normal direction. The correspondingturbulence quantities are deduced from a characteristic length scale and a turbulence intensity. At the cyclone’sexhaust outlet, a radial distribution of the static pressure is defined by

p (r) = ρg(r�)2

2, (7)

where � is the solution dependent rotational velocity of the inner vortex. Since the cyclone is submerged intoa fluidized bed, a corresponding mean hydrostatic pressure is set at the lower dust exit. Thus, a secondary gasflow establishes from the dust exit to the exhaust exit. Nevertheless, this short-cut flow is restricted by theamount of particles that inevitably enter from the surrounding fluidized bed as described later on. At walls themean velocity as well as the Reynolds stresses are set zero. The gas phase boundary layer is not resolved, butmodeled by a standard wall function approach. Thus, the mean velocity distribution at the wall is assumed toobey a logarithmic wall profile

u∗ = 1

κln

(Ey∗) − �B, (8)

with

�B = 1

κln

(1 + 1

2K +

). (9)

accounting for the influence of wall roughness with K + describing the roughness height.

2.2 Lagrangian particle model

The Lagrangian particle model is based on a local force balance for a distinct particle,

d

dtup = 18µg

ρpd2p

CD Rep

24

(ug − up

) + g + fSaff ,

d

dtxp = up,

(10)


Fig. 1 Sketch of the a virtual wall concept and the b roughness shadow effect; the corresponding probability density functionf of the inclination angle ϕ is sketched by the insert

with the drag coefficient CD defined according to Morsi and Alexander [11]. In above equation the particle’sinertia is balanced by a set of forces describing drag, gravity and lift force due to local gradients in the gasphase’s velocity (Saffman force),

fSaff = 2CL√

µgρgdi j

ρpdp (dlkdkl)1/4

∣∣ug−up

∣∣ . (11)

In above force term di j are the components of the gas-phase deformation tensor. In the case of heavysolid particles in an ambient light gas unsteady forces like the added mass force or the Basset force might beneglected. Furthermore, forces due to particle rotation (Magnus force) are excluded from this investigation.

In principal equation (10) considers the influence of the gas mean flow on a particle trajectory. In order toaccount for the influence of turbulent fluctuations on the particle’s trajectories, the local drag force should bebased on the actual gas velocity, Ug = ug + u′

g , rather than on its mean value. In the Discrete Random Walk(DRW) model (e.g., [12]), the fluctuating part of the velocity is derived from the specific turbulent kineticenergy by

u′g =

√2k

3eσ , (12)

with eσ being a randomly orientated vector of unit length. This fluctuating velocity component acts for acharacteristic eddy lifetime,

�tp = 0.3k

ε, (13)

before the next fluctuating velocity component is evaluated.In the context of particle-wall collisions wall roughness plays an important role and should be reflected

by a suitable rebound model. In the following, particle-wall collisions are modeled by a virtual wall conceptas described previously by Tsuji [13] and Sommerfeld [14]. In this approach the wall is virtually inclinedat the place of collision (see Fig. 1a). Thereby the artificial inclination angle is based on a normal Gaussiancontribution. In case of rough walls, this distribution is broad while it degenerates to a Dirac impulse in caseof a perfectly smooth surface. Obviously, particles can only hit restricted regions of the roughness structure asindicated in Fig. 1b. For this reason, the Gaussian distribution function of the virtual angle has to be shiftedtowards positive values [15] (see also insert in Fig. 1b). This “shadow effect” is important if the particles andthe roughness structures are of the same order of magnitude. Once the local wall inclination is determined byabove modeling the particle-wall collision is modeled by a hard sphere model [13]. In this model, sliding andnonsliding friction are considered within the compression and recovery period. As a result of this collisionmodel, the particle’s translational and rotational velocities after the particle-wall rebound are obtained. Detailsof the modeling concepts can be found in the cited literature.

The Lagrangian particles’ properties are defined corresponding to the incoming particle spectrum. At thecyclone’s inlet they are released with the same velocity as the supporting gas phase. At the exhaust outlet and atthe lower dust exit the particles can freely escape the cyclone. Furthermore, the exhausting Lagrangian particlesare monitored in order to obtain the cyclone’s global and fraction separation efficiencies. Note that within thismodeling concept, the Lagrangian particles are traced passively throughout the domain on a temporally fixedgas flow field.


2.3 Eulerian particle model

In the framework of Eulerian particle modeling the particles are pictured as an additional continuous phase.The Eulerian particle phase consequently obeys to a similar set of conservation equations like the gas phase.Thus, in an Eulerian particle simulation mass and momentum conservation can be written in a general formfor both the gas and the particle phase by

∂

∂t

(αqρq

) + ∇ · (αqρquq

) = 0,

∂

∂t

(αqρquq

) + ∇ · (αqρququq

) = −αq∇ pq + ∇ · τ q + αqρqg + Kqp(up − uq

),

(14)

with the phases’ volume fractions summing up to∑

q

αq = 1. (15)

In Eq. (14) the Reynolds stress tensor, the inter-phase momentum exchange as well as the granular pressureand the granular viscosity need further modeling. First, a multi-phase version of the RS model [16] is basedon phase-averaged values for the local density and velocities,

ρ =∑

q

αqρq , u =∑

q αqρquq

ρ. (16)

Next, in case of the particle phase (denoted by subscript s) a granular pressure and a granular viscosity arededuced from kinetic gas theory (e.g., [4]):

ps = pg + ρs�s + 2ρs (1 + ess) αs g0�s, (17)

and

µs = αsdsρs√

�sπ

6 (3 − ess)

[1 + 2

5(1 + ess) (3ess − 1) αs g0

]+ 4

5αsρsds g0 (1 + ess)

(�s

π

)1/2

. (18)

Both terms depend on the radial distribution function

g0 =[

1 −(

αs

αs,max

) 13]−1

(19)

and on the granular temperature �s that obeys to the following transport equation

∂ (αsρs�s)

∂t+ ∇ · (αsρsus�s) = P� + D� − γ� + φsl . (20)

The terms on the right hand side of the above equation describe the production of granular temperature, itsdiffusion as well as dissipation due to collisions and inter-phase energy exchange. Details of the modeling areonce more given in [4]. Recently, van Wachem et al. [17] showed that in many applied particle flow situationsthis transport equation is dominated by its source terms and thus, might be simplified to an algebraic equation:

0 = P� − γ� + φsl . (21)

Finally, the momentum exchange between the continuous gas phase and the particle phase which is expres-sed by the last term in Eq. (14) is modeled by [18]

Ksg = 3αsαgρg

4v2r,sds

CD

(Res

vr,s

) ∣∣us − ug∣∣ (22)

with

CD =(

0.63 + 4.8√

Res/vr,s

)

(23)


and

vr,s = 1

2

(A − 0.06Res+

√(0.06Res)

2 + 0.12Res (2B − A) + A2

), (24)

where the coefficients A and B in the above expression are further detailed in [18].If two particle classes (e.g., large and small particles) are considered, an additional set of conservation

equations has to be added. Thereby, an inter-phase momentum exchange between the two particle classes land s reads [19]

Kls = 3 (1 + els) αsρsαlρl (dl + ds)2 g0

4(ρld3

l + ρsd3s

) |ul − us | . (25)

At the cyclone inlet the Eulerian particle phase is introduced with the same velocity as the gas phase. At thelower dust exit and at the exhaust outlet the Eulerian particle phase can freely escape. At walls a momentumtransfer due to particle-wall collisions is modeled by [20]

τ s = −π

6

√3φ

αs

αs,maxρs g0

√�sus,‖, (26)

with us,‖ denoting the solid phase velocity’s component that is tangential to the wall and φ describing thespecularity of the wall. Note that in this case τ s is a vector rather than a tensor. A value of φ = 0 indicatesthat particle-wall reflections are assumed frictionless and elastic, while φ = 1 describes completely diffusiveparticle reflections.

2.4 Combined approach

If in a virtual one-dimensional experiment, a particle stream is released in quiescent ambient air it will accelerateuntil the drag force balances the gravitational force. Actually this terminal falling velocity depends on theparticle diameter. Larger particles will fall faster than smaller ones. If in a next step two different particleclasses (e.g., small and large particles) are released together, they will inevitably interact by inter-particlecollisions. Thus, the class of large particles will be a little bit retarded and the smaller particles will beaccelerated by the inter-phase momentum exchange [described by Eq. (25)]. Actually, a third substitute classmight be defined that represents the movement of the center of gravity of the two original particle classes. Incase of constant particle density, it can be shown by simple numerically evaluating above one-dimensionalmomentum balances that the substitute particle class can be approximately characterized by the Sauter diameterof the original particle classes,

d32 = d31 + d3

2

d21 + d2

2

. (27)

In the combined Eulerian–Lagrangian approach information of distinct particle diameters is solely carriedby Lagrangian particles that represent the incoming particle spectrum. On their way through the computationaldomain in each cell a mass weighted local Sauter diameter is evaluated by

d32,cell =∑

pm pm p

d3p�tp,cell

∑p

m pm p

d2p�tp,cell

. (28)

In above expression the sums are taken over all Lagrangian particle trajectories and �tp,cell represents thetime-span the particle needs to pass the very computational cell. In the next time-step, the Eulerian particlephase conservation equations are based on this characteristic diameter. Thus, different particle classes are notrepresented by a set of Eulerian particle phases, but by a characteristic substitute class that is based on a locallyvarying mean particle diameter.

Up to now the Lagrangian particles have been traced independently of each others. In case of highlyladen cyclone separators dense particle strands are formed at the outer walls and inter-particle collisions startsdominating the behavior of distinct particles. Thus, at least the mean effect of inter-particle collisions on thetrajectory of a Lagrangian particle has to be accounted for.


Fig. 2 Sketch of the cyclone separator geometry (see also Table 1) and nomenclature for post-processing

Actually, a Lagrangian particle can also be regarded as an additional Eulerian particle phase which normallyrepresents many physical particles. Between this “new” particle phase and the original Eulerian mean particlephase obviously momentum exchange occurs (see Eq. (25)). This momentum exchange might be translatedback to the Lagrangian particle as an additional specific force by

fEu = 3(1 + eps

)αsρs

(dp + ds

)2g0

4(ρpd3

p + ρsd3s

)∣∣up − us

∣∣ (up − us). (29)

In the above equation, the subscript p refers to the Lagrangian particle while the subscript s hints to theEulerian substitute particle class. This mean collisional force tends to catch Lagrangian particles in denseEulerian particle strands, while it is irrelevant in dilute regions where αs → 0. With respect to cycloneseparators this effect is responsible for the increased separation efficiency in case of higher mass loadings. Ifa dense particle strand at the wall is formed even small particles can be captured and thus be separated.

The numerical procedure of the combined Eulerian–Lagrangian approach can be summed up as follows.At first, the Eulerian particle model is applied based on the Sauter diameter of the incoming particle spectrum.Next, in certain time intervals Lagrangian particles are started from the cyclone’s inlet. Based on a fixed gasvelocity field these particles are subsequently traced through the cyclone. Thereby, the particles experience amean collisional force especially in dense particle regions. Simultaneously, the local mean Sauter diameter ofthe Eulerian substitute particle class is updated. After the Lagrangian particle calculation the next computationalcycle starts again with an Eulerian particle simulation that is by now based on the locally varying mean diameter.The combined simulation is assumed converged if characteristic global monitors (e.g., the cyclone’s pressuredrop or separation efficiency) have reached quasi-steady state.

3 Results

The above model has been implemented into the commercial finite volume solver Fluent 6.3 [21]. In thiscode the Lagrangian particle trajectories are integrated by a Runge–Kutta scheme. The diffusive fluxes ofthe conservation equations are discretized by second order central differencing while the convective fluxesare based on the QUICK scheme. Grid sizes range from 80 to 220 k cells and time-steps of 10−3 s are used.Typically, a combined simulation takes about 2 days on a six-processor Linux cluster.

An industrial cyclone sketched in Fig. 2 is chosen as a test case. Actually, this application is quite demandingfor a particle model since the cyclone is submerged into a fluidized bed. Thus, in the lower part of the cyclonea dense particle region has to be accounted for in any case. In order to test the simulations the numerical


Fig. 3 Profiles of a static pressure, and b Eulerian particle volume fraction at the cyclone’s axis (x = y = 0) for a mass loadingof B = 0.01

results are compared to Muschelknautz’ analytic theory on cyclone performance [1]. Although this theorydoes not consider a secondary gas flow through the cyclone’s dust exit, it delivers comparable fundamentaldependencies. Furthermore, real plant measurements round up the picture.

In Fig. 3b, the distribution of the particles’ volume fraction at the cyclone’s vertical axis clearly indicatesthat a fluidized bed establishes in the lower part of the cyclone even in case of low mass loading (B = 0.01).Corresponding to this result the pressure distribution in Fig. 3a shows a nearly linear hydrostatic increase in thatregion. In the upper part of the cyclone the pressure is reduced due to the global vortical flow. The swirling flowis further depicted by the gas-phase tangential velocity in three horizontal axes at different vertical positions inFig. 4. The profiles clearly highlight the influence of the cyclone’s mass loading on the formation of the innervortex. In case of higher mass loading, the supported particles retard the swirling gas flow and thus weakenthe strength of the vortex. Furthermore, due to the asymmetric formation of meandering particle strands atthe cyclone’s walls the axis of the inner vortex axis is bent and the vortex eventually breaks up in the conicalregion of the cyclone. Muschelknautz’ theory offers only one characteristic tangential velocity on an innercylindrical surface (see Table 2) that does not depend on the vertical coordinate. Therefore, a quantitativecomparison should be taken with care. Nevertheless, Muschelknautz’ theory agrees in the principal tendencythat the vortical flow is weakened at higher mass loadings.

As a consequence of this reduced swirling flow the cyclone’s pressure drop is significantly reduced.The simulation results in Fig. 5 show that the cyclone’s pressure loss reduces with increased mass loading.Nevertheless, at excess mass loadings (e.g., B = 10) this trend is reversed. In this case the particle strand atthe wall is partly re-entrained into the inner core and accelerating those particles causes a slightly increasedpressure drop. Since this re-entrainment is inherently unsteady also the time-dependent pressure fluctuationsindicated by the vertical bars in Fig. 5 are significantly increased. While Muschelknautz’ theory agrees withthe first tendency (see Table 2) it cannot picture the effect of particle re-entrainment at excess mass loadings.Nevertheless, recent experiments [22] support this reversion of the pressure drop dependency at higher massloadings.

In case of highly laden cyclones, the formation of a particle strand at the wall significantly influences theseparation efficiency. In Fig. 6 the volume fraction of the particles at three perimeters at different verticalpositions is given. In case of higher mass loadings (e.g., B = 1 or B = 10) the meandering particle strandcan clearly be identified. Once particles enter this dense particle strand, their behavior is governed by inter-particle collisions. In terms of the present mathematical model, Lagrangian particles are thus guided by thecollisional force (29). In Fig. 7 three particles of different diameter are introduced at the same initial inlet


Fig. 4 Profiles of the y-velocity along the x-axis (y = 0) at three vertical positions depicting the inner vortex structure for fourdifferent mass loadings (B = 0.01, 0.1, 1, 10), in addition rough walls are considered for B = 0.01

Table 2 Results of Muschelknautz’s analytic theory [1] as well as real plant measurements

B = 0.01 B = 0.1 B = 1 B = 10 B = 0.01 rough wall B = 1 real plant

vt (m/s) 25.4 24 19.5 16 19.7 –�p (Pa) 3,170 2,729 1,790 1,140 1,350 2,000ηglobal(%) 95.7 98.6 99.7 99.9 89.3 98.5d50 (mm) 4 4.4 6.5 11 14 4

position nearby the outer wall. In case of low mass loading (and thus no particle strand formation), the smallestparticle is exhausted after several revolutions. Medium and large particles experience many revolutions beforethey migrate to the lower part of the cyclone. In contrast to that all particles are captured and are thus separatedby the dense particle strand in case of a high mass loading. In that case the number of revolutions is reducedto approximately one. This tendency is further reflected by Fig. 5 that indicates a correspondingly increasing


Fig. 5 Simulation results of (top) global separation efficiency and (bottom) pressure drop in dependence of the mass loadings;symbols represent mean values, bars denote time-dependent variations, in addition the open symbol depicts the case of roughwalls

global separation efficiency. Only in the last simulation with an excess mass loading of 10 this trend is reversedagain. In that case the particle strand is not completely separated, but partly re-suspends into the inner regionof the cyclone.

Next, the particle spectrum at the cyclone’s exhaust outlet can be evaluated. In Fig. 8 the incoming particlespectrum is given together with the exhausted particle spectra for different mass loadings. The predicted d50diameter of 2mm is slightly smaller than indicated by Muschelknautz’ theory and real plant measurements.That discrepancy might stem from the assumption of perfectly spherical particles. Nevertheless, the exhaustspectra indicate that the separating d50 diameter is highest for the maximum mass loading which again agreeswith Muschelknautz’ theory.

For the case of low mass loading, the effect of wall roughness has been additionally investigated. Anincreased wall roughness has a direct effect on the strength of the cyclone’s inner vortex. Figure 4 clearlyindicates that the corresponding profile of tangential gas-phase velocity is weakened. As a first consequence,the pressure drop is dramatically reduced, which might seem strange at first glance, but agrees very well withmeasurements (e.g., [2]) and with Muschelknautz’ theory. As a second consequence, the simulations predicta slightly reduced separation efficiency as shown in Fig. 5.

4 Conclusion

A combined Eulerian–Lagrangian particle model has been introduced in order to study highly laden cycloneseparators. The numerical model works numerically stable and produces reasonable results at comparable lowcomputational costs.

The simulations show that in case of low or medium laden cyclones the pressure drop decreases withincreasing mass loading. At the same time, the global separation efficiency increases while the separatingdiameter d50 nearly stays the same. Nevertheless, at an excess mass loading these trends are reversed. In that


Fig. 6 Profiles of the Eulerian particle volume fraction at three perimeters for three vertical positions z, in case of higher massloadings (B = 1 or B = 10) particle strands meander down the cyclone’s wall

case the pressure drop increases again, the separation efficiency declines and the corresponding d50 diameteris larger. Furthermore, in case of low mass loading, rough walls cause a reduction of both pressure drop andglobal separation efficiency.


Fig. 7 Characteristic particle trajectories for three different particle diameters and two mass loadings

Fig. 8 Incoming and exhausted particle spectra for four different mass loadings

Although the presented combined model represents a promising simulation candidate, still a trade offbetween feasibility and accuracy had to be made. In that context further research efforts should study the effectof nonspherical particles as well as grid dependencies, especially at the boundary of the outer walls.

Acknowledgments This R&D work has been partly funded by the Austrian Ministry for Economy and Labor in the frame of itsIndustrial Competence Centre Program “KnetMET”.

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Volume 204, Numbers 3-4, 203-216