wall depositions in gas cyclones - tu/e

Wall Depositions in

Gas CyclonesJ.J.H. Houben

May 2006

WPC 2006.06

Technische Universiteit Eindhoven

Technische Universitat Graz

Master Thesiswith titel

Wall Depositions in Cyclones

AuthorJ.J.H. Houben

id 496382Eindhoven University of Technology

May 24, 2006

Supervisors

Prof. dr. ir. J.J.H. BrouwersDr. ir. F.L.A. Ganzevles

Technische Universiteit Eindhoven

O.Univ.-Prof. Dipl.-Ing. Dr.techn. G. StaudingerDipl.-Ing Dr. techn. G. Gronald

Technische Universitat Graz

Contents

Summary 1

Samenvatting 3

Nomenclature 5

1 Introduction 11

2 Cyclone Geometry and Working Principle 13

2.1 Inlet Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Solid Outlet Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Experimental Observations on Wall Depositions 17

3.1 Wall Fouling Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Wall Depositions in Gas Cyclones . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Deposits at the Natural Vortex End . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Particle-Wall Adhesion Criterion 25

4.1 Coefficient of Restitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Sensibility Analysis of the Critical Sticking Velocity . . . . . . . . . . . . . . . 314.3 Influences of the van der Waals and Electrostatic Energies . . . . . . . . . . . . 31

4.3.1 Experimental Observations on the Influences of the van der Waals andElectrostatic Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.4 Influence of the Particle Layer Thickness . . . . . . . . . . . . . . . . . . . . . . 35

5 Resuspension Models 37

5.1 Force Balance Based Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.1.1 The Reeks, Reed and Hall Model . . . . . . . . . . . . . . . . . . . . . . 385.1.2 The Reeks, Reed and Hall Model for Zero Energy Transfer . . . . . . . 385.1.3 Reeks Reed and Hall Model for Non-zero Energy Transfer . . . . . . . . 39

5.2 Moment Balance Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.2.1 The Dynamic Rock’n Roll Model with Resonant Energy Transfer . . . . 405.2.2 ’Quasi-static’ Rock’n Roll Model . . . . . . . . . . . . . . . . . . . . . . 42

ii Contents

5.3 Resuspension of Aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.4 Resuspension by Impacting Particles . . . . . . . . . . . . . . . . . . . . . . . . 45

6 CFD-Results 47

6.1 Comparison of the Experiments and CFD-Simulations . . . . . . . . . . . . . . 476.2 Sub-micron Particle Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.3 Discussion Differences between Measurements and Simulations . . . . . . . . . 48

7 Conclusions and Recommendations 51

Bibliography 55

List of Figures 59

List of Tables 61

A Cyclone Geometry 63

B Particle Distributions 65

B.1 Particle Size Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65B.2 Dimensionless Particle Relaxation Time Distributions . . . . . . . . . . . . . . 66

C Flow Models 71

C.1 Turbulent Flow Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71C.2 Two Phase Flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71C.3 Particle-Particle Collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74C.4 Wall Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

D Particle Layers 77

D.1 Particle Layer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77D.2 Layer Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

E Error Analysis 81

Summary

Gas cyclones are used to separate a dispersed phase (dust) from a gas (e.g. air). The mixtureis brought in rotation because of the geometry. The dispersed phase (which has a larger massdensity) accelerates outside due to the centripetal force. It flows along the wall toward thedust container where it is collected. The gas leaves the cyclone through the vortex finder suchas some small particles.

However, in gas cyclones sometimes particle depositions are built up, which has a negativeeffect on the performance. This can be prevented by coating the steel wall or the particle mustbe removed afterwards, e.g. by a deformable wall. Unfortunately, both methods cost extraoperation time and money.

To gain an insight in the mechanism that causes depositions, a physical model that isbased on energy maintenance has been made. From this, a critical sticking velocity has beencomputed. Beneath this velocity a particle will stick and above it will rebound. From theenergies that are the driving forces behind depositions, the van der Waals energy is the moreimportant one for particles smaller than 2 µm and the electrostatic energy for larger particles.Furthermore, the coefficient of restitution, which describes the energy efficiency of an impact,is an important parameter in the model.

To describe the asymptotic behavior of the layer thickness several resuspension models havebeen discussed. The simplest ones consist out of a force or momentum equilibrium. In the morecomplicated ones statistical flow parameters are needed. It is also possible that agglomerates ofparticles resuspend or that single particles resuspend by the energy of other impacting particles.

The Euler-Lagrange solving method has been used to simulate the particle tracks in theCFD-package. In the computed flow field (steady state) a number of particles has been injectedand their trajectory has been computed. The wall adhesion criterion has been implemented inuser defined boundary functions.

It was found that only very small particles (dP < 1µm) deposited. This in contrast withexperiments with the test cyclone where larger particles were found in the deposits as well.From experiments in literature, it is known that first small particles deposit and after that thelarger ones. In the end the layer approaches an asymptotic thickness. It is also known wherethe depositions are found in gas cyclones.

2 Nomenclature

The cyclone separation efficiency and the fine and coarse dust distribution found in thesimulations correspond with the experiments. The simulated depositions exist out of smallerparticles than measured in the experiments but the places where they are found are similar.

Samenvatting

Gascyclonen worden gebruikt om een disperse fase te scheiden van een continue fase. Hetmengsel wordt in rotatie gebracht ten gevolge van de cycloongeometrie. Doordat de dispersefase een grotere massa dichtheid heeft, wordt deze naar buiten geslingerd. De stroming langsde wand is naar beneden gericht. Hierdoor volgt de disperse fase deze en wordt zij tenslotteopgevangen in een container. Het gas verlaat de cycloon door de ”vortex finder” samen metenkele kleine deeltjes.

Soms ontstaan er echter wandafzettingen in cyclonen die een negatief effect hebben op dewerking. Mogelijkheden om dit te voorkomen dan wel te genezen zijn het coaten van de wandof het achteraf verwijderen van de afzetting door bijvoorbeeld een vervormbare wand. Helaaskosten beide methoden geld en tijd.

Een fysisch model dat gebaseerd is op energiebehoud is gemaakt om meer inzicht te verkrij-gen in het mechanisme dat wandafzettingen veroorzaakt. Hieruit kan de kritische plaksnelheidberekend worden. Een deeltje dat langzamer is dan deze snelheid zal blijven plakken, eendeeltje dat sneller is weerkaatsen. De van-der-Waals-energie en de electrostatische energie zijnde drijvende krachten achter afzettingen. De van-der-Waals-energie is het belangrijkst voordeeltjes kleiner dan 2 µm, voor grotere deeltjes is de electrostatische energie het belangrijkst.Tenslotte is de restitutie-coefficient, die het rendement van de botsing beschrijft, een belangrijkeparameter in het model.

Het asymptotisch gedrag van de wandafzetting kan beschreven worden met resuspensie-modellen. De eenvoudigste bestaan uit een krachten- of momentevenwicht. Voor de gecom-pliceerdere zijn statistische stromingsparameters vereist. Het is ook mogelijk dat agglomeratenals geheel worden teruggebracht in de stroming of dat deeltjes door de energie van anderebotsende deeltjes los worden getrild.

De deeltjesbanen zijn berekend in een CFD-pakket met de Euler-Lagrange methode. Hierbijwordt eerst de continue fase opgelost waarna de deeltjes worden geınjecteerd. De plakvoor-waarde is geımplementeerd in ”user defined boundary functions”.

In de simulaties bleven alleen erg kleine deeltjes (dP < 1µm) plakken. Dit in tegenstellingtot experimenten met de testcycloon waarbij ook grotere deeltjes in de wandafzetting werdengevonden. Uit experimenten vermeld in de literatuur is bekend dat eerst de kleine deeltjes blij-

4 Nomenclature

ven plakken en daarna de grotere. Tenslotte nadert de dikte van de afzetting een asymptotischewaarde. Uit experimenten is ook bekend waar de afzettingen gevonden worden.

Van de CFD-resultaten komt het rendement en de verdelingen van grof- en fijnstof overeenmet de experimenten. De deeltjes zijn weliswaar kleiner maar de plaats waar ze gevondenworden, komt overeen met experimenten.

Nomenclature

A [m2] surfacea [m] height of the inlet of the cycloneb [m] width of the inlet of the cycloneC [-] coefficientc [kg m−3] particle concentrationCF () [-] cumulative distribution functionD [m] diameter of cyclone partd [m] (particle) diameterd′P [m] particle diameter for which CFRRSB(d′P) = 0.632E [J] energye [-] coefficient of restitutionF [N] forcef [-] friction coefficientG [-] geometry factorG(ζi) [-] Gaussian distributiong [m s−2] acceleration due to gravityH [m] heighth [m] depth/thickness~$ [-] Lifshitz-van der Waals constantI [kg m2] moment of inertiaJ [kg m−2s−1] mass fluxk [J K−1] Boltzmann constantL [m] natural length of vortexl [m] distance between particle and wall before collisionm [kg] massN [-] numbern [s−1] maximum resuspension ratenRRSB [-] parameter in RRSB distribution function according to DIN

66145P [-] porosityP () [-] probabilityPDF () [-] probability distribution functionPcol [-] collision probabilityp [Pa] pressure

6 Nomenclature

Q [-] ratio of adhesive force to instantaneous aerodynamic re-moval force

q [C] charger [m] radiusS [m] length of the vortex finders [m] particle track distance / stopping distanceT [K] absolute temperaturet [s] timeu [m s−1] fluid velocityV [m3] volumevdep [m s−1] deposition velocitywP [m s−1] particle velocityx [m] coordinateY [-] dimensionless coordinatey [m] coordinateZ [m−3 s−1] number of collisionsz [m] coordinate in negative gravitational directionz0 [m] distance at contact

Nomenclature 7

Greek

α [◦] wall inclinationβ [s−1] damping constantβ1 [◦] particle impact angleΓ [N ·m] coupleγ [J m−1] adhesive surface energy∆t [s] time step∆~x [m] distance vector∆α [◦] standard deviation wall inclinationδ [m] specimen thicknessε0 [Fm−1] relative permittivityζ [-] factorη [Pa s] dynamic viscosityθ [rad] angle/angular coordinateκ [-] factorλ [m] mean free pathµi [g kg−1] dust loadν [m2 s−1] kinematic viscosityξ [-] energy resonance coefficientρ [kg m−3] mass densityσ [N m−2] stressσ′ [-] standard deviationτP [s] particle relaxation timeτw [Pa] wall shear stressψ [s−1] rate of removalω [rad s−1] angular velocity

8 Nomenclature

Sub/superscripts

A particle Aa adhesionAB reduced particle A and Badh adheringag agglomerateB particle Bb bendingbar barrierC Cunningham correctionC Coulombc conical sectioncb cyclone bodycell cellcol collisioncont contactcrit criticalcycl cycloneD dragd dust exitdamp dampingdep depositiondet detachmente after collision without energy displacementeff effectiveel elasticelst electrostaticeq equilibriumG gash hydraulickin kineticL liftl losslayer layerM Magnusm meanmax maximumn normalnew new valueP particlepl plastic/yieldpore porepot potential

Nomenclature 9

R roughnessr removalrel relativeRRSB Rosin–Rammler–Sperling–Bennett distribution function

according to DIN 66145s smoothSaff Saffmansamp samplet tangentialth thermophoreticvdW van der Waalsw wallx vortex finderθ angular0 zero reference / neutral1 before collision2 after collision50 median value+ positive− negative

Acronyms

Kn Kn = λdP

Knudsen numberReP ReP = ρGdP|−→wP−−→u |

η particle Reynolds numberSc Sc = 3πνηdP

kTCcSchmidt number

τ+P τ+ = τu∗2

ν dimensionless particle relaxation timev+dep v+

dep = vdepu∗ dimensionless particle deposition velocity

Overlines

¯ meanˆ normalized˜ fluctuating (zero mean) component~ vector∗ friction

Mathematical operations

˙ time derivative¨ double time derivative| | absolute value〈〉 average value

1 Introduction

Cyclones are used to separate a dispersed phase from a continuous phase. The two phase flowis brought into rotation and due to the difference in mass density the particles of the heavierphase are forced to accelerate into the direction of the wall. Particles move downwards becauseof gravity and the local flow and are collected in the dustbin at the bottom of the cyclone.The cleaned fluid leaves the cyclone through the vortex finder. However, it might happen thatparticles stick at the wall instead of being collected in the dustbin. In this case deposits at thewall are built up. This has a negative effect on the performance of the cyclone [12].

To prevent deposits, cyclones can be polished or coated with polymers [18]. To removethem, sometimes the wall consists of two layers. A deformable material (e.g. rubber) is usedin the inner layer. The outer layer is usually made of an undeformable material. By changingthe pressure between the layers, the inner layer vibrates and the depositions are removed fromthe wall.

In this study the following aspects of wall depositions will be considered:

� the physical conditions for wall depositions;

� a parameter study to estimate the influence of the different model parameters;

� a model for a CFD-package in order to simulate the (start of) formation of wall depositionsin cyclones;

� the growth of these depositions;

� resuspension of particles back into the flow;

� a qualitative comparison of these results obtained with the CFD-package with formerexperiments.

2 Cyclone Geometry and Working Principle

Cyclones are used to separate a dispersed phase from a continuous phase. The dispersed phase(a solid or liquid) has a larger density than the continuous phase (a liquid or gas). Only solid-gas separation will be considered in this study. A cyclone includes the following parts (figure2.1):

� Inlet (1)

� Gas outlet or vortex finder (2)

� Roof (3)

b

(1)

(2)

(3)

(4)

(5) (6)(6)

(7)

cb

cb

Figure 2.1: Schematic view of a cyclone.

14 2 Cyclone Geometry and Working Principle

� Cyclone body (4)

� Dust outlet (5)

� Vertical tube (6)

� Dust collector (7)

The dimensions of the cyclones considered are presented in Appendix A. The two phase flowenters the cyclone through the inlet. Because of the cyclone shape, it is brought into rotation.The heavier dispersed phase is moved outwards due to the centripetal force. Near the cyclonebody, the axial velocity, i.e. parallel to the cyclone axis, of the flow is pointed downwards.The particles are moved towards the dust outlet where they are collected in a dust collector.The continuous flow turns under in the conical part of the cyclone or in a cylindrically-shapedsection, which may be attached between the conus and the dustbin, and leaves the cyclonethrough the vortex finder in the roof. In section 2.1 and 2.2 several cyclone geometries arepresented.

2.1 Inlet Configurations

To connect a pipe with a filthy fluid to the cyclone, according Hoffmann and Stein [12] sev-eral inlet configurations can be applied. The following types, as presented in figure 2.2, aredistinguished:

a) For the simplest type a circular pipe is used. Since this type does not need a round-to-rectangular transition section it is used in inexpensive cyclones for applications in whichsome sacrifice in separation performance is acceptable; for example in many woodshopand grain processing units.

b) The slotted inlet (sometimes called ’rectangular’ or ’tangential’ inlet) is common used inchemical and petroleum process industries. The inlet is placed at the same radius as thecyclone barrel. From a fabrication/strength point of view it is preferred to construct theinlet slightly under the cyclone roofline rather than at the same elevation. However, inpractice a ring of circulating dust can appear along the inner roofline. Fortunately thisring does not influence the cyclone performance.

c) The third inlet type is the ’wrap-around’ inlet. Since in this type of cyclone the area forflow is decreased, the gas flow undergoes some acceleration upstream. A wrap-aroundinlet produces a larger momentum than a slotted inlet for the same cyclone diameterbecause the inlet is positioned at a larger radius than the cyclone barrel. As the flow willnot collapse easily with the vortex finder, for this part large ones can be used.

2.1 Inlet Configurations 15

Figure 2.2: Side and top views of the four most used inlet configurations (a) circular or ’pipe’ inlet,(b) ’slotted’ (also called ’tangential’) inlet, (c) ’wrap-around’ inlet and (d) axial inlet with swirl vanes[12].

d) For the ’swirl vanes’ the flow enters parallel to the axis of the cyclone. This type is moreoften inserted in cylindrical-bodied cyclones rather than in cylinder-on-cone or conical-bodied geometries. In this case, the separator is called a ’swirl tube’. One advantage ofthe axial inflow geometry is the high axial symmetry in the flow. This eliminates foulingat the backside of the vortex finder.

Of the types mentioned above, in this study, only the slotted inlet cyclone will be considered.

16 2 Cyclone Geometry and Working Principle

2.2 Solid Outlet Configurations

For the connection between the dust outlet and the dustbin several designs can be used [12].In the first place, it is possible to attach the dustbin directly to the dust outlet. This can oftenlead to the end of the vortex. An other option is to place a vertical tube between the dustoutlet and the dustbin. The vortex then may terminate in the tube section or it may extenddown into the dustbin, especially for small wall cyclones. In former experiments performed atthe Technical University of Graz, two geometries for the vertical tube (in figure A.1 indicatedwith C1 and C7) have been used [9]. Both of the tubes had an inner diameter of 180 mm butthe lengths were 300 mm and 500 mm respectively. The last possibility is to install a vortexstabilizer (e.g. of the ’Chinese hat’ type) just under or just above the dust exit. The stabilizerprevents that the vortex ends on a smooth surface and helps to center it.

3 Experimental Observations on Wall

Depositions

Only few experimental reports according wall depositions in gas cyclones are available in litera-ture. Hoffmann et al. [11] observed wall depositions in experiments for determining the naturalvortex length in cyclones. At the Technical University of Graz former work has been performedby Gebhard [9]. He investigated the places, thicknesses and masses of the depositions in twodifferent gas cyclone types for different conditions of volume flow rate and solid loading.

3.1 Wall Fouling Stages

Figure 3.1: Stages of particulate fouling [1].

Abd-Elhady [1] mentions the following stages for particulate fouling in heat exchangers(figure 3.1):

18 3 Experimental Observations on Wall Depositions

(a) At the beginning of operation the heat exchanger tubes are clean and the fouling layergrowth is slow.

(b) Fine particles deposit due to the thermophoresis mechanism. Since in a cyclone the heatgradient is much smaller than in a heat exchanger, the effect of thermophoresis willbe much smaller than in a heat exchanger. The thermophoresis force is presented inappendix C.2.

(c) The thickness of the particle layer grows by inertial impaction. According van Beeket al. [4] the transportation rate by inertia is at least one order of magnitude largerthan by thermophoresis. However, because inertial impaction particles hit the surfacewith a larger velocity, this does not mean that they automatically cause higher particledepositions.

(d) After a certain time the particle layer thickness obtains an asymptotic behavior. Thismeans that at that point the removal rate of particles balances the deposition rate. Thisis due to an increasing removal rate or a decreasing deposition rate.

This study focuses on the beginning of the depositions (i.e. stage (a) and (b) and in some casesstage (c)). Some resuspension models to simulate stage (d) are discussed in chapter 5.

3.2 Wall Depositions in Gas Cyclones

Table 3.1: Material parameters during tests [9].

Trading name OMYACARB 5-GU Chemical compositionMass density 2770 kg/m3 94% CaCO3

Median diameter dP,50 5 µm 4-4.5% MgCO3

Parameters in the RRSB equation after atomizerdP,50 4.5 µm calculated from coarse and fine dustd′P 6.06 µm calculated from median diameternRRSB 1.2

Gebhard [9] mentions wall depositions in his experiments. He used a slotted type cyclonewith two lengths for the vertical tube (as presented in figure A.1) of 500 mm (C1) and 300 mm(C7) respectively. The used solid load was limestone. Its statistical properties are represented intable 3.1. The cumulative (Rosin–Rammler–Sperling–Bennett [31, 9]) probability distributionfor the particle diameter is given by:

CFRRSB(dP) = 1− exp[−(dP

d′P

)nRRSB]

(3.1)

3.2 Wall Depositions in Gas Cyclones 19

in which nRRSB is a parameter which defines the width of the distribution and d′P is the particlediameter for which CFRRSB(d′P) = 0.632. The parameters in the tests varied, for gas and solidmass flow rates are mentioned in table 3.2.

Table 3.2: Volume flow rate and loading conditions during the tests [9].

Test parameters for geometry C1test constant parameter varied parameter

volume flow rate dust loadV [m3/h] µi [g/kg]

C1 V600 600 1.0; 2.0; 3.0; 5.31; 8.0C1 V800 800 1.0; 2.0; 3.0; 5.31C1 V1000 1000 1.0; 2.0; 3.0; 5.31; 7.9C1 V1200 1200 1.0; 2.0; 3.0; 5.31; 6.8

Test parameters for geometry C7C7 V800 800 5.31C7 V1000 1000 1.0; 2.0; 3.0; 5.31C7 V1200 1200 5.31

The deposits on the following cyclone parts were ascertained by Gebhard:

1. Cyclone roof: Only for the tests with cyclone geometry C1 and volume flow rates of600 and 800 m3/h dust deposits appear. They are ring-shaped and flat.

2. Outside of the vortex finder: For all tests a clean ring of 5-10 mm width appearsat the top of the outside of the vortex finder. Under this ring thick dust depositionsof approximately 30 mm width turn up. The rest of the vortex finder is covered withdepositions of 0.2-1 mm thickness. Between the geometries C1 and C7 no large differencesare found. The mean particle diameter of the deposits is constant for the dust load andonly little dependent on the volume flow. The thickness of the particle layer (up to 1.6mm) is nearly constant for varying volume flow rates, although a little higher for V = 600m3/h.

3. Inside of the vortex finder: The depositions at the inside of the vortex finder varyfrom small depositions over depositions with spiral ordered clean spots up to fully withdust covered areas. The masses of the depositions rise for increasing dust loads. Forvolume flow rates of 800, 1000 and 1200 m3/h, the deposits are approximately equal (upto 0.8 wt% of the injected dust). For 600 m3/h the depositions are significantly higherthan for the other volume flow rates with a maximum for µi = 2g/kg. Gebhard suggeststhat this increase is due to better circumstances for particle agglomeration at V = 600m3/h and µi = 2 g/kg. The median particle diameter appeared to be independent for thedust load but decreases for increasing volume flow rates. The thickness of the particlelayer is neither dependent of the volume flow rate nor of the solid mass flow rate.


4. Cyclone body: Mostly flat helix-shaped depositions were observed. For geometry C1and V = 600 m3/h the form was wavy-shaped. By geometry C7 the bottom part of thecyclone body was always covered with dust. The deposits in the cyclone body have thelargest contribution to the total mass of all of the deposits. Also in the cyclone body themass of the depositions increased for increasing dust loads. For the volume flow rates of800, 1000 and 1200 m3/h, the depositions were significantly lower than at 600 m3/h. Themedian particle diameter was independent of the dust load but decreases for increasingvolume flow rates. The thickness of the depositions is only for V = 600 m3/h higher.

5. Vertical tube section: For geometry C1 wall depositions only appear in the bottommost end of the tube. They are up to 2.5 mm thick. For geometry C7 the depositsin the cyclone body continue in the tube section and became thicker (circa 2.5 mm)only for the last 100 mm. The gas velocity decreases from the top to the bottom ofvertical tube according to Obermair et al. [21]. The high velocity in the top possiblyobstructs the formation of deposits. Since the deposits in the bottom part seem to beindependent of the dust load and the volume flow rate, it is plausible that in this sectionthe velocity profile is independent of these two quantities [9]. Unfortunately Zagorski [35]only measured the radial and tangential velocity using tracer particles (figure C.1). Themedian particle diameter appears to have a minimum for dust loads between 3 g/kg and5 g/kg. Dependence of volume flow rate could only be observed for V = 600 m3/h. Thethickness of the depositions is for all volume flow rates in the same order of magnitudeand decreases slightly for a dust load of 5 g/kg.

The cumulative particle size distribution of the experiment with a dust load of 3 g/kg at 1000m3/h is given in figure 3.2. The distribution of the depositions at the cyclone are indeedtranslated to smaller particles. This is in agreement with the assumption that small particlesstick easier.

3.3 Deposits at the Natural Vortex End

Hoffmann et al. [11] investigated the natural vortex length in a gas cyclone. This length is theaxial distance between the vortex finder and the point where the outer vortex flow weakensand changes its direction. Below this point, it is assumed that the gas flow has leaked entirelyto the inner vortex [23]. Hoffmann et al. observed that the position where the vortex ends, wasvisible in the pattern of wall deposits. Below the end of the vortex, the transport of particlesalong the wall towards the dust exit is ineffective. To calculate the natural length of the vortex,two models are mentioned by Hoffmann et al. [11]. First, the relation of Alexander which hasbeen introduced in 1949. It was formulated on the basis of experiments with glass cyclonesof varying diameter from 30.2 to 1200 mm. In his work however, Alexander used cyclones of

3.3 Deposits at the Natural Vortex End 21

Particle size d [µm]P

released dustfine dustcoarse dustvertical tubevortex finder insidevortex finder outsidecyclone wall

released dustfine dustcoarse dustvertical tubevortex finder insidevortex finder outsidecyclone wall

Cum

ulat

ive

dist

ribu

tion

[%

]

Figure 3.2: Cumulative particle size distributions of the experiment C1 V 1000 µ3.0 w1 [9].

geometries which are very different from those used today, and indications are that this relationis no longer sufficient [11]. His model is:

L

Dcb= 2.3

Dx

Dcb

(D2

cb

ab

)1/3

(3.2)

where L is the natural length of the vortex, Dcb the body diameter of the cyclone, Dx thediameter of the vortex finder and a and b are the height and width of the cyclone inlet,respectively. In 1983 Bryant et al. introduced a new relationship [23]:

L

Dcb= 2.26

(Dx

Dcb

)−1(D2cb

ab

)−0.5

(3.3)

The second model mentioned by Hoffmann et al. [11] is the model from Zhongli. It wasintroduced in 1991:

L

Dcb= 2.4

(Dx

Dcb

)−2.25(D2

ab

)−0.361

(3.4)

Suspicious is that the powers of the dimensionless factors in equation (3.2) on the one handand in (3.4) and (3.3) on the other hand have opposite signs. Furthermore, the factors and


Table 3.3: Natural vortex lengths for different models.

Model Natural vortex length smallest range largest rangeL [m] L± 0.5 ·Dx [m] L± 2 ·Dx [m]

Alexander 0.721 0.631 - 0.811 0.361 - 1.081Bryant 0.747 0.657 - 0.837 0.387 - 1.107Zhongli 3.924 3.834 - 4.014 4.564 - 4.284

the powers in equation (3.3) and (3.4) are different. The influence of the bases in equation(3.2)-(3.4) is represented in figure 3.3.

The natural vortex lengths that are predicted for the cyclone types used by Gebhard [9]according to the different models are mentioned in table 3.3. The relations of Alexander andBryant have a significant shorter natural vortex length than that of Zhongli, in whose equationL is even longer than the cyclone length.

Hoffmann et al. [11] found in experiments a positive correlation between the inlet velocityand the vortex length, which is not included in any of the three models. However, they didnotice that for increasing vortex diameters the vortex length has a slightly tendency to belonger. This last observation is qualitatively consistent with the prediction in equation (3.2)and inconsistent with equation (3.3) and (3.4).

0 0.2 0.4 0.6 0.8 110

-1

100

101

102

Dx/D

cb

L/D

cb

Dcb

2/(ab)=9.14

AlexanderBryantZhongli

(a)

0 5 10 15 20 25 3010

-1

100

101

102

Dcb

2/(ab)

L/D

cb

Dx/D

cb=0.375

AlexanderBryantZhongli

(b)

Figure 3.3: Natural vortex lengths, L according equations (3.2)-(3.4) as function of constant (a) DxDcb

and (b)D2

cbab .

In the experiments done by Hoffmann et al. [11] with smoke of condensed vapors of paraffinoil, the walls of the test cyclones were quickly covered with a layer of liquid paraffin. Thestriations in this layer gave a good impression of the velocity profile at the wall. Spiral patternswere visible on the wall of the cyclone separation space and in the top of tube section beneath

3.3 Deposits at the Natural Vortex End 23

(a) (b)

Figure 3.4: (a) The ring formation in the tube section. (b) A sudden transition in the amount ofdust deposits on the wall [11].

the cyclone. However, at the higher inlet velocities a closed ring was observed at a certainposition in the tube section (figure 3.4a). This position was sometimes a few centimeters abovethe smoke plum that was observed in other experiments for locating the turning point of thevortex. The position of the ring varied within a distance of half the tube diameter around themean position (figure 3.5). This range is in accordance to the experiments with vertical-cylindercyclones done by Ji et al. [23]. However, Hoffmann et al. [11] found in experiments with dust ingas cyclones a much larger range, namely one ore two tube diameters. In their experiments, asharp increase of wall deposits was observed at a certain position in the tube section underneaththe cyclone. Using a glass cyclone the phenomenon could be photographed (figure 3.4b). Forthe experiments with paraffin, the position within this range was found to move rapidly andapparently random. Occasionally the position of the ring would move downward when the wallin the tube section dried. This appeared in a sudden jump after which the ring took a newmean position. This indicates that the position of the ring and therefore of the end of thevortex, is apparent of the roughness of the wall, which was confirmed by experiments. In othercases the ring stayed at the same axial position until the tube wall was completely dry.

For cyclones with different geometries the following phenomena were observed [11]:

� The direction of the flow was downwards above the ring and upwards below the ring.This is analogous to the turning of a vortex and the recirculatory flow underneath.

� The vortex length increased for increasing inlet velocity. This phenomena is not includedin any of the models that are proposed in the literature.

� With a larger diameter for the vortex finder, the ring would move downwards but returnagain in the original position within a minute typically.


L

dirtdeposition

++(0.5-2)(0.5-2)DDee

De

Figure 3.5: Schematic reproduction of the ring deposition in the cyclone.

� The vortex length was found to decrease for increasing dust loads in the experimentswith higher inlet velocities and small diameters for the vortex finder.

For the test cyclone of Gebhard [9], the vortex length of Zhonglis model (equation (3.4)) isalmost one order of magnitude larger than of the other models (equations (3.2) and (3.3)).This model also predicts a vortex length which is longer than the cyclone length. Neverthelessall the models only consider the cyclone geometry and neither the influence of the inlet gasvelocity nor that of the dust load.

4 Particle-Wall Adhesion Criterion

Heinl and Bohnet [10] investigated the conveying of quartz powder in a horizontal pipe. Tomodel the particle-wall adhesion they used a suggestion that consists on the energy balance ofa particle-wall collision which is given by:

Ekin,1 + Eelst,1 = EvdW + Eelst,2 + El (4.1)

where Ekin,i is the kinetic energy before (1) and after (2) collision, Eelst,i is the electrostaticenergy and EvdW is the energy due to the van der Waals forces. The energy losses are describedin the term El which is eliminated after introduction of the coefficient of restitution e as:

e2 =Ekin,1 − El

Ekin,1(4.2)

The kinetic energy is written as a function of the particle diameter dP, mass density ρP andthe velocity magnitude wP:

Ekin = ρPπ

12d3

Pw2P (4.3)

The electrostatic energy is given by equation (4.4) according to Israelachvili [14]:

Eelst,i =q2P,i

8πε0 | ∆~x |(4.4)

in which qP,i is the particle charge, ε0 the relative permittivity and | ∆~x | the distance betweenthe particle and the wall. Before collision this distance is called l and after, it is equal tothe sum of half the particle diameter and the distance at contact z0. After substituting thesevalues in equation (4.4) and naming the particle charges before and after collision qP,1 and qP,2

respectively, the following yields:

Eelst,1 =q2P,1

16πε0l(4.5)

Eelst,2 =q2P,2

8πε0(2z0 + dP)

26 4 Particle-Wall Adhesion Criterion

A charged particle that approaches an uncharged wall experiences an attraction force as if aparticle with an opposite charge exists on the other side of the wall [13]. Therefore the factor12 arises in equation (4.5).

For the computation of the particle charge qP,i before (1) and after (2) the wall collision aproposal of Matsuyama and Yamamoto [20] is used:

qP,2 =q0qe

(qe − qP,1) (4.6)

where q0 is the charge of a neutral particle after collision (8 · 10−6 · (dP/m)2 [C] for quartzparticles [10]). The charge, where no charge displacement occurs during a collision, qe isestimated with the potential difference at contact. It is determined with:

qe = 9πε0dP

m(4.7)

The charge before collision was measured by Heinl and Bohnet [10] with a charge spectrometerand found to be 1.5 · 10−11dP[C].

The energy that is stored in the deformation of the particle, the van der Waals energy EvdW,consists out of an elastic part, EvdW,el and a plastic part, EvdW,pl. According to Bruchsal [6]the energy conversion due to elastic deformation is smaller than 1%, so that in approximationthe total dissipation energy is due to plastic deformation:

EvdW = EvdW,el + EvdW,pl ≈ EvdW,pl (4.8)

The van der Waals pressure is given by:

pvdW =~$8πz3

·G (4.9)

in which G is the geometry factor, which is the diameter for spherical particles, and ~$ theLifshitz-van der Waals constant. Smigerski [28] mentions for ~$ values between 3.5 and 6.5eV for quartz powder. The van der Waals energy is calculated by integrating the product ofthe van der Waals pressure and deformed area over the deformed height [6]:

EvdW = −∫ ∞

z0

pvdW · πr2 · dz (4.10)

where r is the radius of the deformed area in figure 4.1. For small deformations (h < dP), r isapproximated in the depth of the deformation h as follows:

r2 = dP · h (4.11)

27

dP

2r

dP

h z0

z

Figure 4.1: Model of the plastic deformed particle.

As remarked earlier, the loss of elastic energy in the van der Waals bounding due to collisionis much smaller than the loss of plastic energy. Therefore, in equation (4.10) pvdW is replacedby ppl, which is the yield stress of the particles material. The depth of the deformation, h,is approximated by the depth of the plastic deformation, hpl. After substitution of equation(4.11) into (4.10) and evaluating the integral the following expression for the van der Waalsenergy is found:

EvdW,pl =∫ hpl

0ppl · π · dp · h · dh (4.12)

=12ppl · π · dp · h2

pl

The depth of the plastic deformation is calculated by putting the expression for Ekin,1 (equation(4.3)) equal to EvdW,pl (equation (4.12)) with the coefficient of restitution e, as in equation(4.2):

hpl = dP · wP

√ρP

6pple2(4.13)

With the assumption of equation (4.8) the result of equation (4.12) is a good approximationfor the total van der Waals energy, EvdW. After substitution of equation (4.11) and (4.13) in(4.10) and evaluating the integral the following expression is found:


Table 4.1: Material properties of limestone [30].

quantity value unity~$ 1.59 · 10−19 [-]ppl 52.5 · 106 N/m2

z0 3.36 · 10−10 mρP 2770 kg/m3

EvdW = −∫ ∞

z0

~$8πz3

· πd2PwP

√ρP

6pple2dz (4.14)

=~$

16πz20

d2PwP

√ρP

6pple2

By substituting equations (4.2), (4.3), (4.5) and (4.14) in (4.1) and realizing that Ekin,2 is zerofor particles sticking at the wall, the critical particle velocity, wP,crit, is derived:

wP,crit =

√√√√( ~$edP4π2z2

0

)2 34pplρP

+3

d3Pπ

2e21

ε0ρP

(2q2P,2

2z0 + dP−q2P,1

l

)(4.15)

The material properties of limestone used in equation (4.15) are represented in table 4.1.Sticking only can occur if the particles absolute velocity is smaller than the critical particlevelocity (equation (4.16)). The condition for adhesion is thus fully independent of the particleconcentration.

| −→wP,1 |5 wP,crit (4.16)

The particle diameter can change due to particle-particle collisions. The wall roughnessinfluences the velocity of the particles and the coefficient of restitution for collisions. Modelsfor these last two phenomena are presented in section C.3 and C.4 respectively. The model ofHeinl and Bohnet [10] does not include that the particles need to have a normal velocity thatis high enough to reach the wall within the particle response time, which is given by [5]:

τP =ρPd

2P

18ηG(4.17)

If ReP � 1 as described in equation (C.3) (i.e. −→wP ≈ −→u ) the Stokes regime yields and thedifferential equation reduces to:

d−→wP

dt= −

−→wP −−→uτP

(4.18)

4.1 Coefficient of Restitution 29

Which has as solution:

−→wP = −→u[1− exp

(− t

τP

)](4.19)

The stopping distance s is found by multiplication of equation (4.19) with τ in equation (4.17):

s = τ−→wP = −→u[1− exp

(− t

τP

)](4.20)

If the stopping distance is larger than the distance of the particle to the wall a particle depositsby ’free flight deposition’. This occurs for particles having large deposition velocities andsmall near wall residence times. Particles with high residence times and negligible wall-normalvelocities are called ’diffusional deposition’ population. Botto et al. [5] showed that for rigidboundaries for τ+

p = 5 particles the ratio diffusional and free flight deposition was 90:10%and for τ+

p = 15 particles 60:40%. The cumulative distribution for τ+ in the experiments ofGebhard [9] is presented in appendix B.

4.1 Coefficient of Restitution

Sommerfeld and Huber [29] found in a detailed experimental analysis in a horizontal channelflow that wall roughness considerably alters the rebound behavior of particles. It caused inaverage a re-dispersion of the particles, i.e. the gravitational settling was reduced.

They also describe an experimental study to provide data for modeling the process of solidparticle impacts at walls. The effects of both the angle, θ1 (between particle trajectory andwall normal vector) and the velocity of the colliding particle on the coefficient of restitutionwere investigated. For hard and soft walls, a very different influence of the angle was found.In the case of a hard wall the coefficient of restitution was found to be as in table 4.2. For asoft wall the coefficient of restitution continuously increases from 0.5 for large angles to 1.0 fornormal impacts. For increasing impact velocity, i.e. for increasing deformation of the wall, thecoefficient of restitution was found to decrease continuously.

The yield strength of steel (over 210 MPa [7]) is much larger than that of limestone (52.5MPa [30]). Therefore it seems to be reasonable to model the wall as being hard if the depositionstarts and soft if already a first layer of limestone has deposited. The nett colliding angle θ1 is

Table 4.2: Coefficients of restitution in the case of a hard wall for several colliding angles [29].

θ1 [o] e

0 0.9745 0.790 1


LR

HR

H - ÄHR R 21

3

13

1WP

Figure 4.2: Model for the wall roughness.

influenced by the particle track (colliding angle β1 in figure 4.2) and the wall roughness. Thelast one can be approximated by the use of statistical models (Appendix D).

Furthermore, e can be divided into a normal and a tangential coefficient of restitution (enand et respectively) which may have a different value. Therefore, the particle velocity beforecollision wP,1 needs to be divided into a normal part, wP,1,n and a tangential part wP,1,t:

wP,1,n = cos θ1 · wP,1 (4.21)

wP,1,t = sin θ1 · wP,1

and after collision:

wP,2,n = − cos θ2 · wP,2 = en · wP,1,n (4.22)

wP,2,t = sin θ2 · wP,2 = et · wP,1,t

After combining equations (4.21) and (4.22), the following expression for the total coefficientof restitution holds:

e =

√(en cos (θ1)wP,1)2 + (et sin (θ1)wP,1)2

w2P,1

=√e2n cos (θ1)

2 + e2t sin (θ1)2 (4.23)

4.2 Sensibility Analysis of the Critical Sticking Velocity 31

Table 4.3: Uncertainty of the parameters in the calculation of the critical velocity.

quantity value unity standard uncertaintye 0.6186 [-] 0.2212l 0.05 m 0.01

√3

3

ppl 52.5 · 106 N/m2 0.1√

33

z0 3.36 · 10−10 m 10−12√

33

~$ 1.59 · 10−19 [-] 10−21√

33

ε0 8.8542 · 10−12 F/m 10−16√

33

ρP 2770 kg/m3√

33

For copper particles with a diameter of 50 µm, Abd-Elhady [1] concluded that en is independentof the impact angle and has a value of 0.37. However, et increases for increasing impact angles.For the computation of et Abd-Elhady proposes the following equation:

et = 1− f(1 + en) cot (θ1) (4.24)

in which f is the friction coefficient. Van Beek [3] gives for copper particles values of 0.17up to 0.2. The coefficients of restitution are plotted in figure 4.3 for f = 0.2. For θ1 < 20o,et becomes negative and is further treated as being zero. Compared with the coefficients ofrestitution in table 4.2, at an impact angle of 45o the value of 0.58 for e is slightly lower. Alarger difference is found at 0o where instead of becoming 0.97, e approaches the value 0.46.The mean value of e is 0.619 and its standard deviation 0.221. The minimum value for e equals0.36 at θ1 = 17.4o.

4.2 Sensibility Analysis of the Critical Sticking Velocity

An error analysis such as subscribed in appendix E is made to estimate the influence of theparameters that must be known to calculate the critical sticking velocity. The graph in figure4.4 is made after applying equation (E.1) on equation (4.15), with the standard uncertaintiesof table 4.3. The coefficient of restitution e has by far the biggest contribution in the totaluncertainty. The influences of the ~$ and z0 are already two orders of magnitude lower. Infigure E.1 wP,crit ± utot is shown such as the relative error. The relative error is very constant(35.8%).

4.3 Influences of the van der Waals and Electrostatic Energies

Equations (4.5) and (4.14) show that Eelst ∝ d−1P and EvdW ∝ d2

P. This means that for smallparticles the contribution of the electrostatic energy is relative large and for large particles


eee

Figure 4.3: Coefficient of restitution e, normal coefficient of restitution en and tangential coefficientof restitution et for f = 0.2 as function of impact angle θ1.

relative small. According to Bruchsal [6] the electrostatic bounding forces are maximal 1% ofthe van der Waals bounding forces for 1 µm particles and even 1� for 2 µm particles. However,Wang et al. [32] report that electrostatic is only important for particles larger than 50 µmwhich is controversy with equation (4.5) and (4.14). Finally according to Wang et al. [32],the influence of electrostatic is negligible small when the channel wall is coated with particlesalready and no tribo-electrification can occur since the flowing particles have identical surfaceproperties as the column wall. This means that only for the particles that build the first layeron the wall, electrostatic is important.

When the two terms beneath the square root sign in equation (4.15) are separated, twotheoretical critical particle velocities exist: one in which influence of the electrostatic energyis zero (wP,crit,vdW) and one in which the van der Waals energy is zero (wP,crit,elst). When

4.3 Influences of the van der Waals and Electrostatic Energies 33

Figure 4.4: Contribution of the parameters in equation (4.15) to the uncertainty budget.

for the computation of the van der Waals energy the properties of table 4.1 are used, for theelectrostatic energy the proposal of Matsuyama and Yamamoto [20] and for e the mean valueas mentioned in section 4.1, the graphic in figure 4.5 arises. The electrostatic contribution inthe total critical velocity is relative constant over the range 0.1µm < dP < 10µm. At a particlediameter a little bit larger than 2 µm the electrostatic energy is as important as the van derWaals energy.

4.3.1 Experimental Observations on the Influences of the van der Waals and

Electrostatic Energies

At the Technical University of Graz Gebhard [9] performed experiments with cyclones with andwithout earth. For the tests with a volume flow rate of 1000 m3/h and a dust concentrationin the inlet of 7.9 g/kg, no optical differences could be observed. However, for the tests with a


Figure 4.5: Critical sticking velocity for zero electrostatic (wP,crit,vdW) and zero van der Waals(wP,crit,el) influence.

lower volume flow rate of 600 m3/h and a dust concentration of 8.0 g/kg was found that thewall depositions in the cyclone were less regular without earth. Furthermore, the depositedmass was slightly higher without earth (4.6% vs. 4% of the mass at the inlet). However, thisresult is in contradiction with equation (4.15). For an earthed cyclone the particle charge aftercollision qP,2 is zero. Therefore the critical velocity decreases and less particles meet the thevelocity condition for adhesion. Nevertheless, Gebhard does not exclude that the difference inthe two tests was caused by a non-optimal reproducibility.

Wang, et al. [32] experimented on the vertical transportation through channels for groupA (66 µm) and group C (20 µm) glass beads, both with a density of 2500 kg/m3. Finally,they estimated the influence of electrostatic by the use of an anti-electrostatic powder. Theyobserved that more and more of the group C particles accumulated on the tube wall when

4.4 Influence of the Particle Layer Thickness 35

the gas velocity was reduced. Although electrostatic charge was expected to occur, hardlyelectrostatic phenomena, such as noise of discharging and electric sparks, were observed.

For the group A particles, little particle deposition was seen. Also clusters were seldom seenfor low solid fluxes and high solid velocities. However, they began to appear as the air velocitywas reduced. Many electrostatic phenomena were observed which reduced significantly afteradding 0.5% anti-electrostatic powder.

The addition of anti-electrostatic powder reduced the pressure gradient for both the par-ticle sizes but relative much more for the large particles. This confirms that the influence ofelectrostatic is much higher for larger particles. Even a larger difference was observed at higherair velocities.

The wall deposition is always an equilibrium of the adhering electrostatic and van der Waalsforces on the one side and the drag and lift forces that work in the opposite direction on theother side. When the air velocity was further decreased from 6.0 to 3.5 m/s, the distributionof particles through the cross section of the channel was found to translate more to the wall.Therefore the drag forces increased although the velocity at the wall decreased. Finally, evenparticles were observed to be peeled of the wall.

After the experiment, the adhered particles were removed first by tapping and then bybrushing. By doing this, two (rather arbitrate) classes of particles were obtained. The brushedparticles were located closer to the wall than the tapped. They were also smaller than thetapped ones. This is not only caused by the fact that the electrostatic charging decreases whenthe wall is fully covered by particles, which have the same surface properties as the particlesin the flow. Also the fact that smaller particles have a relative larger adhesion force and lessmomentum to rebound increases the chance for small particles to stick on the wall.

4.4 Influence of the Particle Layer Thickness

Abd-Elhady et al. [2] mention that the critical sticking velocity is a function of the layerporosity, P . Because P is a continuous decreasing function of the layer thickness, wP,crit is alsoa function of P :

wP,crit = wP,0eCPP (4.25)

in which wP,0 is the critical sticking velocity for a clean wall and CP is a positive constant.Thus, for increasing layer thickness, wP,crit decreases and less particles will meet the stickingcondition. The computation method of the layer porosity from the layer thickness and thederivation of equation (4.25) are explained in appendix D.

5 Resuspension Models

Particles that are adhered to the wall can be resuspended back into the flow by the drag and liftforces of the fluid. Models to simulate this phenomena in a turbulent flow can be categorizedinto [24]:

1. Force balance based models.

2. Energy accumulation based models.

5.1 Force Balance Based Models

For the simplest force balance model a particle is resuspended if the instantaneous aerodynamiclift force is larger than the surface adhesive force. The rate of removal is determined by thefrequency of the turbulent bursts. The distribution of both the aerodynamic and the adhesiveforces are not considered in this model. Wen and Kasper [33] introduced the following modelfor the rate of removal ψ:

ψ = n exp (−Q) (5.1)

in which n is the maximum resuspension rate and Q is the ratio of the adhesive force to theinstantaneous aerodynamic removal force. Jurcik and Wang [16] stated that the rate of removalis the product of the maximum resuspension rate and the chance that Q ≥ 1, i.e. that theremoval forces are larger than the adhesive force Fadh. It is given by:

p = n

∫ ∞

Fadh

P (Fr)dFr (5.2)

where P (Fr) is the removal force distribution. The model of Reeks, Reed and Hall (RRH)[25] describes as threshold that a particle is resuspended when it has accumulated enoughvibrational energy from the flow. The model includes resonant energy transfer, i.e. when thefrequency of the removal force is close to the own frequency of the particle-surface deformation,a particle is removed much easier than applying the same lift force quasi-statically. When thetransfer of resonant energy is zero, the model reduces to the form appropriate to a forcebalance. Ziskind et al. [36] found in experiments that the force of adhesion and the meanand rms lift forces could not explain the high resuspension rate, either for simple force balance

38 5 Resuspension Models

nor for resonant energy transfer. They suggested that rolling could provide a more realisticmechanism for particle removal. In this approach the moments arising from the contact ofa smooth particle on a smooth surface as well as a particle in contact with two and threeasperities were examined.

5.1.1 The Reeks, Reed and Hall Model

Reeks et al. [25] introduced a model in which the resuspension rate is modeled for zero energytransfer and non-zero energy transfer in the form of:

ψ = n exp(− Ebar

2〈Epot〉

)(5.3)

in which n is the typical frequency of the particle surface deformation within the adhesivepotential well, Ebar is the energy barrier which is the difference between the adhesive forceand the mean lift force and 〈Epot〉 is the average potential energy of particles in the well.The model is valid for cases with or without resonant energy transfer. Reeks and Hall [24]did experiments with 10 µm and 20 µm alumina spheres and they found that the influence ofresonance is generally small.

5.1.2 The Reeks, Reed and Hall Model for Zero Energy Transfer

For zero energy transfer equation (5.3) can be written as a function of the (scaled) force ofadhesion Fa, the mean lift force 〈FL〉, and the covariance of the lift force fluctuations 〈F 2

L〉 as:

ψ = n exp

(−κ(Fadh − 〈FL〉)2

〈F 2L〉

)(5.4)

in which κ is a numerical constant depending upon the shape of the surface potential (κ = 1[15] and for a harmonic potential κ = 1

2 [24]) and n is defined by:

n =(

12π

)√〈 ˙F 2

L〉/〈FL2〉 = 0.00658

(u∗2

ν

)(5.5)

where 〈 ˙F 2L〉 is the covariance of the time derivative of the lift force, ν is the kinematic viscosity

and u∗ the friction velocity which is computed from [24, 1]:

u∗ =√τwρG

=

√ρ2G

η

du

dy(5.6)

According Masuda et al. [19] the friction velocity is also represented by a function of the meanvelocity u and the friction coefficient f :

5.1 Force Balance Based Models 39

u∗ = u

√f

2(5.7)

To approximate f , the Blasius equations is used. For a smooth pipe f is given by:

f = 0.0791Re−1/4, 3 · 103 < Re < 105 (5.8)

From equation (5.7) and (5.8) the friction velocity u∗ follows as:

u∗ ≈ 0.2uRe−1/8 (5.9)

An approximation for the mean lift force 〈FL〉 in equation (5.4) for the range 1.8 < rPu∗ < 70

is given by [24]:

〈FL〉 ≈ 4.21ρGν2

(dPu

∗

ν

)2.31

(5.10)

The force of adhesion Fadh, follows from the product of the adhesion force for a perfectlysmooth contact, Fadh,s and the normalized adhesion force Fadh:

Fadh = Fadh,s · Fadh (5.11)

where:

Fadh,s =32πγrP (5.12)

in which γ is the adhesive surface energy of the particle and the substrate. For the normalizedadhesion force Fadh, a log-normal distribution, φ(Fadh) is used:

φ(Fadh) =1√2π

(1

Fadh lnσ′adh

)exp

−12

(ln (Fadh/F adh)

lnσ′adh

)2 (5.13)

in which F adh and σ′adh are the geometric mean and standard deviation of the normalizedadhesive force respectively. Reeks et al. [24] mention that O

(F adh

)= −2 and O (σ′adh) = 1.

5.1.3 Reeks Reed and Hall Model for Non-zero Energy Transfer

For the case that the energy transfer is not equal to zero, an extra resonant term ξ is added inequation (5.4) by replacing the expression for the typical frequency n in equation (5.5) with:

ndamp =( ω

2π

)√ξ + 〈 ˙F 2L〉/(〈F 2

L〉ω2)ξ + 1

(5.14)


in which:

ξ =π

2βω2EL(ω) (5.15)

In equation (5.15) ω is the natural frequency of the particle surface deformation and β isthe combined fluid and mechanical damping of the particle-surface motion. The value inthe normalized energy spectrum of lift force fluctuations around the natural frequency, EL iscomputed with the covariance of the lift force fluctuation 〈F 2

L〉 and the energy spectrum:

EL(n) =EL(n)〈F 2

L〉(5.16)

By doing this, equation (5.4) becomes:

ψ = ndamp exp

(−κ(Fadh − 〈FL〉)2

〈F 2L〉(1 + ξ)

)(5.17)

5.2 Moment Balance Models

5.2.1 The Dynamic Rock’n Roll Model with Resonant Energy Transfer

In the rock’n roll model the criterion for a particle to resuspend is not the force balance betweenthe shear stress of the lift force and the adhesive force (i.e. oscillating the particle vertically).Instead of this the particle will rotate around the point P where it has contact with the surface(see figure 5.1). In figure 5.1, rP is the particle radius and dcont is the diameter of the contactarea. For perfectly spherical particles with the same diameter in a hexagonal closed packedstructure dcont ≈ 1.15 rP. The force needed for rotating the particle is much lower than fordragging it. The equation of motion around point P is given by:

θ + βθθ + ω2θ,0θ = I−1

P Γ(t) (5.18)

where θ is the angular coordinate and IP the moment of inertia about P . Further, in equation(5.18), βθ is the angular damping coefficient and ωθ,0 is the natural angular frequency of thesystem. The couple Γ(t) is the fluctuating part of the couple Γ which is calculated from thelift force FL and the drag force FD as follows:

Γ ∼dcont

2FL + rPFD (5.19)

The mean lift force 〈FL〉 is calculated from equation (5.10). Reeks and Hall [24] computed thevalues for the mean drag force, 〈FD〉 as the mean drag on a sphere near a surface in simpleshear flow. According O’Neill [22] this is 1.7 times the Stokes drag. Then, the mean drag forcebecomes:

5.2 Moment Balance Models 41

È

a

mg

Figure 5.1: Particle-surface geometry for rock’n roll model [24].

〈FD〉 ≈ 8ρGν2

(dPu

∗

ν

)2

(5.20)

Furthermore, for a spherical particle with mass mP, an explicit expression of the moment ofinertia about P is given:

I =57mPr

2P =

20π21

ρPr5P (5.21)

For small rotations θ about P (see figure 5.1) the vertical displacement of point O is dcontθ sothat the restoring couple is given by mPω

2d2contθ. If no other couples are working on the particle

it will oscillate undisturbed with its natural frequency ωθ,0. Combining this with Iω2θ,0θ leads

to the relationship between the natural frequency for vertical oscillations ω0 and for rotatingoscillations ωθ,0:

ω2θ,0 =

5d2cont

7r2Pω2

0 (5.22)

For the damping constant βθ, the drag caused by small oscillations at point O, mPβ is assumedto be the same for a sphere of radius rP oscillating in position with velocity rPθ. Equating themoment of this force gives:


Iβθθ = mPβr2Pθ (5.23)

βθ =57β

For rotation, equation (5.17) becomes:

ψ = nθ exp(

−κΓ2

d2cont〈F 〉2(1 + ξ)

)(5.24)

in which F is the fluctuating component of the force F (t):

F (t) =FL

2+

rPdcont

FD (5.25)

and Γ is the net couple acting at P which exists out:

Γ = −⟨dcont

2FL + rPFD

⟩+ dcontFadh +

dcont

2mgn − rPmgt (5.26)

where gn and gt are the gravitational forces acting on the particle in normal and tangentialdirection, respectively. After substitution of equation (5.26) in (5.24) the resuspension rate isfound:

ψ = nθ exp

−κ(Fadh + 1

2mgn −rP

dcontmgt − 〈dcont

2 FL + rPFD〉)2

〈F 〉2(1 + ξ)

(5.27)

Finally, for computing nθ equation (5.14) becomes:

nθ =(ωθ

2π

)√ξ + 〈 ˙F 2〉/〈F 2〉ω2θ

ξ + 1(5.28)

5.2.2 ’Quasi-static’ Rock’n Roll Model

If the motion of the particles on the surface is not driven by resonance, the equation of motionis approximated by a force balance between the aerodynamic removal forces and the adhesionforces or by a balance of couples. The force balance is written as a balance between theadhesive restoring force as function of the deformation of the particle/surface, Fadh(y) and theaerodynamic removal force as a function of time Fr(t):

Fadh(y) + Fr(t) = 0 (5.29)

5.2 Moment Balance Models 43

When dealing with a balance of couples, Fr(t) becomes the force in equation (5.25). Aftersplitting Fr(t) in a average component 〈Fr〉 and a fluctuating part of zero mean Fr(t), equation(5.29) becomes:

Fr(t) = −Fadh(y)− 〈Fr〉 (5.30)

At the point of detachment y = ydet the following yields:

Fdet = −Fadh(ydet) (5.31)

And after substitution of equation (5.31) in (5.30):

Fr = Fdet − 〈Fr〉 (5.32)

From equation (5.30) the following relations between y and Fr yields:

y(t) = f(F ) (5.33)

y(t) = ˙Ff ′(F )

The resuspension rate ψ gives now:

ψ =

∞∫0

˙FP (Fdet − 〈Fr〉, ˙F )d ˙F

Fdet∫−∞

∞∫−∞

P (Fdet − 〈Fr〉, ˙F )d ˙FdF

(5.34)

in which P (Fr,˙Fr) is the joint probability distribution for Fr and ˙Fr. The fluctuating part

of the removal force Fr and its time derivative ˙Fr have a Gaussian distribution and are fullyindependent of each other (they have zero correlation). For the joint probability distributionP (Fr,

˙Fr) can be written:

P (Fr,˙Fr) = (

¯Fr2

¯F 2r )G(ζ1)G(ζ2) (5.35)

in which G(ζ1) and G(ζ2) are normalized Gaussian distributions:

G(ζ1) = G(ζ2) = (2π)−1/2 exp (−ζ22/2) (5.36)

ψ and ζi are two statistically independent random variables of unit variance. After substitutionof equation (5.35) and (5.36) in equation (5.34) and evaluating the integral an explicit solutionof the resuspension rate is found:


ψ =12π

(〈 ˙F 2

r 〉〈F 2

r 〉

)1/2

exp

(−(Fdet − 〈Fr〉)2

2〈F 2r 〉

)/12

1 + erf

Fdet − 〈Fr〉√2〈F 2

r 〉

(5.37)

The first part in equation (5.37) is equal to the maximum resuspension rate for the rock’n rollmodel with zero energy transfer in equation (5.5):

ψmax = nθ = (1/2π)

(〈 ˙F 2

r 〉〈F 2

r 〉

)1/2

(5.38)

For values of (Fdet−〈Fr〉)/〈F 2r 〉1/2 5 0.75 (i.e. when ψ/nθ < 1) equation (5.38) should be used

and for (Fdet − 〈Fr〉)/〈F 2r 〉1/2 > 0.75 equation (5.37).

5.3 Resuspension of Aggregates

According Masuda and Matsusaka [19] small particles are difficult to resuspend, because theseparation force due to the flow is relatively weak. Then the deposited particles accumulateon the surface and powder layers are formed there. In that case, in stead of single particleswhole aggregates can be resuspended into the flow. An agglomerate of particles is assumed tobe spherical. Then the following equation describes the couple Γ around a rotating point:

Γ =∫ dag

y=z0

(y − z0)dFD(y) (5.39)

in which dag is the diameter of the agglomerate and dFD(y) is the drag force acting on aninfinitesimal small area of the aggregate, which is written as:

dFD(y) = CD(y)ρGu(y)2

2dA(y) (5.40)

=24τwdag

y√y(dag − y)dy (5.41)

Then, the following expression for the bending stress σb is found:

σb =96πτwf(Y ∗) (5.42)

in which f(Y ∗) is the following dimensionless function:

f(Y ∗) =∫ 1

Y ∗

√Y (1− Y )(Y − Y ∗)2dY (Y ∗(1− Y ))−3/2 (5.43)

in which:

5.4 Resuspension by Impacting Particles 45

Y ∗ =z0dag

≈ 0.5 (5.44)

and:

Y =y

dag(5.45)

After substitution of equations (5.44) and (5.45) in (5.43), evaluating the integral and substi-tuting the result in equation (5.42), the following holds:

σb = 3τw (5.46)

which means that the maximum adhesive stress is restricted to the wall shear stress.

5.4 Resuspension by Impacting Particles

Abd-Elhady [1] mentions that if particles collide with a velocity that is larger than a certaincritical velocity (the removal limit), it is possible that particles are removed from the layer thatsticks at a surface. For bronze particles of diameter 54 ± 3 µm the results are represented intable 5.1. The bouncing limit is the velocity at which a particle reflects.

Table 5.1: Bouncing and removal limits for 54 µm bronze particles.

Numerical model ExperimentsCritical sticking velocity 0.15 m/s 0.3 m/s

Bouncing limit 0.1 m/s 0.18 m/sRemoval limit 0.5 m/s 0.6 m/s

6 CFD-Results

The particle sizes in the simulations were divided into two fractions:

1. 0.9µm ≤ dP < 20µm in order to simulate the behavior of the particles measured at theinlet by Gebhard [9].

2. 0.1µm < dP < 0.9µm for the sub-micron particles.

The results of these two simulations are discussed in the following two sections. Accordingthe RRSB distribution function (equation (3.1)) these two ranges correspond with fractions of88.9% and 8.9% respectively.

6.1 Comparison of the Experiments and CFD-Simulations

The RRSB particle size distribution (equation (3.1)) was made discrete within the range ofparticle diameters at the inlet measured by Gebhard [9]. The values for dP and the injectedparticles are shown in table B.1.

In figure 6.1 the results of the CFD-simulations are shown. To compare these results, theexperiments of Gebhard [9] are copied from figure 3.2. The particle distribution of the dust atthe inlet during the experiments of Gebhard [9] has moved towards bigger particles comparedwith the distribution according to equation (3.1) which is used for the CFD-simulations. Thiscould be an explanation for the fact that both the curves for the fine dust and coarse dust aremoved to smaller particle diameters in the simulations compared with the experiments (figure6.1a). Nevertheless no particles were observed to stick to the wall. Also for a lower volumeflow rate of 600 m3/h, no particle deposition occurred during simulations (figure 6.1b). Asexpectations are, more particles are in the fine dust fraction and less in the coarse dust fractionat 600 m3/h than at 1000 m3/h (figure 6.2).

A possible explanation for the fact that no single particle deposited during simulation isthat the fraction dP < 0.9µm is not negligible. This is in agreement with the experiments ofWang et al. [32] (section 4.3) that small particles form the layer closest to the wall.

48 6 CFD-Results

100

101

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Particle size dP [ m]

in sim

fine dust sim

coarse dust sim in exp

fine dust exp

coarse dust exp

1000 m³/h

µ

(a)

100

101

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Particle size dP [µm]

in simfine dust sim

coarse dust simin meas

600 m³/h

(b)

Figure 6.1: (a) Cumulative particle size distributions calculated with the results of experiments andCFD-simulations at 1000 m3/h and 3 g/kg. (b) Cumulative particle size distributions of the CFD-simulations at 600 m3/h (legends correspond with curves from top to bottom).

6.2 Sub-micron Particle Behavior

The distribution of the particles injected in the simulations has been extrapolated to smallerparticles with the use of equation (3.1). The injected particles are given in table B.2. Theresults of the depositions are plotted in figure 6.3. The particle size distribution (figure 6.3a)has moved about two orders of magnitude to smaller particles compared with the measurementsof Gebhard [9] (figure 3.2). However, the order of the depositions on the various parts (fromsmall to large particle size: vortex finder, vertical tube section and cyclone body) correspondswith the measurements. Most of the deposited particles stick in the cyclone body with a peakat 0.05 µm. Under this size, particles follow the flow for a longer time (table 6.1). Above,particles have a larger momentum to rebound.

The distance from the bottom of the vortex finder to the deposition was found to have anaverage value of 0.52 m with a standard deviation of 0.53 m. This means that the average valueis smaller than the uncertainty. Although the mean value is smaller than predicted with themodels for the natural vortex length, the ring position in the models of Alexander and Bryantpartly falls within the largest range for these two models (table 3.3). The model of Zhongli isnot usable for this situation.

6.3 Discussion Differences between Measurements and Simulations

Possible explanations for the fact that the particle size distribution of the measured depositionshas moved to larger particles are:

6.3 Discussion Differences between Measurements and Simulations 49

100

101

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Particle size dP [µm]

inlet dust

fine dust 1000 m3/h

coarse dust 1000 m3/h

fine dust 600 m3/h

coarse dust 600 m3/h

Figure 6.2: Cumulative particle size distributions of the inlet, fine and coarse dust fractions (legendcorresponds with curves from top to bottom).

Table 6.1: Mean time for a particle to deposit.

dP [µm] 0.0125 0.020 0.050 0.100 0.125 0.250 0.500mean time till deposition [s] 1.29 0.37 0.26 0.16 0.25 0.11 NaN

1. Imperfections in the measurements of the particle size distributions during experiments.

2. A physical not realistic sticking criterion.

3. A not realistic flow field for the CFD-simulations.

The computed flow field does not seem to be a reasonable explanation since even at a volumeflow rate of 600 m3/h in the simulation no depositions were found within the range where theywere found during experiments at 1000 m3/h.

It is more plausible to state that the differences between experiments and simulation liewithin a combination of a not physical realistic sticking criterion and the limited range inwhich the particles were measured during the experiments. In the simulations particles with

50 6 CFD-Results

(a)

0 0.1 0.2 0.3 0.4 0.5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Particle size dP

[µm]

Fra

ctio

n [-]

fine dust

coarse dust

cyclone bodyvertical tube

vortex finder

(b)

Figure 6.3: (a) Cumulative distributions for sub-micron particles at 1000 m3/h. (b) Fractions of thedepositions with respect to the released dust (legend corresponds with curves from top to bottom).

dP < 0.1µm were found to deposit. According equation (3.1) and the information from themanufacturer (table 3.1) this fraction corresponds with 0.7% of the particles. The stickingbehavior for particles at a clean wall and a covered wall may be different, e.g. due to a differentLifshitz van der Waals constant ~ω. Finally, it is possible that particles, that are removed fromthe wall after experiments, exist out of several small particles that have agglomerated.

7 Conclusions and Recommendations

A physical model to simulate the sticking of limestone particles in a CFD-package has beendeveloped. In this model the critical velocity is the criterion whether a particle sticks orrebounds. The two driving energies behind the critical sticking velocity are the van der Waalsenergie and the electrostatic energy. For particles smaller than 2 µm the van der Waals energyis the most important, for bigger particles the electrostatic energy. The van der Waals energyhas been computed with the material properties of limestone, the electrostatic energy witha proposal for quartz powder. However, no difference has been made yet in the behaviorof impacting particles at a clean wall or at a wall that has been covered with a layer ofparticles already. The coefficient of restitution is the factor that describes the efficiency of animpact. It can be dependent on the impact angle and the walls/layers material. Its averagevalue was used in the simulations. This is the reason why the coefficient of restitution has thelargest contribution in relative uncertainty of the critical sticking velocity (which has an almostconstant value of 35.8%).

Several models for resuspension are found in literature. The resuspension by the impactof other particles (above the removal limit) is the simplest model. Further the force balancemodel (Reeks, Reed and Hall without resonant energy transfer) and the moment balance modelswithout resonant energy transfer (resuspension of aggregates, quasi-static rock’n roll model)are relatively simple to apply. Nevertheless statistical flow properties are needed for thesemodels. More complicated are the models for which energy transfer properties are needed(Reeks, Reed and Hall model for non-zero energy transfer, rock’n roll model with resonantenergy transfer). From literature it is known that energy transfer does not play an importantrole. For the models that describe an equilibrium between the sticking force/momentum andthe removal forces/momentum applied by the flow, an adjustment of the flow field (by solvingit once again) is needed when a particle layer builds up. Therefore resuspension of (single)particles by other impacting particles seems to be the most realizable. However, for this modelno specific measurements/physical models for limestone are available yet.

In the simulations only particles smaller than 0.1 µm were found to stick. In former exper-iments, the range of measured particles starts at 0.9 µm. However, the fraction of particlessmaller than 0.9 µm is about 10%, which is in agreement with the analytical particle size dis-tribution function given by the manufacturer of the dust. Because of this and because of theobservation that particles in this fraction easier stick, it is assumed that this fraction is notnegligible. The form of the cumulative distribution curves of the fine and coarse dust fit to

52 7 Conclusions and Recommendations

the experimental data. That of the several depositions are moved to smaller particle sizes buthave the same shape and order mutually.

For the future work the following recommendations should be considered:

� Particles in the class smaller than 0.9 µm have to be divided into smaller subclasses inexperiments to observe their behavior more precise.

� In the equation for computing the critical sticking velocity a distinction in the coefficientof restitution and/or the Lifshitz van der Waals constant between a clean wall or a (partly)covered wall has to be made to allow larger particles to stick at a layer of small particles.

� More tests with an earthed cyclone should be performed to get an estimation of theinfluence of the electrostatic energy. The use of anti-electrostatic powder is less suitablebecause it influences the particle size distribution of the limestone. If the electrostaticenergy turns out not to be negligible, the model for quartz powder can be replaced by amodel that is suitable for limestone. If necessary relevant charges should be measured inexperiments.

� A larger number of particles with a continuous distribution will give a more realisticview of the injected particles in the CFD-simulations. Various mass flow rates can besimulated by the the injection of particles in the right ratios.

� The further growth of the depositions can be simulated with the adjustment of the compu-tation of the critical sticking velocity. To model the asymptotic behavior of the thicknessof the depositions resuspension (starting with the simplest) models can be implemented.

Acknowledgement

After almost a year of hard labor, my master thesis is finished. I would like to thank all thepeople who have supported me during this time.

First of all, I would like to thank my parents, my brother and the rest of my family for theirsupport they have been given me since the first day of my life. Without them I would not havereached what I have reached now.

Further I would like to thank Ao. Univ.-Prof. Dipl.-Ing. Dr. techn. Michael Narodoslawskyand Hesi who pointed me the opportunities of the Institute for Chemical Apparatus Design,Particle Technology and Combustion at the Graz University of Technology for doing my masterthesis. I am most grateful for their advices.

I would like to render thanks to O. Univ.-Prof. Dipl.-Ing. Dr.techn. Gernot Staudinger forgiving me the opportunity to study at his institute and for his scientific support. For the dailysupport I found a great help in Dipl.-Ing. Dr.techn. Gunter Gronald who had always time forme although he was self busy completing his PhD thesis.

At my home university Prof. Dr. Ir. Bert Brouwers was always interested in my work suchas Dr. Ir. Erik van Kemenade and Dr. Ir. Frank Ganzevles who helped me finishing my work.I would like to thank them all for their efforts.

For the non-scientific support I would like to thank all my colleagues in Graz and in Eind-hoven for their contribution in making the work climate pleasant for me.

Finally, I would like to thank Dipl.-Ing. Dr.techn. Martin Pogoreutz of the Austrian Energyand Environment for providing me the facilities of his company.

Bibliography

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[2] M. Abd-Elhady, C. Rindt, J. Wijers, and A. v. Steenhoven. Particulate fouling in wasteincinerators as influenced by the critical sticking velocity and layer porosity. Energy, 30:1469–1479, 2005.

[3] M. v. Beek. Gas-Side Fouling in Heat-Recovery Boilers. PhD thesis, Eindhoven Universityof Technology, 2001.

[4] M. v. Beek, C. Rindt, J. Wijers, and A. v. Steenhoven. Analysis of fouling in refuse wasteinclinerators. Heat Transfer Engineering, 22:22–31, 2001.

[5] L. Botto, C. Narayanan, M. Fulgosi, and D. Lakehal. Effect of near-wall turbulenceenhancement on the mechanism of particle deposition. International Journal of MultiphaseFlow, 31:940–956, 2005.

[6] R. Bruchsal. Der Einfluß von Partikelstoß und Partikelhaftung auf die Abscheidung inFaserfiltern. PhD thesis, Technische Universitat Karlsruhe, 1981.

[7] W. Callister. Material science and engineering an introduction. Wiley International, 2003.

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[9] J. Gebhard. Einfluss der Beladung und des Gasdurchsatzes auf den Gesamtabscheide-grad und den Druckverlust eines Gaszyklons mit Fallrohr. Master’s thesis, TechnischeUniversitat Graz, 2004.

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[11] A. Hoffmann, R. de Jonge, H. Arends, and C. Hanrats. Evidence of the ’natural vortexlength’ and its effect on the separation efficiency of gas cyclones. Filtration and Separation,32:799–804, September 1995.

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[15] K. Johnson, K. Kendall, and A. Roberts. Surface energy and the contact of elastic solids.Proceedings of the Royal Society of London A, 324:301–313, 1971.

[16] B. Jurcik and H. Wang. The model of particle resuspension in turbulent flow. Journal ofAerosol Science, 22:S149–S152, 1991.

[17] H. v. Kemenade. Aerosols and particle transport. In Aerosols in Biomass Combustion,volume 6, pages 107–118, 2005.

[18] W. Krambrock. Die Berechnung des Zyklonabscheiders und praktische Gesichtspunktezur Auslegung. Aufbereitungstechnik, 7:391–401, 1971.

[19] H. Masuda and S. Matsusaka. Powder Technology Handbook, chapter Particle depositionand reentrainment. Marcel Dekker Inc. New York-Basel-Hong Kong, 2nd edition, 1997.

[20] J. Matsuyama and H. Yamamoto. Electrification of single polymer particles by successiveimpacts with metal targets. IEEE Tranactions on Industry Applications, 31:1441–1445,1995.

[21] S. Obermair, J. Woisetschlager, and G. Staudinger. Investigation of the flow pattern indiffertent dust outlet geometries of a gas cyclone by laser doppler anemometry. PowderTechnology, 138:239–251, 2003.

[22] M. O’Neill. A sphere in contact with a plane wall in a slow linear shear flow. ChemicalEngineering Science, 23:1293–1298, 1968.

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List of Figures

2.1 Schematic view of a cyclone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Side and top views of the four most used inlet configurations (a) circular or

’pipe’ inlet, (b) ’slotted’ (also called ’tangential’) inlet, (c) ’wrap-around’ inletand (d) axial inlet with swirl vanes [12]. . . . . . . . . . . . . . . . . . . . . . . 15

3.1 Stages of particulate fouling [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Cumulative particle size distributions of the experiment C1 V 1000 µ3.0 w1 [9]. 213.3 Natural vortex lengths, L according equations (3.2)-(3.4) as function of constant

(a) DxDcb

and (b) D2cb

ab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4 (a) The ring formation in the tube section. (b) A sudden transition in the

amount of dust deposits on the wall [11]. . . . . . . . . . . . . . . . . . . . . . . 233.5 Schematic reproduction of the ring deposition in the cyclone. . . . . . . . . . . 24

4.1 Model of the plastic deformed particle. . . . . . . . . . . . . . . . . . . . . . . . 274.2 Model for the wall roughness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3 Coefficient of restitution e, normal coefficient of restitution en and tangential

coefficient of restitution et for f = 0.2 as function of impact angle θ1. . . . . . . 324.4 Contribution of the parameters in equation (4.15) to the uncertainty budget. . 334.5 Critical sticking velocity for zero electrostatic (wP,crit,vdW) and zero van der

Waals (wP,crit,el) influence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.1 Particle-surface geometry for rock’n roll model [24]. . . . . . . . . . . . . . . . 41

6.1 (a) Cumulative particle size distributions calculated with the results of experi-ments and CFD-simulations at 1000 m3/h and 3 g/kg. (b) Cumulative particlesize distributions of the CFD-simulations at 600 m3/h (legends correspond withcurves from top to bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6.2 Cumulative particle size distributions of the inlet, fine and coarse dust fractions(legend corresponds with curves from top to bottom). . . . . . . . . . . . . . . 49

6.3 (a) Cumulative distributions for sub-micron particles at 1000 m3/h. (b) Frac-tions of the depositions with respect to the released dust (legend correspondswith curves from top to bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . 50

60 List of Figures

A.1 Geometries of the test cyclones C1 and C7. . . . . . . . . . . . . . . . . . . . . 63

B.1 (a) Cumulative particle size distribution function. (b) Particle size distributionfunction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

B.2 Cumulative distribution of the dimensionless particle relaxation times for V=(a) 600 m3/h, (b) 800 m3/h, (c) 1000 m3/h and (d) 1200 m3/h. . . . . . . . . 68

B.3 Particle deposition from fully developed turbulent pipe flow: a summary ofexperimental data [34]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

C.1 (a) Tangential velocity plot (left simulation, right LDA measurements). (b)

Axial velocity plot (left simulation, right LDA measurements)[35]. . . . . . . . 72C.2 (a) Magnus-force for a rotating particle in a flow. (b) Saffman-force in a flow

with a velocity gradient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72C.3 Model for the wall roughness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

D.1 Growth of the adhering particle layers [10]. . . . . . . . . . . . . . . . . . . . . 78D.2 Porosity versus particle layer thickness [2]. . . . . . . . . . . . . . . . . . . . . . 79

E.1 (a) Critical sticking velocity ±u. (b) Relative uncertainty. . . . . . . . . . . . 82

List of Tables

3.1 Material parameters during tests [9]. . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Volume flow rate and loading conditions during the tests [9]. . . . . . . . . . . 193.3 Natural vortex lengths for different models. . . . . . . . . . . . . . . . . . . . . 22

4.1 Material properties of limestone [30]. . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Coefficients of restitution in the case of a hard wall for several colliding angles

[29]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3 Uncertainty of the parameters in the calculation of the critical velocity. . . . . 31

5.1 Bouncing and removal limits for 54 µm bronze particles. . . . . . . . . . . . . . 45

6.1 Mean time for a particle to deposit. . . . . . . . . . . . . . . . . . . . . . . . . 49

B.1 Discrete particle size distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . 66B.2 Discrete particle size distribution of sub-micron particles. . . . . . . . . . . . . 66B.3 Reynolds numbers for the various volume flow rates through the test cyclone. . 67

E.1 Calculation uncertainty budget. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

A Cyclone Geometry

Figure A.1: Geometries of the test cyclones C1 and C7.

B Particle Distributions

In the experiments performed by Gebhard [9] the particle size distribution was measured.From the PSD, the distribution for the dimensionless particle relaxation time is calculated fordifferent volume flow rates through the test cyclone.

B.1 Particle Size Distribution

The particle size distribution of the limestone used in the experiments of Gebhard [9] isdistributed according the RRSB (DIN 66145) cumulative particle size distribution function(CPDF) which is given by:

CPRRSB(dP) = 1− exp[−(dP

d′P

)nRRSB]

(B.1)

The CPDF is plotted in figure (B.1a) for the parameters as mentioned in table (3.1). Afterdifferentiating equation (B.1) to dP, the particle size distribution (PDF) is found (figure B.1b).The mode value of the particle size distribution is 2 µm and the median 5 µm.

(a)

-1

(b)

Figure B.1: (a) Cumulative particle size distribution function. (b) Particle size distribution function.

66 B Particle Distributions

Table B.1: Discrete particle size distribution.

dP 0.9 µm 1 µm 1.25 µm 1.5 µm 1.75 µm 2 µm 3 µm 4 µmPDF (dP) 0.0936 0.0943 0.0952 0.0951 0.0944 0.0933 0.0857 0.0760

dP 5 µm 6 µm 7 µm 8 µm 9 µm 10 µm 20 µmPDF (dP) 0.0660 0.0563 0.0475 0.0397 0.0329 0.0271 0.0029

Table B.2: Discrete particle size distribution of sub-micron particles.

dP 0.0125 µm 0.02 µm 0.05 µm 0.1 µm 0.125 µm 0.25 µm 0.5 µmPDF (dP) 0.0303 0.0333 0.0399 0.0456 0.0476 0.0540 0.0603

In table B.1 for particles in the range of 1-20 µm the fraction of a mixture existing out of the15 mentioned sizes is shown and in table B.2 7 extra particle diameters smaller than 1 µm.

B.2 Dimensionless Particle Relaxation Time Distributions

According to Masuda and Matsusaka [19] the deposition velocity is defined by:

vdep =J

c(B.2)

where J is the mass flux to the wall per unit wall area and c is the mean particle concentration inthe flow. The deposition velocity is made dimensionless by scaling it with the friction velocity:

v+dep =

vdep

u∗(B.3)

u∗ is approximated by the following function of the mean velocity u and the Reynolds number:

u∗ = 0.2uRe−1/4, for 3 · 103 < Re < 105 (B.4)

As approximation the Reynolds number in the cyclone will be based on Re in the inlet:

Re =uDh

ν(B.5)

where Dh is the hydraulic diameter:

Dh =4ab

2(a+ b)(B.6)

in which a and b are the hight and the width of the cyclone inlet (0.175 m and 0.100 mrespectively). For various volume flow rates at 293 K (ν = 18.24·10−6 m2 s−1), Re is representedin table B.3. It is noticed that for the lower volume flow rates, the condition for Re in equation

B.2 Dimensionless Particle Relaxation Time Distributions 67

Table B.3: Reynolds numbers for the various volume flow rates through the test cyclone.

V Dh Re[m3/h] [m] [-]

600 0.127 6.65 · 104

800 0.127 8.86 · 104

1000 0.127 1.11 · 105

1200 0.127 1.33 · 105

(B.4) is met and that for the higher Re numbers is slightly exceeded. The particle relaxationtime is given by:

τ =(ρP − ρG)d2

PCc

18µ(B.7)

in which Cc is the Cunningham correction factor which is a function of the Knudsen numberKn [17]:

Cc = 1 + 2Kn[1.257 + 0.4 exp

(−1.1

1Kn

)](B.8)

The Knudsen number is the ratio of the mean free path of the continuous phase and themolecular diameter of the dispersed phase (Kn = λ/dP). For air at 293 K and 1 bar, λequals 0.066 µm. From equation (B.4) and (B.7) the dimensionless particle relaxation time iscalculated:

τ+ =τu∗2

ν=

0.04τRe1/2ν

(B.9)

From this particle size distribution, the distributions of dimensionless particle relaxationtimes is determined by using a Monte Carlo method. Since τ+ is a function of Re, it is afunction of the volume flow rate as well. For V = 600, 800, 1000 and 1200 m3/h the distributionfunctions are plotted in figure B.2.

In figure B.3 the results of five experiments are shown. Papavergos and Hedley divided v+dep

in the following three ranges according to Masuda an Matsusaka [19]:

vdep+ = 0.065Sc−2/3, τ+ < 0.2vdep+ = 3.5 · 10−4τ+2, 0.2 < τ+ < 20vdep+ = 0.18, τ+ > 20

(B.10)

in which Sc is the Schmidt number:

Sc =3πνηdP

kTCc(B.11)

68 B Particle Distributions

(a) (b)

[-]

(c) (d)

Figure B.2: Cumulative distribution of the dimensionless particle relaxation times for V= (a) 600m3/h, (b) 800 m3/h, (c) 1000 m3/h and (d) 1200 m3/h.

Figure B.2 shows that, depending on the volume flow rate through the cyclone, the fractionparticles for which 0.2 < τ+ < 20 lies between 60% and 80%. The measured values for thedimensionless deposition velocities in figure B.2 can differ up to two orders of magnitude forthe same value of τ+. This makes the behavior of the particle mixture difficult to predict.

B.2 Dimensionless Particle Relaxation Time Distributions 69

Figure B.3: Particle deposition from fully developed turbulent pipe flow: a summary of experimentaldata [34].

C Flow Models

In order to simulate the particle tracks with Fluent, the Euler-Lagrange solving method is used.This implies that first the solution of the continuous phase is computed (and, if necessary,compared with LDA-experiments). In section C.1 these results are discussed. The calculationof the particle tracks of the dispersed phase is discussed in section C.2.

C.1 Turbulent Flow Models

Zagorski [35] computed the velocity profile of cyclone type C1 and compared the results withLDA-measurements. The solutions of the CFD-simulations were reached by applying the fol-lowing steps:

� The simulation was started with the laminar model.

� When the residuals flattened out, the model was switched to the standard k-ε model.

� When these residuals flattened out as well, the Reynolds Stress Model (RSM) was applied.

� At the end, the inlet velocity was raised (6-9-12.69 m/s) to reach the final volume flowrate of 800 m3/h.

The velocity profiles for the under part of the conical section, the vertical tube and the dustbinare shown in figure C.1. The tangential velocity is underestimated in this simulation. For theaxial velocity the simulation correspond better with the measurements. No results are availablefor the upper part of the cyclone. Exact values for the measurement points are given in themaster thesis of Zagorski [35].

C.2 Two Phase Flow Model

The particles are injected into the solution of the continuous phase and their trajectories arecomputed. To compute the path of a particle with diameter dP, density ρP and current velocity−→wP in a fluid with local velocity −→u and dynamic viscosity η, Newtons second law is applied:

d−→wP

dt=

18ηρPd2

P

CDReP24

(−→u −−→wP) +~g(ρP − ρG)

ρP+ Fi (C.1)

72 C Flow Models

(a) (b)

Figure C.1: (a) Tangential velocity plot (left simulation, right LDA measurements). (b) Axial velocityplot (left simulation, right LDA measurements)[35].

in which the first term on the right-hand side represents the drag force, the second the grav-itation force and the third the additional forces. These forces can be the Magnus-, Saffman-,thermophoretic and Coulomb-force.

WP

FM

(a)

FSaff

(b)

Figure C.2: (a) Magnus-force for a rotating particle in a flow. (b) Saffman-force in a flow with avelocity gradient.

C.2 Two Phase Flow Model 73

In a flow, a rotating particle experiences a relative fluid velocity (decreased velocity at oneside and an increased one at the other side (figure C.2)). This causes a pressure gradientover the particle. Therefore the particle experiences a force into the direction of this gradient.This phenomena is called the Magnus-effect (Magnus-force

−→F M). In a flow with a velocity

gradient, a pressure gradient exists as well, which induces the so called Saffman-force,−→F Saff .

Both forces induce an acceleration perpendicular to the original particle path by which this ischanged. According the Fluent manual [8], the Saffman-force is only important for submicronparticles. The Coulomb-force

−→F C only yields for charged particles and the thermophoretic

force−→F th for flows with a temperature gradient. The forces are computed as follows:

−→F M = CMπ

(dP

2

)3

ρG(−→ω P −∇×−→u )× (−→wP −−→u ) (C.2)

−→F Saff = 6.46ρν1/2d

2P

4(−→u −−→wP)

∣∣∣∣d−→ud−→x∣∣∣∣1/2

−→F th = −Cth

1mPT

∂T

∂x

|−→F C| =

qPq04πε0l2

The drag coefficient for the Magnus-force according to Heinl and Bohnet [10] equals CM = 2.The equation to compute the Saffman-force is based on a laminar flow around a particle. Inthe equation for computing the thermophoretic force, the derivative of the temperature T withrespect to the position ~x is a linear factor. Since the uncleaned gas is at room temperature,it is plausible to make the assumption that it stays at the same temperature in the cyclone(∂T

∂~x = 0). Therefore the temperature field is homogeneous and Fth is negligible. For theCoulomb-force it is assumed that the wall is uncharged. In this case, a particle will detect afictitious particle with the same charge (qP = q0) and the direction of

−→F C is to the wall.

The particle Reynolds number in equation (C.1), ReP is defined as:

ReP =ρGdP|−→wP −−→u |

η(C.3)

and the drag coefficient CD as [10]:

CD =

{316 + 24

ReP≈ 24

RePfor ReP<0.01;

24ReP

(1 + 0.1315Re0.82−0.05 log10 ReP

P

)for 0.01<ReP<20.

(C.4)

In order to calculate the angular velocity in equation C.2 the following model is used [10]:

−→ω P,new =12(∇×−→u )(−→ω P,old −

12(∇×−→u )) exp

(− 60ηd2

PρP∆t)

(C.5)

74 C Flow Models

C.3 Particle-Particle Collision

The probability for a collision between two particles Pcol according to the kinetic gas theory isgiven by [10]:

Pcol = 1− e−|wP,rel|

λ∆t (C.6)

which depends on the relative velocity between particles, wP,rel, and the mean free path lengthλ which is λ = f(dP,i, dP,j, ρP, c).

The mean relative velocity can be written as function of the temperature T and the Boltz-mann constant k as:

|wP,rel| =√

8kTπmAB

(C.7)

in which mAB is the reduced mass, which is defined as [27]:

mAB =mAmB

mA +mB(C.8)

with mA and mB the masses of the two colliding particles. The number of collisions betweenmolecule A and B per unit of volume and per unit of time, ZAB, follows from multiplicationof equation (C.7) with the molecule concentrations NA and NB and the collision cross sectionAcol which is:

Acol = πd2col,AB = π(rA + rB)2

so that ZAB becomes:

ZAB =(

8πkTmAB

)1/2

d2col,ABNANB (C.9)

And finally, for the mean free path λ is found:

λ =|−→wP|ZAB

=|−→wP|

d2col,ABNANB

·(

8πkTmAB

)−1/2

C.4 Wall Roughness

Heinl and Bohnet [10] modeled the surface roughness. They adjusted an existing model forparticles smaller than the wall roughness. A schematic model for the wall roughness is presentedin figure C.3. A particle moves into direction of a well at the wall with velocity −→wP underan angle β1 (with respect to the mean wall inclination). The typical size of the well is LR

C.4 Wall Roughness 75

LR

HR

H - ÄHR R 21

3

13

1WP

Figure C.3: Model for the wall roughness.

in the length direction and of the depth for both sides HR and HR −∆HR respectively. Thetwo inclinations α1 and α3 are determined from a standard normal deviation with standarddeviation ∆α:

∆α = ζαw,max (C.10)

in which ζ is a factor, which is 1/6 for 40 µm particles according Sommerfeld and Huber [29].Heinl and Bohnet [10] used the value of 0.5 as well. The maximum wall inclination γW,max isgiven by:

αw,max = arctan(

2HR

dP

)(C.11)

for the case no particles stick on the wall. For the wall roughness HR, Heinl and Bohnet [10]used values of Triesch and Bohnet [31]. In a smooth steel pipe HR is equal to 5 µm. For thecase a layer of particles is sticking to the wall already, in stead of the wall roughness HR, theparticle diameter of the sticking particles dP,adh should be used. In this situation equation(C.11) becomes:

αw,max =π

2arccos

(dP,adh

dP,adh + dP

)(C.12)

The standard normal probability function of the wall inclination then becomes:

P (∆α, α) =1√

2π∆α2exp

(− α2

2∆α2

)(C.13)

76 C Flow Models

There exists a so-called shadow effect for small particle impact angles. This implies that whenthe impact angle β1 becomes smaller than the absolute value of a negative wall inclination,| α− | a particle may not hit the lee side (indicated by ”1” in figure C.3) of a roughness structure.However, with decreasing chance to hit the lee side, the chance to hit the luff side (indicatedby ”3” in figure 4.2) increases. Thus, for a given β1 and α the following three regimes of theeffective roughness are identified [29]:

1. The particle cannot hit a roughness structure with | α− |> β1 where the probabilitybecomes zero:

P (β1, α) = 0 (C.14)

2. The probability to hit a roughness structure with a negative inclination in the interval0 <| α− |< β1 is smaller than to hit a horizontal wall by the factor:

P (β1, α) =sin (β1 + α−)

sinβ1(C.15)

3. The probability to hit a positive inclined wall roughness structure (i.e. α = α+ > 0) ishigher than to hit a horizontal wall by the factor:

P (β1, α) =sin (β1 + α+)

sinβ1(C.16)

When a particle does not hit the lee side, it has a higher probability to hit the luff side.The effective probability function of the wall roughness inclination seen by the the particle,Peff will therefore be larger than that of the wall itself (equation (C.13)). It is calculated bymultiplication of equation (C.13) with the factor in equation (C.16):

Peff(β1,∆α, α) = P (∆α, α)f(β1, α) =1√

2π∆α2exp

(− α2

2∆α2

)sin (β1 + α+)

sinβ1(C.17)

The effective mean roughness angle is calculated by integrating the product of Peff and α fromα = β1 to α→∞:

αeff(∆α, β1) =∫ ∞

β1

αPeff(β1,∆α, α)dα (C.18)

D Particle Layers

The growth of particle layers on the wall is considered. The way in which particles are stackedin the cells is considered as well. In addition, the influence of the layer thickness on the stickingcriterion is discussed.

D.1 Particle Layer Model

To simulate the particle flow the Lagrange treatment is used by Heinl and Bohnet [10] inwhich not the single particles but particle packages are calculated. The number of particlesper package is rather arbitrair. When a package has reached the wall, not the entire packageis considered to adhere at the wall but a specific fraction. By doing this, the influence of thepackage size is reduced. When the velocity condition (equation (4.15)) that is required foradhesion is met, the number of particles after collision NP,2 is calculated from:

NP,2 = NP,1 −NP,adh (D.1)

where NP,1 is the number of particles before collision and NP,adh is the number of particlesthat has adhered to the wall, which is calculated from:

NP,adh =4Acell

πd2P

(D.2)

in which the porosity due to the holes between the particles that stick on the cell with surfaceAcell is neglected. The remaining particles in the packages, NP,2, are transported further inthe flow. When a package reaches the cell that is closest to the wall, the number of particleswithin the package that stick at the wall will be computed again. The particles will only enterthe closest wall cell if:

HP,layer 5 Hcell (D.3)

The particles will adhere in the second closest cell to the wall if the the particle layer is thickerthan the cell (figure D.1).

78 D Particle Layers

Figure D.1: Growth of the adhering particle layers [10].

D.2 Layer Porosity

Abd-Elhady et al. [2] describes a model for the formation of particle layers at heat transfersurfaces in which the critical sticking velocity is given by a function that includes the influenceof the current particle layer thickness. The coefficient of regularity of a porous structure Cpore,is introduced as the ratio between the pore diameter, Dpore, in a specimen of thickness δ andthe equilibrium pore diameter Dpore,eq.

Cpore =Dpore

Dpore,eq=(

δ

20dP,m

)−0.282

(D.4)

For layers thicker than or as thick as the the equilibrium thickness δeq, Dpore becomes equalto the equilibrium pore diameter, Dpore,eq. For bronze particles this equilibrium thickness δeqwas determined in experiments to be 20 dP,m. The pore volume, Vpore, is proportional to thethird power of the pore diameter and therefore the following holds:

Vpore

Veq=(

Dpore

Dpore,eq

)3

= C3pore (D.5)

where Veq is the pore volume for the reference diameter Dpore,eq. For two sintered layers ofthickness δ and δeq the following relation for the porosity P yields:

P =Vpore/Veq

(Vpore/Veq − 1) + 1/Peq(D.6)

in which P is determined from a sample with mass msamp, volume Vsamp and particle densityρP from:

D.2 Layer Porosity 79

Figure D.2: Porosity versus particle layer thickness [2].

P = 1− (msamp/Vsamp)ρP

(D.7)

And after substitution of equation (D.5) in (D.6):

P =C3

pore

(C3pore − 1) + 1/Peq

(D.8)

In equation (D.4) it is seen that for increasing specimen thickness δ the coefficient Cpore

decreases and therefore P decreases in equation (D.8) until the equilibrium porosity, Peq, isreached for δ ≥ 20 dp,m. In figure (D.2) this relation is plotted for a particle layer that consistsof bronze particles with a diameter of 54 µm, which has a final porosity of 0.54. An explanationfor this behaviour can be found in the energy loss (El in equation (4.2)) during collisions. Thisloss is probably due to the breakage of adhesion energy between the moved particles and itsneighboring particles. For increasing porosity, more particles are initiated to move and so theenergy loss increases as well. This leads to an increasing critical velocity. The critical particlevelocity wP,crit was found to be a function of the porosity as:

wP,crit = wcrit,0eCPP (D.9)

80 D Particle Layers

In which the factor wcrit,0 is the critical particle velocity at zero porosity, which can be calcu-lated with equation (4.15). The factor CP has to be determined experimentally. Abd-Elhadyet al. found for copper particles of 54 and 42.4 µm respectively values of 6.72 and 6.07.

E Error Analysis

An error analysis is made as in equation (E.1) [26]:

u2M =

∑i

([∂M

∂mi

]Mµ

)· u2

mi(E.1)

In table E.1 this is written in tabular form. The error analyses is adopted on equation (4.15)and table 4.3. The range for critical sticking velocity plus/minus one standard deviation isshown in figure E.1a. The relative error (σ/wP,crit) is presented in figure E.1b. The relativeerror has a constant error (in 4 significant digits) of 35.76%.

Table E.1: Calculation uncertainty budget.

variable value of standard sensitivity contribution tovariable uncertainty coeffiecient standard uncertainty in M

Mi mi u(mi) ci = ∂M∂dmi

ci · u(mi) = Ui(M)M1 m1 u(m1) c1 u1(M)M2 m2 u(m2) c2 u2(M)...

......

......

MN mN u(mN) cN uN(M)

M m total : u(M) =√u2

1 + . . .+ u2N

82 E Error Analysis

(a) (b)

Figure E.1: (a) Critical sticking velocity ±u. (b) Relative uncertainty.

wall depositions in gas cyclones - tu/e

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