Title

Forces on Particles and Bubbles

M. Sommerfeld

Mechanische Verfahrenstechnik Zentrum für Ingenieurwissenschaften Martin-Luther-Universität Halle-Wittenberg

D-06099 Halle (Saale), Germany www-mvt.iw.uni-halle.de

Martin-Luther-Universität Halle-Wittenberg Content of the Lecture

 BBO equation and particle tracking  Forces acting on particles moving in fluids  Drag force  Pressure, virtual mass and Basset  Transverse lift forces  Electrostatic force  Thermophoretic and Brownian force  Importance of the different forces  Particle response time and Stokes number  Behaviour of bubbles and forces  Particle response to oscillatory flow filed

Martin-Luther-Universität Halle-Wittenberg Equation of Motion 1  The equation of motion for particles in a quiescent fluid was first derived by Basset (1888), Bousinesque (1885), and Oseen (1927) BBO-equation.  A rigorous derivation of the equation of motion for non-uniform Stokes flow was performed by Maxey and Riley (1983). The BBO-equation without the Faxen terms (due to curvature of the velocity field) is given by:    d u 18 µ   m   Du d u  P = F − − P − ∇ + ∇τ +  F − P  mP 2 mP (uF uP ) ( p ) 0.5 mF   d t ρP DP ρP  Dt d t    Importance of  d u d u  F P the different  t * − *    ρ µ m d t d t * (u − u )  forces ??? + 9 F F P  dt + F P 0  + m g  ∫ * 1 2  P π ρP DP 0 (t − t ) t    

Accounts for d Derivative along D Substantial : : initial condition dt particle path Dt derivative

1: drag force 2: pressure term 3: added mass 4: Basset force (with initial condition) 5: gravity force Martin-Luther-Universität Halle-Wittenberg Equation of Motion 2 The calculation of particle trajectories requires the solution of several partial differential equations: particle location particle velocity particle angular velocity    d x p  d u  d ω  = u p p = p mp = ∑ Fi Ip T dt dt dt

 The consideration of heat and mass transfer requires the solution of two additional partial differential equations for droplet diameter and droplet temperature.  Analytical solutions for the different forces acting on the particle are only available for small particle Reynolds numbers (Stokes regime).  For higher particle Reynolds numbers empirical correlations are needed to calculate the different forces.  In the flowing the different forces for describing the particle motion are introduced and discussed (see also Sommerfeld 2000, 2008 and 2010).

Martin-Luther-Universität Halle-Wittenberg Drag Force 1  The drag force is the most important one and includes the friction and form drag or pressure drag.  For highly viscous flow (low particle Reynolds number) an analytic solution can be obtained for the drag force (Stokes 1851):    F = 3 π µ D (u − u ) D P F P  At higher particle Reynolds numbers however, empirical correlations are needed which are expressed as a drag coefficient: F C = D D ρ v2 A 2 rel P  This results in the following expression for the drag force:

 ρ π     F = F D2 C (u − u ) u − u D 2 4 P D F P F P

Martin-Luther-Universität Halle-Wittenberg Drag Force 2

 Fitting of drag coefficient data for spherical solid particles:

Stokes regime Standard drag correlation (Stokes 1851): (Schiller and Naumann, 1933): 24 C = 24 24 D = + 0.687 = ReP CD (1 0.15 ReP ) fD ReP ReP

2 measurements 10 Stokes regime correlation Newton regime: 101 Newton regime

[-] CD ≈ 0.44 D C 100 Critical Reynolds number

-1 10   -1 0 1 2 3 4 5 6 ρ D (u − u ) 10 10 10 10 10 10 10 10 F P F P ReP = Re P [-] µF Martin-Luther-Universität Halle-Wittenberg Drag Force 3  Summary of the drag coefficient obtained in different experimental studies (Crowe et al. 2012):

Martin-Luther-Universität Halle-Wittenberg Drag Force 4 The average drag coefficient for non-spherical particles (for a certain stable orientation) is determined from empirical correlations fitted to experimental data (Haider and Levenspiel 1989; Thompson and Clark 1991).

1000 24 B C CD = (1+ A ReP )+ ReP D 1+ 100 φ = 0.025 ReP φ = 0.05

[ - ] φ = 0.1 D φ = 0.2 c 10 φ = 0.3 φ = 0.5 φ = 0.7 1 φ = 0.9 A = exp (2.3288 − 6.4581 φ + 2.4486 φ2 ) φ = 1.0

-1 0 1 2 3 4 5 B = 0.0964 + 0.5565 φ 10 10 10 10 10 10 10 Re P [ - ] C = exp (4.905 −13.8944 φ +18.4222 φ2 −10.2599 φ3 )

D = exp (1.4681+12.2584 φ − 20.7322 φ2 +15.8855 φ3 ) sphericity S Martin-Luther-Universität φ = V Halle-Wittenberg SPartikel Drag Force 5

RarefactionThe importance effects: of rarefaction effects may be estimated on the basis of the Knudsen number (Crowe 2006):   λ π γ Ma (u − u ) Kn = = P = F P P Ma P Molecules DP 2 ReP a  Mean free path of the gas molecules:

µF  8 p  λ = c =   Particle Mol  π ρ  0.499 cMol ρF  F   Classification of the different regimes:

Continuum: KnP < 0.015 Slip flow: 0.015 < Kn < 0.15 P Transitional: 0.15 < KnP < 4.5 Free molecule: KnP > 4.5

Martin-Luther-Universität Halle-Wittenberg Drag Force 6

Cunningham correction for rarefaction effects (Davies 1945):

C 1 1 D = = < CD,Stokes Cu   0.55  ReP 0.25 1+ Kn 2.514 + 0.8 exp−  P    0.1< Kn <1000   Kn P  P

1

[ - ] 0.1

D, Stokes 0.01 / C D C 1E-3

1E-4 1E-3 0.01 0.1 1 10 100 1000 Kn [ - ] Martin-Luther-Universität Halle-Wittenberg Drag Force 7  Other effects on the drag force:

Turbulence of the surrounding fluid reduces the critical Reynolds number to about 1000.

Surface roughness also causes a reduction of the critical Reynolds number.

The porosity of particles results in a reduction of the drag coefficient.

With increasing particle concentration the drag is considerably increased (hydrodynamic interaction).

Particle motion in the vicinity of walls (normal and parallel).

Martin-Luther-Universität Halle-Wittenberg Pressure Force  Force on the particle due to a local pressure gradient and the shear stress in the flow:  mP  FP = (− ∇ p + ∇ τ) ρP

 With the Navier-Stokes equation one finds that:    D u   − ∇ + ∇ τ = ρ  F −  p F  g  D t   This yields the total pressure force as:

  ρ  D u   F = m F  F − g The second term is p P   the buoyancy force ρP  D t 

Martin-Luther-Universität Halle-Wittenberg Virtual Mass and Basset Force 1  For higher particle Reynolds numbers the virtual (added) mass and the Basset (history) force may be expressed as:

mP d   FA = 0.5 CA ρF (u F − u P ) ρP d t

 d    The second term in the (u − u )   ρ µ m  t d t F P (u − u )  Basset force accounts = F F P τ + F P 0 FB 9 CB  1 2 d  for the initial slip (Reeks π ρ D ∫ (t − τ) t P P  0  & McKee 1984)  

 Coefficients given by Odar and Hamilton (1964):

0.132 Acceleration number C = 2.1− A A2 + 0.12 C   2 ReP < 60 u F − u P 0.52 AC =   CB = 0.48 + d u − u (A +1)3 D F P C P d t

Martin-Luther-Universität Halle-Wittenberg Virtual Mass and Basset Force 2  Recent studies of Michaelides and Roig (2011) showed that the added mass

coefficient is constant (CA =1) and the Basset coefficient should be expressed in dependence of the Strouhal number:

0.82 2.5 CB = 2.0 −1.0533 [1− exp(− 0.14 ReP Sr ) ]

 The Strouhal number describes the behaviour of oscillatory flows. In this context the fluid time scale is the reciprocal value of the characteristic fluid oscillations.  In turbulent flows this time scale corresponds to the integral time scale of turbulence. 1 T Sr = = I

St τP

 Consequently the Strouhal number is the reciprocal value of the particle turbulent Stokes number.

Martin-Luther-Universität Halle-Wittenberg Other Forces Other important forces (field or external forces) are:  Gravity force:   = Fg mp g  Centrifugal and Coriolis force (appears only if the equation of motion is written in polar co-ordinates):

2 vP,ϕ vP,r vP,ϕ FZ = mP FC = mP rP rP

 Electrostatic force:   Fe = qP E

 Thermophoretic force  Magnetic force

Martin-Luther-Universität Halle-Wittenberg Electrostatic Forces 1  In an electrostatic precipitator, for example, the particles are charged by an ion-bombardment created by a negative corona discharge in the vicinity of a charging wire (Löffler 1988).  The charging of the particles is caused by two mechanisms: t Field charging occurs 2 qp (t) = n ⋅e = π ε0 p DP E1 due to the convective t + τq

motion of the ions and is 2 relevant for particles larger qmax = n e = π ε0 p DP E1 than about 0.5 to 1 µm. D − 1 (conducting particles: p = 3, D  ∞) p = 1+ 2 D + 2 (non-conducting particles p = 1.5 ÷ 2) D: relative dielectric constant of particle

ε0: absolute dielectric constant in vacuum Diffusion charging is the e: elementary charge

result of the thermal motion E1: field strength in the charging region of the ions and is relevant for particles with a diameter k ⋅T D  D c N e2 t  = π ε P + P 0 smaller than about 0.2 µm. qP (t) 4 0 ln 1  e 2  8 ε0 k T  Martin-Luther-Universität Halle-Wittenberg Electrostatic Forces 2 Calculated flow field, electrostatic potential and particle trajectories in an electrostatic precipitator (Böttner and Sommerfeld 2003)

Uav = 0.5 m/s

VH = 60 kV Particle charge:

0.66 ⋅ qmax Channel: 300 mm x 600 mm 0.15 Separation wires: 150 mm 0.10

Y [ m ] 0.05 Particles 12 – 20 µm 0.00 0.0 0.2 0.4 0.6 0.8 1.0 0.15 X [ m ] 0.10 Particles 0.9 – 1.5 µm

Y [ m ] 0.05

0.00 0.0 0.2 0.4 0.6 Martin0.8 -Luther1.0 -Universität X [ m ] Halle-Wittenberg Thermophoretic Force  Thermophoretic force due to temperature gradient (LBM):

 κF  6 π 1.17  + 2.18 Kn  κP  2 1 FT = dPart ρF νF ∇TF κ κ  κ  T F, P: thermal (1+ 3.42 Kn) 1+ 2 F + 4.36 Kn F conductivity of  κ   P  fluid and particles

dT/dh=1K/mm hot cold

Molecules

dT/dh=10 K/mm

Particle

DP = 100 nm

Martin-Luther-Universität Halle-Wittenberg Brownian Motion 1

 Brownian particle motion (LBM):

216⋅k ν T = ζ Boltz F F FB mP 2 5  ρ  ρF dP ∆t ζ Random number (0 ≤ ζ ≤ 1) π  P  i   CCun  ρF 

Only drag force

drag force and Brownian motion

Martin-Luther-Universität Halle-Wittenberg Brownian Motion 2  Lagrangian simulation of particle Brownian diffusion in homogeneous isotropic turbulence:

2.0x107 Tracking a large number of dP = 5nm d = 10nm particles and sampling the 1.5x107 P dP = 50nm displacement

dP = 100nm 1.0x107 d = 400nm P

PDF [1/m] PDF -5 5.0x106 10 Stokes-Einstein 10-6 ρ ρ P / F = 833 [m²/s]

0.0 D ρ / ρ = 500 0.0 -7 -7 -7 -6 10-7 P F 2.5x10 5.0x10 7.5x10 1.0x10 ρ / ρ = 100 Particle displacement, |x| [m] P F 10-8

10-9

k Boltz Tf 10-10 DS-E = Diffusion coefficient, 3πμf dp 10-11 10-9 10-8 10-7 10-6

Particle diameter, dP [m] Martin-Luther-Universität Halle-Wittenberg Slip-Shear Lift Force 1 Illustration of slip-shear lift force:

FLS uP

 Analytic expression of Saffman (1965, 1968) for small particle Reynolds numbers:

2 0.5 D 0.5 ∂ u F = 6.46 P (ρ µ ) F (u − u ) LS,Saff 4 F F ∂ y F P

Martin-Luther-Universität Halle-Wittenberg Slip-Shear Lift Force 2  Slip-shear lift for higher particle Reynolds numbers:

 Rotation of the fluid: ρF π 2    FLS = Dp CLS DP ((u F − u p )×ωF )    2 4 ωF = rot u F = ∇× u F

 Lift coefficient: Shear Reynolds Number:  4.1126 2 ρF Dp ωF CLS = f (Rep ,Res ) 0.5 ReS = ReS µF

 Correlation for higher Reynolds numbers (Mei 1997):

1  Rep  1 = − β 2 −  + β 2 ≤ Re f (Rep ,Res ) (1 0.3314 )exp  0.3314 for : Rep 40 β = S  10  0.5 ReP 1 2 f (Rep ,Res )= 0.0524 (β Rep ) for : Rep ≥ 40

Martin-Luther-Universität Halle-Wittenberg Slip-Shear Lift Force 3 Lift coefficient as a function of particle Reynolds number with the non-dimensional shear rate as a parameter (Sommerfeld 2010):

100 Saffman ReS β = 0.05 β = 0.5 β = 0.1 ReP β = 0.3 10 β = 0.5 [ - ] LS C 1 Straight lines: Stokes regime

0.1 0.01 0.1 1 10 100 1000

ReP [ - ]

Martin-Luther-Universität Halle-Wittenberg Slip-Shear Lift Force 4  Importance of slip-shear lift force compared to transverse drag force (Sommerfeld 1996):

2 0.5 D 0.5 ∂ u   DP [µm] 6.46 P (ρ µ) u − u du/dy F 4 ∂ y P LS = [1/s] Air Water F π 2   Dr ρ D C (v − v ) u − u 8 P d P P 1 2310 588

0.5 0.5   10 730 186 F  ρ  ∂ u u − u 1 LS = 0.17 D   P P   1 0.66 100 231 59 FDr  µ  ∂ y (v − vP ) 1+ ReP 6 1000 73 19  Stokes drag  Slip velocity ratio u/v: 10

0.5 0.5 FLS  ρ  ∂ u =1.7 DP   =1 Limiting particle diameter: FDr  µ  ∂ y −0.5 −0.5  ρ  ∂ u DP > 0.588    µ  ∂ y Martin-Luther-Universität Halle-Wittenberg Slip-Rotation Lift Force 1  Illustration of slip-rotation lift force:

Analytic expression FLR Rubinow and Keller (1961)     3   FLR = π R P ρF (Ω× V) V =(u F − u P )

 1   Particle rotation ωP Ω = ∇× u F − ωP relative to the fluid 2

 Lift force for higher particle Reynolds numbers:     ρ π   Ω×(u − u ) F = F D2 C u − u F P LR 2 4 p LR F P Ω  2 Reynolds number ρF Dp Ω of rotation: ReR = µF Martin-Luther-Universität Halle-Wittenberg Slip-Rotation Lift Force 2  Lift coefficient with non-dimensional relative rate of rotation as a parameter (Sommerfeld 2010):

Rubinow and Keller (1961) Oesterle and Bui Dinh (1998)  for ReP < 140 (2000) Dp Ω Re = ⋅ γ = = R CLR 2   0.4 0.7 u F − u p ReP CLR = 0.45 + (2⋅ γ − 0.45)exp (− 0.075⋅ γ ⋅ReP ) 10 γ = 0.5 γ = 1 γ = 3

[ - ] Straight lines: LR

C Stokes regime 1

0.1 1 10 100 1000 Re P [ - ] Martin-Luther-Universität Halle-Wittenberg Slip-Rotation Lift Force 3  Importance of slip-rotation lift force (Sommerfeld 1996):

π 3 1  ∂ v ∂ u  z  ρ DP   −  − ωP  (u − u P ) 8 2  ∂ x ∂ y  FLR     = DP (µm) F π 2 Dr ρ D c (v − v ) u − u 8 P d P P z Air Water ωP [1/ s]

1000 194 45  no rotation of the fluid 2000 137 32  Stokes drag 5000 87 22 F ρ D2 ωz (u − u ) LR = P P P 10000 61 15 FDr 24 µ (v − vP )

FLR ρ 2 z = 0.417 DP ωP =1 Limiting particle diameter: FDr µ µ 1 > DP 2.4 z ρ ωP Martin-Luther-Universität Halle-Wittenberg Torque Angular velocity of particles and torque:  Rubinow and Keller (1961):  Korrelation Gl. 41 und 42  1000 dωP 3  Rubinow und Keller (1961) IP = T = −π µF DP ωP Dennis et al. (1980) dt 100 Sawatzki (1970)  R Generalised torque: C 10 5  ρ  D    = F  p  Ω ⋅Ω T   CR 1 2  2 

0.1 0.1 1 10 100 1000 Re  The torque-coefficient was found from: R

Experiments by Sawatzki (1970)

Direct numerical simulations by Dennis et al. (1980) π 12.9 128.4 64 = + < < CR = for : ReR < 32 CR 0.5 for :32 ReR 1000 ReR ReR ReR

Martin-Luther-Universität Halle-Wittenberg Effect of Lift Forces  Calculation for a particle-laden vertical pipe flow (diameter = 20 mm) with

DP = 400 µm (Lee und Durst, 1982):

Importance of transverse lift forces

Without lift forces With lift forces

Martin-Luther-Universität Halle-Wittenberg Particle Response 1 Derivation of the particle response time based on the equation of motion:

du p ρ π du p 18 µ CD Rep = F 2 − − = F − mp Dp CD u F u p (u F u p ) 2 (u F u p ) dt 2 4 dt ρp Dp 24

du p 1 2 = (u F − u p ) 4 ρ D dt τ τ = p p p 24 p C = f 3 µF CD ReP D Re D  Particle response time: P 0.687 fD = (1 + 0.15 ReP ) ρ D2 p p τp = 18 µF fD u F

 Integration yields: uP 0.632 uF   t  u = u 1− exp−  P F   τ    p  time τP Martin-Luther-Universität Halle-Wittenberg Particle Response 2  The response of particles to velocity changes in the surrounding fluid may be characterised with the so-called Stokes number (ratio of particle response time to relevant time scale of fluid): τ St = p τF  Analysis of particle dispersion in a plane mixing layer (Crowe et al. 1996): Time scale of vortex development in a shear layer D τ = F U

ρ D2 U St = p P 18 µF fD D

Martin-Luther-Universität Halle-Wittenberg Particle Response 3

St << 1 (the particles almost completely follow the vortex structures)

St ∼ 1 (the particles become more inertial and accumulate at the rim of the vortex)

Martin-Luther-Universität Halle-Wittenberg Particle Response 4  St >> 1 (the particles are not able to follow the vortex structure completely)

Summary of particle response behaviour

Martin-Luther-Universität Halle-Wittenberg Bubble Behaviour 1

 In most of the technical applications bubbles are relatively large (DB > 2 mm) and therefore are generally non-spherical and oscillating.  The bubble shapes can be characterised by representative non-dimensional parameters (Clift et al. 1978): ρ V d Re = F B e µ

We g ρ − ρ d2 Eo = = F B e Fr σ

g µ4 ρ − ρ = F B Mo 2 3 ρF σ

2 2 VB de ρF − ρB V We = Fr = B σ g de

Response time (ρ + 0.5 ρ ) D2 τ = B F B for bubbles B 18 µF fD Martin-Luther-Universität Halle-Wittenberg Bubble Behaviour 2  The drag coefficient of bubbles was obtained from single bubble rise experiments and was found to be strongly dependent on the liquid type and the contamination of the liquid.

Purified liquids: fluid bubble

Contaminated liquids: rigid bubble

The drag coefficient is an average value

Martin-Luther-Universität Halle-Wittenberg Bubble Behaviour 3

Rigid bubble Drag coefficient 24 c = (1+ 0.15 Re 0.687 ) → Re <1000 D Re B B B rigid bubble − c = 9.5⋅10 5 Re1.397 → 1000 < Re <1530 D B B 100 fluid bubble (present) c = 2.61 → Re >1530 fluid bubble Tomiyama D B 10

Tomiyama et al. (1998) rigid: - [ ] D

 24 8 Eo  c = + 0.687 CD max (1 0.15 Re ),  1 Re 3 Eo + 4

Fluid bubble (Lain et al. 2002) 0.1 16 cD = → ReB <1.5 0.1 1 10 100 1000 ReB 14.9 Re [ - ] = → < < B cD 0.78 1.5 ReB 80 ReB 48  2.21  =  −  + ⋅ −15 4.756 → < < cD 1 0.5  1.86 10 ReB 80 ReB 1530 ReB  ReB  g (ρ − ρ )D2 c = 2.61 → Re >1530 f g b D B Eo = Tomiyama et al. (1998) fluid: σ    16 0.687 48  8 E 0 MartinCD = max-Luthermin-Universität (1+ 0 .15 Re b ), ,  Re Re 3 E + 4 Halle-Wittenberg  b b  0  Bubble Behaviour 4  Bubble slip velocity: comparison of experiments with results from different

correlations for the drag coefficient 0.4 correlations (Bröder and Sommerfeld 2007) rigid bubble fluid bubble 0.3 Tomiyama [m/s] b

0.2 , v' b Experiments , u'

0.4 b loop facility U 0.1 Ub bubble column 0.3 Ub 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.2 diameter Db [mm] [ m/s ] [ m/s B

V Haberman & Morton (1956) 0.1 rigid bubble fluid bubble (present) fluid bubble Tomiyama 0.0 0 1 2 3 4 5

D B [ mm ] Martin-Luther-Universität Halle-Wittenberg Bubble Behaviour 5  In bubbly flows the added mass force is of great importance since especially wobbling bubbles never exhibit stationary rise behaviour (Tomiyama 2004)

Spherical bubbles ReB < 1: CVM = 0.5

For ellipsoidal bubbles the added mass coefficient is a tensor :

Cvm,h 0 0    CVM =  0 Cvm,h 0     0 0 Cvm;h 

For oblate bubbles (aspect ratio: E < 1) an analytic solution for the coefficients in horizontal (h) and vertical (v) direction were provided by Lamb (1932):

E cos−1 E − 1− E2 cos−1 E − E 1− E2 = C = for E <1 CVM,v VM,h −1 2 −1 E2 1− E2 − E cos−1 E (2 E − E) 1− E − cos E

Martin-Luther-Universität Halle-Wittenberg Bubble Behaviour 6 The magnitude and direction of the transverse lift force strongly depends on the bubble size (Tomiyama 2004):

3 π De F = − C ρ (U − U )x rot U Spherical bubbles ReB >> 1: C = 0.5 A Lift L 6 B L L Lift

min (0.288⋅ tan h (0.121 ReB )), f (Eoh ) for : Eoh < 4 Tomiyama- CLift =  f (Eoh ) for : 4 ≤ Eoh correlation

3 2 f (Eoh ) = 0.00105 Eoh − 0.0159 Eoh − 0.0204 Eoh + 0.474

ρ ∆ρ D2 Eo = h h σ

Dh = a

small large bubbles bubbles

Martin-Luther-Universität Halle-Wittenberg Bubble Behaviour 7  Influence of shear flow on bubble migration calculated by a VOF approach (Tomiyama et al. 1993)

Eo = 1, Mo = 10-3 Eo = 10, Mo = 10-3

g (ρ − ρ )D2 Eo = f g b σ

g µ4 (ρ − ρ ) = f f g Mo 2 3 ρf σ

Determination of lift forces

Martin-Luther-Universität Halle-Wittenberg Bubble Behaviour 8  Flow around nearly spherical and large non-spherical bubbles simulated by VOF (Bothe et al. 2007)

g (ρ − ρ )D2 Eo = f g h h σ Tomiyama correlation

min (0.288⋅ tan h (0.121 ReB )), f (Eoh ) for : Eoh < 4 CA =  f (Eoh ) for : 4 ≤ Eoh

3 2 f (Eoh ) = 0.00105 Eoh − 0.0159 Eoh − 0.0204 Eoh + 0.474

Martin-Luther-Universität Halle-Wittenberg Bubble Behaviour 9  Lift force on a spherical bubble obtained by Legendre and Magaudet (1997) from resolved numerical simulations. Bubble Reynolds-number Non-dimensional d UL ρF VB DV shear rate DB Reb = dy µ Sr = F UL − UB  0.1 ≤ Reb ≤ 500  Sr ≤ 1 10

1 [ - ]

L,b C 0.1 Leg & Mag, Sr = 0.01 Leg & Mag, Sr = 0.2 2 2 Tomiyama et al. CL,b = CL,low + CL,high 0.01 0.1 1 10 100 1000 Re [ - ] 2 2 b  6 2.255   1+16 / Re  =   +  b  CL,b 2 1/ 2 Martin3/ 2 -Luther0.5 -Universität  π (Re Sr) (1+ 0.2 Re Sr)  1+ 29 / Re  b b Halle -Wittenbergb  Particle Response 1 Analysis of particle response in turbulent flow and the importance of the different forces (Hjelmfelt and Mockros, 1966). Simplified equation of motion for the Stokes regime: d u d u − P d u 18 µ ∂ u  d u d u  ρ µ m t d τ d τ m P = m (u − u ) − m + 0.5 m  − P  + 9 P dτ P 2 P P F F   ∫ 1 2 d t ρP DP ∂ t  d t d t  π ρP DP (t − τ) t0 Rearrangement of the equation:

t t d u d u d τ ∂ u d u d τ P + + P τ = + + τ a u P c ∫ 1 2 d a u b c ∫ 1 2 d d t t (t − τ) ∂ t t (t − τ) 0 0 with: 18 µ ρ = 3 9 1 µ a 2 b = c = (ρ ρ + 0.5) D ρ ρ + P P 2 ( P 0.5) (ρP ρ + 0.5) DP π ρ

Martin-Luther-Universität Halle-Wittenberg Particle Response 2 The velocities of fluid and particles are expressed by Fourier integrals: ∞ ∞ = ς ω + λ ω ω = σ ω + ϕ ω ω u ∫ ( cos t sin t) d u P ∫ ( cos t sin t) d 0 0

Introducing the velocities into the equation of motion yields the amplitude ratio and the phase angle:  f  2 2 β = −1 2 η = (1+ f1 ) + f2 tan   1+ f1  With: ω (ω+ c 0.5 π ω)(b −1) ω (a + c 0.5 π ω)(b −1) = f1 = 2 2 f2 2 2 (a + c 0.5 π ω) + (ω+ c 0.5 π ω) (a + c 0.5 π ω) + (ω+ c 0.5 π ω) As a parameter a modifies Stokes number is used (different from Hjelmfelt and Mockros, 1966) : ρ D2 St = τ ω = P P ω P 18 µ

Martin-Luther-Universität Halle-Wittenberg Particle Response 3

 Droplets in air (ρ /ρ = 1000, D = 50 µm): p P 102 SD+PG+AM+BH (I) SD+PG+AM (II) 1 [-] 10 SD+PG (III) η SD (IV) Amplitude

100 Phase

-1

Amplitude ratio 10 1.5 Type II, III, IV SD+PG+AM+BH SD+PG+AM -2 1.0 10 SD+PG 10-3 10-2 10-1 100 101 102 103 0.5 SD Stokes number St [-] [radians] β 0.0

-0.5 Increasing ω

Phase anglePhase -1.0

-1.5 10-3 10-2 10-1 100 101 102 103 Stokes number St [-] Martin-Luther-Universität Halle-Wittenberg Particle Response 4 ρ ρ µ  Particles in liquid ( p/ = 2.5, DP = 200 m): 102 SD+PG+AM+BH Amplitude SD+PG+AM 1 [-] 10 η SD+PG SD

100

-1

Amplitude ratio 10 1.5 SD+PG+AM+BH -2 SD+PG+AM 10 1.0 10-3 10-2 10-1 100 101 102 103 SD+PG 0.5 SD

Stokes number St [-] [radians] β

2 0.0 ρ D St = τ ω = P P ω P 18 µ -0.5

Phase anglePhase -1.0

-1.5 Phase 10-3 10-2 10-1 100 101 102 103 Stokes number St [-] Martin-Luther-Universität Halle-Wittenberg Particle Response 5

 Air bubbles in water air (ρp/ρ = 0.001, DP = 500 µm):

3 10 Amplitude 2 SD+PG+AM+BH [-] 10 η SD+PG+AM SD+PG 1 10 SD

100

Amplitude ratio -1 10 1.5 10-2 -3 -2 -1 0 1 2 3 1.0 10 10 10 10 10 10 10 Stokes number St [-] 0.5 [radians]

2 β ρP DP 0.0 St = τ ω = ω P 18 µ SD+PG+AM+BH -0.5 SD+PG+AM SD+PG -1.0 SD Phase anglePhase Phase -1.5 10-3 10-2 10-1 100 101 102 103 Stokes number St [-] Martin-Luther-Universität Halle-Wittenberg Literature • Crowe, C.T., Sommerfeld, M. & Tsuji, Y.: Fundamentals of Gas-Particle and Gas-Droplet Flows. CRC Press, Boca Raton, USA (1998) • Crowe, C.T., Schwarzkopf, J.D., Sommerfeld, M. and Tsuji, Y.: Multiphase Flows with Droplets and Particles. 2nd Edition, CRC Press, Boca Raton, U.S.A. (2012), ISBN 978-1-4398-4050-4 • Crowe, C.T. (Ed): Multiphase Flow Handbook. CRC Press, Taylor & Francis Group, Boca Raton, USA (2006) • Michaelides, E.E.: Particles, Bubbles and Drops. World Scientific Publishing, Singapore (2006) • Sommerfeld, M: Modellierung und numerische Berechnung von partikelbeladenen turbulenten Strömungen mit Hilfe des Euler/Lagrange- Verfahrens. Habilitationsschrift, Universität Erlangen-Nürnberg, Shaker Verlag, Aachen (1996) • Sommerfeld, M., van Wachem, B. and Oliemans, R.: Best Practice Guidelines for Computational Fluid Dynamics of Dispersed Multiphase Flows. ERCOFTAC (European Research Community on Flow, Turbulence and Combustion, ISBN 978-91-633-3564-8 (2008)

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