inter-particle collision phenomena in · inter-particle collision phenomena in turbulent particle...

Martin-Luther-Universität Halle-Wittenberg

Title

Inter-Particle Collision Phenomena in Turbulent Particle-Laden Flows

M. Sommerfeld, M. Ernst and S. Lain Zentrum für Ingenieurwissenschaften Martin-Luther-Universität Halle-Wittenberg D-06099 Halle (Saale), Germany www-mvt.iw.uni-halle.de

Particle dispersion in swirling flow


Content of the Lecture

Inter-particle collision effects in dispersed multiphase flows

Preferential concentration in homogeneous isotropic turbulence (Analysis by the Lattice-Boltzmann method)

Euler/Lagrange approach and modelling inter-particle collisions

Inter-particle collisions in a horizontal channel flow

Particle collision effects in pneumatic conveying through a bend

Conclusions/Outlook


Transport phenomena in dispersed gas-solid flows:

Dilute two-phase flow aerodynamic transport one- or two-way coupling

Dense two-phase flow particle-particle interaction four-way coupling

Classification of Multiphase Flows 2 Introduction 1

Upward gas-solid flow (DPM simulation)

(Helland et al. 2000)


Introduction 2 Classification of dispersed multiphase flows

The importance of interaction between particles may be estimated with the inter-particle spacing for a certain particle arrangement:

Cubic arrangement 3/1

PP 6DL

απ

=

Modification of flow by particles

Interaction between particles

No effect of particles on the flow

5236.0max, =Pα1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01 0.1

volume fraction [-]

100 10 1inter-particle spacing L / DP

Dilute DispersedTwo-Phase Flow

Dense DispersedTwo-Phase Flow

Two-WayCoupling

One-WayCoupling

Four-WayCoupling


Introduction 3 The importance of particle collisions may be estimated by comparing the

particle response time with the inter-particle collision time (Crowe 1981).

Collision frequency according to kinetic theory:

1c

p <τ

τ1

c

p >τ

τ

t p t c

up

cc f

1=τ ( )∑

=

−+π

==classN

1jjji

2ji

i

ijc nuuDD

4nN

f

Dilute two-phase flow

t pt c

up

Dense two-phase flow


Introduction Particle Collisions 1 Collisions between particles occur under the following conditions:

Effects of solid particle collisions:

A relative motion between the particles is caused by the following effects:

high number concentration of particles high relative velocity between the particles

Brownian or thermal motion of particles Laminar or turbulent shear Particle inertia in turbulent flow Mean drift between particles of different size

Momentum transfer between particles Induce particle rotation Agglomeration of particles Breakage of particles (grinding)

Collision modelling steps Occurrence of a collisions

Outcome of a collision


Introduction Particle Collisions 2 A number of theoretical results are available for the collision rate in two-phase

flows derived for different conditions. Collision rate due to Brownian motion (Smoluchowski 1916):

Collision rate due to turbulent shear (Saffman & Turner 1956):

Collision rate due to particle inertia in turbulence (Saffman & Turner 1956):

Collision rate due to differential sedimentation:

( )ji

2ji

Fij DD

DD3

Tk2N+

µ=

( )2/1

F

3jiji

2/1

ij RRnn158N

νε

+

π

=

( ) ( ) ( )2/1

F

322

j2iFpji

2ji

F

2/1

ij DDnnDD18

3.12

N

νε

−ρ−ρ+µ

π=

( ) ( ) 2j

2iFpji

2ji

Fij DDnnDD

72gN −⋅ρ−ρ+µ

π=


Introduction Particle Collisions 3 A correlation combining turbulent inertia and differential sedimentation was

introduced by Gourdel et al. (1999):

Collision frequency according to theory of kinetic gases:

( ) ( )zGUUnnDD4

N jiji2

jiij −+π

= ( ) ( ) zerfz2

11zexpz

1zG

++−

π=

( )ji

2ji

kkUU

43z

+

−= ( )2'

p2'p

2'pp wvu

21k ++=

( ) jiji2

jiij nnuuDD4

N −+

π=

L

1

2

u rel

collision cylinder

2

1


Introduction Particle Collisions 4 In turbulent flows the velocities of colliding droplets may be partially

correlated, since they are moving in the same eddy upon collision.

The degree of correlation depends on the turbulent Stokes number:

For completely correlated velocities (St → 0) the result of Saffman and Turner

(1956) is valid:

For completely uncorrelated velocities (St → ∞) the result of Abrahamson (1975) is valid:

t

p

TSt

τ=

( )2/1

F

3jiji

2/1

ij RRnn158N

νε

+

π

=

( ) 2j

2i

2jiji

21

23

ij RRnn2N σ+σ+π=

Limitation: no external forces mono-disperse particles

Reduced collision rate


Introduction Particle Collisions 5 Comparison of point-particle DNS by Lattice-Boltzmann method with theories in dependence of particle Stokes number (Ernst & Sommerfeld 2012):

Parameter ValueSpatial discretisation 0.6 mmFluid density 1.17 kg/m3

Kolmogorov length scale 0.29 mmIntegral length scale 12.5 mmKolmogorov time scale 5.78 msIntegral timescale 81.0 msTaylor Reynolds number 82.14

Parameter ValueSpatial discretisation 0.6 mmFluid density 1.17 kg/m3

Kolmogorov length scale 0.29 mmIntegral length scale 12.5 mmKolmogorov time scale 5.78 msIntegral timescale 81.0 msTaylor Reynolds number 82.14

Saffman and Turner (1956) only valid for St → 0

Cubic box of 643 cells

LF

pp

TD

Stµ

ρ18

2

=0 1 2 3 4 5 6 70

50

100

150αP = 0.01

Abrahamson (1975):Kinetic theory

DNS by LBM

Collis

ion

frequ

ency

, N *

V Box

Stokes number, St

Saffman & Turner (1956)


Lattice-Boltzmann Method 1 Lattice-Boltzmann equation: Behaviour of fluids on mesoscopic level

Key variable: Discrete distribution function fσi

Discretization of space by a regular grid

Discretization of the velocity space: D3Q19

Macroscopic parameters (density, momentum): Derived as moments of fσi

Iteration loop: Relaxation (t+) & Propagation (t+∆t)

( ) ( ) ( ) ( )( ) iσ0iσiσiiσi Fttftfttftttf ∆+−

∆−=−∆+∆+ ,,,, xxxx σσ τ

ξ

Point of time: t Point of time: t+ Point of time: t+∆t


Modeling Fluid Flow: Spectral Forcing of Turbulence o Direct numerical simulation of homogeneous isotropic turbulence HIT

o Stochastic modeling: Pseudo-spectral method by Eswaran & Pope (1988)

o Spectral space: Mapping of the Fourier transforms (shell model)

Lattice-Boltzmann Method 2

NCells = 1283; ReT = 29.9; η / ∆x = 0.4


• Transport of mono-disperse, spherical particles by the Lagrangian approach

<uP uP>

<uF uF>

time

St1

St2

Initialization Fluid

Statistics Fluid

Initialization Particles

Statistics Particles

Agglomeration

Ref.: Wunsch (2009)

Particle translational velocity:

( )24

182

Pd

PP

PPDrag

P

PP

Recd

mdtdm

ρ−µ

==uuFu

Particle location:

PP

P

dtd ux

=

• ODE`s are solved simultaneously: Calculation of min. particle time step, ∆tP

( )CollisionPEddyCrossCVLBMP tttt ττ⋅=∆ ,,,,min25.0

Lagrangian Approach: Equations of Motion

Drag coefficient: Correlation*

( )687.015.0124P

Pd Re

Rec +=

*Schiller, L., and Naumann, A. (1993). Ver. Deut. Ing., 44:318-320.



Lagrangian Approach: Collision Algorithm

*Sundaram, S., and Collins, L.R. (1996). J. Comput. Phys., 124:337-350.

Deterministic collision model*

xP (t+∆t), uP (t+∆t)(Without collisions)

Calculate collision times

Sort collision times in descending order

Move overlapping pair backward to collision time

Move post-colliding particles forward to t+∆t

Check for new overlaps

yes

xP (t), uP (t)

Any overlaps

left?

yes

no

no

Add/ Delete collisions

xP (t+∆t), uP (t+∆t)(With collisions)

Overlaps?

Calculate post-collision velocities

xP (t+∆t), uP (t+∆t)(Without collisions)

Calculate collision times

Sort collision times in descending order

Move overlapping pair backward to collision time

Move post-colliding particles forward to t+∆t

Check for new overlaps

yes

xP (t), uP (t)

Any overlaps

left?

Any overlaps

left?

yes

no

no

Add/ Delete collisions



Overlaps?

Overlaps?

Calculate post-collision velocities

Calculation of collision time:

Collision criterion:

( ) ( )PjPiPij ddtt +≤∆+ 5.0x

j

i ∆xt

∆xC ∆x∆t

t t + ∆t t + ∆t - ∆tC

( ) ( ) ( )PjPicPijPij ddttttt +=∆⋅∆++∆+ 5.0ux

Prevent interpenetrations:

PPmaxP, dt <<∆⋅u


)(,

)()(,)(,jPim

Juu jPi*

jPi+−=

Post-collision velocities:


HIT-Results 1 Effect of Stokes number on preferential concentration (αP = 1.0 x 10-3)

StK = 1.25 StK = 9.67 StK = 0.1

without collisions

with collisions


HIT-Results 2 Global particle accumulation in dependence of particle Stokes number;

effect of inter-particle collisions


Euler/Lagrange Approach 1

The fluid flow is calculated by solving the Reynolds-averaged conservation equations (steady or unsteady) by accounting for the influence of the particles (source terms).

Turbulence models with coupling:

k-ε turbulence model Reynolds-stress model

The Lagrangian approach relies on tracking a large number of representative particles (point-mass) through the flow field accounting for rotation and all relevant forces like:

drag force gravity/buoyancy slip/shear lift slip/rotation lift torque on the particle

Particle properties and source terms result from ensemble averaging for each control volume

Two-way coupling iterations

Models elementary processes: turbulent dispersion particle-rough wall collision inter-particle collisions


Particle-Phase Modelling The solution of the particle equations of motion require the generation of the

instantaneous fluid velocity along the particle trajectory using stochastic approaches, e.g. the isotropic Langevin model (e.g. Sommerfeld 1996).

For modelling the interaction time of particles to turbulence information on time and length scales of turbulence is needed

Calculation of inelastic wall collision process by solving the

momentum equations in connection with Coulombs law of friction. sliding collision non-sliding collision

Modelling of wall roughness: The particle collision angle is composed of the trajectory angle and a stochastic contribution: ξγ+α=α′ ∆11

α γ1

Shadow-Effect The roughness angle follows a normal distribution


Effect of Wall Roughness 1

Calculated particle trajectories in a horizontal channel, effect of wall roughness, Uav = 18 m/s :

30 µm St = 1.7

with wall roughness

110 µm St = 17

with wall roughness

Channel length/height: 6 m / 35 mm


Effect of Wall Roughness 2

0.0 0.5 1.0 1.5 2.0 2.50.0

0.2

0.4

0.6

0.8

1.0 DP = 195 µm η = 1.0 without wall roughness with wall roughness

y / H

fP / fP,av

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.10.0

0.2

0.4

0.6

0.8

1.0

DP = 195 µm η = 1.0

without WR with WR

y / H

UP / Uav

0.00 0.05 0.10 0.15 0.20 0.250.0

0.2

0.4

0.6

0.8

1.0DP = 195 µm; η = 1.0

no roughness with roughness

y / H

u´p / Uav

0.00 0.05 0.10 0.150.0

0.2

0.4

0.6

0.8

1.0

y / H

v´p / Uav


Inter-Particle Collision Model 1 Stochastic inter-particle collision model

In the trajectory calculation of the considered particle a fictitious collision partner is generated for each time step. The properties of the fictitious particle are sampled from local distribution functions:

In sampling the fictitious particle velocity fluctuation the correlation of the fluctuating velocity is respected (from LES, Simonin):

Calculation of collision probability between the considered particle and the fictitious particle: A collision occurs when a random number in the range [0 - 1] becomes smaller than the collision probability

⇒ Particle diameter ⇒ Particle velocities

( ) ( ) n2

LPii,realLPi,fict T,R1uT,Ru ξτ−σ+′τ=′ ( )

τ−=τ

4.0

L

pLp T

55.0expT,R

( ) tnuuDDtfP PPPPPc ∆−+=∆= 212

214π


Inter-Particle Collision Model 2 The collision process is calculated in a co-ordinate system where the fictitious particle is stationary. Generation of impact point

by a random process:

Calculation of new velocities of the considered particle (translation and rotation).

Re-transformation of the velocities (considered particle) !!!

L

1

2L

u rel

φ

collision cylinder

2

1

Ψ2

1

( )Larcsin1L:withZYL 22

=φ≤+=

π<Ψ< 20

Non-sliding collision

solution of the impulse equations Coulomb`s law of friction oblique inelastic collision (Hard Sphere Model)

Sliding collision


Inter-Particle Collision Model 3 Consideration of fluid dynamic effects for the interaction of particles of

different size (impact efficiency; Ho & Sommerfeld 2002).

The occurrence of agglomeration may be decided on the basis of an energy balance (only Van der Waals forces):

Boundary particle

Stream lines

Separated particle

dp

DK

collector

Yc

La

U0

For the inertial regime the impact efficiency may be calculated from (Schuch and Löffler 1978):

b

i

i

K

c

aDY2

+ΨΨ

==ηfor: Rep < 1 a = 0,65 b = 3,7

K

2p2p1pp

i D18duu

µ

−ρ=

Ψ

dvdw2k1k EEEE +∆+=

( )ppl

2o

2pl

2/12pl

1kr P6z

Akk1

R21U

ρπ

−=

Agglomeration if:

krrel UcosU

≤φ

Collision occurs: Ca YL ≤


Inter-Particle Collisions in Wall Bounded Flows Inter-particle collision mechanisms in pneumatic conveying:

Very high mean relative velocity in the vicinity of walls

High local particle concentration due to inertial segregation: gravitational settling; centrifugal segregation in bends

Shear flow


Horizontal Channel 1 Particle trajectories in a horizontal channel; effect of inter-

particle collisions, Uav = 18 m/s (35 mm height and 6 m length), (Sommerfeld 2003)

30 µm St = 1.7

30 µm St = 1.7

With wall roughness

no collisions

with collisions


Horizontal Channel 2 Particle trajectories in a horizontal channel effect of inter-particle

collisions, Uav = 18 m/s (Sommerfeld 2003) :

110 µm rough

110 µm St = 17

110 µm St = 17

no collisions

with collisions


Horizontal Channel 3 Influence of inter-particle collisions on mass flux profiles:

Channel height: 35 mm Channel length: 6 m Uav = 18 m/s 30 µm

Smooth wall

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.0

0.2

0.4

0.6

0.8

1.0

without collisions η = 0.1 η = 1.0 η = 4.0

y / H

[ - ]

fP / fP,av [ - ]

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

0.8

1.0 without collisions η = 0.1 η = 1.0 η = 4.0

y / H

[ -

]

fP / fp, av [ - ]

110 µm


Horizontal Channel 4 Influence of inter-particle collisions on particle velocity profiles:

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.20.0

0.2

0.4

0.6

0.8

1.0

gas phase without collisions η = 0.1 η = 1.0 η = 4.0y

/ H [

- ]

Up / Uav [ - ]

0.00 0.05 0.10 0.15 0.200.0

0.2

0.4

0.6

0.8

1.0

without collisions η = 0.1 η = 1.0 η = 4.0

y / H

[ - ]

u´p / Uav [ - ]0.00 0.02 0.04 0.06 0.08 0.10

0.0

0.2

0.4

0.6

0.8

1.0

without collisions η = 0.1 η = 1.0 η = 4.0y

/ H [

- ]

v´p / Uav [ - ]

110 µm


Pneumatic Conveying 1 Analysis of operational conditions on pneumatic conveying through a pipe

system consisting of a 5m horizontal pipe, a 90°-bend and 5m vertical pipe

• Particle rope disintegration • Secondary flow effects

• Gravitational settling • Turbulent dispersion

• Inertial particle separation • Rope formation • Secondary flow modification

5m

5m

• Pressure drop of the pipe system

2.54⋅Dpipe

Dpipe = 0.15 m

Dpipe = 0.08 m

27, 14, 21 m/s

25 blocks

568,000 CV`s

Tracking of 200,000 parcels

Lain and Sommerfeld 2013 and 2014


Pneumatic Conveying 2 Summary of flow conditions for different pipe diameter (Huber and

Sommerfeld 1994 and 1998)

Pipe diameter 0.08 m 0.08 m 0.15 m

Bulk velocity 14 m/s 21 m/s 27 m/s Mass loading 0.5 0.5 0.3 Repipe 74,667 112,000 270,000 Debend 33,144 49,716 63,890

ρ = 1.2 kg/m3, µ = 18.0 10-6 N s/m2

pipebend

pipebend

avpipepipe RR

ReDe

UDRe =

µ

ρ=


Pneumatic Conveying 3

0 20 40 60 80 1000

5

10

15

PDF [%]

Particle Diameter [µm]

bend

av2PP

90,bend

av2PP

bend

RU2

18d

CU

18dSt

o

πµρ

=

µρ

=

TL,cl = 10.3 ms

S

2PP

P V18

DDµ

ρ=λ<

(ρp = 2,500 kg/m3)

D = 0.15 m

Diameter [µm]

Turbulent Stokes 1 [ - ]

Response distance [mm]

Bend Stokes 2 [ - ]

15 – 85 (40) 0.169 – 5.412 0.03 – 30.49 0.078 – 2.515 20 0.300 0.093 0.139 40 1.199 1.5 0.557 80 4.794 23.9 2.228 135 13.653 194.0 6.344 30 + 60 (40)3 0.674 + 2.697 0.47 + 7.57 0.313 + 1.253


Pneumatic Conveying 4 Influence of inter-particle collisions on concentration, size effect

two-way coupling four-way coupling

40 µm

20 µm


Pneumatic Conveying 5 Influence of inter-particle collisions on concentration, size effect

80 µm

two-way coupling four-way coupling

135 µm


Pneumatic Conveying 5 (D = 150 mm, hor. 5m, bend and vert. 5m, Uav = 27 m/s, η = 0.3, ∆γ = 10°).

particle size 20 µm

four-way coupling two-way coupling


Pneumatic Conveying 6 • (D = 150 mm, hor. 5m, bend and vert. 5m, Uav = 27 m/s, η = 0.3, ∆γ = 10°).




Pneumatic Conveying 7 (D = 150 mm, hor. 5m, bend and vert. 5m, Uav = 27 m/s, η = 0.3, ∆γ = 10°).




Pneumatic Conveying 8 (D = 150 mm, horizontal 5m, bend and vertical 5m, Uav = 27 m/s, η = 0.3, ∆γ =

10°, particle size distribution 15 – 85 µm).



Mass Loading Effect 1 D = 150 mm, Uav = 27 m/s, ∆γ = 10°, particle size distribution 15 – 85 µm

η = 0.3 η = 1.0


Comparison with Measurements 1

-0.5 0.0 0.50.5

1.0

1.5

2.0

-0.5 0.0 0.50.5

1.0

1.5

2.0

-1.0 -0.5 0.0 0.5 1.00.5

1.0

1.5

2.0

y = 1.1 m experiment calculation 4-w calculation 2-w

D P, m

ean /

DP,

0 [

- ]

y = 0.6 m experiment calculation 4-w 2-w

D P, m

ean /

DP,

0 [

- ] y = 0.1 m

experiment calculation 4-w 2-w

D P, m

ean /

DP,

0 [

- ]

z / R [ - ]

-0.5 0.0 0.50

1

2

3

4

-0.5 0.0 0.50

1

2

3

4

5

-1.0 -0.5 0.0 0.5 1.00

5

10

15

20

25


c P [ k

g/m

3 ]


c P [ k

g/m

3 ]


c P [ k

g/m

3 ]

z / R [ - ]

D = 80 mm, Rbend = 0.203 m, Uav = 14 m/s, η = 0.5, ∆γ = 10°, particle size distribution 15 – 85 µm, Stbend = 0.076 – 2.445

y/D = 13.75

y/D = 7.5

y/D = 1.25


Comparison with Measurements 2 D = 80 mm, Rbend = 0.203 m, Uav = 14 m/s, η = 0.5, ∆γ = 10°, particle size

distribution 15 – 85 µm, Stbend = 0.076 – 2.445

-0.5 0.0 0.50.6

0.8

1.0

1.2

-0.5 0.0 0.5

0.6

0.8

1.0

1.2

-1.0 -0.5 0.0 0.5 1.00.2

0.4

0.6

0.8

1.0

1.2


U P / U

0 [ -

]


U P / U

0 [ -

]


U P / U

0 [ -

]

z / R [ - ]

-0.5 0.0 0.50.0

0.1

0.2

0.3

-0.5 0.0 0.50.0

0.1

0.2

-1.0 -0.5 0.0 0.5 1.00.00

0.05

0.10

0.15


u P,rm

s / U

0 [ -

]


u P,rm

s / U

0 [ -

] y = 0.1 m

experiment calculation 4-w calculation 2-w

u P,rm

s / U

0 [ -

]

z / R [ - ]

y/D = 13.75

y/D = 7.5

y/D = 1.25


Conclusions

Collisions between particles are induced by locally high concentrations and high instantaneous relative velocity.

An instantaneous relative velocity is produced by various mechanism, Brownian motion, turbulence, shear flows and differential settling.

Estimates for the importance of inter-particle collisions were provided. Preferential concentration in turbulence structures occurring at Stokes

numbers around one are only slightly affected, at least at the considered concentration

Inter-particle collisions have a large influence on the development of particle-laden flows through process equipment, also at moderate concentration.

This is mainly caused by inertial segregation of the particles. Under certain conditions particles are trapped in regions of high concentration

due to the reduction of particle collision mean free path.


Agglomeration Models for Solid Particles

Agglomerate structure model

Location vectors

Convex hull

Agglomeration models

Agglomerate structure Effective surface area Volume of convex hull Porosity of the agglomerate

Volume equivalent sphere

Simple agglomeration model

Number of primary particles

Penetration depth

Point-particle assumption

Hull

Part

VV1−=ε

Sequential agglomeration model

Number of primary particles Hull volume/diameter Porosity of hull Contact forces

inter-particle collision phenomena in · inter-particle collision phenomena in turbulent particle...

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