Inter-Particle Collision Phenomena in · Inter-Particle Collision Phenomena in Turbulent Particle...
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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
Martin-Luther-Universität Halle-Wittenberg
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
Martin-Luther-Universität Halle-Wittenberg
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)
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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
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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
Martin-Luther-Universität Halle-Wittenberg
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
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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=
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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
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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
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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)
Martin-Luther-Universität Halle-Wittenberg
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
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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
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• 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.
Lattice-Boltzmann Method 3
Martin-Luther-Universität Halle-Wittenberg
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
xP (t+∆t), uP (t+∆t)(With collisions)
xP (t+∆t), uP (t+∆t)(With 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
Lattice-Boltzmann Method 4
)(,
)()(,)(,jPim
Juu jPi*
jPi+−=
Post-collision velocities:
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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
Martin-Luther-Universität Halle-Wittenberg
HIT-Results 2 Global particle accumulation in dependence of particle Stokes number;
effect of inter-particle collisions
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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
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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
Martin-Luther-Universität Halle-Wittenberg
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
Martin-Luther-Universität Halle-Wittenberg
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
Martin-Luther-Universität Halle-Wittenberg
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
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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
Martin-Luther-Universität Halle-Wittenberg
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 ≤
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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
Martin-Luther-Universität Halle-Wittenberg
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
Martin-Luther-Universität Halle-Wittenberg
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
Martin-Luther-Universität Halle-Wittenberg
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
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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
Martin-Luther-Universität Halle-Wittenberg
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
Martin-Luther-Universität Halle-Wittenberg
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=
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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
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Pneumatic Conveying 4 Influence of inter-particle collisions on concentration, size effect
two-way coupling four-way coupling
40 µm
20 µm
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Pneumatic Conveying 5 Influence of inter-particle collisions on concentration, size effect
80 µm
two-way coupling four-way coupling
135 µm
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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
Martin-Luther-Universität Halle-Wittenberg
Pneumatic Conveying 6 • (D = 150 mm, hor. 5m, bend and vert. 5m, Uav = 27 m/s, η = 0.3, ∆γ = 10°).
particle size 40 µm
four-way coupling two-way coupling
Martin-Luther-Universität Halle-Wittenberg
Pneumatic Conveying 7 (D = 150 mm, hor. 5m, bend and vert. 5m, Uav = 27 m/s, η = 0.3, ∆γ = 10°).
particle size 80 µm
four-way coupling two-way coupling
Martin-Luther-Universität Halle-Wittenberg
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).
four-way coupling two-way coupling
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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
Martin-Luther-Universität Halle-Wittenberg
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
y = 1.1 m experiment calculation 4-w calculation 2-w
c P [ k
g/m
3 ]
y = 0.6 m experiment calculation 4-w calculation 2-w
c P [ k
g/m
3 ]
y = 0.1 m experiment calculation 4-w calculation 2-w
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
Martin-Luther-Universität Halle-Wittenberg
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
y = 1.1 m experiment calculation 4-w calculation 2-w
U P / U
0 [ -
]
y = 0.6 m experiment calculation 4-w calculation 2-w
U P / U
0 [ -
]
y = 0.1 m experiment calculation 4-w calculation 2-w
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
y = 1.1 m experiment calculation 4-w calculation 2-w
u P,rm
s / U
0 [ -
]
y = 0.6 m experiment calculation 4-w calculation 2-w
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
Martin-Luther-Universität Halle-Wittenberg
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.
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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