Dense suspensions
the richness of solid-liquid interactions at
the particle scale
Jos Derksen
Chemical & Materials Engineering
University of Alberta
Canada
[email protected]
www.ualberta.ca/~jos/home.html
Alberta: oil sands
mining
tailings
Non-Newtonian
liquids with solid
particles
reclaimed
land
clay suspension
Oil sands processing – e.g.: “Hydro-transport”
Slurry pipeline with
oilsands, water, air
• wildly turbulent flow
• very wide particle size
distribution: from bricks to
clay platelets („nano
particles‟)
• inhomogeneous
multiphase flow
• pipe-wall erosion and
sedimentation (formation
of stagnant layers) are
issues
the good thing about simulations:
easy to look inside
simulation – side view
experiment
simulation – cross section
Solid particles in (turbulent)
flow
Lagrangian solid-liquid simulations
unresolved particles versus resolved particles
dp<

particle size < grid spacing
flow around particles is not resolved
particle are moved around by drag
forces & more exotic forces
particles collide
up to 108 particles
dp  
particle size > grid spacing
hydrodynamics and hydrodynamic forces
are fully resolved
direct coupling between particle motion
and liquid flow
up to 104 particles
My typical SL parameter space
solids volume fraction 
> 0.1
solid over liquid density ratio
2
Stokes number St 
particle time scale
flow time scale
Reynolds number (based on particle size)
We do not get away with
• one-way coupling
• drag-only
• no collisions
On the positive side
• non-Brownian particles
p
10
O(1)
anywhere, mostly Re1
Meso-scale simulations
dp  
O(103) particles in a 3D
periodic domain
direct simulations resolving the solid-liquid interfaces
Our interests
• see what happens at the meso-scale
• feed back meso-scale insights as (subgrid) models to the macro-scale
A few words on modeling & numerics
fluid
flow
particle
motion
scalar
dp  
Lattice-Boltzmann method for solving the flow
of interstitial fluid
3D, time-dependent
Explicitly resolve the solid-liquid interface:
immersed boundary method
particle diameter typically 12 grid-spacings
Solve equations of linear and rotational motion
for each sphere
forces & torques:
directly (and fully) coupled to
hydrodynamics
plus gravity
hard-sphere collisions
NB: in this talk: particles are spheres
Some validation material
single particle validation
dp
PIV experiment of a falling
sphere in a closed box*
U p, d p
Rep 
 1.5.... 32

dp
 0.15
L
multiple particle validation:
liquid-solid fluidization
g
1 cm
L
Rep  32
void formation
particle velocities
LB simulation PIV exp
*Ten Cate et al., Phys. Fluids (2002)
experiment**
simulation***
** Duru & Guazzelli JFM (2002)
*** Derksen & Sundaresan JFM (2007)
Rest of this talk: sample applications
what to learn from meso-scale simulations?
erosion & sedimentation
aggregation / flocculation
Hydro-transport
Back to the slurry
pipeline
Erosion & sedimentation
Start simple: laminar flow, spherical particles, monosized
w
u0
u0
W
Shields number
shear stress over net gravity
g
g
p
2a
Reynolds number
Re
p
 2.5
glass beads in water

Movie: courtesy François Charru (IMFT)
a2
Density ratio
minor role for if Re is modest
Experimental data
c
critical Shields number
onset of bed erosion
Quite some scatter
systems are hard to control at
the particle-scale
• non-sphericity
• friction coeffs
• short-range interactions
Ouriemi et al., PoF 19 (2007)
Back-of-envelope analysis for low Re
for the bed to start moving we need a
vertical hydrodynamic force that
c
overcomes net gravity  at
4 3
ag
3
Fvert ,hydro
c
g
p
2a
32.1
Fz
9.22
2
a (*)
2
a4 (**)
(*) O‟Neill, CES 23 (1968)
a
2
Fnet grav
4 2 1
a
3
c
independent of Re
Fvert ,hydro
Fx
p
Fvert ,hydro
Re
a2
OK,
except for the fact that Fx is
horizontal
(**) Leighton & Acrivos, ZAMP 36 (1985)
A single-sphere may be too simple
to understand the critical Shields number*
u0
beds of fixed spheres
H
monolayers
double layers
Surface occupancy  monolayers
n a2 0.4
0.5
0.7
triple layers
a2
Re
u0
H
cross section
through typical triple
layer bed
* Derksen & Larsen, JFM 673 (2011)
3
0.40
2
0.70
1
0.70
Average drag and lift - Monolayers
Re
0.05
Re
0.05 FD*
FD
a2
FL*
FL
2 4
a
remember,
single sphere
color: velocity magnitude
FD*
32.1
FL*
9.22
Sphere-to-sphere force variation - Mono
Re
0.05
single sphere
peaks at
FD*
32.1
FL*
9.22
1
0.06
Lift turns into vertical viscous drag
vertical velocity one radius (a) above the wall
1
0.04
0.02
0.0
-0.02
0.25
-0.04
1
uz
a
0.06
linear trend
rms FL*
rms uz*
550
if rms FL
rms uz 6
then
6
Re
380
a
Lift as vertical viscous drag - Monolayers
rms-drag, viscous scaling
rms-lift, inertial scaling
rms-lift, viscous scaling
for the critical Shields number to be independent of Re we
needed a vertical force that scales like
Fvert ,hydro
a2
here you have it in terms of an rms vertical force level
Fvert ,hydro
Fnet grav
RMS of drag
and lift forces
double
double & triple
layers
as a function of  of
the top layer
pressure
p
triple
0.5
0.1
-0.1
-0.5
Let‟s try moving spheres
add liquid and
then shear the
liquid above
the bed
up
a
0.1
blue: mobile
red: fixed
up
a
0.05
initialization: create a granular
bed of equally sized spheres
0.025
0.05
up
a
p
0.025
up
a
the Shields number
g
0.80
0.1
2a
0.40
Critical Shields number?
average solids flux per unit width
0.15
up
a
0.1
0.10
0.1
up
a
0.05
0.025
0.05
up
a
up
a
0.025
lo and behold: 0.1
c
0.15
critical features: lubrication
forces & friction coefficient in
p-p collisions
Sample applications
what to learn from meso-scale simulations?
erosion & sedimentation
aggregation / flocculation
Aggregation at the meso-scale
a little bit of recent work
sticky particles in shear flow
A closer look at aggregation
at the meso-scale
attractive interaction between particles
defined by a
square-well-potential
two parameters:  and vc
if vr < vc particles stick
dimensionless numbers
v c
a

a
in energy terms: ESqWP
E pot 
2
1
 mp  v c 
2
r
ESqWP
Flocculation
aggregation enhanced settling
computational approach
solid, sticky spheres in a
3D periodic domain
fp     p  m  ge z
ff   m    ge z
m   p  1    
Flocculation, Global Poly-Glu Co., Ltd 
the main dimensionless variables are
 (solids volume fraction)
0.12 – 0.32
Re 
2aU

6 – 72
U
v c
U
0.005 – 0.03
SqWP

a
0.025
Computational flocculation
  0.12
v c
 0.025
U
Re  24
color: flock size
nagg
1
nagg
2
nagg
3
nagg
4
  0.2
for reference
Re  6
Settling velocities
average
settling
velocity as a
function of
time
at t=0 we
switch on the
SqWP
Average settling velocities
as a function of the strength of the SqWP
average
settling
velocity as a
function of
time
??
(more data points coming – simulations running as we speak)
Sticky particles in turbulence
flocculation is now called aggregation
generate homogeneous
isotropic turbulence in
(again) a fully periodic, 3D
domain – through linear
forcing*
& add solid, spherical sticky
particles
key dimensionless variables
 : solids volume fraction
K
: Kolmogorov scale over sphere radius 0.15 - atypical
a
v c
: SqWP depth over Kolmogorov velocity scale 0.3
K
* Rosales & Meneveau, PoF 17 (2005)
Aggregation in turbulence
  0.08
K
a
 0.13
v c
K
 0.3
color: aggregate size
primary sphere
2
nagg
5
5
nagg
8
nagg
8
the 4 largest aggregates at
some moment
Domain size quickly becomes an issue
• for representative aggregate size distributions
• to create well-developed turbulence
 20a 
 60a 
3
3
too small
gets
computationally
challenging
aggregate size distributions
Some preliminary results
aggregate size distributions
effect of K
(the smaller K, the more power input)
turbulence modulation by the solids
Summary & Perspective
• Dense SL systems, lots of SL interactions
• Computational approach
minimal modeling
small (meso-scale) systems
Perspective
• Erosion & sedimentation
lift & drag  critical Shields number More complexity
non-spherical particles
• Aggregation / flocculation
soft (deformable) particles
effect of flocculants depends on
Erosion
Reynolds number
towards turbulence*
Aggregation
what happens if
K
a
 1  ish
Acknowledgements
(former) students
Andreas ten Cate
Eelco van Vliet
Orest Shardt
sponsors
Lattice-Boltzmann method
Particles move from one lattice
site to the other and collide:
fi x  ei , t  1  fi x, t   i f x, t 
   fi
i
u   fi ei
i
Space, time, and velocity
are discretized:
local operations:
good parallel efficiency
uniform, cubic lattice
2nd order (space and time) representation
of a Navier-Stokes-like equation, e.g.:
u

1 
  u u 

uu  


 f


t
x
3 x
x  x
x 
this is incompressible Navier-Stokes if
velocity/
physical time-step
constraint
u2  csound 2

1
p
 csound 
3
3