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Discrete element method-based models for the consolidation of particle packings in paper-coating applications.

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ASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING
Asia-Pac. J. Chem. Eng. 2011; 6: 44–54
Published online 20 May 2010 in Wiley Online Library
(wileyonlinelibrary.com) DOI:10.1002/apj.447
Special Theme Research Article
Discrete element method-based models for the
consolidation of particle packings in paper-coating
applications†
G. Pianet,1 F. Bertrand,1 * D. Vidal1,2 and B. Mallet1
1
2
École Polytechnique de Montréal, P.O. Box 6079, Stn. CV, Montréal, Quebec, Canada H3C 3A7
FPInnovations – Paprican, Pointe-Claire, Quebec, Canada H9R 3J9
Received 2 October 2009; Revised 15 February 2010; Accepted 23 March 2010
ABSTRACT: This paper concerns the use of the discrete element method (DEM) for the three-dimensional simulation of
the consolidation of particle packings in a surrounding liquid such as those in paper-coating applications. The accuracy
of DEM is first assessed using X-ray microtomography experiments in the case of polydisperse particle distributions. It
is shown that simulations that only account for gravitational and contact forces are in excellent agreement with literature
data. However, paper-coating applications involve more complex mechanisms governing the transient particle/pigment
consolidation: compression effects during metering and calendering operations, and drag forces during the drainage of
the liquid into the basesheet. To simulate the deposition of particles on the basesheet under drainage conditions more
realistically, three modelling strategies are considered in this work, which respectively take into account: (1) gravity,
(2) mechanical compression and (3) uniform drag. The results show that the first two strategies yield stratified structures
and similar bulk porosities, and that the use of a uniform drag leads to significant changes in the particle dynamics
and packing properties. Finally, exploratory results with a model for the two-way coupling between the dynamics of
the liquid and solid phases reveal its potential for predicting the compression of wet granular media.  2010 Curtin
University of Technology and John Wiley & Sons, Ltd.
KEYWORDS: particle packings; liquid; compression; discrete element method; lattice Boltzmann method; two-way
coupling
INTRODUCTION
The numerical investigation of particulate/granular flow
is a challenging research field in both engineering and
computer science. The static and dynamic characterisation of powders, slurries and pastes is a key step
in many industrial applications, including papermaking
and paper-coating processes. The numerical approaches
to simulate this type of flow are commonly classified
into continuum and discrete models. Continuum models, which have been used for industrial scale systems,
have proven inaccurate in the case of dense particulate
flows owing to their inability to account for particle segregation and particle collisions. On the other hand, the
discrete or particulate models, which can be of deterministic or probabilistic nature, simulate the motion of
*Correspondence to: F. Bertrand, École Polytechnique de Montréal,
P.O. Box 6079, Stn. CV, Montréal, Quebec, Canada H3C 3A7.
E-mail: francois.bertrand@polymtl.ca
†
Presented at the 2008 TAPPI Advanced Coating Fundamentals
Symposium.
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Curtin University is a trademark of Curtin University of Technology
each particle individually and are more appropriate for
investigating collisional phenomena such as avalanches
or segregation mechanisms.[1,2] Among these methods,
the discrete element method (DEM) has proven to be
both accurate and flexible when explicit dependency to
time and non-elastic conservative particle contacts are
required.
In this work, a variant of the DEM model of Zhou
et al .[3] is used, which is based on a soft-particle springand-dashpot model to account for energy dissipation due
to inter-particle contacts, and the use of the Coulomb
law of friction, to simulate the deposition of particles
on the basesheet of paper-coating applications. In such
a process, the time scale of buoyancy effects inherent
to gravitational forces is usually much larger than that
related to the compression during the metering and
calendering operations, and to the drag forces during
the drainage of the liquid into the basesheet, which
are the two main competing mechanisms governing the
transient particle/pigment consolidation. As a result, the
effect of gravity is assumed to be negligible here, and
the particle deposition is considered to be driven by two
Asia-Pacific Journal of Chemical Engineering
DEM-BASED MODELS FOR THE CONSOLIDATION OF PARTICLE PACKINGS
other mechanisms related to the drag-driven settling of
particles and their mechanically driven compression. It
is the first time that such models are being used to
simulate the settling and the compression of particle
packings within the context of paper coating.
The second mechanism is modelled by a downwardmoving solid wall so that the particles are progressively
packed against the basesheet, while periodic conditions
are imposed on the side boundaries. The integration
of forces due to the contact between the top layer
of particles and this moving solid wall is done at
each time step. More specifically, at each of these
iterations, the resulting pressure is compared to a
reference pressure and the plane is moved downwards
when this pressure is too low or upwards when it is too
high. This mechanism was designed so as to reproduce
the mechanical constraints of the calendering process.
The first mechanism is much more intricate to model
since the viscous drag is related to solid–liquid friction,
the extent of which can be deduced from the relative
local velocity of the liquid phase. As a first approximation, one may consider a uniform velocity field.
More accurate strategies exist, all of which account
explicitly for the complex solid–liquid interactions. For
a comprehensive account on discrete particle models
and the most popular solid–liquid coupling techniques,
the reader is referred to the thorough review of Zhu
et al .[4] In the current work, the local mean velocity
of the liquid phase is solved numerically with the lattice–Boltzmann method (LBM). To this end, a two-way
coupling approach is introduced to handle fluid–particle
interactions and unusually high solids volume fractions
(∼60%). More precisely, instead of using traditional
point-force models to consider the effect of suspended
particles on fluid flow, a force term is introduced, which
is based on multi-scale porous media modelling, and
whereby the time- and space-dependent porosity and
permeability values are updated according to the instantaneous local particle distribution. In contrast to twoway coupling approaches for dilute suspensions, this
method is well suited to dense particulate flow, and
designed to capture the main mechanisms of momentum
transfer for both particle–particle and fluid–particle
interactions.
First, a brief overview of DEM and underlying collisional models is presented. Second, a DEM-based
model is introduced and its accuracy is assessed by
simulating the packing of polydisperse spherical particles under gravity and comparing the results with X-ray
microtomography (XMT) experimental data. Next, a
model that explicitly takes into account fluid and solid
dynamics is presented. It is based on the DEM-based
model for the motion of the pigments and the LBM for
the flow of the liquid phase. Results of various packing simulations obtained with both the uncoupled and
coupled models are then presented to investigate global
packing properties and, in an exploratory manner, the
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
role of the liquid phase on particle migration. Finally,
concluding remarks discuss some of the current limitations of the hybrid LBM–DEM modelling strategy
proposed in this work.
THE DISCRETE ELEMENT METHOD
Model description
DEM is a time-dependent Lagrangian soft-particle
method (SPM)[5] for which particle overlapping is
allowed. In this method, it is possible to constrain overlaps with appropriate physical models in such a way
as to recover elastic collision and particle deformation behaviour (Fig. 1). The characteristic deformation
parameters are established by closure relationships that
depend on particle structural properties. This additional
set of physical parameters opens a whole spectrum
of applications that simpler hard-sphere models cannot
deal with.
In DEM, the position xi = (xi , yi , zi ) of the centre of
a particle i of mass mi and its velocity at given time t is
computed by numerically integrating Newton’s second
law of motion:
mi
d2 xi
dt 2
= Ftotal,i
(1)
The rotation of each particle is obtained in a similar
fashion through an angular momentum balance. In
Eqn (1), Ftotal,i is the total force acting on particle
i . In a general framework, it comprises three distinct
components:
Ftotal,i = Fhydrodynamic,i + Fnon−hydrodynamic,i
+
Fcontact,ij
(2)
j =i
The first term gathers all hydrodynamic forces such
as buoyant, drag, pressure gradient, lubrication and
lift forces (see Zhu et al .[4] for more details). Particular attention is given to hydrodynamic forces in the
following section. The second term is related to nonhydrodynamic forces as those prevailing, for example,
in colloidal systems. These hydrodynamic forces, which
are rather straightforward to include but rather intricate
to validate when coupled with hydrodynamic forces, are
not considered in the current work. The last term stands
for the contact forces between particle i and neighbouring particles j involved in an (i , j ) collision. They
are evaluated here by resorting to the so-called generalised spring-and-dashpot model, which is a variant of
the classical linear spring-and-dashpot model introduced
by Cundall and Strack.[5] Contact forces are projected
on both normal and tangential directions of a local
Asia-Pac. J. Chem. Eng. 2011; 6: 44–54
DOI: 10.1002/apj
45
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G. PIANET ET AL.
Asia-Pacific Journal of Chemical Engineering
→
t
particle j at time (t + 1)
→
n
dt
particle i
dn
particle j at time (t)
DEM representation of two colliding particles. This figure is available in colour online at
www.apjChemEng.com.
Figure 1.
Figure 2. Sketch of the springand-dashpot model.
referential (Fig. 2) where the magnitude and direction
of particle overlap, δ, rules the behaviour of repulsive
forces.
The model used in the present work is similar to that
of Zhou et al .,[3] in which shear and normal forces are
connected via the Coulomb law of friction. Equations
reported in Table 1 detail how stiffness coefficients kn
and kt and damping coefficients Cn and Ct are brought
into play in the collision rules. In this model, they are
expressed as functions of Young’s modulus E , Poisson
ratio υ and friction coefficient µs . More details about
this model and the setting of the coefficients can be
found in Bertrand et al .[6] It is sufficient to say that it
has proven efficient for the simulation of many granular
flow problems by our research group, mainly in the field
of mixing[7] and coating.[1] Values of the coefficients
used in this work are similar to those used in Ref. [7],
and have led to overlaps smaller than 1% of the particle
diameter. Eqn (1) was integrated using the half-step
leapfrog Verlet integration scheme coupled with an
appropriate time-stepping strategy.
Assessment of the DEM-based model through
XMT measurements
Considering the large number of physical and numerical parameters to be set in a DEM-based model, proper
validation is an important task. For instance, the assessment of our DEM code in a dynamical context has been
Table 1. DEM model of Zhou et al.[3] (α = 3/2, β = 1,
and δn,ij and δt,ij are the normal and tangential
components of the overlap between particles i and
j).
Fcontact,ij = Fcontact,n,ij + Fcontact,t ,ij
α
+ Cn δ̇n,ij
Fcontact,n,ij = kn δn,ij
β
Fcontact,t ,ij = kt δt ,ij + Ct δ̇t ,ij
kn = kn (E , υ)
Cn = cn (E , υ)
3/2 kt = kt µs kn δn,ij 3/2 Ct = ct µs kn δn,ij  2010 Curtin University of Technology and John Wiley & Sons, Ltd.
done with the flow through an orifice and the Beverloo correlation.[6] Also, in the field of powder mixing
in pharmaceutical blenders, our code has proven accurate by means of comparisons with experimental data.[7]
In the context of particle-settling simulations, the final
packing state can be easily analysed. For instance, this
can be done by means of the so-called radial distribution
function (RDF) that can be readily obtained from simulation results or from experimental samples scanned
with XMT.[8] The RDF provides a way to quantitatively
characterise the structure that is present in a collection
of particles. A comparison of numerical and experimental RDF functions is given in Fig. 3, in the case of a
gravity-driven packing of 2000 polydisperse glass particles whose particle size distribution (PSD) is given
in Fig. 4. For instance, the peak of the RDF g(r) at
normalised inter-particle spacing r/ < D >= 1, where
< D > denotes the average particle diameter, represents the nearest-neighbour contacts, whereas that at
r/ < D >= 2 denotes the contacts involving particles
separated by two particle diameters. This clearly shows
an excellent agreement between the simulation results
obtained with our DEM-based model and the experimental data from Ref. [8]. These results demonstrate
the adequacy of our DEM-based model for the simulation of the gravity-dominated packing of polydisperse
particles. Nevertheless, as was mentioned in Ref. [1],
interstitial fluid flow may play a significant role in the
packing of pigments in paper-coating applications. The
next section addresses this important issue by introducing a realistic model for the simulation of the flow of
wet granular media.
FLUID–SOLID COUPLING MODEL
In this work, the influence of the fluid phase on particles
is considered from two different points of view. The
first and simplistic approach[1] supposes that the fluid
velocity u = UF is uniform in the whole domain.
In this case, each particle i experiences a drag force
Fd = 6π µai (UF − dxi /dt), where µ is the fluid
Asia-Pac. J. Chem. Eng. 2011; 6: 44–54
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
DEM-BASED MODELS FOR THE CONSOLIDATION OF PARTICLE PACKINGS
Figure 3. (a) Radial distribution function (RDF) corresponding to (b) 2000
gravity-packed polydisperse particles. Color indicates particle size, from
blue (small) to red (large). This figure is available in colour online at
www.apjChemEng.com.
14
10
5
7
11
2
0
4
3
1
12
8
Z
6
Y
9
13
X
Figure 4. Particle size distribution for the gravity-driven
packing of 2000 particles.
Figure 5. D3Q15 lattice. This figure is available in
colour online at www.apjChemEng.com.
dynamic viscosity and ai the particle radius. This is a
crude approximation since the Stokes law for the drag
force is valid for isolated spherical particles. The second
approach is elaborated in this section, and relies on
the use of a Navier–Stokes solver for calculating the
velocity field accurately. The main difficulty then lies
in devising an adequate method for coupling the fluid
and solid phases.
The lattice Boltzmann method
Conventional computational fluid dynamics (CFD)
methods (e.g. finite element, finite volume and finite
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
difference methods) have been proven to have limitations in solving the Navier–Stokes equations in porous
media. On the contrary, the LBM, developed in the early
1990s, is now considered by many researchers[9] as a
method of choice for the simulation of single-phase or
multiphase flows in complex geometries. LBM can be
viewed as a particulate method inasmuch as it ‘simulates’ the macroscopic behaviour of fluid molecules in
motion. More precisely, it is based on the discretisation
of the Boltzmann equation from kinetic gas theory and
the propagation and collision of populations of particles
on the nodes of a lattice representing the computational
domain. In practice, these populations of particles propagate and collide at every time step δt on a lattice with
Asia-Pac. J. Chem. Eng. 2011; 6: 44–54
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G. PIANET ET AL.
Asia-Pacific Journal of Chemical Engineering
spacing δx and along ei velocity directions, where the
number of directions i depends on the lattice type. A
15-velocity-direction D3Q15 lattice is used in this work,
as described in Fig. 5.
The collision step is given by
can be solved with an explicit scheme, which makes
it a good candidate to be combined with DEM for the
development of a fully coupled fluid–solid model. The
reader is referred to the book of Succi[9] for more details
on LBM and its implementation.
eq
fi (x, t) − fi (x, t)
fi (x, t) = fi (x, t) −
τ∗
∗
(3)
and is followed by the propagation step
∗
fi (x + ei δt , t + δt ) = fi (x, t)
(4)
where fi (x, t) is the particle probability distribution
function (the so-called population) in the direction of
the velocity ei at position x and time t, and τ ∗ is
the dimensionless relaxation time. The second term
of the right-hand side of Eqn (4) approximates the
collision process through the Bhatnagar, Gross and
Krook (BGK) relaxation procedure, towards a local
equilibrium population, which for the D3Q15 lattice is
given by
2
δt
eq
fi (x, t) = wi ρ 1 + 3(ei · u)
δx
2 4
δt
9
3
δt
2
(5)
+ (ei · u)
− (u · u)
2
δx
2
δx
1 for
where w0 = 92 , wi = 91 for i = 1 : 6 and wi = 72
i = 7 : 14. In this equation,
fi
(6)
ρ = ρ(x, t) =
i
and
u = u(x, t) =
1
fi ei
ρ i
One of the objectives of this ongoing work is to develop
a simulation tool for investigating the consolidation of
pigments in coating/calendering operations. More generally, we also intend to use this computer model to
investigate other solid–liquid flow problems, such as
the ones related to the mixing of dispersions. Coupling
techniques based on direct numerical simulation (DNS)
are a priori attractive because they rely on an explicit
discretisation of the particles. Unfortunately, these techniques are known to be excessively expensive both in
time and memory, and are still limited to the simulation
of a few hundred monodisperse particles.[4]
In the present context, the so-called two-way coupling
techniques represent the best alternative owing to their
capability of resolving the impact of the particles
on fluid motion. The novel technique introduced in
this paper is restricted to the case of concentrated
suspensions, the behaviour of which is that of a timeand space-dependent porous medium. To this end,
a porous medium model was implemented into our
LBM code, which brings into play a forcing term, Fp ,
following along the lines of Guo and Zhao.[10] Note that
this time- and space-dependent forcing technique was
proposed and validated by Guo et al .[11] It takes into
account the drag due to the flow of particles making
up the porous medium by means of the Forcheimerextended Darcy’s drag force:
(7)
Fp = −
represent the local fluid density and the local fluid
velocity, respectively, and the dimensionless relaxation
time τ ∗ is related to the fluid kinematic viscosity ν
through
ν
1
(8)
+
τ∗ =
2
2
δt cs
where cs is the speed of sound defined as
3(1 − w0 ) δx
cs =
7
δt
Fluid–particle coupling
(9)
LBM is used in this work for two reasons: first, in
addition to being straightforward to implement, it has
already proven to be most efficient for solving a large
variety of fluid mechanics problems in significantly
smaller computational times as opposed to classical
CFD methods and, second, the Boltzmann equation
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
1.75
µε
|u|u + εG
u−√
ρK
150K ε
(10)
where ε is the porosity, K is the permeability, ρ is the
fluid density and G is an additional body force if needed.
Note that an extrapolation method was also used for
an accurate imposition of the boundary conditions[12]
delimiting the porous medium. With this technique,
second-order accuracy in space can be achieved for
a large set of boundary conditions, including sliding
boundaries and curved walls, which are often encountered in complex applications.
The validity of this LBM-based model for the solution
of the generalised Navier–Stokes equations describing
(isothermal) incompressible fluid flow in porous media
was assessed by simulating the porous plane Couette
and porous Poiseuille flows. In the first case, the
forcing term is induced by a vertical velocity gradient
originating from the top sliding wall. In the second case,
the forcing term is induced by a uniform body force,
Asia-Pac. J. Chem. Eng. 2011; 6: 44–54
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
DEM-BASED MODELS FOR THE CONSOLIDATION OF PARTICLE PACKINGS
(a)
(b)
Figure 6. (a) Porous plane Couette flow. The plain lines represent the finite reference
difference solutions, and the squares the solutions obtained with the LBM-based
model and its forcing term; (b) porous Poiseuille flow. The plain lines represent the
finite difference reference solutions, the triangles the solutions obtained with the
LBM-based model and its forcing term. Note that Re = ρHuref /µ is the Reynolds
number, where H is the width of the channel and uref the characteristic velocity (that
of the sliding wall for the Couette flow of that at the centre of the channel for the
Poiseuille flow), and Da = K/H2 is the Darcy number.
whereas the top and bottom walls are fixed. As seen in
Fig. 6, the reference solutions obtained with a secondorder finite difference solver are precisely predicted for
both problems and different values of the Darcy number
Da (different levels of permeability), which shows the
accuracy of the proposed approach for simulating fluid
flow in porous media.
Two-way coupling algorithm
The two-way coupling fluid–solid model proposed
in this work combines the DEM-based and LBMbased models described above. The overall algorithm
is transient and requires initial conditions for both the
particles and the fluid. At every time step, local porosity
values are first obtained for each lattice cell, and local
permeability values are estimated by means of the
following Carman–Kozeny correlation for particles of
diameter d:
K (x) = ε3 (x)d 2 /150(1 − ε(x))2
(11)
The resulting porosity and permeability distributions
are then introduced into the forcing term [Eqn (10)]
of the LBM model. Next, the velocity and pressure
distributions are computed with LBM, in which each
lattice cell is considered as a macroscopic representative
elementary volume (REV; see for instance Ref. [10])
of the porous medium. The following step consists of
performing a number of DEM iterations, typically from
100 to 1000, to move the particles. For each particle i ,
the hydrodynamic forces, that is the term Fhydrodynamic,i
in Eqn (2), are evaluated locally from the fluid velocity,
pressure gradient, porosity and distance to the nearest
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
particles, all of which were previously computed. The
resulting particle distribution is then used to move
on to the next time step. Two remarks are in order.
First, numerical experiments (not described here) have
revealed that performing many DEM iterations within
one single LBM time iteration has shown to be a good
strategy to reduce the computational cost while yielding
results that do not depend significantly on this number
of iterations. Moreover, Fhydrodynamic,i corresponds in
this work to the viscous drag force, which is the most
important force related to the fluid–particle interactions
involved in coating applications. Note that the viscous
drag is evaluated here from classical correlations due to
Ergun[13] and Wen and Yu.[14] Alternatively, one could
resort to recent correlations based on averaged results
from the DNS of randomly dispersed particles.[4]
RESULTS
In this section, three different packing techniques
described in Fig. 7 are investigated, compared and discussed within the context of pigment consolidation in
paper-coating applications.
The first case is the gravity-driven packing technique.
A typical DEM simulation is typically rather fast in
the first steps owing to the uniform acceleration of
gravity that leads to the bulk motion of the particles
with very few collisions until the lowermost particles
reach the bottom wall. A quick compression phase then
occurs, which leads to a loose packing porosity lying
within the range delimited by the loose random packing
limit, εLRP ∼ 44%, and the close random packing limit,
εCRP ∼ 36%, depending on both the friction parameter
and the weight of the bed of particles. We revert to
Asia-Pac. J. Chem. Eng. 2011; 6: 44–54
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G. PIANET ET AL.
Asia-Pacific Journal of Chemical Engineering
(a)
(b)
(c)
Figure 7. Schematics of the three particle packing techniques: (a) gravity
driven, (b) mechanically driven (compression) and (c) drag driven.
this point below. As shown in Fig. 3 of the section
on Assessment of the DEM-Based Model Through
XMT Measurements, gravity-driven packings are well
predicted by DEM, although these results are of limited
use for paper-coating applications where gravity effects
are known to be small.
The second case concerns the mechanical compression of particles. As already discussed in the introduction, the particles are compressed by a downwardmoving solid wall so that they progressively pack
against the bottom plane. At each time iteration, the
pressure resulting from the contact between the top layer
of particles and this moving plane is compared to a reference pressure, and the plane is moved downwards
when this pressure is too low or upwards when it is too
high. This mechanism in a way mimics the mechanical constraints of the calendering process. Figure 8
displays the initial and final states of the packing of
46 339 polydisperse particles as predicted with the
DEM-based model and mechanical compression only,
that is, without gravitational and drag forces. The initial mean porosity ε = 45% (∼1.02 × εLRP ), which is
close to that for a monodisperse loose random packing, εLRP , reduces to 38% (∼1.06 × εCRP ), which is
slightly above εCRP , the porosity of a monodisperse
close random packing. In this case, the percolation of
the particles are driven by the contact forces and, as a
result, the porosity levels off near but not lower than
the monodisperse close random packing value (εCRP ),
due to the barrier effect of the largest particles. From
a computational point of view, the first DEM iterations
of this mechanical compression packing simulation are
stable and rather similar to the ones for a gravity-driven
packing technique. Things are, however, much different in the last iterations, which may become unstable if
the total pressure on the moving plane is not carefully
chosen and controlled.
Note that, in the context of the calendering process, it
is likely that these mechanically driven particle packings
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
are more realistic than those obtained with gravity only.
In most paper-coating processes, buoyancy is often
negligible due to the high fluid viscosity and small
density differences between the particles and the fluid.
However, we believe that an enhanced percolation of
small particles and a further decrease in porosity can be
obtained by bringing into play liquid drainage and the
corresponding viscous drag.
For comparison purposes, 2014 monodisperse particles were packed using the gravity-driven, mechanically driven and drag-driven techniques. Initially, the
particles of diameter d = 800 µm were randomly dispersed in a rectangular domain of size (Lx , Ly , Lz ) =
(30 000, 3000, 3000 µm), with periodic boundary conditions along the y and z directions and solid walls
at x = 0 and 30 000 µm. Five initial particle configurations corresponding to random dispersions with a
prescribed 75% porosity were generated. Each time,
the overlapping particles were subjected to a fictitious
repulsive potential to displace them towards nearest free
spaces. The RDFs of these initial packings revealed that
peaks due to reorganisation were almost undetectable,
so that this dispersion technique can be considered as
quasi-stochastic. Note also that the simulation parameters were tuned so that the relevant driving forces
were equivalent in magnitude for all three packing techniques. This was done a posteriori by verifying the
time-dependent force exerted on a few probe particles.
All simulations were stopped when the packing heights
levelled off to some constant value.
Figure 9 shows the initial particle configuration and
the final packings for all three techniques. Figure 10
contains a graph of the porosity vs the vertical position.
The porosity values are in fact averaged values coming from the five different initial particle configurations
described above. Note, however, that these configurations had a small impact on the resulting porosity
curves. As seen in Figs 9 and 10, the gravity-driven
Asia-Pac. J. Chem. Eng. 2011; 6: 44–54
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Asia-Pacific Journal of Chemical Engineering
DEM-BASED MODELS FOR THE CONSOLIDATION OF PARTICLE PACKINGS
Figure 8. Mechanical compression of 46 339 polydisperse particles:
(a) initial state and (b) final state. Colour indicates particle size, from
blue (small) to red (large). This figure is available in colour online at
www.apjChemEng.com.
technique yields the loosest packing. The porosity oscillates around 40%, and thus falls right between the CRP
and LRP bounds as has already been observed above
for a different gravity-driven packing. The mechanically driven packing technique is observed to yield a
global decrease in porosity, close but slightly above on
average to the CRP bound. No concentration gradients
are found across the height of such packings, mostly
because of the monodisperse nature of the particles, so
that the compression effect is mechanically transmitted
across the whole packing.
In the case of the drag-driven packing technique,
a uniform drag was imposed, and significant differences with respect to the other two cases can be readily
noticed as regard the global porosity structure. First,
the packings are much tighter, the final porosity ranging between the close packing bound (εCP ∼ 26%) and
the CRP bound. Second, the slope of the corresponding
porosity curve in Fig. 10 reveals a noteworthy concentration gradient, which is likely due to the effect of drag
and induced compression. Once again, different initial
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
packings were tested with no major impact on the global
porosity profiles and microstructure. Increasing (respectively decreasing) the value of the friction parameters
resulted in a global shift of the porosity profiles towards
larger (respectively smaller) values.
A tool was developed for characterising the instantaneous pore size distribution. It is based on an erosion–dilatation method proposed by Toivakka and
Nyfors[15] to detect the real pores of a porous medium
and determine the position of their centres. Lateral periodic boundary conditions were considered and the pore
size distribution was calculated between the centres of
the lowest and highest particles in order to limit vertical
end effects. Figure 11 shows a graph of the pore size
distribution with respect to deq /d, where the pore volumes are expressed as spheres of equivalent diameters
deq . As can be seen, the gravity-driven and mechanically
driven packing results exhibit similar distributions, both
in terms of shape and magnitude, with one single peak
in the small pore size region at around deq /d = 0.2. The
drag-driven packing profile is significantly different,
Asia-Pac. J. Chem. Eng. 2011; 6: 44–54
DOI: 10.1002/apj
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G. PIANET ET AL.
Asia-Pacific Journal of Chemical Engineering
(a)
(b)
(c)
(d)
Figure 9.
Packing of 2014 monodisperse particles: (a) initial particle
configuration, and final packings for (b) gravity-driven, (c) mechanically driven
and (d) drag-driven techniques. Colour indicates the initial vertical position of
the particles. This figure is available in colour online at www.apjChemEng.com.
since many of the largest pores observed at deq /d > 0.4
with the other two packing techniques are converted
to smaller ones. Moreover, there is one large peak at
deq /d = 0.2 and a smaller one at 0.3 and, as a consequence, the relatively important number of pores with
similar properties, reveal that important particle reorganisation occurs during the drag-driven packing of particles. Another comment concerning the mechanically
driven packing results is of interest. The fact that, over
the course of the simulation, the total pressure applied
by the particles on the compression plane increased
monotonically to a user-defined maximum value (see
section on Results) indicates that the particle consolidation and the apparent structural reorganisation of the bed
of particles may be affected in some way by the interparticle penetration inherent to DEM. Note that this may
also be attributed to the speed at which the compression plane is pushed downwards because, if this speed
is too much, the particles will not have sufficient time
to reorganise. Future work will address this issue.
As discussed in the section on Fluid–Solid Coupling
Model, we are developing an LBM–DEM two-way
coupling algorithm for simulating the consolidation of
packings of particles under nonuniform drag forces. One
first exploratory result obtained with this hybrid model
is presented here. A pressure drop was imposed as the
driving force for the flow. The same initial configuration
of 2014 monodisperse particles was considered and the
apparent viscosity of the solid–liquid suspension was
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Figure 10. Graph of the porosity vs the vertical position for
all three packing techniques.
calculated so as to match as closely as possible the
viscosity used in the uniform drag packing simulation
discussed above. The local porosities were evaluated
analytically in spheres of volumes equal to the ones
in the lattice Boltzmann cell. Resorting to a larger
investigation zone would yield increased porous matrix
homogeneity and coarser flow details. Figure 12 shows
the fluid velocity and porosity fields at different times
during the particle consolidation. It is first noticed
Asia-Pac. J. Chem. Eng. 2011; 6: 44–54
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
DEM-BASED MODELS FOR THE CONSOLIDATION OF PARTICLE PACKINGS
Figure 11.
Pore size distribution. Pore volumes are
expressed as spheres of equivalent diameter deq .
that the fluid velocity is strongly correlated to the
local packing structure characteristics, with a significant
velocity decay in the consolidated zone near the bottom
plane. These results are more consistent with drainage
conditions than those obtained in the case of a uniform
drag packing technique for which the very large fluid
velocities at the bottom induce a very quick and very
strong compression. Fluid momentum in the lateral
directions is such that the particle kinematics is much
more complex than that related to the quasi-onedimensional and subsequently stratified bulk behaviour
observed with the previous packing methods. It can
also be seen that the homogeneous initial configuration
rapidly changes to a three-part flow comprising a free
fluid zone, a fluidised particle bed zone and a fixed
bed zone in which little structural changes occur.
It is also interesting to note that the two interfaces
between these three zones are characterised by different
propagation speeds, in analogy with other settling
processes. However, it must be mentioned that the local
recirculations with particle resuspension occurring at
the top are probably overestimated in this simulation.
Experimental data are required, which would help
assess the accuracy of these results and help improve
the hybrid LBM–DEM model proposed in this work
for simulating the compression of particle packings.
CONCLUSION
The objective of this work was to compare different
ways of using the DEM for consolidating and compressing particle packings in the context of paper-coating
applications. It was shown that simple methods, namely
the gravity-driven and mechanically driven techniques,
yield stratified structures all along the deposition process and similar final bulk porosities. The mechanically
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Figure 12. Results obtained with the LBM–DEM twoway coupling algorithm. Particle packing consolidation
under nonuniform drag forces at different times (from left
to right). Fluid velocity field (arrows) and local porosity
distribution. This figure is available in colour online at
www.apjChemEng.com.
driven packing technique was designed so as to reproduce the mechanical constraints of calendering processes in a simple way, but no significant changes in
the resulting microstructures were observed compared
to gravity-driven packings. Simulations with a uniform
viscous drag led to significant changes in the particle
dynamics and packing properties. This has provided the
impetus to develop an LBM–DEM two-way coupling
algorithm in which particle motion is coupled to fluid
dynamics. The advantage of such a complex model lies
in the fact that it gives access to heretofore inaccessible
information such as the local hydrodynamic constraints
on the particles. Independent assessments of both fluid
(LBM) and particle (DEM) models were performed,
and the results obtained were shown to be in very
good agreement with experimental data or theory. In
the context of high solid volume fractions, the coupling
principle considers the solid phase as a time- and spacedependent porous matrix in LBM, the porosity and permeability of which are recovered from DEM results.
One first exploratory result was presented and showed
Asia-Pac. J. Chem. Eng. 2011; 6: 44–54
DOI: 10.1002/apj
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G. PIANET ET AL.
Asia-Pacific Journal of Chemical Engineering
the potential of this hybrid LBM–DEM model for predicting the behaviour of wet granular media such as the
ones in coating applications. Future work will focus
on the validation of this LBM–DEM model against
experimental data and on its extension to the case of
polydisperse systems of nonspherical particles.
Acknowledgements
The financial contributions of NSERC and the SENTINEL Bioactive Paper Network are gratefully
acknowledged.
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