An extension of the hard-sphere particle-particle
collision model to study agglomeration
Pawel Kosinski, Alex C. Hoffmann
The University of Bergen, Department of Physics and Technology
Bergen, Norway
Chemical Engineering Science 65 (2010), pp. 3231-3239, doi:10.1016/j.ces.2010.02.012
Abstract
When performing Eulerian-Lagrangian simulations of particle-fluid flows
collisions between the particles need to be accounted for. One of the
methods used for this is the hard-sphere model. This model, however,
does not take into account cohesive forces between the particles, and for
this reason it is not able to simulate many aspects of real flows, such as the
formation of agglomerates. There have been some attempts in literature
to treat cohesive forces in simulations of particulate flows but none of
these methods were actually implemented directly into the hard-sphere
model but rather have been solved separately as a part of the numerical
scheme. In this paper we show how the standard hard-sphere model may
be extended to include these important interactions in an efficient and
proper way. The extended model is presented in detail and some numerical
results are shown.
Keywords
fluid-particle flows; hard-sphere model; numerical simulation; cohesion; agglomeration.
1
Introduction
Modelling of particle agglomeration is crucial in many processes. The formation of plugs in pipelines and process equipment due to solid particles such
as hydrates is one example. This paper introduces an extension to the hardsphere particle collision model to account for particle agglomeration in EulerianLagrangian simulations of particle-fluid flows.
With the field of Eulerian-Lagrangian computational simulation of fluidparticle (we are in this paper concerned only with solid particles) flows rapidly
expanding, there is an increasing need for reliable models for particle-particle
and particle-wall collisions and interactions.
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Chemical Engineering Science 65 (2010), pp. 3231-3239,
doi:10.1016/j.ces.2010.02.012
In general, collision models are classified as hard-sphere or soft-sphere, depending on whether particle deformation during a collision is explicitly incorporated in the model or not. The focus of this paper will be on hard-sphere
collision models. Hard-sphere models may be formulated on basis of concepts
involving deformation of the colliding particles, although deformation, as mentioned, is not involved in the final model equations.
Hard-sphere models are used extensively in the numerical simulation of particle or fluid-particle systems. Particularly popular has been the simulation of
fluidized beds (see, for instance, references [1, 2, 3], this topic was recently reviewed by Deen et al. [4]) and two or three phase flows in pipes or reactors (e.g.
references [5, 6, 7, 8, 9]).
Two hard-sphere collision models are widely used, that of Hoomans et al.
[10], and that of Crowe et al. [11].
One issue, which as yet has not been addressed in any formal way in these
collision models, is that of cohesion between particles or adhesion of a particle
to a solid wall. However, cohesion and adhesion are essential elements in a
wide range of processes of considerable interest to the scientific and engineering
communities. Examples of industrial processes where this is of interest are the
formation of hydrate or wax plugs in pipelines in the oil and gas industries,
defluidization in fluidized bed processes, or the formation of plugs in pneumatic
conveying lines.
Much of the work in the published research literature aimed at modelling
particle cohesion has, till now, been dedicated to the simulation of fluidized beds.
Cohesion in fluidized beds has been a research focus for two reasons: one is to
resolve a long-standing controversy about the role of interparticle cohesive forces
in stabilizing “particulate” (bubble-less) fluidization; the other is to determine
the role of cohesive forces in causing defluidization of a bed, which is a problem
in fluidized-bed operation. In the former case the cohesion is considered to arise
from van der Waals forces, while in the latter the focus is on cohesion arising
from liquid-bridging. However, in addition to fluidized beds, some articles have
also discussed interparticle cohesion in granular flows, such as the flow of the
particles in a rotary dryer.
Basically cohesive forces can be included in Eulerian-Lagrangian simulations
in two ways:
1. by including the cohesive force in the particle equation of motion and thus
simply include it as an extra force acting on the particle in the numerical
scheme, or
2. by incorporating the cohesive force in an impulse-based collision model.
Most of the published literature has been focused on the former using soft-sphere
collision models, while, for reasons given below, we are here concerned with the
latter option. We do, however, briefly review also some of the papers discussing
the former option below.
Reviews of the role played by interparticle cohesion in fluidized beds have
been written by Visser [12] and Seville et al. [13].
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Chemical Engineering Science 65 (2010), pp. 3231-3239,
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Mikami et al. [14] extended the classical soft-sphere collision model of Tsuji
et al. [15] with an extra non-linear spring and a rupture joint to account for the
cohesive force acting between particles connected by a stretching liquid bridge
which ruptures at a given critical surface separation. The non-linear spring
force and the rupture distance were tuned by comparison with solutions of the
Laplace-Young equation and experimental data. In another context, Lian et
al. [16] studied the collision of pendular-state agglomerates, building on their
earlier work [17] to characterize cohesive forces arising from pendular bridges.
We now turn to studies of the behaviour of powders that are dry, but so
fine that cohesive forces nevertheless play a role in their macroscopic behaviour.
Baxter et al. [18] studied powder flow problems arising from van der Waals
forces, modelled by a Lennard-Jones type potential, including short-range repulsion, but modified to allow for some particle deformation—and the associated
cohesion—in the contact point. Ye et al. studied the fluidization of fine powders
(Geldart Group A) modelling particle cohesion with a Hamaker-type interaction, but without including short-range repulsion. To avoid the singularity at
particle contact, the force was cut-off at a finite surface separation, which is chosen to be consistent with the physical surface sepation in a “contact point” [19].
Pandit et al. [20] also studied the particulate fluidization of Geldart Group A
powders using a Hamaker-type particle interaction.
The most relevant papers for the present article are those of Weber and coworkers [21, 22]. Rather than using the Hamaker interaction, they introduced a
simple “square-well potential” between the particles, by allowing a dirac-delta
type attractive force to act between two particles at a certain surface separation, creating a square-well potential from which two colliding particles might
or might not escape, depending on the velocity of impact and the coefficient
of restitution. One reason for using this simple interaction law is to have a
law which is suitable for implementation in impulse-type, hard-sphere particle
collision models.
Such hard-sphere models, e.g. the one of Crowe et al. [11], which we will be
workingR with here, are based on estimating the impulses, i.e. the time-integrated
forces, Fdt, acting on particles during specified periods during the collision,
due, for instance, to elastic/plastic deformation, and thus estimate changes in
the translational velocity:
Z
m∆v = J ≡ Fdt,
(1)
with a similar time-integrated equation describing the change in the rate of
particle rotation incurred by a rotational impulse (see below).
As Hrenya and coworkers point out, implementation in such a model of
a given functional form for F as a function of surface separation, D, is not
straight-forward since the time-integral depends on the time that the particle
spends within the force field, which depends not only on the approach and
departure velocities to and from the the collision but also, due to the particles’
reaction to F, on F itself. This makes it impossible to find an analytical solution
to Equation (1) even for the simple form of Hamaker interaction without the
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Chemical Engineering Science 65 (2010), pp. 3231-3239,
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repulsion term. Hrenya and co-workers overcame this problem by tweaking the
depth of their square-well potential so that it matches numerical simulations of
Hamaker interactions.
In fluidized systems or granular flow, which most of the above references
are concerned with, it may be convenient to incorporate the particle interaction force in the particle equation of motion, and thus directly in the numerical
scheme, which is option 1 above. Complexity in the interaction force law, F(D),
is then less of a problem. However, in more dilute flows, where the particle displacement within one time-step may be much larger, it is not computationally
efficient to simulate the interaction directly, rather incorporating the interaction
in the collision model is more desirable and in many cases even the only practicable option. Another advantage of incorporating interparticle or particle-wall
interaction in an impulse-based collision model is, as pointed out by Weber et
al. [21] that such a collision model may be incorporated in continuum schemes
for the simulation of the dynamics of a dispersed phase.
In this paper our objective is to incorporate cohesion forces into a hardsphere model directly, i.e. the model is modified and extended. All the mathematical expressions are analytical and make it possible to find the post-collisional
particle velocities directly. In the following chapter we show the derivation of
the model.
2
Derivation of the model
The standard hard-sphere model is based on writing the impulse equations for
two colliding rigid spheres. Details can be found in various references, we will
refer to the description in Crowe et al. [11] and base our derivation on this model
using their notation.
Figures 1 and 2 show the situation of a collision with the main variables. All
the impulses are defined as those acting on particle 1, and the relative velocities
are calculated as e.g. v1 − v2 .
The impulse equations can be written as (see also Figs. 1a-b and 2):
(0)
m1 (v1 − v1 ) = J
(0)
m2 (v2 − v2 ) = −J
I1 (ω 1 −
I2 (ω 2 −
(0)
ω1 )
(0)
ω2 )
(2a)
(2b)
= r1 n × J
(2c)
= r2 n × J
(2d)
where subscripts 1 and 2 identify the colliding particles, n is the unit vector from
particle 1 to particle 2, J is the impulse that is acting on the particles during a
collision, mi and Ii are the mass and the moment of inertia for both particles,
respectively, vi and ω i are the linear and the angular velocity, respectively
and ri is the particle radius. The superscript (0) denotes the state before the
collision.
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Chemical Engineering Science 65 (2010), pp. 3231-3239,
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The impulsive force J in Eq. 2 represents, in the standard hard-sphere model,
the mutual repulsion of the colliding particles due to their elastic deformation.
This leads to their bouncing off after the collision. Since this paper also deals
an attractive force between the particles, the impulse J has to include such
interactions as well.
Following the derivation in ref. [11], we define the restitution coefficient:
n · G = −e(n · G(0) )
(3)
where G = v1 − v2 is the relative velocity between the particle centres after the
collision, G(0) is the relative velocity before the collision, and e is the restitution
coefficient.
The restitution coefficient is a property of the material of the particles and
describes how much absolute momentum the particles lose during the collision.
The value is often assumed as constant and is estimated by empirical observations. It can also be evaluated by detailed studies of the dissipative processes
in the particle material and the fluid between the two colliding particles by, for
example, Finite Element Method techniques.
The derivation of the hard-sphere model in [11] leads to the following value
of the normal component of the impulse:
Jn,a = −
m1 m2
(1 + e)(n · G(0) )
m1 + m2
(4)
where we, for reasons given below, have added the subscript a to denote that this
includes only the repulsive interaction due to the deformation of the particles.
On this equation we begin to extend the model. To make the addition of
cohesive impulse clear in this paper we suggest to write the normal component of
the impulse as two contributions, one repulsive, due to mechanical deformation,
as mentioned above, and one due to cohesive forces, whatever their origin:
Jn = Jn,a + Jn,c
(5)
Jn,a is calculated as in the standard hard-sphere model. Jn,c , on the other
hand, has to be modelled separately taking into account what kind of cohesive
interaction is relevant for the specific application. This is described in more
detail in Section 3. Note that we used subscript t to denote the adhesive force
in a previous paper [23] on particle-wall interaction. However, in this paper the
subscript t is reserved to denote “tangential”.
We rewrite Eqs. 2 solving for the final translational and rotational velocity
components, and writing J out in its components:
(0)
v1 = v1 + (Jn n + Jt t)/m1
v2 =
ω1 =
ω2 =
(0)
ω1
(0)
ω2
(0)
v2
(6a)
− (Jn n + Jt t)/m2
(6b)
+ r1 n × (Jn n + Jt t)/I1
(6c)
+ r2 n × (Jn n + Jt t)/I2
(6d)
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Equations (6) can be treated as the final formulae for finding the new velocities after particle-particle collision with cohesion between the particles. The
repulsive part of the normal component of the impulsive force, Jn,a , can be
found from Eq. (5), while, as mentioned, the attractive part must be quantified considering the specific system, which we will discuss below. Jt is, in the
classical model and in this extension, dependent on the particles’ behaviour as
they are in contact. There are two cases possible (see [11]): the case when
the particles stop sliding sometimes during the collision and the case when the
particles continue to slide. These cases have to be treated separately and this
is considered below.
3
3.1
Model for Jt and final expressions for the postcollisional velocities
Particles slide throughout the collision
If one assumes that the particles slide during the collision, the tangential component of the impulsive force, Jt can, assuming Coulombian fraction with dynamic
friction coefficient f , be written as:
Jt = (Jn,a − Jn,c )f
(7)
where the minus sign signifies that the attractive impulse acting in the contact
point, Jn,c , (the value of which is positive, while Jn,a is negative, see Figure 2)
acts with the normal impulse on the particle, Jn,a , to increase the friction in the
point. The same argument was made for a particle-wall collision in ref. [23].
Substituting the expressions for Jn and Jt in Eq. (6) gives the final expressions for the post-collisional velocities:
Jn,c
m2
(n − f t) −
(1 + e)n · G(0) (n + f t)
m1
m1 + m2
Jn,c
m1
(0)
v2 = v2 −
(n − f t) +
(1 + e)n · G(0) (n + f t)
m2
m1 + m2
·
¸
5
Jn,c
m2
(0)
(0)
ω1 = ω1 +
(n × t)f −
−
(1 + e)n · G
2r1
m1
m1 + m2
·
¸
5
Jn,c
m1
(0)
(0)
ω2 = ω2 +
(n × t)f −
−
(1 + e)n · G
2r2
m2
m1 + m2
(0)
v1 = v1 +
(8a)
(8b)
(8c)
(8d)
This reduces to the standard hard-sphere model (one sign has been corrected in
each of the two translational velocity equations from ref. [11]) for Jn,c = 0.
According to the standard hard-sphere model (see details in [11]), the case
when the particles slide during the collision occurs when:
Jt > −
2 m1 m2
(0)
|G |
7 m1 + m2 ct
6
(9)
Chemical Engineering Science 65 (2010), pp. 3231-3239,
doi:10.1016/j.ces.2010.02.012
(0)
where Gct is the tangential component of the particles relative velocity and is
given by:
(0)
(0)
(0)
Gct = G(0) − (G(0) · n) · n + r1 ω 1 × n + r2 ω 2 × n.
(10)
The condition in the form of Eq. (9) is also the same in our extended model,
because its derivation remains the same. However, the criterion is actually more
restrictive in the extended model due to the difference in Jt , as can be seen from
rewriting the criterion as follows. In the original model, Eq. (9) can be rewritten
to:
n · G(0)
2
1
,
(11)
<
(0)
7
f
(1
+ e)
|Gct |
while in the extended model it becomes the more restrictive:
2
m1 + m2 Jn,c
1
(0)
n · G(0) < −
+
|G |,
m1 m2 (1 + e) 7 f (1 + e) ct
(12)
which reduces to the criterion of the original model for Jn,c = 0.
3.2
Particles stop sliding during collision
The second case describes the situation where the particles stop sliding during
the collision. This case is described in detail in [11]. The derivation presented
in this reference is also valid for the case when there is cohesion between the
particles.
As indicated by Figure 2, Jt is always negative, and the criterion in Eq. (9)
is derived from the result that once the tangential impulse given by the righthand-side of the equation has acted on the particles they have reached such rates
of rotation that they no longer slide but roll over each other for the rest of the
collision. During this rolling period no tangential force acts in the contact point
and there is therefore no additional tangential impulse. The value of Jt for the
case where the particles stop sliding during the collision is therefore given by:
Jt = −
2 m1 m2
(0)
|G |
7 m1 + m2 ct
(13)
Again using Eq. (6), as well as Eq. (13), yields:
·µ
¶
¸
Jn,c
m2
2
m2
(0)
(0)
v1 =
+
−
(1 + e)n · G
n−
|G |t
m1
m1 + m2
7 m1 + m2 ct
(14a)
¶
¸
·µ
m
2
m
J
1
1
n,c
(0)
(0)
v2 = v2 −
−
(1 + e)n · G(0) n −
|Gct |t
m2
m1 + m2
7 m1 + m2
(14b)
m
5
2
(0)
(0)
ω1 = ω1 −
(n × t)|Gct |
(14c)
7r1
m1 + m2
m1
5
(0)
(0)
(n × t)|Gct |
(14d)
ω2 = ω2 −
7r2
m1 + m2
(0)
v1
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4
Condition for agglomeration
The model described above works correctly for the case when the particles will
bounce off after the collision. Nevertheless, if the cohesive force is high enough
the particles will not bounce off. The presence of a cohesive force causes the colliding particles to enter a potential well. The particles loose mechanical energy
during the collision due to the dissipative processes involved in the collision,
such as plastic particle deformation, viscous dissipation in the fluid present between the particles and any dissipation associated with the separation of the
surfaces during the latter part of the collision [19]. This loss of mechanical energy may lead to the particles not managing to escape the potential well after
the collision.
While for the collision of a particle with a wall we know that the particle
has to rise from the wall to escape after the collision, and the condition for
agglomeration can thus be written in terms of the vertical component of the
particle’s impulse [23], it is not so clear-cut for a particle-particle collision, or
for the collision between a particle and an asperity on a wall.
The hard-sphere model for collisions does not envisage any change in the
plane of contact during the collision. However, physically any rolling or sliding
of the particles during the collision (which is conceptually a part of the model)
will lead to such a change. This is illustrated in Figure 3.
The consequence is that the component of the outgoing impulse normal to
the plane of contact is higher than that envisaged by the hard-sphere model. In
fact, as long as the particles are of similar sizes, it seems intuitively reasonable
to formulate the criterion for agglomeration in terms of the resultant of the relative velocity of the particles after the collision rather than only the component
normal to the plane of collision. We grant, however, that in the limit of a very
small particle colliding with a very large one, the agglomeration criterion should
in principle be the same as that for a particle-wall collision, which is in terms
of the normal component of the velocity [23].
The impulse equations for the two particles are:
(0)
m1 (v1 − v1 ) = J
(0)
m2 (v2 − v2 ) = −J.
Setting the condition that for agglomeration v1 = v2 gives:
´
m1 m2 ³ (0)
m1 m2
(0)
Jl =
v2 − v1
G(0)
=−
m1 + m2
m1 + m2
(15a)
(15b)
(16)
where the subscript l denotes that this is the limiting impulse for agglomerate
formation.
We can form the dot products of this equation with n and t, respectively,
to obtain conditions for the normal and tangetial components of the impulse:
n · Jl = Jnl = −
m1 m2
m1 m2
(0)
(0)
n · G(0) =
(v − vn1 )
m1 + m2
m1 + m2 n2
8
(17)
Chemical Engineering Science 65 (2010), pp. 3231-3239,
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t · Jl = Jtl =
m1 m2
m1 m2
(0)
(0)
t · G(0) =
(v − vt1 ).
m1 + m2
m1 + m2 t2
(18)
We first consider Equation (17) and the expression for Jnl . In the standard
hard-sphere model the absolute value of this parameter is always smaller than
the absolute value of the impulsive force Jn,m (defined by Eq. (4)) except for
the case when the restitution coefficient is equal to 0. This confirms the fact
that the particles always bounce off for e > 0.
In the extended model presented in this paper, this is not always the case
due to the presence of the cohesive force Jn,c . Thus agglomeration will take
place if:
Jnl < Jn,m + Jn,c
(19)
where we take into account the fact that both Jn,m and Jnl are, in principle,
negative. This inequality can be rewritten to:
Jn,c >
m1 m2
e n · G(0) .
m1 + m2
(20)
Equation (18) constitutes the condition for agglomeration that:
Jt < Jtl .
(21)
This condition does not refer to Case 2, where it is already assumed that the
particles stop sliding and that the relative velocity of the contact point is zero.
In that case, the condition described by Eq. 19 is sufficient.
5
Translational and rotational velocity of the
agglomerate formed
While the hard-sphere model takes into account the translational and rotational
motion of two individual particles before and after a collision not leading to
agglomeration, and it is therefore appropriate to formulate the condition for
agglomeration as the condition that the relative translational velocity is zero as
we did above, an agglomerate, if formed, will normally rotate around its center
of mass. In this section we derive the translational and rotational velocities of
an agglomerate after formation.
The translational velocity of the formed agglomerate, vagg , is given by the
conservation of momentum:
(0)
(0)
m1 v1 + m2 v2 = (m1 + m2 )vagg
(22)
We find the rate of rotation, ω agg , from the conservation of moment of
momentum. We thus obtain the angular velocity of the agglomerate as:
(0)
(0)
Iagg ω agg = M1 + M2
(0)
⇒ ω agg =
9
(0)
M1 + M2
Iagg
(23)
Chemical Engineering Science 65 (2010), pp. 3231-3239,
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where Iagg is the moment of inertia of the agglomerate around its centre of mass
(0)
(0)
(COM) and M1 and M2 are the moments of momenta of the two colliding
particles about the agglomerate COM just before the collision.
We know from the standard two-body problem that the centre-of-mass (COM)
or “barycenter” of the formed agglomerate is on the tie-line between the centres
of the two (spherical) particles, and that the distance to the center of each of
the particles is:
ai = (ri + rj )
mj
where i, j = 1, 2
mi + mj
(24)
We need the moment of inertia of the two-sphere assembly around an axis
through the COM of the agglomerate and perpendicular to the tie-line between
the particle centres, Iagg . For this we use the fact that the moment of inertia of
a composite body around a given axis is the sum of the moments of inertia of
the individual bodies around the same axis. To calculate the moments of inertia
of the two particles around the COM of the agglomerate we use the parallel axis
theorem:
I = Ii + mi a2
(25)
where Ii is the moment of inertia of the particle around its own centre and a is
the distance of the particle centroid from the COM of the agglomerate. Since
for each particle I = (2/5)mr2 we obtain:
Iagg
·
¸2
·
¸2
m2
2 2
m1
2
= (r1 m1 +r2 m2 )+m1 (r1 + r2 )
+m2 (r1 + r2 )
5
m1 + m2
m1 + m2
2 2
m
m
1
2
= (r1 m1 + r22 m2 ) +
(r1 + r2 )2 (26)
5
m1 + m2
We now find the total moment of momentum of the particles around the
COM of the agglomerate just before the collision. As the particles impact
they may both be rotating and translating, and we can calculate the moment
of momentum of each around the COM of the agglomerate as the sum of a
“remote” term describing the moment of momentum of the translating particle
mass concentrated at the particle center around the agglomerate COM and
a “local” term describing the moment of momentum of the rotating particle
around its own center:
(0)
(0)
(0)
M1 = (−a1 n) × m1 v1 + I1 ω 1
(0)
(0)
(0)
M2 = (a2 n) × m2 v2 + I2 ω 2
(0)
ai,i=1,2 , Iagg , M1
(0)
M2
(27a)
(27b)
Inserting the expressions for
and
given by Equations
(24), (26), (27a) and (27b), respectively, in Equation (23) gives the rate of
roation of the agglomerate, ω agg .
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6
Model for Jn,c
In the model above we have made use of the impulse Jn,c due to any particleparticle attractive forces. Although quantifying Jn,c is not the focus of this
paper, we need some reasonable approximation to show results from the model.
The literature dedicated to modeling or measuring the adhesive interaction
between particles is extensive. Here we only mention the classical book of Israelachvili [19] and the recent work by Zhou and Peukert [11].
Attractive forces due to Hamaker interaction or liquid bridging are, as long
as the particles are not in contact, known functions of theR particle surface separation, D. To work out the impulse, however, Jn,c = Fn,c dt, we need an
expression for the force as a function of time. A change of variable is therefore
necessary. For example during the approach to the collision:
Z
Z
t
Jn,c =
Dc
Fn,c (t)dt =
0
Da
dt
Fn,c (D)
dD =
dD
Z
Dc
Da
Fn,c (D)
dD (28)
vn1 (D) − vn2 (D)
where Da and Dc are the surface separations at which the interaction force is
negligibly small at at contact, respectively. The subscript n denotes an axis in
the direction of the tie-line between the particle centres, and vn1 and vn2 are
velocity components for the particles along the n-axis.
This change of variables is the central problem in quantifying Jn,c , since
vn1 and vn2 depend on Fn,c (D). Finding analytical expressions for Jn,c may be
difficult or impossible depending on the complexity of the form of Fn,c (D). Only
the correct functional form for Fn,c (D) will give universally valid Jn,c . Another
conceptual problem with this approach is, as mentioned, that the cohesive force
acts also during the period of contact.
Weber et al [21], in their implementation of particle cohesion in a numerical
scheme, used a simple “square-well” potential generated by a Dirac delta-type
force acting between the particles at a particular separation. They then fitted
this potential using numerical simulations to obtain approximately correct Jn,c
for a range of particle properties and approach velocities.
In our previous paper, [23], we made use of an approximation making it
possible to carry out the change of variables in Equation (28), namely that the
force Fn,c is constant and equal to the average value of the Hamaker force over
the surface separation interval in which the Hamaker force is significant.
In this paper, however, the above approach would be very complicated, since
both of the particles are moving and their trajectories will be altered due to the
existence of cohesive forces acting over a distance.
For this reason we estimate Jn,c using another approach: Rather than taking
the “contact period” to include the periods before and after the collision during
which Fn,c is significant and neglect the actual period of contact, as we did in
[23], we try to estimate the actual period of contact (the sum of the compression and recovery periods) from the literature and assume Hamaker interaction,
calculated for two perfect spheres in contact, to be acting during this period
only.
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Chemical Engineering Science 65 (2010), pp. 3231-3239,
doi:10.1016/j.ces.2010.02.012
In fact, the most physically realistic may be to sum the contributions to Jn,c
estimated by both approaches, but this is, as mentioned, beyond the scope of
this paper.
The duration of impact for two elastic bodies is well-known in literature (see,
for instance [24, 25, 26]):
Ã
tc = 1.43
!1/5
m2
,
E∗2 r(n · G(0) )
(29)
where:
• m is the effective mass defined as m ≡
m1 m2
m1 +m2 ,
• E∗ is the effective Youngs modulus defined as: E∗ ≡ [(1 − ν12 )E1−1 + (1 −
ν22 )E2−1 ]−1 , with νi being the Poisson ratios and Ei being the particles’
Youngs moduli,
• r is the effective radius defined as r ≡
r1 r2
r1 +r2 .
As mentioned, we assume that during this impact, the interaction force is
that of Hamaker interaction between two spheres [19], which, of course, represents an approximation since the particles are flattened against each other:
A
r,
6D2
(30)
where A is the Hamaker constant. We assume that the separation between
the particles during contact is D = Dc = 0.2 nm, as is often done [19]. This
value corresponds to the distance between centres of atoms of diameter ∼0.2
nm.
With these assumptions the force Fn,c becomes constant for a given collision
and the impulse becomes:
Z
Jn,c =
Ã
tc
Fn,c dt = 0.238
0
!1/5
m2
E∗2 (n · G
(0)
)
Ar4/5
.
Dc2
(31)
In this section we presented two possible approaches for estimating the value
of the impulsive force Jn,c . The first approach, defined by Eq. (28), was not
used here, partly due to its complexity since our focus in this paper is on the extension of the hard-sphere model. We therefore chose another, simpler, method,
defined finally by Eq. (31), a method which is, at least in part, supported by
experiment. A full description of Jn,c will involve interaction both between the
separated particles during approach and departure and between the particles
during contact. Important further work will be to quantify Jn,c in an optimal
way, but the method used in this paper can serve as a first approximation.
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Chemical Engineering Science 65 (2010), pp. 3231-3239,
doi:10.1016/j.ces.2010.02.012
7
Some results
In the following we demonstrate the performance of the extended hard-sphere
model method by examining the behavior of particles both during a single collision and in a larger numerical simulation.
In order to better illustrate the results and their physical meaning we are
using dimensional parameters for description of the systems and the particleparticle collisions.
In the first example we show snapshots of two particles colliding without
and with cohesion taken into account. In this example the two particles have
diameters of 15 and 10 µm and both have a density of 2200 kg/m3 . The particles
move toward each other in the x-y plane with velocities of (0.07, 0, 0) and (0.035, 0, 0) m/s, respectively, and collide, the collision plane having an angle
of 45◦ to the x-axis. Snapshots showing the positions and orientations of the
particles are shown in Fig. 4 at three points in time (the same for the two cases):
before, just after and longer after the collision. In the left three snapshots
the classical hard-sphere model is used, and in the right the extended model
as described in this paper with a Hamaker constant of 1 × 10−19 J, a particle
Youngs modulus of 70 GPa, a Poisson ratio of 0.16. In both cases the restitution
and friction coefficients were taken as 0.9 and 0.15, respectively. The effect of
the cohesion on the post-collisional velocities is clear.
In the second example of results from the model we show a numerical simulation of particles advected in a circular pipe flow with a parabolic laminar
velocity profile.
In the simulation the computational domain was rectangular and two-dimensional with the ratio between the length and height equal to 4. The boundary conditions at the two ends were periodic. The solid walls were assumed
to be smooth (in real applications the walls would be rough, this is crucial
especially for small particles and has for such particles been dealt with e.g.
by randomly changing the slope of the wall within a certain range during the
particle-wall collision [27]). The domain was divided into 128 by 32 grid cells.
Grid-independency was tested by using different meshes.
The fluid Reynolds number was 1000 and during the start-up phase we
ran CFD simulations until the flow was established. Subsequently 20000 solid,
spherical particles were introduced into the channel at random positions. The
details are shown in Fig. 5.
The simulation code, the mathematical model and the numerical scheme
were the same as in our previous work [28] and therefore the description is not
repeated here. The only modification in the algorithm is the implementation of
the extended hard-sphere model, such that the post-collisional particle velocities
were different from those in simulations using the classical hard-sphere model.
The possibility of agglomeration made it necessary to account also for the
behaviour of agglomerates after their formation, this means not only finding
the post-collisional velocities, which we give above, but also their longer-term
behaviour, which involves their interaction with the fluid. In fact, two connected particles form a more complex structure than a single spherical particle,
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Chemical Engineering Science 65 (2010), pp. 3231-3239,
doi:10.1016/j.ces.2010.02.012
a structure that moves in response to the interaction with the fluid and possible
collisions with other particles, the wall or other agglomerates. A complete description of such phenomena, e.g. the modelling of the drag force, is challenging.
Therefore we introduce the following assumption: if a collision between two
particles should lead to the formation of an agglomerate, a new particle is introduced into the system the volume of which is equal to the sum of the volumes
of the two agglomerating particles. The linear and the angular velocities of
this new particle are equal to vagg and ω agg , respectively (their derivations are
shown above).
The particle size was 10µm and their density was 2200 kg/m2 . The particle
Youngs modulus was 70 GPa and the Poisson ratio was 0.16. These correspond
to very fine quartz particles. The Hamaker constant was set to 10−19 J, the
restitution coefficient was 0.9 and the friction coefficient was 0.15. In order to
describe the particle-wall collisions, the restitution coefficient was modified to
1.0 as these interactions were not studied.
We emphasize that these parameters may be used for fitting of the results
to experiments, if necessary.
In Figure 6 we show one snapshot of particle positions in a small, zoomed-in
region in the middle of the computational domain. The white dots represents
“bigger” particles, i.e. agglomerates.
Not all the collisions lead to formation of agglomerates: this depends on
many factors, among others the relative velocity and the size of the particles.
This we illustrate in Fig. 7 where the accumulated number of collisions during
the simulation process is shown. The solid line is the total number of collisions
that result in bouncing off or agglomerate forming. The dashed line is the
number of collisions that led to the formation of agglomerates. It is interesting
to note that the number of collisions resulting in agglomerates is significantly
large in this example, constituting about 25% of the collisions as seen in Fig. 7.
We note that the initial configuration and particle concentration as well as
the physical parameters, such as the particle properties, used in this example
are common in real industrial applications. Thus one important conclusion is
that the extended model presented in this paper should describe the phenomena
taking place better than the standard hard-sphere model.
8
Concluding remarks
In this paper an extension of the standard hard-sphere model, being one of the
most popular techniques for simulating flows with particles, has been presented.
The new model takes into account the cohesion between the particles that is
modelled as an impulse of force attracting them, and the extension is therefore
conceptually consistent with the original hard-sphere model.
To extend the hard-sphere model in this way is of importance in many contexts, since such an approach is fast and efficient, while the incorporation of an
attractive force in the actual numerical scheme is only practicable in very few
systems where the particles are very closely spaced, such as in fluidized beds.
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Chemical Engineering Science 65 (2010), pp. 3231-3239,
doi:10.1016/j.ces.2010.02.012
To quantify the force in a preliminary way, we assume that Hamaker interaction between two contacting spheres acts only during the actual contact, and
only acts in this period. Further work to quantify the force and resulting impulse
may be done, also extending the work to forces arising from liquid bridging or
electrostatic forces.
Some results of the model are shown to illustrate the effect of the cohesive
force on a single collision, and the formation of agglomerates in a system of
many particles advected in a fluid.
Both the extended and the standard hard-sphere models describe only binary collisions. In dense flows the particles may be subject to multiple-particle
interactions. However, as discussed in the Introduction, modelling of such flows
(e.g. fluidized beds) may more efficiently be done by incorporating particle
interactions in the numerical scheme, while this work focuses on more dilute
systems, where this is not practicable.
Acknowledgments
This research was financed through the Hyperion project by the Research Council of Norway and StatoilHydro.
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Figure 1: Graphical 2-dimensional presentation of a particle collision showing
the translational velocities, vi and the impulse vector, J, resolved in components
normal (Jn ) and tangential (Jt ) to the plane of collision.
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Chemical Engineering Science 65 (2010), pp. 3231-3239,
doi:10.1016/j.ces.2010.02.012
Particle 2
n
P
Jt
t
Jn
J
Plane of/
collision
Particle 1
Figure 2: Illustration of the geometry of a collision, showing the point, P, and
plane of collision, the unit normal pointing away from particle 1, n, the unit
tangential vector pointing in the direction of the relative velocity between the
two surfaces at the point of collision, t, and the impulse vector, J (which by
convention is acting on particle 1) resolved in components normal, Jn = Jn n,
and tangential, Jt = Jt t, to the plane of collision.
v1
v1
v1(0)
v2(0)
A
v2
v2
B
C
Figure 3: Figure illustrating the effect of the particles rolling 10◦ during the
collision. The component of the outgoing impulse normal to the plane of contact
is higher than envisaged in the hard-sphere model. A: Incoming particles. B:
Particles leaving the collision as envisaged by the hard-sphere model with the
hard-sphere post-collisional velocities indicated. C: Particles leaving after some
rolling with the hard-sphere post-collisional velocities indicated.
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Chemical Engineering Science 65 (2010), pp. 3231-3239,
doi:10.1016/j.ces.2010.02.012
0.00002
0.00002
- 0.00006
- 0.00006
0.00006
0.00006
- 0.00006
- 0.00006
0.00002
0.00002
- 0.00006
0.00006
- 0.00006
0.00006
- 0.00006
- 0.00006
0.00002
0.00002
- 0.00006
- 0.00006
0.00006
0.00006
- 0.00006
- 0.00006
A
B
Figure 4: Snapshots of the positions and the orientations of two colliding particles at three given points in time. Particle diameters: 15 and 10 µm density:
2200 kg/m3 . Initial velocities: (0.07, 0, 0) m/s and (-0.035, 0, 0). The collision plane makes an angle of 45◦ to the x-axis. A: the classical hard-sphere
model B: the extended model with cohesion; Hamaker constant: 1 × 10−19 J;
Youngs modulus: 70 GPa; Poisson ratio: 0.16. The coefficients of restitution
and friction were 0.9 and 0.15, respectively in both cases.
Figure 5: The scheme of the computational domain used for illustration of the
extended hard-sphere model. The flow is laminar and a number of particles is
introduced into the system at random positions.
19
Chemical Engineering Science 65 (2010), pp. 3231-3239,
doi:10.1016/j.ces.2010.02.012
Figure 6: A snapshot of particle positions in a zoomed region in the computational domain. The black dots represent the primary particles, while the white
dots are the particles formed after agglomeration. Please note that the size of
the dots does not correspond to their real physical size.
20
Chemical Engineering Science 65 (2010), pp. 3231-3239,
doi:10.1016/j.ces.2010.02.012
Figure 7: The accumulated number of particle-particle collisions. The solid line
is the total number of collisions, while the dashed line the number of collisions
resulting in agglomerate formation
21