Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 10.7.1 - Validating colloids agglomeration with existing theoretical Kernel
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 Material Information
Title: 10.7.1 - Validating colloids agglomeration with existing theoretical Kernel Collision, Agglomeration and Breakup
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Mohaupt, M.
Taniere, Anne
Minier, J.-P.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: colloid
Brownian motion
collision algorithm
kernel
 Notes
Abstract: The temporal tracking of the agglomeration of very small particles up to very large-inertia particles is investigated considering spherical particles. A deterministic collision algorithm capable of handling the multi-scale aspect underlying this study is developed in this paper. Both the collisions due to random motion of Brownian particles and those induced by the deterministic movement of large-inertia particle must be treated which means strict conditions on the choice of time step for the simulation are required. The total cost of computation would be dramatically increased which is not in accordance with the main goal of the present study. Two different limit behaviours are examinated here to study the efficiency of the present algorithm through the calculation of collision kernels according to existing proposals in research literature on the subject.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Resource Identifier: 1071-Mohaupt-ICMF2010.pdf

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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Validating colloids agglomeration with existing theoretical Kernel


Mikael MOHAUPT*, Anne TANIERE* and Jean-Pierre MINIERt

LEMTA UMR 7563 CNRS, ESSTIN, Universit6 Henry Poincar6 Nancy 1,
2 rue Jean Lamour, F-54529 Vandoeuvre-Lbs-Nancy, France
Electricity de France, Div. R&D, MFTT, 6 Quai Watier, 78400 Chatou, France
mikael.mohaupt@esstin.uhp-nancy.fr, anne.taniere@esstin.uhp-nancy.fr and jean-pierre.minier@edf.fr
Keywords: colloid, Brownian motion, collision algorithm, kernel




Abstract

The temporal tracking of the agglomeration of very small particles up to very large-inertia particles is investigated
considering spherical particles. A deterministic collision algorithm capable of handling the multi-scale aspect
underlying this study is developed in this paper. Both the collisions due to random motion of Brownian particles and
those induced by the deterministic movement of large-inertia particle must be treated which means strict conditions on
the choice of time step for the simulation are required. The total cost of computation would be dramatically increased
which is not in accordance with the main goal of the present study. Two different limit behaviours are examinated
here to study the efficiency of the present algorithm through the calculation of collision kernels according to existing
proposals in research literature on the subject.


Nomenclature


Roman symbols
B diffusion coefficient (m s3/2)
dp diameter of particles (nm)
9 gravitational constant (m s 1)
l~ (t) Stochastic integral on position (m)
1v (t) Stochastic integral on velocitiy (m s1)
KB Boltzmann constant (m2 kg s 2 K 1)
L length of the numerical control domain (m)
Mnp mass of particles (kg)
N particles number (-)
Nc computed number of collisions (-)
P pressure (N m 2)
r radius of particles (m)
T temperature (K)
Tp particle agitation (m2 s 2)
t time (s)
to initial time (s)
tma, total simulation time (s)
v velocity (m s-1)
W(t) Wiener process (s1/2)
x position (m)


collision time step (s)
mean free path (m)
standard Gaussian random variable (-)
standard Gaussian random variable (-)
medium fluid viscosity (Pa s)
density (kg m 3)
relaxation time (s)


Subscripts
eq
max
m 1, 2,3
p
f
Superscripts
i

ij
Special notation


d- (t)


equilibrium
maximum
space direction
discrete phase (particle)
continuous phase (fluid)


particle i
particle j
relative value (j i)
n
Euclidean norm
mean value, i.e. (1/N) 1(-)
time increment


Greek symbols
At
Atini


time step (s)
initial time step (s)











Introduction

Particle deposition processes can be divided into several
steps (or elementary mechanisms): dispersion, single-
particle deposition, agglomeration and clogging. The
numerical simulation of the agglomeration mechanisms
for colloids transported in particle-laden flows is a key
issue for a wide range of engineering processes and
applications, such as nuclear safety, food technology or
aerosol concerns. The complexity of these simulations
stems from the fact that widely different scales are
involved in the overall physical phenomena. At a
microscopic level, colloids moving under a Brownian
influence can agglomerate due to electro-chemical
forces between them. At a macroscopic level, these
particles are affected by hydrodynamic forces and
transported by the fluid turbulence. These two levels
are not independent both because particles may grow in
time (due to agglomeration) which can modify the way
they are transported and because different macroscopic
agitations can result in different agglomeration rates.
To study agglomeration phenomena, it is first nec-
essary to define an algorithm for the detection of
particle collisions. Contrary to probabilistic methods
to treat particle-particle collisions (Oesterle & Petitjean
(1993)), we chose to adopt a deterministic point of
view as previously done by Chen & Kontomaris &
McLaughlin (1998) in a turbulent particle laden flow,
for instance. However, the smallest time scale encoun-
tered in this kind of study is the turbulent time scale
which is, for particles large enough, of the same order of
the particle relaxation time (denoted Tp). This imposes
therefore that the time step of their numerical simulation
is reasonable in the sense that At < Tp. In the present
study, we want to simulate the Brownian movement
of particles. The constraint At < 7p considerably
increases the total computational cost since the colloid
relaxation time is smaller than the particle relaxation
time considered in the above mentioned turbulent
studies. More precisely, our objective is to establish an
algorithm of collision capable of treating both Brownian
collisions and collisions of large-inertia particles with a
reasonable time step, thus decreasing the computation
total cost. The case of large-inertia particles is not
in itself a problem since the relaxation time scale is
naturally large with respect to the time step. What we
need is a model for the Brownian case which bypasses
this limitation of the time scale. The recent work
of (Peirano et al. (2006)) focusing on this problem
proposed numerical modelling and associated schemes
allowing the simulation of a Brownian trajectory in a
discrete sense without respecting the severe condition
At < Tp.


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


The present study will thus focus on the multi-scale
nature of these collisions which will be treated by a
single deterministic algorithm without any constraint
or threshold on the time step. We propose here to
study different test cases corresponding to two limits
of particle behaviour. The first is related to the motion
of small particles which move rapidly according to
Brownian motion and where trajectories are mainly
random diffusivee limit). The second limit case deals
with very large inertia particles. These particles move
in straight lines with constant velocities between colli-
sions. In order to validate these physical configurations,
we choose to check the efficiency of the algorithm
of collisions by calculating collision rates for which
the theoretical expressions are available in the previ-
ous works (Smoluchowski (1917); Friedlander (2000)).

This paper presents our work as follows. First
we shall describe the details of the present collision
algorithm. Then, the testing of the algorithm for the two
limits cases is discussed and the validation is carried out
comparing the computed results of collision rates and
those issued from theoretical expressions. An analysis
and discussion of the obtained results are given in
conclusion.



1 Particle equation movement and numerical
integration scheme.

We shall approach the Brownian movement of particles
from the Langevin point of view. For simplicity's sake,
we will consider motion in one dimension. Newton's
equation of motion for the particle (radius r, mass mp,
position xp(t), velocity vp(t), particle relaxation time
Tp, and fluid medium viscosity /) is:

( dxp(t)

d(t) F(t)
S(dt1)

where the force, F(t), due to the interaction of the Brow-
nian particle with the surrounding medium, thus con-
tains both frictional force proportional to the velocity
of the Brownian particle and random force. Therefore,
Langevin's model reads:


dxp (t)W
dvp(tW


p (t)dt
vP(t)dt + BdW(t),
Tp


B is the diffusion coefficient and dW(t) is the Wiener
process whose increments over small time steps are
stationary and independent, < dW(t) > 0, and <
dWj(t)dWj(t') = 0 > with t / t'. This system be-
ing physically valid when dt < Tp.







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


If we neglected the random force in Eq.2, the latter equa-
tion would become:


Sdxp(t)
Sdvp(t)
dt


Vp(t)dt
vP(


which has the familiar solution

xp(t) xp(O) + o vp(s)ds
Vp () = e'vp (0)


(3)


given by


< 1 >-


< IV >
< Ixl >-


(BTp)2 {At


Att]
OTp]


-A;]}t
C 1P ]


{BTp,[1 c- e


(10)
Therefore, relations for position and velocity of a
(4) Brownian particle given by the system (7) associated
with relations (9) and (10) are:


predicting that the velocity of Brownian particle decay to
zero at long times which cannot in fact be true. By con-
sidering in equilibrium where KB is Boltzmann constant
and T is the temperature of fluid where the particles are
immersed, the application of equipartition theorem stip-
ulates that


KBT


< -/) >eq


while the expression of the velocity from the system (4)
gives
2t
< I) >eq= e (6)
< v (0) >eq 0.

Therefore, the random force is necessary to obtain a
finite value of the velocity variance corresponding to
the correct equilibrium. The stochastic calculus gives
the correct solution. The recent work of Peirano et al.
(2006) helped us to apply the rules of stochastic calculus
in Ito' sense. The solution to system (2) at t to + At
is therefore given by :


{ xp(t)
vp(t)


At
x~p(to)(o)+ )Tp(1 -e 1 )+ I(t),
o At + ().
vp(to)c P 14(t).


where the stochastic integrals Ix (t) and I, (t) are defined


I(t)
1- W


BTp tt dW(s) BTpe ftt e dW(s),
Be f e dW(s).


(8)
In order to facilitate numerical applications, Peirano et
al. (2006) applied the Choleski algorithm to (I I,) as:


I1(t)
Ift(t)


( / <2 < W<2 >
<12 > C
J~p,,


(9)
where (C and &C are two independent standard Gaus-
sian random variables (zero mean and variance equals
to unity). The components of the covariance matrix are


At
axp(t) = p(to) + p(to)Tp(l1 c )
SAt 112
A+BTt [At 2 (1 j
a(t1e e )
] B r>(1 ) [27 (1 2 n 1)] ,

v(t) -vp(to)e 1 +[ (-e 2 A .
(11)
This system of equations can be used directly in discrete
time which amounts to treating the model and its numer-
ical scheme without distinction. In addition, to be con-
sistent with analytical solutions of the system (7), the
numerical schemes established by Peirano et al. (2006)
can provide the correct system limit of EDS. They are
linked with the physical behaviour of particles although
the separation of scale, 0 < dt < Tp, imposed by the
system (7) was not respected. In other words, these
numerical schemes allow to perform simulations with
a time step At such as At > Tp but also in the case
At < Tp as shown by Fig. displaying the particle vari-
ance position in the case where At < T7 and in Fig.2
when At > Tp. Taking the following value of the dif-
fusion coefficient (12) obtained from the equipartition
theorem for translational kinetic energy, i.e.


B KBT (12)
V rnpTp

the variance of the particle position as well as the vari-
ance of the velocity can be obtained. As can be seen
in Figs. 1 and 2, the calculated values and those given
by the exact solution correspond well. Moreover, the
physics of a Brownian particle is respected since the val-
ues of particle variance positions reach a state of equi-
librium (Fig.2). We also observed that the mean parti-
cle position evolves around zero (results which are not
shown here). Therefore, the above results are consis-
tent with Einstein's observation, i.e. for long diffusion
times, one has < x(t) >~ (BT)2t. Concerning the re-
sults on the particle velocity, same remark can be made
regarding to the consistency of the computed results and
those which came from exact solutions (see Figs.3 and
4) whatever the value of At with respect to Tp.


i








7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


times) times)


Figure 1: Brownian particle position variance. N =
10000 particles, case where At < Tp, At 1 x 10 0 s,
t,,m = 1 x 105s, Tp 3 10 6.




6E-08

5E-08 computed
5E-08 exact solution
thermal equilibrium

4E-08

A
V%'3E-08
V

2E-08


1E-08


0 200 400 600 800 1001
times)



Figure 2: Brownian particle position variance. N
10000 particles, case where At > Tp, At 1 x 10 2s,
t,,m = 1000s, Tp 3 x 106s.



To sum up, the use of relations 2 and the associated
numerical model allows us to study the diffusive limit
case corresponding to Brownian motion whatever the
value of At with respect to Tp. This kind of model
includes the drag force as well as the Brownian force
which both acting at the same level on colloids. For the


Figure 3: Brownian particle velocity variance. N
10000 particles, case where At < Tp, At 1 x 10 ls,
tm = 1 x 10 5s, Tp 3 x 10 6.










8E-05 6


6E-06
A
V
v
4E-06



2E-06



0


-- computed
exact solution
thermal equilibrium






200 400 600 800
times)


Figure 4: Brownian particle velocity variance. N
10000 particles, case where At > Tp, At 1 x 102s,
t,,m = 1000s, Tp 3 x 106.





smallest dimension in the range of diameter of colloids,
the particle inertia effect is negligible with respect to the
Brownian one. Therefore, stochastic integrals (9) pre-
vail. The system of equations (11) is thus approximated


6E-16


5E-16


4E-16

A
%1-3E-16
v

2E-16


1E-16











by neglecting the drag force:


xp (t) = xp(to + BTp At 2 (1
(1 L (1 e
+BT(1 e )2 [2Tp(1 -e

Vp(t)= Y- [1e ]


At) 1/2
1P )/

-1/2,


(13)
In the other limit case corresponding to the large-inertia
particle, the stochastic effect disappears and the parti-
cle displacement is in a straight line during a time step.
Under this assumption, the system (11) yields:
A,
{ Xp(t) = X(to) + Vp(to)Tp(l (14)
t (14)
VP (t) = Vp(to)e .

Thanks to the system of equations (11), the diffusive
behaviour of particles can be simulated whatever may
be values of the particle relaxation time with respect to
the time step. In the Brownian case (the most restrictive
because it corresponds to the smaller value of Tp) this ad-
vantage can help us our aim was to devise an algorithm
that can simulate collisions of Brownian motion without
respecting the constraint (At < Tp). For large-inertia
particle corresponding to the ballistic limit, this condi-
tion is by no means a restriction (the extreme case being
that Tp -- oc). Between these two limit cases, which
may reflect a complex particle laden flow problem, the
upper limit of the time step of the simulation will be im-
posed by the physical phenomena involved (turbulence,
for instance). Based on the idea developed by Peirano


Table 1: Considered test cases.
Test Type of particle motion Time step condition
case 1 Ballistic limit At < 7T
case 2 Diffusive limit At < 7p
case 3 Diffusive limit At > Tp


et al. (2006),our aim is to construct a collision algo-
rithm including both the effects of Brownian simulation
and effects associated with large inertia of the particles.
Some test cases (described in Table. 1) will be examined
in a later section after a presentation of the details of the
collision algorithm.

Algorithmic details

A brief review.

A large number of collision algorithms have already
proved their consistency in one physical case or an-
other, corresponding specifically to diffusive behaviours


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


or ballistic behaviours of particles. The large multi-scale
character (from diffusive limit to ballistic one) seems
not to be taken into account in existing deterministic
algorithms of collision. The only way to do this is to
adapt the time step simulation to keep it much smaller
than the particle relaxation time in order to obtain parti-
cle displacement in a straight line. Therefore, whatever
the diameter of particles and forces acting upon them,
the free-molecular regime up to the ballistic one can be
simulated. However, the major drawback lies in the im-
portant computational cost. This can be reduced how-
ever if a probabilistic collision algorithm is used. For
instance, Oesterle & Petitjean (1993) proposed a La-
grangian simulation technique of a non-dilute gas-solid
suspension flow where a probabilistic method is used to
simulate particle-to-particle collisions during the trajec-
tory calculation of a particle. The principle consists of
introducing artificial inter-particle collisions during the
trajectory calculation, the probability of these collisions
depending on the local concentration and velocity of the
solid phase. The main advantage of this method lies in
the fact that dense or moderately dense flows can be sim-
ulated by computing a reasonable number of simultane-
ous trajectories. This simulation technique is not appro-
priate for our study since our main goal is to analyse
the physical phenomenon of the agglomeration process
and even to develop new agglomeration kernels. Thus
a collision algorithm of deterministic nature is essential
to achieve our goal. As stated earlier, this type of al-
gorithms is often time-consuming because the time step
is usually very short compared to the relaxation time of
the particle. Moreover, these algorithms are often based
on the overlapping detection. It consists of regarding
each pair of particles at each time step in the compu-
tation domain and testing if the distance between two
particles is smaller than the sum of their radii. An exam-
ple of schematic of overlapping algorithm is presented
in Fig.5. An expanded study on this simple kind of al-
gorithm (Wunsch & Fede & Simonin (2008)) stipulates
that to obtain good results on this particular case, severe
conditions have to be respected, the most important be-
ing based on the ratio between 6- avoiding thus to miss
overlapping. The relation is:


61 3 At
-dp= 2 Tp
-l fT~
1 1p


0.13,


where 61 is the mean free path of particles during one
time step (in m) and Tp 1/3(v1 + vp2 + v3) is
the particle agitation. This is a severe limit for the
maximum time step allowed when dp is very small, as
in the case of a colloid particle. Unfortunately, there is
no generalization on the time step which has to be used
for a given physical factor. Algorithms appearing in the
literature are elaborated for a given physical case. For







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Figure 5: A simple collision algorithm based on over-
lapping detection.


example, when studying turbulent flow in a dispersed
phase with larger particles whose relaxation times are
greater than unity, we can give more flexibility to the
value of time step, thereby reducing the total numerical
time cost. Another way to reduce this computational
cost while keeping At < 7, is to use the kind of
geometrical tests proposed by Lavieville & Deutsch &
Simonin (1995). They carried out geometrical tests
on the mesh in order to limit the number of possible
neighbours of a given particle, restricting therefore the
number of numeric tests from N2 to N log(N).

The present collision algorithm.

An overview of the elaborated algorithm is given
in Fig. 6. The key issue in a collision algorithm
considering spherical solid particles is to focus on the
behaviour of each couple of particles in the flow. This
is the most inefficient process since no collisions are
missed but it remains the simplest way of making sure
all collisions are covered. We are aware that better
numerical costs could be obtained by focusing on the
definition of the vicinity and by using a new criterion
which define possible neighbours of a given particle.
However, this is not our main goal as we seek to reduce
computational costs by working on numerical schemes
associated with the physics of colloids rather than on a
technical definition of the vicinity of a given particle.

When considering perfect elastic rebounds, two dif-


Figure 6: Limit case collision algorithm based on
straight line displacement.


ferent ways can be used in order to study the collision
detection for a given pair of particles. The first is to de-
tect and to treat an overlap as shown in Fig. 5. The sec-
ond one lies in detection of a collision during a time step
which is based on criteria derived from the molecular
dynamic theory. These criteria are explained below. The
first criterion defines the space domain covered during
one time step by the relative movement of both particles.
This domain is given by a cylinder defined as (Eq. 16)
and presented in Fig. 7 which is defined for a particle
couple (i, j).

base = 7(ri + r3)2


length

direction


11 ,i, v 1 '


:-(Vi Vi


volume = 7(r' + r3- || ;.|| t.. (16)

One necessary condition for detecting a collision at a
given time between a pair of particles (i, j) is that an-
other particle (denoted j) is located in the cylinder de-
fined above (Eq. 16). This condition is not sufficient
because it does not suggest that particle trajectories are
crossed. According to that, the possible collision can
be given using the second criterion defined by relation
(Eq. 17)
wX3 V3 < 0, (17)
which consists of observing whether the two particles











yo x2


zorx3


II V" At


Figure 7: Collision algorithm. First detection collision
criterion.


are moving towards each other (condition is true) or if
they move away from each other (condition is false).


Figure 8: Collision algorithm. Second detection colli-
sion criterion.


Once these two stages are successfully passed we
need to calculate the particle collision time step (At')
which defines the necessary time for the particle denoted
i to enter in collision with another (denoted j here). The
calculus of such a time step is well-known and a good
explanation is available in the work of Sigurgeirsson et
al. (2001). Consequently, we have to solve (Eq. 18)
defined as:

||xJ(t + At,) x(t + At,)| =r' + r, (18)


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


which is equivalent to solve (19) :

(v' (t))2At +2(v (t) x (t))Atc+(x' (t))2 (r'+r')2.
(19)

Atc will be deduced by taking the minimum of positive
solution of 19. Finally, the end of the process will take
the minimum of Atc among those calculated for each
particle i. This is a key point in our algorithm and
could be seen as a highly strict. However, this path is
the best way to approach processes of agglomeration
that may occur at different scales (from electrochemical
forces for the smallest scales up to hydrodynamic
forces for the largest ones). The choice of a minimum
collision time step allows us to bring the numerical
representation of the flow to a state approaching nearer
to a collision event from any state of a particle laden
flow. The agglomeration process can thus be treated
with the utmost precision contrary to the most common
treatment based on a probability of agglomeration.



Testing the algorithm : ballistic regime

This kind of behaviour corresponds to a case when par-
ticles move in straight lines with constant velocities dur-
ing one time step. This is typically the case of kinetic
theory of gases in which the only way to change particle
direction is the collision with another.
Kinetic gas theory considers what is happened in a per-
fect gas, focusing on how molecules hit each other to
define the pressure in the gas. This is dependent on
the number of collisions encountered, and thus depen-
dent on molecule concentration. The objective of this
section is to test the collision algorithm in the case of
gases by calculating the well-known collision frequency
of molecules according to the kinetic theory of gases
given by the following relation (Reif (1965))


N 2 16KBTN
Kth = d
L w -rmp 2


The idea is to compare this kind of theoretical frequency
with the measured frequency which is simply deducted
from the the number of collisions Nc per unit volume
per time during the time simulation, i.e.


2NcL3
N(N 1)t


where L is the dimension of the numerical domain.

Simulation overview.











The numerical domain is a cubic control volume with
periodic conditions in each direction. The length of the
cubic volume is taken by 10nm. Particles i,-'"'I are
uniformly distributed in the numerical domain with an
initial distribution of molecules velocities obeying to a
Maxwellian distribution in a directional sense (Fig. 9) :


F(vpm) rn xp
2KBT


2KBT


written in a normalized sense (Fig. 9), it yields :


F(iivp) I 47r (2MPT) 3 Vpl 2exp
2K T


0.0008



0.0006



S0.0004

0- 2
0.0002


2
2KBT )
(23)


Figure 9: Initial distribution of molecules velocity. 250
helium molecules.

Since particles have a constant velocity and are dis-
placed in a straight line, particle positions are advanced
in time following this numerical scheme derived from
relations (14) using mathematical limited development
at A around zero.
5,

xp(t) = x(to) + V(to)At. (24)

The physical conditions used herein correspond to
ideal gases under a 4 bar pressure which allows us to
obtain a sufficient concentration of molecules of gases
and therefore to expect more particles collisions in the
numerical domain. Theses gases are listed in Table.2
where details of their molar mass are given along with
details of their Van Der Waals's diameter, including


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010



Table 2: Tested ideal gases and used properties.
Gas Molar mass (g) VdW radius (pm)
Helium 4 140
Neon 20 154
Argon 40 188
Krypton 83.3 202
Xenon 131.3 216


totally the spatial geometry of molecules according
to the hard sphere modelling of molecules. This was
thus a perfect case to validate the collision detection
process. The time step used is At 1014s and the
total time of simulation is taken equal to tm,,a 10 s,
corresponding to a number of encountered collision
between 500000 and 1500000 depending on the ideal
gas considered.

Results.

Results are presented for some of the gases listed
(Table. 2) and are displayed in Fig.10 as a function
of the current time. The ratio between the theoretical
collision frequency and the computed one from sim-
ulation evolves around unity whatever the gas tested
attesting that the collision process is validated. In order
to complement this validation, we have carried out
simulations by varying simulation's parameters such as
the time step At. Results are reported in Fig.11 where
only the Helium gas is tested. Good results were also
obtained.

We can thus conclude that the collision detection is
validated since no dependency on particle diameters or
time steps is observed. The next step of validating col-
lision algorithm is to test the detection collision process
in the other limit case, i.e. for a diffusive movement of
particles, typically the Brownian case.


Testing the algorithm: diffusive regime


The two limit diffusive cases corresponding to Cases 2
and 3 are used here in order to calculate the Brownian
kernel with the present collision algorithm. For simplic-
ity's sake and to keep an assumption of monodispersed
flow, perfect elastic collisions are considered. The goal
is to compute Brownian kernels issued from numeri-
cal simulation (21) and to compare them with those ob-
tained from literature (Smoluchowski (1917)):


s 8KBT
3p














14 1 1 1

1 2'
E




'A
.A
08


C 06 -
06


Helium
2 A krypton
02- Neon
Xenon

a 5E-08 1E-C
time (s)


Figure 10: Dimensionless collision kernel in kinetic gas
theory 250 molecules.


S4

1 2



.2
?08


o06
.2
S04 dt=105 s
S
E dt=103 s
02 -- dt10 14s
02


0 5E-09 1E-0
time (s)


Figure 11: Dimensionless collision kernel in case of
Helium: time independency. 250 molecules.



Recently, Trzeciak et al. (2005) raises the issue of treat-
ment of elastic collisions when simulates a random mo-
tion of particles. Indeed, after the collision, particles can
remain close to each other and overestimating the re-
collision delay depending on the pair distribution func-
tion. If particles stay in each other's vicinity after the
collision, a completely different pair distribution func-
tion can be obtained, hence different coagulation rate.


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


The simple way to overcome this problem, in our case,
is to treat collision events by disappearing one of the
collision partners and to redistribute it in the numerical
domain thanks to the initial distribution of particle posi-
tions (uniform distribution). Indeed, the pair distribution
function is not affected since Brownian motion admits a
zero mean value for the particle position. Details of sim-
ulation parameters are available in Table.3.
Whatever the cases (2 and 3), the system of equations


Table 3: Summary of gas and particle properties used in
computations.
Variable Value Units

KB 1.3806503. 10 23 m3 kg K- s-2
7 T 296.15 K
p 1.83245 10-5 Pa s
pp 1000.00 kg m3
pf 1.00 kg m-3
tmax 100.00 s


(11 is used in order to obtain positions and velocities. In
Case 2, we integrate the system (11) in time on a time
step which is considered very small compared with the
physical phenomenon response. Therefore, the particle
displacement could be seen as a straight line during a
time step. Such an assumption is not valid in Case 3
where At > Tp. Nonetheless, according to Figs.2 and
4, the statistical Brownian motion is well reproduced.
The results of Case 2 and Case 3 are reported respec-
tively in Fig.12 and in Fig.13 versus the current time.
Figures 12 and 13 show the ratio between the calculated
Brownian Kernel and the theoretical kernel as a func-
tion of the current time. 50 particles are used to pro-
ceed to the computation of the ratio in Case 1 whereas
in Case 2 1000 particles are considered. Therefore these
results are preliminary. These results required relatively
8 short computation time (about a week) and nonetheless
show whether the present collision algorithm is capa-
ble of calculating the frequency of Brownian collisions.
Two types of results are given. The first type of results,
as shown in Fig. 12, are carried out when At < Tp
whereas in Fig. 13, the calculations were made for the
case where At > Tp. These two opposite cases are
not incidental to our approach. It should be noted of
course that our ultimate goal is to follow the formation
of a cluster in time and for a reasonable cost of compu-
tation time. For a lower cost, it can be simulated with a
time step such that At > Tp while respecting the physi-
cal phenomena, i.e. the diffusive behaviour of Brownian
particles. This is what we attempted to achieve by pre-
senting a case where At < 7p where we are sure that
physics is respected. The results (Fig. 12) show that our















1 8
16
14
" 12
E

E 08
m 06
04
02
0


time (s)


Figure 12: Brownian kernel, case where At < Tp.
N = 50 particles in a control cubic volume (edge length:
0.5pim ). At 3 x 10 s, 7p 3.03 x 10 6s for
dp = 10 and 7- 7.58 x 107s ford = 10 m.


E


Figure 13: Brownian kernel, case where At > Tp. N
1000 particles in a control cubic volume (edge length:
1mm ), At 102s, Tp 3.03 x 106s for dp
10 ',, and Tp 7.58 x 107s for dp 10 m.



algorithm gives good results, since the ratio of kernels
evolves around the unit. Results seem not to depend on
particle diameter. However, results should improve by
taking more than 50 particles and currently we are work-


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


ing along these lines. The results presented for the other
case (Fig. 13) defined as At > Tp are inconclusive. The
ratio can reach values 30 times greater than a unit and
depends on the value of particle diameter.
As exposed above, the used model given by relation (11)
can predict the position and the velocity of a particle
whatever the value of the time step with respect to Tp.
Nevertheless, when the time step is chosen much higher
than the particle relaxation time (At > Tp), the given
results must be interpreted with care since during a time
step the particle displacement is not a straight line. The
probabilistic nature of Brownian motion has to be taken
into account. A simple illustration of this problem is de-
scribed in Fig. 14. Moreover, for a given pair of particle
(i, j), exposed criteria do not necessarily lead to a col-
lision detection in the case where At > Tp (Fig. 15(b))
rather than in the case where At < Tp (Fig. 15(a)) for
which straight line approximation is acceptable inducing
thus a collision.




-----Q

(xp(l), Vp()) (xp(t + Al), vp( + At))


r \ t / I

\ I I I

(XW(t). vpt)) ( (X,(t + At), V,(t + At))


Figure 14: Opposition between particle real displace-
ment and numerical displacement in the two studied
cases for Brownian motion.

At first sight, our algorithm and more precisely the
conditions of collisions (Fig. 7 and Fig. 8) highlight the
problem, i.e. the detection of collision is elaborated
only for a straight displacement during one time step.
The collision cylinder (Eq. 16) must be redefined to
enlarge the vicinity of the pair of considered particles
taking their random trajectories into account. Moreover,
the calculus of Atc (Eq. 19) is also distorted for the
same reasons. These two steps have to be re-examined
from another perspective. The idea is to include the ran-
dom nature of displacement between t and t + At in the
collision algorithm. The calculus of the collision time
step for a rectilinear motion applied to Brownian ran-
dom walk, without any particular adaptation, conducts
to an excessive number of collisions in mismatch with
the simulated physical phenomenon.











S |At < Tp (a)

-4-

(*'(t + At)
S< AtI (b)
,-, / -'

S I '
(t) r -

t- .,(t + At)


Figure 15: Possible relative displacement of a pair of
particles.


Further study could be devoted to a special aspect of
Brownian theory, i.e. the Brownian bridge which could
be seen as a valid possible solution for this problem.
This probabilistic tool offers the possibility to restitute
a possible random trajectory between t and t + At in ac-
cordance with statistical properties of Brownian motion.

Conclusion

In a context of a study devoted to the particle agglom-
eration process, it is necessary to first establish a colli-
sion algorithm which could be applied to a large range of
physical phenomena such as Brownian motion of parti-
cles as well as the motion of very large-inertia particles.
This kind of multi-scale approach to treating collisions
is present in an agglomeration process since the particle
diameters can grow in time due to the agglomeration.
It seems therefore imperative to include this multi-scale
dimension directly in the collision algorithm. This algo-
rithm must be able to handle both collisions at very small
particle relaxation scales (like Brownian collisions) and
collisions between very large-inertia particles. The main
goal of the present work was therefore to present prelim-
inary steps in constructing of such a collision algorithm
of particles while respecting the following constraint of
a reasonable numerical time step in order to not to in-
crease too much the total computational cost. To vali-
date this approach, some tests were carried out for the
two limit physical cases. The first corresponds to the
motion of large-inertia particles which have a straight
line displacement during one time step and the second
is devoted to Brownian particles where trajectories are
mainly random between t and t + At. The efficiency
of the algorithm was tested on the calculation of colli-
sion kernels by comparing results issued from numeri-
cal computations to the well-known theoretical relations


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


which are available in the literature.
The problem seems to be easier when a time step is cho-
sen to be smaller compared to the particle relaxation
time. In such a case, whatever Brownian motion or the
movement of heavy particles, the time step chosen was
so small that we can consider that particles move in a
straight line. However, this kind of simulation is ex-
tremely expensive for the Brownian case. It is not dif-
ficult to reduce the total computational cost in the case
of a hard sphere model in the kinetic theory of gases,
the particle relaxation time Tp being considered as infi-
nite. Choosing a reasonable time step is therefore easier
in this limit physical case. The collision algorithm was
tested for this case and a good level of compliance with
theoretical results was obtained. The collision kernel ra-
tio evolves around unity whatever the diameter of tested
particles and also whatever the chosen time step. This
case is not really problematic in contrary to Brownian
motion since the latter varies extremely rapidly between
two time steps and the well-know statistics, as the vari-
ance of the position of particles at long times (the result
of Einstein), could be not reached if it is observed at
At > Tp. However, recently Peirano et al. (2006) pro-
vided an opportunity to solve Brownian motion even if
At > Tp adapting the analytical solution of Langevin
equation in a discrete sense. We used their numerical
schemes and good results on the physics of Brownian
motion were obtained whatever the value of the time step
with respect to the particle relaxation time. Nonethe-
less, when the Brownian collision kernels are calculated
in case of At < Tp, the ratio evolves around unity con-
trary to the case for which At > Tp. For this case, we
think that the Brownian character has to be directly in-
cluded in the collision algorithm. Nevertheless, the dif-
ficulty is that between two time steps, the trajectories of
a pair of Brownian particles vary rapidly. The more the
time step is large, the more the difficulty of predicting
the displacement of the particle pair in a cylindrical do-
main, as it has been defined, seems arduous. However,
the literature on the Brownian bridge theory could be a
mean to get other results. That is the next step of the
present study.

Acknowledgements

We thank Pascal Fede and Tomasz Trzeciak for their nu-
merous useful remarks and discussions about this work.

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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


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