Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Lagrangian modelling of agglomerate structures in a homogeneous isotropic turbulence
Sebastian Stubing and Martin Sommerfeld
Lehrstuhl Mechanische Verfahrenstechnik, MartinLutherUniversitat HalleWittenberg, 06099 Halle (Saale), Germany
sebastian.stuebing@iw.unihalle.de
Keywords: homogeneous isotropic turbulence, Lagrangian simulations, agglomeration, morphology, porosity, spray
drying.
Abstract
Spraydrying is a widely used technique in different industries like food and flavour or pharmaceutical industry to produce
powder materials. The construction and operation of spray dryers is mainly done by try and error, hence, a lot of experience
and testing time is required for finding proper operational conditions. Computational Fluid Dynamics (CFD) can help to
reduce this effort by simulating the entire process. Here, a numerical method in the frame of the EulerLagrangeapproach has
been used for the simulation of particle motion and agglomeration in turbulent flows. In order to model the structure of
agglomerates all position vectors of the primary particles that form the agglomerate are stored. By calculating the volume of
the convex hull of any agglomerate and the real volume of the particles, the porosity of the agglomerates can be estimated
(Stuibing and Sommerfeld 2007). Furthermore, the fractal dimension is determined by the box counting method. Other
structure parameters like the freesurface area and the radius of gyration are provided by the model as well.
In this work the new structure model was tested in a box with periodic boundary conditions that encloses a homogeneous
isotropic turbulence field. The evaporation of solvent was neglected and the dryingstate of the particles was fixed by
specifying different viscosities. The results of these test case calculations show that the model is capable to be used in
industrial spraydrying applications for calculating the product properties like porosity and freesurface area in advance. For a
detailed validation of the introduced model experimental studies have to be conducted and analysed.
Introduction
In the drying industry spray drying plays a major role. It is
applied in various branches like the food and flavour or the
pharmaceutical industry. For producing dry powders with
the spray drying process a liquid feed is converted into a
spray by suitable atomizers. The created droplets get in
contact with a hot air stream and the solvent evaporates. At
the end of the drying process the powder has to be separated
from the gas stream by means of an appropriate separator.
During spray drying the particles undergo different drying
states, from liquid droplets via viscous and high viscous
particles to a completely dry powder. Calculating the entire
process numerically involves diverse subprocesses. On the
one hand side the initial velocity and size distribution of the
injected droplets are crucial for the particle concentration
inside the spray dryer. The atomization may be simulated by
using an appropriate model. Alternatively, the measured
droplet size distributions downstream of the atomization
region may be used as an input to the numerical
calculations.
Another important aspect of spray dryer simulation is the
calculation of the fluid flow. Not only the particle motion
but also the evaporation of solvent is strongly influenced by
the flow field of hot gas. For the calculation of the fluid
flow numerous turbulence models are available, but the ke
and the ReynoldsStress models have been well established
in the simulation of spray drying applications (Langrish
2007). Fluid flow simulations can be used for the
optimisation of spray dryer operation. Stibing et al. 2007
analysed the fluid flow with respect to particle wall
depositions. By changing the air inlet conditions the
production of waste material that resulted from thermal
degradation of deposited particles was minimised.
The benefit of spray dried powders strongly depends on the
drying process. A very fast solvent evaporation may lead to
a crust formation at the particle surface that hinders further
particle drying. The diffusion time of residual moisture from
the particle core to the surface increases by crust formation.
That will lead to particles with a dry and hard shell but
liquid core. Reliable simulation of the evaporation process
help to optimise the dryer operation.
Final powder properties not only depend on the evaporation
process but also on the occurrence of agglomeration.
Agglomeration is introduced by particleparticle collisions
and depends on the particle drying state. While collisions of
liquid droplets lead to coalescence or the creation of
satellite droplets caused by splashing, highly viscous
particles build structured agglomerates. The penetration
depths determines the resulting agglomerate morphology
and strenght. Particles of very high viscosities slightly
penetrate into each other which will result in high
agglomerate porosities. Decreasing particle viscosity leads
to higher penetration depths and lower agglomerate
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
porosities.
In this paper the authors present a Lagrangian model to
mimic agglomeration in spray dryers. It is a further
development of a model for the simulation of collisions and
agglomeration between particles of different drying states
(Blei and Sommerfeld (2007)). By analysing the agglo
merate morphology different structure parameters are pro
vided by the model. The model was tested in a simple test
case and the results are presented here.
Nomenclature
acceleration (ms2)
area (m2)
dimension ()
diameter (m)
force (N)
turbulent kinetic energy (m2s2)
mass (kg)
number ()
radius (nm)
temperature (K)
volume (m3)
Greek letters
e energy dissipation (m2s3), porosity (), size (m)
g dynamic viscosity (kgm 's 1)
 sphericity ()
Subscripts
i index
G gyration
hull convex hull
p particle
s surface
sol solid
t total
r remaining
Numerical Scheme
The calculation of agglomerate structures requires informa
tion about the particle trajectories, whether primary particle
or agglomerate. Besides the initial conditions of particle
movement the trajectories are affected mainly by the sur
rounding fluid field. Therefore, the determination of the
fluid field is the basis of simulating particle trajectories.
In a spatially fixed coordinate system (Eulerian point of
view) the solution of the timeaveraged NavierStokes equa
tions determines the continuous fluid phase properties. On
this basis the particle movement is simulated in a
Lagrangian point of view, i.e. particle coordinate system.
Particle motion is affected by fluid dynamic and external
field forces. For the latter, the local fluid velocity at each
particle location along its trajectory has to be considered to
calculate the affecting force. In this paper the local fluid
velocity was not calculated but specified by the turbulent
kinetic energy k and the dissipation rate E (see section 'test
case'). Using an isotropic Langevin dispersion model
(Haworth and Pope (1986)) the local particle velocity is
reconstructed from k and e. By solving Newton's law of
motion,
dii
m~
cit
(eq.1)
all particle trajectories can be determined.
A requirement for the creation of agglomerates are
particleparticle collisions. For the simulation of inter
particle collisions the stochastic model by Sommerfeld
(2001) was used. To determine two colliding particle a
fictitious particle is generated for each real particle. All
particle properties of this fictitious particle are sampled out
of the local real particle properties in the considered cell.
Agglomerates are not considered in the PDF, hence only
collisions between two primary particles or between agglo
merates and primary particles may occur. By means of the
real and fictitious particle properties a collision probability
can be calculated. To decide whether or not a collision
occurs an equally distributed random number in the
interval [0,1] is generated. A collision between the real and
fictitious particles takes place, if the random number is less
than the calculated collision probability.
Depending on the particle character, colliding particles
penetrate differentially into each other. While dry particles
penetrate merely little, collisions between viscous or liquid
particles result in various penetration depth up to complete
coalescence. Agglomerates consisting of dry particles stick
together mainly by vanderWaals forces. Due to high
penetration depths, viscous particles build concrete and,
hence, stable agglomerates. A detailed description of
different particle states and consequent collision regimes is
given by Blei and Sommerfeld (2007).
According to Blei (2006) the penetrating particle is
decelerated by the Stokesian resistance force in radial
direction and by shearing forces in the tangential
directions. The particle movement of the penetrating
particle is considered in a spherical coordinate system, in
which the particle with the lower viscosity is considered to
be fixed. In the model there is no volume expansion of the
low viscous particle due to the penetration of the high
viscous particle. The density in the overlapping region is
higher due to neglecting the penetration volume.
In the following a closer explanation of the model
advancement regarding the structure characterisation is
given.
Modelling agglomerate structure
To describe the structure of agglomerates the location of
each primary particle inside the agglomerate has to be
known. Therefore, the particle position vectors related to a
reference particle are stored in a data tree. Using this data
structure enables not only to characterize the agglomerate
structure but also to consider agglomerate breakup.
Disaggregation of agglomerates caused for example by
wall collisions or strong shear, can be handled by
partitioning the data tree at the pertinent location where the
bonding forces are weaker than the forces acting onto the
agglomerate. The elements (particles) of the data tree are
bonded by pointers. In the case of agglomerate breakup the
pointer between the respective elements is erased and two
separate data trees agglomeratess) remain (Lipowsky and
Sommerfeld 2008).
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Paper No
Characterisation of agglomerate structure is carried out by
means of various parameters. For the description of
agglomerate porosity the convex hull is considered. The
surface of each primary particle is discretized by a certain
amount of equally distributed points. The convex hull is
calculated by means of an incremental algorithm
(O'Rourke 2001) out of the set of surface points.
Since the porosity is defined as the ratio of pore volume to
total volume, the total volume is given by the convex hull.
The pore volume is the difference between total Vt and
particle volume minus the volume of the overlapping
region V,. Hence, the agglomerate porosity is calculated by
using equation 2.
E = 
V.
(eq. 2)
On the basis of the convex hull not only the porosity but
also the agglomerate sphericity is determined. The
sphericity of a body is defined as the ratio of surface area
of a volume equivalent sphere to the actual surface area of
the considered body. Here, the actual surface area is the
surface area of the convex hull. In the calculation of the
volume of the volume equivalent sphere two different
volumes can be considered. On the one hand the volume of
the convex hull Vhull and on the other hand the solid
volume Vso, which is the volume of all primary particles
minus the volume of the overlapping region. In the first
case the convex hull is considered to be a rigid body, by
relating the surface area of the hull volume equivalent
sphere to the surface area of the hull. In the second case,
the inner structure of the agglomerate by means of the
solid volume that is considered in the calculation of the
surface area of the volume equivalent sphere, is connected
with the convex hull. Hence, the void fraction is included
into the calculation of the sphericity. By means of equation
3 the sphericity is determined to:
6V
sol
2 = hA
As.hull
1 =
(eq. 3)
Beside the calculation of agglomerate porosity the surface
discretisation by equally distributed points is also used for
the evaluation of the free surface area. The total amount of
surface points nt on each primary particle equates to the
total surface area. In regions of overlapping particles all
surface points that are located inside another particle have
to be deleted. The remaining number of surface points n,
enables to calculate the free surface area of each primary
particle (eq. 4).
A = d2
(eq.4)
To decide whether an agglomerate is more compact or
chain like in shape, the radius of gyration can be used. It
indicates the distribution of mass inside an agglomerate.
Therein, the distance between each particle r, and the
centre of gravity rG is calculated. By averaging over all
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
primary particles n, an equivalent diameter, i.e. the radius
of gyration RG, is calculated using equation 5.
(eq.5)
1 ~ 
n
Another alternative of describing agglomerate structure is
the usage of the fractal dimension. In this paper the fractal
dimension is determined by using the box counting
method. Thereby, the agglomerate is covered by a certain
number of boxes with decreasing side length little by little.
The number of not empty boxes, i.e. boxes that include any
part of the agglomerate, is counted. The procedure is
repeated for decreasing box sizes e. Finally, the number of
not empty boxes N(e) is plotted versus the reciprocal box
size 1/e in a double logarithmic diagram. The slope of the
resulting straight line gives the fractal dimension of the
considered agglomerate (eq. 6).
logN(e )
D logic
e log1/
Test Case
The model was tested in a homogeneous isotropic turbu
lence. In a cubic box of edge length 0.2 m 10,000 parcels
were tracked sequentially up to 1 s with a maximum number
of 1,000 time steps per particle trajectory by applying
periodic boundary conditions. The fluid inside the
calculation domain was air under atmospheric conditions
(f = 1.806105 kgm's1, T = 293.15 K).
Table 1: Turbulence characteristics and particle properties.
PROPERTY VALUE
Gas phase rms velocity 0.3 m/s
Turbulent kinetic energy k 0.135 m2/s2
Dissipation e 6.84 m2/s3
Longitudinal Eulerian length 7.25 10 m
scale
Integral Eulerian time scale 0.023 s
Particle density p, 1200 kg/m3
Particle diameter d, 100. 106 m
Volume fraction 0.0525 m3/m3
Particle relaxation time t, 0.037 s
Stokes number for primary 1.61
particles
(eq. 6)
Paper No
The particle movement is solely influenced by turbulence
through the drag force and interparticle collisions, because
all additional forces including gravity force are not
considered. Particle Stokes numbers of St = 1.61 indicate
that the particles are able to follow turbulent structures in a
short time.
Table 1 shows the characteristics of the homogeneous
isotropic turbulence field as it was used in the test case cal
culations (Sommerfeld 2001).
To simulate different particle drying states as they occur
during the spray drying process, various viscosities of the
calculated particles were applied. During each calculation
the viscosity was kept constant which enables a comparison
of the viscosity effect on agglomerate structure.
The particles were injected to the calculation domain
homogeneously distributed over the entire box volume. The
initial particle velocity was set to 90% of the local
instantaneous fluid velocity. The initial particle size distri
bution was monodisperse with a diameter of 100 uim. All
structure parameters were calculated at the end of the
calculation.
Results and Discussion
Homogeneous isotropic turbulence shows transient and
rotational invariance of the flow. Therefore, impacts of
primary particles at an agglomerate occur from any direc
tion. The estimated agglomerate shape is spherical and more
or less compact dependent on the viscosity of the primary
particles. Figure 1 shows an image of an agglomerate that
was created in the test case calculations, with a particle
viscosity of 10 Pas. The depicted agglomerate shows a
porous and spherical morphology with some dendritic
branches including 98 primary particles. The porosity of this
agglomerate calculated with the convex hull is quite high,
namely e = 0.727.
Figure 1: Agglomerate consisting of 98 primary particles
with a viscosity of 10 Pas created in the test case calcula
tions (df= 2.2, e = 0.727, W, = 0.923, '2 = 0.388,
RG = 0.23103m).
Due to the high viscosity, the velocity of interacting
particles is reduced fast, hence the penetration depth is only
small. Particles with a low dynamic viscosity are expected
to perform collisions resulting in high penetration depths.
Due to a low deceleration of colliding low viscous particles,
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
they are able to interpenetrate completely and separate
again. This separation of particles leads to a low number of
primary particles per agglomerate. Increasing viscosity
causes higher decelerations and, hence, lower penetration
depths.
20000
18000 A
S00 dyn. viscosity
16000
1 0.2 Pas
14000 o 0.5 Pas
S12000 A 1.0 Pas
000 v 5.0 Pas
S10000 10.0 Pas
" 8000 W7
6000 v
4000 
2000 .
20 30 40
rel penetration depth [%]
Figure 2: Probability density distribution of penetration
depth related to the diameter of penetrating particle depen
ding on the dynamic viscosity.
By varying the viscosity of colliding particles, the shape of
agglomerate structure is denser at lower viscosities and will
become more porous with increasing viscosity while the
penetration depth is decreasing. Figure 2 shows the distri
butions of penetration depths dependent on the particle
viscosity. Each PDF is obtained from one simulation and
was averaged over all collision events. Particles of high
viscosities are able to penetrate only slightly into each other,
while particles with lower viscosities show a broader
distribution of penetration depths. The penetration depth is
normalised to the diameter of the penetrated particle
resulting in 100% for complete coalescence and 0% for
point contact between the particles.
dyn. viscosity
500  0.2 Pas
0.5 Pas
A 1.0 Pas
400 v 5.0 Pas
S 10.0 Pas
LL 300
200
100
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
porosity []
Figure 3: Probability density distribution of agglomerate
porosity depending on the particle dynamic viscosity.
As the penetration depth influences the resulting
agglomerate porosity, agglomerates consisting of high vis
cous primary particles are expected to show high porosity
Paper No
values. Figure 3 depicts the distribution of calculated agglo
merate porosities at different dynamic viscosities of the pri
mary particles. The PDFs in figures 3 to 8 were averaged
over all agglomerates (10,000) at the end of the calculation.
The agglomerate porosity increases with increasing particle
viscosity, as expected.
Low viscous particles perform collisions that result in
deeper penetration depths compared to high viscous
particles (cf. Figure 2). Particles which are able to penetrate
deeply into each other build very compact agglomerates
resulting in both, low porosities and small free surface areas.
At high penetration depths the overlapping particle surface
area is high, thus, most of the particle surface area is located
inside another particle. Only a very little part of the
agglomerate surface area stays in contact with the surroun
ding fluid.
350
300
250
dyn. viscosity
S200 0.2 Pas
0.5 Pas
1.0 Pas
S150 5.0 Pas
 10.0 Pas
o 1
50 
0 1 2 3 4 5
free surface area A.su [106 m2]
Figure 4: Probability density distribution of agglomerate
free surface area depending on the particle dynamic visco
sity.
Figure 4 shows the free surface area depending on the
particle viscosity. The free surface area decreases while the
particle viscosity is decreasing. Porosity and free surface
area are important properties of powder products as they
provide information about the amount of fluid that may get
into contact with the agglomerate surface. High free surface
areas may lead to higher dissolution rates of spraydried
powder, for instance. That can be an important product
property of instant food like milk powder. The trend towards
lower free surface areas with decreasing viscosities is
correctly captured by the model.
To classify the shape of particles, the sphericity is often
used. Nearly spherical particles show sphericity values close
to one. More elongated particles will have lower values of
sphericity. Two sphericities may be identified for the
characterisation of agglomerate structure. On the one hand
side the sphericity is related to the particle volume and on
the other hand to the volume of the convex hull. Figure 5
shows the probability distribution of agglomerate sphericity
related to the convex hull. Therein, the volume equivalent
sphere is calculated from the volume of the convex hull.
In homogeneous isotropic turbulence, spherical shapes are
estimated due to a transient and rotational invariance of the
flow. Primary particles may collide with agglomerates
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
coming from any direction, as there is no preferred direction
in the flow field. As can be seen from Figure 5 the cal
culated agglomerate sphericity is very close to unity, indi
cating almost spherical shapes. Since all simulations have
been done in the same turbulence field, very similar shapes
of the convex hulls are expected, consequently no tendency
in the sphericity distribution is discernible.
300,
150
LL
100.
50.
.r .
Em%
0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96
sphericity referring to hull volume []
Figure 5: Probability density distribution of agglomerate
sphericity calculated with surface area of the volume equi
valent sphere of the convex hull volume depending on the
particle dynamic viscosity.
LL
o 300
200
100
0
dyn. viscosity
0.2 Pas
0.5 Pas
 1.0 Pas
v5.0 Pas
10.0 Pas
0.3 0.4 0.5 0.6 0.7
sphericity referring to particle volume []
Figure 6: Probability density distribution of agglomerate
sphericity calculated with surface area of the volume equi
valent sphere of the total primary particle volume depending
on the particle dynamic viscosity.
This method of calculation characterises merely the outer
shape of the convex hull. The convex hull describes the
agglomerate as a rigid body. In order to get information of
the inner structure of the agglomerate the volume of primary
particles is taken into account. Hereby the outer shape of the
convex hull is connected with the inner shape, i.e.
agglomerate structure. The sphericity is calculated with
equation 3. The surface area of the volume equivalent
sphere that has the solid volume (the volume of all primary
particles minus the volume of the overlapping regions) is
divided by the surface area of the convex hull. Under this
Paper No
assumption, the outer shape (convex hull) is connected to
the inner shape (agglomerate primary particles). Figure 6
shows the distribution function of the sphericity calculated
based on the total particle volume. Agglomerates that
consist of primary particles with low viscosities are more
compact and hence more spherical than those with higher
viscosities. Consequently, the spherecity is higher for more
compact agglomerates (Figure 6). As the total particle
volume is used to calculate the surface area of the volume
equivalent sphere, the sphericity measures the void fraction
inside the hull.
The radius of gyration is an important measure in charac
terising the spatial extension of nonregular shaped
particles. It considers the distribution of mass inside the
agglomerate. For very compact agglomerates, the mass
distribution is close to the agglomerate centre. Thus, the
radius of gyration becomes low. In direct comparison, as
shown in Figure 8, agglomerates consisting of particles with
a viscosity of 0.2 Pas hold lower radii of gyration than
agglomerates of higher viscosities.
600
500 
cP A dyn. viscosity
400 A 0.2 Pas
A o 0.5 Pas
300 A 1.0 Pas
300 5.0 Pas
So10.0 Pas
200 
0 q4, .. ,
0.5 1.0 1.5 20 2.5 30 3.5
radius of gyration Rg [104 m]
Figure 7: Probability density distribution of agglomerate
radius of gyration depending on the particle dynamic visco
sity.
Agglomerates that consist of less than a few hundreds of
primary particles are not fractals in the reasonable sense.
Hence, the fractal theory is not valid. That will lead to errors
in the determination of the fractal dimension of quasifractal
agglomerates. Nevertheless, the fractal dimension was
calculated to characterise agglomerate morphology in this
paper. Many authors (Brasil et al. 2001, Bushell et al. 2002,
Schmid et al. 2006, Shen et al. 2008) give some typical
values of the fractal dimensions of chainlike agglomerates
(Df = 1.7 to 1.9) and compact ones (Df = 2.1 to 2.75).
Figure 8 shows the fractal dimensions of agglomerates that
were generated in the test case calculations. The values of
the distribution maxima range between 2.2 for the viscosity
of 10 Pas and 2.4 for 0.2 Pas that indicate a compact and
spherical structure of the agglomerates. The shift between
the maxima from 2.2 to 2.4 with decreasing viscosity
implies that the agglomerate structure becomes more
compact with decreasing primary particle viscosity. This
trend was shown by all calculated structure parameters in
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
the introduced test case simulations.
400
350
%T
300 %, dyn. viscosity
~e 0.2 Pas
250 o 0.5 Pas
SA 1.0 Pas
LL 200 5.0 Pas
S 10.0 Pas
150
100 j,
50
1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
fractal dimension D, []
Figure 8: Probability density distribution of agglomerate
fractal dimension depending on the particle dynamic
viscosity.
Conclusions
In this paper a Lagrangian approach for modelling
agglomerate structures as they occur in spray dryers was
described and tested. For that purpose a homogeneous iso
tropic turbulence field in a box with periodic boundary con
ditions was used as a test case. While tracking the particles
through the calculation domain the viscosity was kept
constant. For a comparison of agglomerate structures
resulting from different particle drying states, several
simulations with viscosities from 0.2 to 10 Pas have been
performed.
For the characterisation of agglomerate structures, the
position vectors of all involved primary particles are stored
in a data tree structure. By means of calculating the convex
hull of each agglomerate the porosity and sphericity were
determined. The free surface area is calculated by a
discretisation of the surface of all primary particles inside
the agglomerate. Mass distribution in agglomerates is deter
mined by calculating the radius of gyration. The spatial
structure is characterized by the fractal dimension which is
calculated by the box counting method.
Viscosity of primary particles strongly influences the
resulting agglomerate structure. Collisions between particles
of low viscosities result in a variety of outcomes. Due to the
low viscosity particle penetration is very high and may also
lead to a separation of the particles. Agglomerates that
consist of low viscosity particles are compact in their shape.
An increasing viscosity results in more porous
agglomerates. Consequently, the free surface area declines
with decreasing particle viscosity.
In spray drying the product properties have to fulfil different
customer demands. The suggested structure parameters are
able to be used for the characterisation of agglomerate
morphologies and may help to adjust the drying process.
For a further validation of the Lagrangian model, experi
mental data of real spraydrying processes have to be
compared with the simulations. Particularly, the penetration
of one particle into the other has to be investigated to get
Paper No
reliable data comparable to model assumptions. In future
work, an industrial spray dryer will be simulated and com
pared to available experimental data of the powder proper
ties.
Acknowledgements
The financial support of German Research Foundation DFG
(grant So 20430/1) is gratefully acknowledged. The fluid
flow calculation could have been done by courtesy of Prof.
Schifer, TU Darmstadt, permitting to use the fluid solver
Fastest 4.0.
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