Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
EulerianLagrangian Methods for Modeling
of Gas Stirred Vessels with a Dynamic Free Surface
Jan Erik Olsen and Schalk Cloete
SINTEF Materials & Chemistry
7465 Trondheim, Norway
Jan.E.Olsen @tsintef.no
Keywords: bubble column, Lagrangian, modeling
Abstract
Modeling of as stirred vessels is the subject of this and a vast number of scientific papers. In this paper three
EulerianLagrangian models have been compared with experiments to investigate their differences and their potentials of
predicting the flow in gas stirred vessels. The models have been applied to gas rates yielding void fractions far above their
assumed validity. Still, the comparison with experiments is reasonable. The paper discusses the results and possibilities for
future enhancements of the modeling approach.
Nomenclature
Successful modeling of bubble plumes in bubble columns
and vessels is important for the understanding and
improvement of many applications and processes. Often a
Lagrangian approach to modeling of the gas bubbles has
been assumed to be computationally expensive.
Development in computing capabilities and numerical
algorithms has made modeling of bubble plumes with
Lagrangian tracking affordable and (sometimes) preferable.
The method has successfully been applied to study mixing
in ladles (Cloete, Eksteen & Bradshaw, 2009) and plume
behavior in the ocean due to breakage of gas pipelines
(Cloete, Olsen & Skjetne, 2009).
The authors have previously applied a coupled
DPM and VOF model in which the continuous phases are
described by the Eulerian VOF model and the bubbles are
tracked as Lagrangian particles with the discrete phase
model (DPM). The Eulerian phases and the Lagrangian
bubbles are coupled through the momentum equations only.
Since the modeling approach have been applied to scenarios
with high gas fractions, it has been questioned whether the
modeling assumptions are valid. The lacking coupling in the
continuity equations might be a limiting factor of the model
applicability. There might also be additional dispersion
between the bubbles at higher void fractions. Based on this,
a study of different EulerianLagrangian modeling
approaches has been conducted to verify the importance of
the coupling in the continuity equations and bubble
dispersion at high void fractions.
constant
constant
drag coefficient
diameter (m)
force (N)
gravitational constant (ms1)
pressure (Nm2)
Reynolds number
time (s)
velocity (m/s)
Greek letters
a volume fraction
e turbulent energy dissipation rate
g viscosity (Pas)
p density (kg/m3)
C surface tension (Nm)
T time constant (s)
Subsripts
b bubble
d drag
eq equilibrium
1 liquid
1 lift
m molecular
t turbulent
td turbulent dispersion
q phase index
vm virtual mass
Introduction
Paper No
Model Description
To study the hydrodynamics of a vessel, one need to account
for the behavior of the liquid in the ladle, the gas above the
liquid and the bubbles in the liquid. Sometimes the vessel
may contain secondary liquid phases or the top gas may be
ignored and a degassing boundary condition applied. To
study the mixing behavior of a vessel, it is not
recommended to apply a degassing boundary condition
(Olsen & Cloete, 2009) and we will therefore only consider
models with a continuous top gas. The continuous phases
are modeled as Eulerian phases, either with a VOF model or
a multifluid model. A Lagrangian method, DPM, is used to
track the bubbles. The Lagrangian bubbles are connected to
the Eulerian phases with a twoway coupling through
interchange terms such as the drag force in the respective
momentum equations.
Eulerian Phases
Three different approaches to modeling of the Eulerian
phases have been considered: a VOF model with no
coupling to the Lagrangian bubbles in the continuity
equation, a multifluid model with no coupling in the
continuity equation and a multifluid model with coupling in
the continuity equation. For short we will call these
approaches DPM/VOF model, DPM/multifluidmodel and
DDPM/multifluidmodel respectively, where DDPM refers
to dense discrete phase model.
In the DPM/VOFmodel, the continuous phases, i.e.
gas above the surface and liquid below the surface, is
mathematically described by the VOF model which is based
on the standard single fluid Eulerian mixture model where a
single set of governing equations is shared between different
phases. The distinction of the VOF model, however, is that
it places strong emphasis on accurate tracking of the
interfaces between various phases that might be present in
the domain. The VOF model solves for conservation of
mass
S(1)
a ukp )+ V (apq 0 1) 0
and momentum
(pi)+V(piF) =
ct (2)
Vp + V [ u(V + V T+ pg +
The momentum contribution from the gas bubbles is
provided as input to the external force F. The mixture
density is given by the phase densities
p = C ,qp (3)
The viscosity is the sum of turbulent viscosity and
molecular mixture viscosity
/t =/, +/m (4)
where the molecular mixture viscosity is given by the
properties of the phases equivalent to the mixture density.
Turbulence is modeled by the standard ke turbulence model
with default model constants (Launder & Spalding, 1972).
In addition to solving the continuity, momentum and
turbulence equations, the VOF model also performs
procedures for sharpening the interface between the phases.
The DPM/multifluidmodel applies the
EulerianEulerian multifluid model (Drew & Passman,
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
1999) for the continuous phases. The model solves the
continuity equation
t (op)+V (ao pq )= 0 (5)
for all secondary phases. The primary phase is calculated
from
S= 1 (6)
q
which enforces the sum of volume fractions to unity. Note
that the volume fraction of the Lagrangian bubbles which are
tracked with the DPM is neglected in Eqs. (5) and (6). Three
momentum equations for each phase are solved:
(qP, )+ v( ,pqqq)=
at (7)
aVp + V [ a, ( + Vu)]+ aqp + F
As for the DPM/VOFmodel, the momentum contribution
from the gas bubbles is provided as input to the external
force F. The turbulence modeling is equivalent to
DPM/VOFmodel.
The DDPM/multifluidmodel distinguishes itself
from the DPM/multifluidmodel by including the
contribution of the Lagrangian bubbles in Eqs. (5) and (6).
Because of this, the model is expected to perform better than
the other models at higher volume fractions of bubbles.
Lagrangian Phase
The discrete phase model is used to track the movement and
influence of the bubbles. The bubbles are tracked in groups
called parcels. Each parcel may contain a vast number of
individual bubbles. The bubbles in the same parcel are
treated as if they have the same diameter, density, velocity
and other properties. This makes the method affordable
compared to a more classical Lagrangian method which
tracks all bubbles. The total number of parcels in a
simulation may be limited to typically 50000100000, while
the total number of bubbles may easily exceed 1 million.
The bubbles (or parcels) are mathematically tracked
with Newtons second law
dub, (p ~p
ub b + + Fd b+F +F +Fd (8)
dt Pb
where the bubble acceleration is governed by such forces as
buoyancy, drag, turbulent dispersion, virtual mass and lift. In
this study we will neglect the lift forces since they are
insignificant (Olsen & Cloete, 2009). The more significant
forces are buoyancy, drag and turbulent dispersion. The drag
force is defined as
18u2 CDRe
Fd 2 24 (9)
Pbd 24
where CD and Re are the drag coefficient and the Reynolds
number respectively. It is important to use a drag coefficient
valid for bubble plumes and not for single bubbles only. The
applied drag formulation is the formulation of Kolev (2005)
which accounts for bubble size and bubble volume fraction.
Turbulent dispersion is a force due to the velocity
fluctuations which is not captured by the mean velocity field
and thus not accounted for in the drag force which is based
on the mean velocity. The model presented here uses a
stochastic formulation for the turbulent dispersion (Cloete,
Eksteen & Bradshaw, 2009). Note that the dispersion force is
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
sometimes neglected by authors and an assumed dispersion
is established by tuning of an artificial lift coefficient.
The DDPM/multifluidmodel failed to produce
reasonable results initially. The outcome of the simulations
was a plume tunneling straight up with excessively high void
fractions and no dispersion. Additionally, the
DDPM/multifluidmodel was very unstable, since the
Eulerian phase volume fraction was limited to 1, but a cell
could actually contain particles summing to a volume greater
than the cell volume. This meant that the volume of the
discrete phase could vanish into such a cell and emerge at
some later stage, leading to large pressure fluctuations. It
was therefore necessary to implement an additional
dispersive force which pushes the Lagrangian particles away
from each other if the void fraction exceeds 0.6. This was
done only to prevent the number of particles in a cell
exceeding the cell volume itself. From a theoretical
viewpoint it may be argued that a pressure will build up
between bubbles approaching each other once the bubbles
come close enough, causing a dispersive force at high void
fractions. No quantitative derivation of this force has been
made, however, and the implementation is simply an adhoc
procedure which moves particles against the discrete phase
volume fraction gradient when the cell volume fraction
exceeds 0.6. This additional dispersive force is applied to the
DDPM simulations only.
The bubble size governs the drag forces (and mass
transfer rates whenever that is considered), and is thus an
important quantity in the model. It is estimated from an
interfacial area transport equation which calculates a mean
bubble size in any grid cell based on the material properties
of the gas and liquid and the void fraction and turbulent
dissipation rate (Laux & Johansen, 1999). The local mean
bubble size db is modeled by a Lagrangian transport equation
a(pdb)
at
deq d
Pb
Here Trel is the relaxation time and db is the mean
equilibrium diameter. The equilibrium diameter is the
diameter a bubble will achieve if it resides sufficiently long
at the same flow conditions. The term on the right hand side
forces the mean bubble diameter towards its equilibrium
diameter db during a timeframe given by the relaxation
time. The relaxation time is given by the turbulence
dissipation rate and kinetic energy. The equilibrium diameter
is calculated as follows
S/\ 6
eq C 025
d = Cab 0.4 2 +C2
lbp
The coefficients C1 and C2 are tuning parameters.
C2 is often taken as the smallest possible bubble size, while
C1 is the more significant of the two parameters for which
most of the tuning procedure evolves around. For airwater
systems the following is often used: C1=4.0 and C2=200m.
More details of the bubble size model are given by Laux and
Johansen (1999). The model gives good predictions of
bubble size compared with experiments (Cloete, Olsen &
Skjetne, 2009).
6cOmm
Figure 1: Geometry of validation experiments
Implementation and Numerical Schemes
The models have been implemented in the commercial CFD
software Fluent 12.1 with user defined functions (UDFs) for
drag coefficient and bubble size. Higher order numerical
schemes are applied. The pressurevelocity coupling for the
DPM/VOFmodel is PISO, and for the other models the
phase coupled SIMPLE algorithm is used. The mesh is
based on a crude mesh which is refined in the plume and
surface region until no further changes are seen in the
results. In the simulations described below, a mesh with
126474 cells is used. The time step is chosen so that
convergence is achieved at all times.
Model Validation and Comparison
The models described above are compared against the
experiments of Engebretsen et al. (1997) which were
designed to study the bubble plumes from a subsea gas
release. A series of experiments were conducted in a
rectangular basin with a depth of 6.9 m and a surface area of
6 x 9 m as seen in Figure 1. The basin was filled with water
and air was released at the bottom at gas rates of 83, 170
and 750 NI/s (equivalent to 50, 100 and 450 1/s referred to
the state at the inlet). The inlet was comprised of a release
valve with a rapidly acting piston injecting gas vertically
with arrangements in front of it to reduce the vertical
momentum. Because of this momentum breaker, the
fluctuations in the gas flow and the length of the inlet jet
were minimized.
All models were run in full 3D at the medium (170
Nl/s) and maximum (750 Nl/s) gas rates. The resulting
bubble fields are seen in Figure 2 and 3. The bubbles are
colored by bubble diameter ranging from 0.5 mm to 10mm
with a typical average value around 5 mm which was also
observed in the experiments. The snapshots are taken 20 s
after the initial release and quasi steady state is obtained. We
see that that the highest gas rate yields the highest fountain
height and gas hold up.
Paper No
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
0 025 05 075 1 125 15
Radial Distance (m)
175 2
Figure 4: Velocity profiles from DPM/VOFmodel at
different heights for a gas rate of 170 NI/s.
Figure 2: Bubbles and free surface resulting from a gas
release of 170 NI/s.
2" Mode138m
2 A Mode1588m
15
, 1
05
0
0 025 05 075 1 125 15 175 2
Radial Distance (m)
Figure 5: Velocity profiles from DPM/Multifluidmodel
at different heights for a gas rate of 170 NI/s.
Figure 3: Bubbles and free surface resulting from a gas
release of 750 NI/s.
The resulting velocity profiles of the plumes after a
quasi steady state is reached for the gas rate of 170 NI/s are
seen in Figure 4 to Figure 6. They show that all models
compare well with the experimentally measured velocity
profiles. There is a slight discrepancy at the upper position.
This might be explained by some near surface modeling
issues, like the turbulence model which actually do not
account for the presence of a large scale interface. Note also
that the DPM/VOFmodel and DPM/Multifluidmodel yield
practically identical results and thus figures and plots for the
DPM/Multifluidmodel are omitted below. The results
show that the medium gas rate itself is higher than gas rates
used in most other modeling studies, and that the resulting
void fractions violate the model assumptions of volume
fractions below 15% for more than half the length of the
plume. This is seen in Figure 7.
1 2
S15
1U
> 1
0 025 05 075 1 1 25 15
Radial Distance (m)
1 75 2
Figure 6: Velocity profiles from DDPM/Multifluid
model at different heights for a gas rate of 170 NI/s.
Still the model performs well. This may be
explained by the interaction between the overpredicted
void fractions (Figure 8) and the negligence of the dispersed
phase volume fraction in the DPM/VOF and
DPM/Multifluidmodel approaches. Since the whole plume
is assumed to be 100% water, there is a larger mass to be
accelerated than would actually be the case. It would
therefore be expected that the model would underpredict
the velocities in this region. The overpredicted void
fractions, however, show that a larger amount of gas is now
Paper No
Paper No
present to accelerate this larger mass of water. These two
effects seem to cancel each other out almost perfectly.
It is shown that the volume fraction plot for the
DDPM/Multifluidmodel (Figure 9) is in somewhat better
agreement with the experimental data. This is due to the
extra particle dispersion force implemented with this model.
Since this extra spreading force was only implemented
when the voidage exceeded 60%, this improvement is only
slight, but still significant. It is interesting to note, however,
that the velocity at the lowest measuring point (Figure 6) is
now somewhat overpredicted. This can be explained by
noting that the volume fraction of bubbles at this point is
still overpredicted (Figure 9), but the actual volume of
bubbles is now taken into account in the mass balance.
Therefore, a larger mass of bubbles will accelerate a lower
mass of water, leading to larger velocities. It is encouraging
to see that the prediction is still adequate though.
Since the DPM formulation inherently treats the
dispersed phase as particles (bubbles in this case) it can
never be expected to accurately capture jetting behavior that
would be observed at the start of really strong plumes. The
experimental data does show, however, that such a jet will
not form easily. The bubbles seem to experience a strong
dispersive force with an increase in the volume fraction as
can be seen from the lowest measurement in Figure 9. When
no dispersive force is included, the volume fraction
prediction is more than double that of the experimental
measurements. The model prediction is very good at the
highest measuring point where the voidage is below 10%,
however. This finding suggests that an interparticle
dispersion force becomes highly significant at voidages
above 10%.
In general, the volume fractions shown in Figure 8
and Figure 9 show a large overprediction in the inlet region,
a medium overprediction in the middle region and a good
prediction towards the free surface. The particleparticle
interaction force discussed previously explains this trend. At
higher volume fractions, this force cannot be ignored any
longer and needs to be modeled. This is certainly a topic for
further research.
5.00e01
4.75e1l
4.50e01
4.25e01
40.e01
3.75eO1
3.50e01
325e01
3.00e01
2.75e01
2,50901
225e01
2.00Ol1
1 75e01
1. 50e01
1.00e01
7.50e02
2.50"o2
Figure 7: Void fractions (modelled by DDPM/
Multifluidmodel) for a gas rate of 170 NI/s.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
45
40 Experiment 1 75 m _
Experiment 3 8 m
35 A Experiment 5 88 m
_30 Model 175 m
Model3 8 m
g25 Model 5 88 m
20
vO 20
> 15
10 .
0 025 05 075 1 125 15 175 2
Radial distance (m)
Figure 8: Void profiles from DPM/VOFmodel at
different heights for a gas rate of 170 NI/s.
45 
Experiment 1 75 m
40 Experiment 3 80 m
35 A Experiment 5 88 m
DDPM/Multifluid 1 75 m
'30 DDPM/Multifluid 3 80 m
S25 DDPM/Multifluid 5 88 m
20
vg 20  V
> 15
10
5 A
0 025 05 075 1 125 15 175 2
Radial distance (m)
Figure 9: Void profiles from DDPM/Multifluidmodel
at different heights for a gas rate of 170 NI/s.
When the strength of the plume is increased to 750
Nl/s, the effect of the particleparticle interaction force
becomes even more apparent. It is shown in Figure 11 that
the DPM/VOFmodel now predicts unphysical volume
fractions in the lower regions of the vessel. When the
additional dispersive force is included with the
DDPM/Multifluidmodel approach, however, the volume
fraction prediction is more than halved. Unfortunately, no
experimental data was available for comparison, but the
comparison for the medium gas rate shown in Figure 9
suggests that the experimental gas volume fraction might
be even lower.
The velocity profiles for the large gas flow rate
shown in Figure 10 indicate that the plume created by the
DDPM/Multifluidmodel becomes wider at the bottom and
slightly narrower at the top when compared to the
DPM/VOF predictions. The wider bottom region can be
directly related to the additional dispersive force that is
included when the voidage exceeds 60%. This force would
push particles outwards and therefore create a wider
plume. The slight narrowing of the plume towards the
upper regions is interesting to observe and shows that the
DPM/VOF model actually creates a somewhat stronger
dispersion in these regions. Since the extra particleparticle
dispersion force implemented only activates at volume
fractions above 60%, it does not play any role in these
Paper No
regions. Thus, the plume dispersion in this region is
controlled only by the random walk model used to model
the turbulent dispersion. It seems that the modelled
strength of the eddies is slightly decreased with the
decrease in the mixture density when the dispersed phase
volume fraction is considered in the DDPM/Multifluid
model approach. The effect seems to be only slight,
however, and does not merit further investigation at this
point. Note also from Figure 12 that the void fractions
modelled for the highest gas rates exceed 15% in a
significantly large region of the plume.
In spite of the high void fractions for which the
models should not be applicable, the models seems to do a
fairly good job at predicting velocity and void profiles with
the exception of void profiles near the inlet. Table 1
summarizes some global observations from the models and
the experiments. These include the only reported
observations at the highest gas rate. The rise time is the
time it takes between the start of the release until the first
bubble surfaces. We see that the all models predict this
quite well. The fountain height or surface elevation is the
elevation of the surface at steady state conditions due to
the bubble plume. It is predicted well for the lowest gas
rate, and overpredicted at the highest gas rate.
Initially, when only DPM/VOF simulations were
completed, the overpredicted fountain heights for the large
gas flow rate was assumed to be caused by the model not
accounting for the gas volume fraction in the continuity
equations. It was reasoned that the upward moving plume,
modelled to comprise of 100% water, would contain an
overpredicted amount of momentum and therefore lead to
overpredicted fountain heights. It is shown in Table 1,
however, that the DDPM/multifluidmodel, which
accounts for the discrete phase volume fraction, creates
virtually the same overprediction of the fountain height as
that observed with the DPM/VOF model. It is therefore
proposed that the overpredicted plume heights is not the
result of an overpredicted plume mass, but rather an
underpredicted plume dispersion. The dispersion force
lacking in this case would be the interparticle dispersion
force present at higher volume fractions. As reasoned
earlier, this force might already become significant at
voidages as low as 10%. The voidage with the large gas
rate is more than 20% even in the regions close to the free
surface, meaning that negligence of this additional
dispersive force could lead to a large underprediction of
the plume diameter. The resulting narrower plume would
reach the free surface in a more concentrated fashion and
make a more pronounced fountain. More detailed
experimental measurements at high gas flow rates are
needed for confirmation.
Table 1: Rise time and surface elevation from models
and experiments for gas rates of 170 and 750 Nl/s
Rise Time s Elevation cm
170N1/s 750N1/s 170N1/s 750N1/s
DPM/VOF 4.9 3.0 28 69
DPM/Multifluid 4.7 2.8 29 71
DDPM/Multifluid 4.7 3.1 32 63
Experimental 4.8 3.1 30 45
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
5
45 DPM/VOF 1 75 m
SDPM/VOF 3 8m
4 DPM/VOF 5 88m
 DDPM/Multfluid 1 75 m
 '4 DDPM/Multfluid 3 8 m
E 3  DDPM/Multfluid 5 88 m
t25
0 2
>15
S1 5
05
0 025 05 075 1 125 15 175 2
Radial Distance (m)
Figure 10: Comparison of velocity profiles from
DPM/VOFmodel and DDPM/Multifluidmodel at
different heights for a gas rate of 750 NI/s.
0 025 05 075 1 1 25
Radial distance (m)
15 1 75 2
Figure 11: Comparison of void profiles from
DPM/VOFmodel and DDPM/Multifluidmodel at
different heights for a gas rate of 750 NI/s.
9.50e1
8.01
750e01
7550e01
7.00e01
6.50e01
5.00e01
5.50e01
BOOeO1
4o50e1al
4,00e01
4L50.1
3.00"01
1.50e01
5,00e02
0.00e +w
Figure 12: Void fractions (modelled by DDPM/
Multifluidmodel) for a gas rate of 750 Nl/s.
Paper No
Computational time
The computational cost is an important consideration to be
made when choosing between these three modelling
approaches. In order to evaluate this, the simulations were
timed to estimate the number of second that can be
completed in a single day of processing on 8 processors.
All simulations were run at a constant timestep of 0.002 s
to enable a direct comparison and use identical
underrelaxation factors for each simulation.
When timed, the DPM/VOFmodel could simulate
16.2 s/day, the DPM/Multifluidmodel 15.7 s/day and the
DDPM/Multifluidmodel 11.4 s/day. The DPM/Multifluid
model is somewhat slower than the DPM/VOFmodel
since an extra set of transport equations now has to be
solved for the gas phase also. The DDPM/Multifluid
model is substantially slower because smaller particle
timesteps had to be implemented in order to ensure model
stability.
Under practical operation, however, the
DPM/VOFmodel can safely be run at timesteps up to one
order of magnitude greater than the present setting. It is a
very stable formulation and can use very high
underrelaxation factors thanks to the PISO
pressurevelocity coupling scheme. The
DDPM/Multifluidmodel cannot be pushed to larger
timesteps though. The inclusion of the particle volume
fraction in the continuity equations quickly creates
instabilities and requires very conservative solver settings.
When all is considered, the DPM/VOFmodel is
probably in the order of 10 times faster than the
DDPM/Multifluid model, which is certainly highly
significant.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
enhancement of this modeling approach would be to
develop a dense discrete phase model coupled with the VOF
model, i.e. a DDPM/VOFmodel. For higher gas rates it
would also be wise to develop a proper dispersive force for
the bubblebubble interactions.
References
Cloete, S., Eksteen, J.J. and Bradshaw, S.M. "A
mathematical modeling study of fluid flow and mixing in
full scale gas stirred ladles", Progress in Computational
Fluid Dynamics, Vol.9 (No.6/7):345356 (2009)
Cloete, S., Olsen, J.E. and Skjetne, P., "CFD modelling of
plume and free surface behaviour resulting from subsea gas
release.", Applied Ocean Research, Vol.31, 220225 (2009)
Drew, D.A. & Passman, S.L.: Theory of Multicomponent
Fluids, InApplied Mathematical Sciences, Vol.135, Springer
(1999)
Engebretsen, T, Northug, T, Sjoen, K., and Fannelop, T.K.,
Surface flow and gas dispersion from a subsea release of
natural gas, Seventh Int. Offshore and Polar Engineering
Conference, Honolulu, USA (1997)
Olsen, J.E. & Cloete, S. "Coupled DPM and VOF Model for
Analyses of Gas stirred Ladles at Higher Gas", Proceedings
of the 7h International Conference on CFD in the Minerals
and Process Industries, CSIRO, Melbourne (2009)
Kolev, N.I., Multiphase Flow Dynamics 2: Thermal and
Conclusions Mechanical Interactions, 2nd ed. Springer, Germany (2005)
Three EulerianLagrangian models have been compared
with experimental data. They have been applied to gas
stirred vessels with void fractions exceeding the model
assumptions, although one model have incorporated a
bubble coupling to its continuity equation and an additional
dispersive force to account for dispersion due to
bubblebubble interactions. Two gas rates have been
considered and all models yield good results for the lowest
gas rates. For the highest gas rates, little data is available for
comparison. Even if the comparison with the experiments is
not perfect, the results are still good considering the
complexity of the problem. Thus further research and
improvement of the modeling approach should be well
worth the effort.
The effect of including the mass of the Lagrangian
bubbles in the continuity equation of the Eulerian phases
does not seem to have a major effect. The effect of the
additional dispersive force is more dominant. This is an
adhoc force and further research should be conducted on
this matter.
The CPU cost of the models is considerable, but not
discouraging for industrial applications. However, the
DPM/VOFmodel is by far the most efficient method,
mainly because of the PISO scheme associated with the
method, but also because of the lesser amount of equations
being solved. The authors believe that the most appropriate
Launder, B.E. and D.B. Spalding, Lectures in Mathematical
Models of Turbulence, Academic Press, London (1972)
Laux, H. and Johansen, S.T., "A CFD Analysis of the Air
Entrainment Rate due to a Plunging Steel Jet Combining
Mathematical Models for Dispersed and Separated
Multiphase Flows. ", Fluid Flow Phenomena in Metals
Processing (1999)
