Content uploaded by Wayne Strasser
Author content
All content in this area was uploaded by Wayne Strasser on Apr 25, 2019
Content may be subject to copyright.
1
Discrete Particle Study of Turbulence Coupling in a
Confined Jet Gas-Liquid Separator
Wayne Strasser
Eastman Chemical Company, strasser@eastman.com
Abstract
A 3-D CFD investigation of particle-induced flow effects and liquid entrainment from an
industrial-scale separator has been carried out using the Eulerian-Lagrangian (E-L) two-way coupled
multiphase approach. A differential Reynolds stress model (RSM) was used to predict the gas phase
turbulence field. The dispersed (liquid) phase was present at a relatively high mass loading (0.25) but
low volume fraction (0.05). A discrete random walk (DRW) method was used to track the paths of the
liquid droplet sampling bursts. It was found that gas phase deformation and turbulence fields were
significantly impacted by the presence of the liquid particles; These effects have been parametrically
quantified. It was also found that a very large number of independent steady-state DRW burst samples
was needed (1000) to make conclusions about entrainment. Known plant run conditions and
entrainment rates validated the numerical method.
1. Introduction and Method
Background
Gas-liquid, gas-solid, and liquid-solid separators are integral operations in many industrial
processes. The goal of the gas-liquid separator, for example, is to separate a mixture of gas and liquid
into two distinct, essentially pure, streams. A gas stream exiting with no liquid is said to contain no
carryover, or entrainment. The presence of dispersed particulate material in a continuous stream can
influence the dynamics of the continuous stream (two-way coupling). An experimental cyclone study
by Obermair et al. [1] discussed the effects of particles on vortex flow at a loading of 6.5E-4. They
found that the presence of the particles greatly increased turbulence and reduced the strength of the
vortex in the cyclone downcomer. Similarly, the reduction of the tangential velocity in the bottom of a
cyclone by the presence of solids was discussed by Gil et al. [2]. The authors showed that, for a solids
loading in the range of 0.1, the solids-induced swirl reduction lowered the overall pressure drop of the
unit. Vandu and Krishna [3] found that the mass transfer coefficient in a bubble column is reduced by
the presence of solids at a loading of 0.25. They proposed that this is a result of increased gas bubble
coalescence caused by solids-induced turbulence dissipation reduction. Faeth [4] found that
anisotropy was increased in gas jets involving liquid sprays. Ahmadi and Abu-Zaid [5] performed a
numerical analysis of anisotropic dense two-phase flows with a similar density ratio as in the present
work. They showed significant mixture normal stress differences for shear rates two orders of
magnitude greater than those of the present study. The turbulence production and dissipation were
governed by the continuous phase, generally, but there was significant fluctuation energy transfer from
the particulate phase to the continuous phase for a particulate volume fraction > 0.5. Graham [6]
discussed a numerical study of particles in a shear flow. He found the carrier phase
production:dissipation ratio and anisotropy increased while its turbulence is attenuated if the particle
diameter is low compared to turbulence scales (wake-less). He showed that continuous phase
turbulence enhancement is only possible for very high mass loadings (>1) and Stokes numbers above
~5. Mostafa et al. [7] showed a reduction in turbulence and increase in anisotropy by particulates.
They expanded on previous work showing that the attenuation is greater, the smaller the ratio of
particle to gas phase velocity fluctuations. Lastly, a thorough study of interparticle collisions in
particle-laden tube flow was provided by Boree and Caraman [8]. They found a departure from Tchen
2
theory even at mass loadings as low as 0.11. They also found gas phase turbulence attenuation and a
flatter mean velocity profile, but the local damping was dependent on mass loading. At higher
loadings (1.1) the damping was only near the walls, while at 0.6 loading, damping occurred at the axis.
They proposed that near-wall anisotropy caused by particle collisional effects is influential on the
radial tube flow profile down to volume fractions lower than 0.01. To the author's knowledge, no
study exists in the open literature that quantifies particle-induced anisotropy in parametric terms. The
goal of this study is to use a numerical method in a proven, established commercial solver backbone,
Fluent 6.2.16, to quantify the effects of a dispersed liquid on the turbulence and flow of its gas phase
carrier in an industrial separator.
Physics
Thorough comparisons of typical methods for numerically treating fluid-particle flows were
given in Crowe et al. [9]. It has been argued (Heinl et al. [10] and Fluent, Inc. [11]) that it is more
advantageous to treat the dispersed phase via Lagrangian tracking (as opposed to Eulerian) methods
for particle volume fractions below ~0.1. In the E-L method, paths of individual particles, or statistical
groups of particles, are tracked through the continuous phase via a Newton's 2
nd
law force balance:
LjgjDj
pj
FFF
dt
du ++=
(Forces due to drag, gravity, and lift, respectively) (1)
( )
cjpj
p
pD
Dj uu
C
F−=
τ
24
Re
,
c
pp
p
d
µ
ρ
τ
18
2
=
,
c
psc
p
du
µ
ρ
=Re
,
(
)
2
cjpjs
uuu −=
(2)
(
)
p
cpj
gj
g
F
ρ
ρ
ρ
−
=
and
( )
pici
l
ck
k
cl
pp
j
ci
c
Lj uu
x
u
x
u
d
x
u
F−
∂
∂
∂
∂
∂
∂
=
4
1
2
1
19.5
ρ
ρυ
(3)
A series of fixed monodisperse particle diameters is tested in the present study. Considering the
Morton, Eotvos, and particle Reynolds numbers for these droplets, it is estimated they are spherical in
shape (Clift et al. [12]). The drag coefficient can, therefore, be approximated by the correlation of
Morsi and Alexander [13] for rigid spherical particles. Although droplet internal circulations and
surface tension/ contamination effects are known to have some influence on drag coefficient (Clift et
al. [12], Krishna and van Baten [14], others), the rigid particle assumption captures the essential
physical mechanisms and is expected to yield useful results for the purposes of this study. The lift
force correlation is valid only for relatively low particle Reynolds numbers, which is the case in the
present work.
DRW is a stochastic subset of discrete particle modeling (Crowe et al. [9]). Both phases are
fully coupled through mean and fluctuating velocities. It is assumed that particles and fluid elements
follow different trajectories and only interact for finite periods of time. The particles interact with
eddies whose scales are resolved by the continuous phase turbulence model. When using DRW in
equations 1-3, the continuous phase velocity is expressed as the mean plus the fluctuating value. The
fluctuating velocity values are discrete piecewise functions of time. They are assumed to remain
constant as long as the particle is under the influence of a given eddy structure. This length of
interaction time is the minimum of two values, either the randomized characteristic eddy lifetime or
the eddy crossing time scale shown below:
( )
r
k
el
log3.0
ε
τ
−=
or
−
−−=
pcp
pec
uu
l
τ
ττ
1ln
with
ε
2
3
09.0 k
l=
(4)
3
The 0.3 in the first equation can be a function of mass loading (Graham [6]), but that aspect has been
ignored here in that this study involves much lower loadings that than of Graham [6]. Typical
Reynolds-averaged Navier-Stokes (RANS)-resolved length scales are two orders larger than the
particle diameters of the present work, so the particles would classify as wake-less according to
Graham. When the smaller of these two time scales is exceeded, a new random fluctuating continuous
phase velocity component is generated via
2'
cjcj
uu
ζ
= (5)
A known shortcoming (Crowe et al. [9]) of methods that superimpose chaotic fluctuations on the mean
is the inability to predict anti-dispersion of particles in the peripheries of vortical structures for
intermediate (near unity) Stokes numbers (Zheng et al. [15]). The continuous phase time scale in the
Stokes number calculation is that of the turbulence and not the mean velocity field as discussed in
Sommerfeld [16]. For the range of particle diameters studied, the particle Stokes numbers are in this
intermediate regime; However, the DRW method includes enough physics to be a useful
approximation to the real particle spreading in the separator for this design study. The maximum
diameter for which inter-particle collisions can be ignored can be deduced from discussions in Crowe
et al. [9] and Soo [17]:
pc
c
pZ
d
σρ
µ
33.1
<
,
02.0≈
c
p
σ
σ
, and
k
c
3
2
=
σ
(6)
Monodisperse particle sizes for this CFD study are near or below this critical value. As a result, the
ignoring of inter-particle collisions is reasonable.
Turbulence modulation by Lagrangian particles involves additional terms be included in the
turbulence equations as discussed in Faeth [4], Amsden et al.[18], Xiong et al. [19], Mostafa et al.[7],
and Graham[6]. All except Graham [6] framed the additional terms as sources/sinks in the typical
eddy-viscosity k and transport equations. Fluent, for example, follows the methods of Amsden et al.
[18] in which the particles act to extract turbulence kinetic energy from the continuous phase as long
as the particle diameter is larger than 10% of the largest turbulent length scale ("l" in Eq. 4); Below
this, the reverse occurs. To the author's knowledge, Graham [6] is the most recent to discuss particle
source terms for Reynolds stress components in a Lagrangian framework. He developed these sources
for an algebraic stress model based on the Stokesian drag assumption. Reynolds stress source terms
are not available in the current release of Fluent in the Lagrangian framework, but might be included
in future work. For Eulerian-Eulerian simulations, these interphase turbulence transport terms have
been given for a differential Reynolds stress model by Cokljat, et al. [20].
The steady incompressible RANS Eulerian linear momentum balance in Cartesian coordinates
for the continuous phase is shown in equation 7.
S
x
p
x
uu
xx
u
x
uu
icj
cjci
jj
ci
c
j
cicj
+
∂
∂
−
∂
∂
−
∂∂
∂
=
∂
∂
ρ
ν
1
''
2
(7)
S is a source term containing the particle velocities, which are influenced by all forces shown in
equations 1-3. Notice that the volume fraction occupied by the dispersed phase is not explicitly
accounted for in Equation 7, which is the reason for the ~0.1 upper limit on the dispersed phase
volume fraction.
The advantages to using the RSM approach in modeling continuous phase turbulence include
natural realizability and, to a large extent, the ability to capture the effects of streamline curvature,
turbulence anisotropy, and rapid changes in strain rate. The Reynolds stress tensor components are
obtained by solution of the steady, incompressible differential equation:
4
(
)
( )
k
cj
k
ci
c
i
cj
j
ci
ck
ci
ckcj
k
cj
ckci
cjikcikj
c
ckcjci
k
cjci
c
kk
cjcick
x
u
x
u
x
u
x
u
p
x
u
uu
x
u
uu
uupuuu
x
uu
xx
uuu
∂
∂
∂
∂
−
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
−
+−−
∂
∂
∂
∂
=
∂
∂
'
'
'
'
''''
'''''
''''
2
1
1
ν
ρ
δδ
ρ
ν
(8)
The terms on the RHS are as follows: diffusion (molecular, followed by turbulent), production,
pressure-strain, and dissipation. Production and molecular diffusion need no modeling, but the others
do. The LLR-IP (Launder [21] and Fluent, Inc. [11]) approach for modeling Reynolds stresses is used
for the present work. The turbulent diffusive stress transport is treated with the gradient-diffusion
hypothesis and the dissipation is modeled assuming isotropy. Pressure strain is well known to be one
of the most important terms; It has zero trace and serves to redistribute the stresses. In the present
work, the pressure-strain model incorporates a linear return-to-isotropy term (or slow), rapid pressure-
strain term, and a wall-reflection term. The SSG (Speziale et al. [22]) approach offers a quadratic
pressure-strain approach with no wall-reflection. Employment of the SSG model was attempted on
multiple occasions using various "coaxing" techniques (including temporarily adjusting the non-linear
slow term's model coefficient), but the commercial solver continually diverged. Hanjalic [23]
proposed that RSM convergence difficulty is a fundamental problem with the typical Navier-Stokes
solver and outlines improvements to alleviate these problems. A non-equilibrium wall function
approach was used for the near-wall cells as discussed in Fluent, Inc. [11]. This method involves
typical wall functions that have been modified to relax the production=dissipation assumption. Given
that this geometry has no severe adverse pressure gradients, this option seems reasonable for the intent
of the present work.
The author is keenly aware that the RANS approach cannot resolve turbulence scale
information below that of the largest scale, and results in the smearing out of most
transients/unsteadiness (Speziale [24], many others). Pope [25] presented that the normalized
Reynolds stress tensor is not enough to completely describe the evolution of turbulence. He showed
DNS simulations resulting in two differing trajectories on the - plane (definitions forthcoming) with
the same starting RANS conditions. In the present work, the shedding of vortices where the entry jet
meets the counter flow in the vessel should produce a rich, time-dependent, 3-D solution. However,
because the overall effort aims at eventual optimization of the real industrial scale process, it was
desired to focus the investigation on steady, Reynolds-averaged methods more suitable for design
applications.
Numerics and Boundary Conditions
Second order upwinding with linear reconstruction (Fluent, Inc. [11]) was used for all spatial
variables except turbulence dissipation rate, as second order upwinding dissipation inhibited
convergence. A third-order MUSCL scheme is available in this commercial release but does not
contain flux limiters and could cause local overshoots where the flow is not aligned with the grid.
Derivatives were discretized using the nodal method (weighted by nodal values on surrounding faces
instead of simple arithmetical grid cell center averages). The solver was run in double precision mode
due to expectations of low values of carryover. The SIMPLE algorithm was used for pressure-velocity
coupling via the segregated implicit solver. An advanced multidimensional slope limiting scheme
(total variation diminishing) was utilized to prevent variable overshoot. Pressure checker-boarding
was prevented using a second order Rhie-Chow method. Algebraic multigrid was used to reduce large
wave error propagation. The author is aware of the proliferation of CFD "abuse" (quotation from
Hanjalic [23] regarding LES), so more details on the importance and ramifications of sound numerics
is discussed in Strasser [26].
5
The mesh, involving prisms, tetrahedra, hexahedra, and pyramids, was built with the intent to
balance computational load and accuracy. The smallest grid lengths scales were near the inlet and
outlet. The expansion of cells away from the inlet and outlet to areas with lesser gradients was carried
out in such a way as to minimize cell aspect ratios, centroid shifts, and skewness. More on these
concepts can be found in the transonic gas turbine blade passage research of Strasser et al. [27]. Wall
resolution was set such that the first cell was within the log-law region. More will be discussed on
grid dependence in upcoming sections. Particle trajectory ODEs are solved using a high-order Runge
Kutta with embedded error control as discussed in Fluent [11]. As previously discussed, the present
computations ignore the volume of the particles. The smallest computational cell length scale is,
therefore, kept much larger than the particle diameter.
The diverging gas jet inlet is given a fixed uniform velocity, while the particles injected at this
inlet are given the same uniform velocity. The outlet is a fixed pressure outlet. It is known from
internal testing that the liquid droplets will adhere to any wall whenever they come into contact inside
the separator. All of the wall boundaries were, therefore, set to "escape." In other words, the particles
were no longer accounted for after they contacted a wall. Only those particles that were found in the
outlet at the top of the vessel were considered to be carried over (entrainment). Wall effects, such as
adhesion, electrostatics, particle-particle effects, wall roughness (Heinl [10]), and particle-collision
induced anisotropy (Boree and Caraman [8]) have been ignored in the present work since they do not
apply.
Measures
Turbulence anisotropy is important for a number of reasons, one of which being that normal
turbulent stresses only contribute to production of turbulence kinetic energy and momentum transport
when they are unequal (Pope [25]). Quantification of anisotropy will be carried out using the Lumley
triangle (from Pope [25 and Lumley [28]) two-parameter approach.
NRATcjicij
IIbb 26
2
−==
η
and
NRATckicjkcij
IIIbbb 36
3
==
ξ
(9)
Each normalized Reynolds stress anisotropy tensor (NRAT) component is found by:
k
kuu
b
ijcjci
cij
2
3
2
''
δ
−
=
with determinant
32
54271
ξη
+−=F
(10)
The triangle is not a perfect triangle, but it is a nearly triangular map (upside down) bound by stress
realizability centered about = 0. The boundary curves represent special states of turbulence, with the
point farthest away from the origin being one-component turbulence (one fluctuating velocity
component dominates) at = = 1/3. All values of are positive. Positive values of have been
shown to exist in near-wall flow, while negative values of have been found in a turbulent mixing
layer. A determinant of zero indicates anything on the uppermost bounding curve of the triangle (two-
component and one-component), while a value of unity indicates isotropic turbulence. Also from a
mixing study of Galletti et al. [29], a single parameter is proposed for measuring the deviation from the
origin on the Lumley triangle.
( )
2
2
NRATNRAT
IIIIIL −+=
normalized by
34.0
max
=
L
(11)
Deformation classification is addressed by Pirozzoli and Grasso [30] for both the incompressible flow
of the present study and compressible flow. They characterize flow based on the three principal mean
deformation tensor invariants (Aris [31]):
6
(
)
321
λ
λ
λ
+
+
−
=
−
=
DT
IP
,
323121
λλλλλλ
++==
DT
IIQ
,
321
λλλ
−=−=
DT
IIIR
,
and
−+
−+=∆
22332
4
1
2
9
4
27 QPQPQPRR
, which is the "discriminant" (12)
The authors also discuss the relative size of the eigenvalues to one another, with typical values being
related by the ratio 1:-4:3.
2. Results
Overall Flow Field and Carryover
The separator geometry is a vertical cylindrical vessel, half of which is shown in Figure 1.
The gas-particle mixture enters from the top of the vessel through a divergent entry (gas Reynolds
number of order E7) that extends into the vessel about a third of its total height. The feed mixture
disperses into a turbulent jet that interacts with a reverse mixture flow returning upwards from the
vessel bottom. Jet spreading is coupled with stagnation. Gravity, dispersion, and residence time allow
most of the droplets to segregate toward the bottom of the vessel, while the gas escapes through an exit
near the top. Figure 1 depicts axial velocity contours from a preliminary gas-only analysis. Red on
this plot indicates any velocity that is greater than the theoretical particle slip velocity. The vessel
average velocity magnitude is larger than the theoretical particle slip velocity, so there is a propensity
for liquid material to be carried over. Only the bottom section of the vessel is "safe" (blue) for
droplets. Both 3-D and axisymmetric models were run for this study, and preliminary gas-only results
indicated the two are nearly indistinguishable. Based on early work, DRW results from both types of
models are presented where possible. All of the contour plots shown will be from the axisymmetric
models to help visually accent and distinguish particle coupling effects, not flow and vessel
asymmetry.
A typical scatter sample series of carryover (liquid exiting through gas outlet) for the "base"
particle diameter is shown in Figure 2. These represent 1000 sequential independent steady-state
DRW burst samples. There is a fixed number of continuous phase iterations between each burst
(discussed more later). The DRW approach allows for randomization within the turbulence field
computations, so a true random "blast" of particles is considered. The average is ~900 parts per
million by weight (PPMW) based on inlet liquid particle feed rate. The standard deviation is ~1000
PPMW. With a standard deviation larger than the mean, it is expected that many samples have to be
run before confidence can be gained in the answer. For example, two pairs of tests were run with
identical CFD setups. When 70 DRW bursts were used, the two resulting means were 660 PPMW and
820 PPMW, respectively. For 1000 DRW bursts, the mean carryover rates were 920 and 940 PPMW,
respectively. Because the last pair's means were close enough for purposes of this study, a minimum
data set size of 1000 samples will be used from here forward.
Carryover rates were sensitive to fluid-particle coupling. For example, if the particles are just
released into the fluid with no feedback onto the continuous phase, the carryover rates are more than 3
times the coupled carryover value. Coupling allows more to droplets accumulate in the bottom instead
of getting carried over. Figure 3 shows typical instantaneous discrete phase concentration (consistent,
but arbitrary, range to visualize differences) contours without (left) and with (right) coupling. It
appears that more particles are headed towards the bottom without coupling, but that is just a visual
artifact. The radial particle spreading is much greater without coupling, so more of the particles get
caught in the gas upwash. The flow characteristics that contribute to this will be discussed later.
The carryover results (average and standard deviation) were extremely sensitive to assumed
particle size, as one might expect. When the mean particle diameter was increased 50% above the
"base" particle size, the mean carryover rate fell to 0. Not one of the 1000 burst samples resulted in
any liquid carryover. On the other hand, when the particle size was reduced to ~50% of base value,
7
the mean carryover rose from 900 PPMW to nearly 5,000 PPMW. The standard deviation fell from a
value larger than the mean to a value which was about half of the mean. Experimental data were
provided for a real production unit of the same dimensions, feed rates, and properties of those modeled
here. The experimental carryover rate was 5,200 PPMW. It is known through internal plant testing
(and results from external consultant experimentation with plexiglass units) that the three CFD particle
size values considered in this work are within a reasonable range for the real industrial mean particle
size. The "base" size was simply chosen, early on, as a value in the known industrial size range to
begin to study the method. The standard deviation is not able to be obtained from the experimental
data due to the fact that each experimental data point represents an average of 120 vessel residence
times worth of continuous particle feed data. There are no independent bursts to measure in a
production unit.
Numerical Tests and Grid Independence
There remains some question in the open literature as to the importance of various features of
the E-L approach. In the present study, E-L features were evaluated for their effects on the results,
both in terms of the mean and standard deviation liquid carryover. It is has been proposed in Faeth [4]
that approximately 5000 particle groups are needed per injection for a stochastic two-way coupled
approach. Li and Wang [32] proposed that 1250 particle groups are required for independent results in
their gas turbine film cooling effectiveness study. They also propose that the number of particle
groups per grid point plays little importance in the outcome and that higher turbulence intensity
augments droplet dispersion, as one would expect.
Table 1: Numerical test effects on liquid droplet carryover
Range
Item Tested Tested Mean Standard Deviation
N Continuous Phase Iterations Between Particle Bursts 10 500 +10% Not Significant
N Particle Groups Injected Per Inlet Cell 10 20 Not Significant -30%
N Particle Groups Tracked 1000 2000 Not Significant -30%
Characteristic Eddy Lifetime Constant 0.15 0.3 -25% Not Significant
Momentum Coupling Underrelaxation 0.1 0.3 Not Significant Not Significant
Feed Turbulence Intensity 5% 10% -10% Not Significant
Particle Lagrangian Steps Per Cell Length Scale 5 10 Not Significant Not Significant
Grid Resolution 1X 3X -40% +20%
Effect On
Table 1 summarizes the findings in the present set of studies. "Not significant" is defined as a change
3%. The number of groups tracked per burst, number of groups per inlet grid cell, momentum
underrelaxation, and particle steps per cell length proved not to be an important consideration in the E-
L setup, in terms of the mean. More groups tracked per burst resulted in reduced variability. For
reference, each particle group of the 1000 contains O (E5) particles. Substantially increasing the
number of continuous phase iterations per particle burst only changed the mean 10%. Increasing
intensity reduced carryover by spreading particles out more radially and azimuthally. The eddy
lifetime constant had an inverse effect on carryover. Increasing this constant increases the length of
time a continuous phase flow structure interacts with relatively small particles, holding them back. It
makes sense that this would reduce the number of particles carried out of the vessel. Lastly, grid
resolution was increased in all three dimensions until a grid-independent solution was reached.
Increased resolution gives the particles less resistance to radial and azimuthal dispersion. Less liquid
is carried over as a result. Since the flow resistance is lessened, burst-to-burst uniqueness is cultivated,
and the standard deviation is increased. The E-L findings from the above tests had been incorporated
in the run that produced the 5000 PPMW results mentioned in the previous experimental validation
discussion.
8
Particles Effects on the Continuous Field
The turbulence field is sensitive to the presence of the particles. Figure 4 shows a typical
instantaneous continuous phase turbulent viscosity ratio (TVR) without (left) and with (right) fluid-
particle coupling for the base particle size. The range on this figure is 0 (blue) to the peak ratio seen in
the uncoupled case (red). The peak viscosity region is much larger and pulled axially downward in the
vessel due to particle drag and fluid-particle shear. This effect is consistently shown in Figure 5,
offering contours of the ratio of production-to-dissipation (PD) of turbulence kinetic energy. The
range on these plots is 0 (blue) to 2 (red), so anything blue represents a negative production.
Production is increased both at the inlet jet interface and the near-bottom centerline area by the
inclusion of particle-fluid feedback. The area of negative production near the top has increased for the
coupled case. Turbulence intensity (normalized by superficial velocity) is shown in Figure 6 over the
range of 0 to 100%. Intensity appears to only be increased by particle feedback, the opposite of the
attenuation spoken of in the Background section.
The graphs in Figures 7, 8, and 9 depict these same effects, respectively, for a few CFD cases.
Each quantity is mass-weighted area-averaged (MFWAA) on axial planes beginning at the tip of the
inlet divergent cone. Axial cutting planes extend more than 10 jet diameters down from the cone, with
the bottom of the vessel near 13 diameters. The graphed value is a ratio of the value calculated in the
coupled (E-L) model to the value calculated in the uncoupled (gas-only) model. Starting in Figure 7, it
is seen that the TVR increases dramatically axially as a result of particle inclusion. The base particle
diameter axisymmetric model, smaller diameter axisymmetric model, and the smaller diameter 3-D
model all respond in about the same manner. The "smaller" diameter here is ~50% of the base
diameter as discussed in the Carryover section. Coupling has little effect at the beginning of the jet,
but increases TVR by a factor of ~15 near the plane at 10 jet diameters. Figure 8 displays PD effects.
PD is increased substantially at most axial planes, with the peak increase differing between models.
Particle diameter has little effect, but the 3-D model shows a more rapid increase with increasing
distance from the cone. Lastly, axial turbulence intensity (normalized by superficial velocity)
dependence is shown in Figure 9. Intensity is increased substantially as the flow proceeds along the jet
and interacts with the reverse flow/stagnates. Again, particle size and 2-D/3-D effects play almost no
role in the results.
The next four figures help explain the turbulence anisotropy fields, with and without coupling
for the base particle size. In all cases, blue is lowest value, and red is the highest value mentioned
in the corresponding text discussion. Figure 10 shows contours of the first stress anisotropy tensor
parameter, , with a range of 0 to 0.125. It can be seen that there is an increase in overall anisotropy
by coupling, especially along the vessel center near the bottom. The second stress anisotropy tensor
parameter, , effects can be seen in Figure 11 for the range of -0.1 to 0.1. There is a range of states
present, but the coupling increases the tendency towards one-component in the bottom half. Figure 12
depicts contours of the determinant (Eq. 10) of the NRAT over the range of 0 to 1. Just as in the other
cases, there is more anisotropy (lower values of determinant) in the lower vicinity of the vessel caused
by particles. Contours of the normalized distance (Eq. 11) from the Lumley origin (0 to 0.2) are
shown in Figure 13. These plots look similar to those in Figure 10, where coupling causes an increase
at the jet interface and jet impingement zone.
Figures 14 and 15 show MFWAA coupled:uncoupled ratios for both the normalized length
(Eq. 11) and determinant (Eq. 10), respectively, for the same three pairs of cases discussed in previous
figures along the vessel axial direction. Both show increasing departure from isotropy with distance.
The axisymmetric results did not depend heavily on particle diameter. One would expect these two
figures to be nearly mirror images of one another since they measure anisotropy in opposite directions.
The normalized distance appears to be a more sensitive measure (than F) of the departure from
isotropy, as it shows more relative deviation from 1.0. As in previous and upcoming figures, 3-D
effects appear to play a role in the results. Neither of the 3-D model plots follows the exact trend of
the axisymmetric results. The axisymmetric results show further and more smoothly trending
deviation between the coupled and uncoupled cases.
9
Figure 16 shows azimuthally-averaged radial values of anisotropy (again, summed up in a
single parameter from Eq. 11), intensity, and axial velocity for a case involving the base particle
diameter. The ratio of coupled:uncoupled values were averaged in a plane that was 1 jet diameter
away from the cone exit. From previous figures, it is expected that planar average of both anisotropy
and intensity is near 1.0, but it was desired to look for the radial variability. The jet interface at this
axial location is about 0.3 vessel radii. It can be seen that, in general, the turbulence attenuation
decreases with distance out from the centerline, while the anisotropy increases. Both the intensity ratio
and anisotropy ratio are typically above 1.0 outside the jet and below 1.0 inside the jet. The
azimuthally-averaged axial velocity at this location was affected by about the same ratio as the other
measures, but there was no general trend. Most of the deviation was seen outside the jet. Lastly, the
overall static pressure drop (not shown) of the separator was lowered about 5% by particles due to
reduced turbulence.
A Lumley plot of versus is given in Figure 17 for the 3-D case at the plane 1 jet diameter
from the cone. This area would include near-wall flows (at the separator walls), as well as a turbulent
shear layer from the jet. All of the values are bound correctly by the positive and negative
axisymmetric lines. No values in either the coupled or uncoupled approach are very close to the upper
triangle boundary two-component curve (not shown) starting at = 1/6. Some of both cases approach
the axisymmetric curves. The average values between the two cases are nearly identical, but the
average is slightly shifted to the right by coupling.
Contours of the second deformation tensor invariant for the coupled and uncoupled base
diameter cases are shown in Figure 18. The range is -1 (blue, regions of high strain) to 1 (red,
regions of high vorticity). With the inclusion of particles, the flow field has changed from a localized
region of high vorticity near the jet interface to axially alternating regions of high vorticity and high
strain along the jet centerline. The changes in the mean deformation discriminant, , are given in
Figure 19. The range is -1 (blue, nonfocal turbulent structures with three real deformation
eigenvalues) to 1 (red, focal turbulent structures with only one real eigenvalue). The inclusion of
particle effects has caused the flow to exhibit much more of a focal character. To further illustrate this
effect, Figure 20 shows contours (same range as Figure 19) of the discriminant on a cross section of
the vessel 1 jet diameter from the cone exit. Overall, there is more of a focal nature to the coupled
flow with some traces of nonfocal both within the jet and around the vessel periphery.
Similar to previously discussed graphs, Figures 21 and 22 show the mass-weighted area-averaged
planar values of the second invariant and the discriminant, respectively. The values plotted here are
expressed in ratios of the value calculated in the coupled model to that in the uncoupled model as
before. Because of the nature of these calculations, values are more sensitive to local gradients, and
their ranges are much greater than in previous figures. To simplify the discussion, the plots are
marked with "Pos." indicating large positive ratios and "Neg." indicating large negative ratios. In
Figure 21, like in Figure 8, there is some deviation activity around the 1 or 2 jet diameter distance
position followed by no activity and then stronger deviation toward the vessel bottom. One striking
character in these results is that particle size appears to make a large difference for the axisymmetric
runs. The base particle diameter actually has the opposite effect (similar in magnitude) as the smaller
diameter run on the second invariant. For the base diameter, the flow is more strain-driven, while for
the smaller particle diameter the flow is more vorticity-driven. The 3-D model shows mixed results.
In some ways, it behaves like the smaller diameter axisymmetric case (trending up at the right), but in
other ways it trends like the other case (an upward blip early in the jet wake and moving negative on
the right). The view in Figure 22 of the axisymmetric cases is rather uneventful compared to the 3-D
case, which shows a large increase in the focal measure toward the end of the jet wake. 3-D coupling
effects obviously play a much larger role in the focal aspect of turbulent structures.
There are two items not shown graphically here, but are worth mentioning. The third invariant
measure, R, did not change much as a result of particle coupling with the continuous phase. The
overall vessel averages were negative numbers (similar in magnitude) for the base particle size
comparison, indicating both coupled and uncoupled flows were typically stable, focus-stretching.
Also, not shown are some typical mean deformation eigenvalue combinations found in the flow fields.
Time did not permit gathering enough data to make global conclusions, but a few nonfocal (three real
10
tensor eigenvalues) points were found on the vessel cross-sectional plane 1 jet diameter from the cone.
In the uncoupled case, a typical eigenvalue ratio was 1:-3:2 (compared to 1:-4:3 previously mentioned
in the Measures section). In a coupled case on the same plane, a typical ratio was 1:-11:11. The
author does not propose to build a theory on a few CFD point values from a steady RANS model, but
the possibility of a coupling effect here could be worthy of further pursuit with LES/DNS.
3. Conclusions
A series of steady-state RSM DRW models have been used to attempt to quantify the effects of
particle coupling at 0.25 mass loading on the continuous phase mean deformation and turbulence
fields. Numerical sensitivity tests, grid independent solutions, and data from an experimental
counterpart helped establish the method. The basis and caveats for various aspects of modeling were
also presented. Certain aspects of the Eulerian-Lagrangian approach were shown to be very important
in ensuring an accurate, statistically sound outcome. The following continuous phase features were, in
general, increased as a result of particle coupling (feedback) effects: turbulence viscosity, turbulence
production:dissipation, turbulence intensity (normalized by superficial velocity), turbulence
anisotropy, vorticity-dominant regions, and the focal aspect of the turbulent structures. Parametric
quantification was given for these various measures. None of the turbulence attenuation that is
discussed in the open literature was found for the stokes number and other conditions of the present
work. Preliminary results indicate that there may be an important effect of coupling on the mean
deformation tensor eigenvalues. For some measures, particle size and 3-D effects appear to play an
important role.
4. Future Work
The ultimate goal is to make mechanical modifications to the separator geometry and reduce the
carryover rate. Now that a reasonable method has been established, mechanical change numerical
evaluations will be possible. A number of items could be considered to improve the method. Liquid
droplet coalescence and breakup effects could be incorporated only with an experimental counterpart.
The use of particle-induced Reynolds stress source terms could offer advancements in the physical
soundness of related studies. More time could be spent further investigating 3-D, particle size, and
mean deformation eigenvalue effects. Also, the potential for false migration of particles due to kinetic
energy gradients could be explored as discussed in Strutt and Lightstone [33]. In addition, other
measures of anisotropy such as Taylor's anisotropy coefficient utilized in Keirsbulk et al. [34] could be
pursued. Lastly, the use of a transient, non-RANS approach could provide a deeper understanding of
the phenomena discussed in this work.
5. Nomenclature
b Normalized anisotropy tensor
d Particle diameter
F Determinant of NRAT
g Gravity
I Turbulence intensity (super. vel.)
k Turbulence kinetic energy
l Turbulent length scale
L Distance from Lumley origin
PR Invariants used in discriminant
r Uniform random number (01)
S Source term
T Time
11
u
'
Fluctuating velocity component
Z Particle mass loading,
c
p
mass
mass
IIII Tensor invariants
Time-averaged quantity
Greek
Response time
Kronecker Delta
Turbulence dissipation rate
, Lumley triangle parameters
Kinematic viscosity
Normally distributed random number
Molecular viscosity
RMS fluctuation velocity
Eigenvalues
Deformation discriminant
Subscripts
c Continuous phase
D Drag
DT Deformation tensor
el Eddy lifetime
ec Eddy crossing
i,j,k,l Tensor indices
L Lift
NRAT Norm. Rey. stress anisotropy tensor
p Dispersed phase
s Slip
6. Acknowledgements
The author would like to recognize those who contributed to this effort. Ken Dooley and Mary Lamar,
both of Eastman Chemical Company, provided experimental data and particle fate output file
statistical analysis.
7. References
[1] Obermair, S., Gutschi, C, Woisetschlager, J., and Staudinger, G., "Flow Pattern and
Agglomeration in the Dust Outlet of a Gas Cyclone Investigated by Phase Doppler
Anemometry," Powder Technology, 156, pp. 34-42, 2005.
[2] Gil, A., Cortes, C., Romeo, L., and Velilla, J., "Gas-particle Flow Inside Cyclone Diplets with
Pneumatic Extraction," Powder Technology, 128, pp. 78-91, 2002.
[3] Vandu, C. and Krishna, R., "Volumetric Mass Transfer Coefficients in Slurry Bubble
Columns Operating in the Churn-turbulent Flow Regime," Chemical Engineering and
Processing, 43, pp. 987-995, 2004.
[4] Faeth, G., "Spray Atomization and Combustion," AIAA Paper No. 86-0136, 1986.
12
[5] Ahmadi, G. and Abu-Zaid, S., "Anisotropic Model for Dense Two-phase Flows," ASME-
ASCE-SES Joint Summer Meeting, Mechanics of Deformation and Flow of Particulate
Materials, pp. 308-320, 1997.
[6] Graham, D., "The Influence of Small Particles on the Structure of a Turbulent Shear Flow,"
Journal of Fluids Engineering, 126, pp. 613-619, 2004.
[7] Mostafa, A., Mongia, H., McDonell, V., and Samuelsen, G., " Experimental and Numerical
Study of Particle-laden Coaxial Jet Flows," International Journal of Heat and Fluid Flow, 11,
pp. 90-98, 1990.
[8] Boree, J. and Caraman, N., "Dilute Bidispersed Tube Flow: Role of Interclass Collisions at
Increased Loadings," Physics of Fluids, 17, p. 055108-055, 2005.
[9] Crowe, C., Sommerfeld, M., and Tsuji, Y., Multiphase Flows with Droplets and Particles,
CRC, 1998.
[10] Heinl, E. and Bohnet, M., "Calculation of Particle-Wall Adhesion in Horizontal Gas-Solids
Flow Using CFD," Powder Technology, 159, pp. 95-104, 2005.
[11] FLUENT, Inc., FLUENT 6.2 User's Guide, Lebanon, NH, 2005.
[12] Clift, R., Grace, J., and Weber, M., Bubbles, Drops, and Particles, Academic Press, 1978.
[13] Morsi, S. and Alexander, Al, "An Investigation of Particle Trajectories in Two-Phase Flow
Systems," Journal of Fluid Mechanics, 55, pp. 193-208, 1972.
[14] Krishna, R. and van Baten, J., "Simulating the Rise Characteristics of Gas Bubbles in Liquids
Using CFD," Chemical Engineering Technology, 24, pp. 427-430, 2001.
[15] Zheng, Y., Fan, J., Lou, K., and Cen, K., "Direct Numerical Simulation on the Particle
Dispersion by the vortex Structures in a Three-dimensional Mixing Layer; the Roll-up,"
Powder Technology, 146, pp. 102-110, 2004.
[16] Sommerfeld, M., "Validation of a Stochastic Lagrangian Modelling Approach for Inter-
particle Collisions in Homogeneous Isotropic Turbulence," International Journal of
Multiphase Flow, 27, pp. 1829-1858, 2001.
[17] Soo, S., Fluid Dynamics of Multiphase Systems, Blaisdell, 1967.
[18] Amsden, A., O'Rourke, P., and Butler, T., "KIVA-II: A Computer Program of Chemically
Reactive Flows with Sprays," Los Alamos Report LA-11560-MS, 1989.
[19] Xiong, Y., Zhang, M., and Yuan, Z., "Three-dimensional Numerical Simulation Method for
Gas-Solid Injector," Powder Technology, 160, pp. 180-189, 2005.
[20] Cokljat, D., Slack, M., Vasquez, S., Bakker, A., and Montante, G., "Reynolds-Stress Model
for Eulerian Multiphase," Progress in Computational Fluid Dynamics, 6, pp. 168-178, 2006.
[21] Launder, B., Reece, G., and Rodi, W., "Progress in the Developments of a Reynolds-Stress
Turbulence Closure," Journal of Fluid Mechanics, 68, pp. 537-566, 1975.
[22] Speziale, C., Sarkar, S., and Gatski, T., "Modelling the Pressure-Strain Correlation of
Turbulence: An Invariant Dynamical Systems Approach," Journal of Fluid Mechanics, 277,
pp. 245-272, 1991.
[23] Hanjalic, K., "Will RANS Survive LES? A View of Perspectives," Journal of Fluids
Engineering, 127, pp. 831-839, 2005.
[24] Speziale, C., "Turbulence Modeling for Time-Dependent RANS and VLES: A Review,"
AIAA Journal, 36, pp. 173-184, 1998.
[25] Pope, S., Turbulent Flows, Cambridge University Press, 2000.
13
[26] Strasser, W., "CFD Investigation of Gear Pump Mixing Using Deforming/ Agglomerating
Mesh," Journal of Fluids Engineering, Accepted for April 07 publication.
[27] Strasser, W., Feldman, G. , Wilkins, C., and Leylek, J., “Transonic Passage Turbine Blade Tip
Clearance with Scalloped Shroud: Part II – Losses with and without Scrubbing Effects in
Engine Configuration,” ASME Paper No. IMECE2004-59116.
[28] Lumley, J., "Computational Modeling of Turbulent Flows," Advances in Applied Mechanics,
18, pp. 123-176, 1978.
[29] Galletti, C., Brunazzi, E., Pintus, S., Paglianti, A., Yianneskis, M., "A Study of Reynolds
Stresses, Triple Products, and Turbulence States in a Radially Stirred Tank with 3-D Laser
Anemometry," Chemical Engineering Research and Design, 82, pp. 1214-1228, 2004.
[30] Pirozzoli, S., and Grasso, F., "Direct Numerical Simulations of Isotropic Compressible
Turbulence: Influence of Compressibility on Dynamics and Structures," Physics of Fluids, 16,
pp. 4386–4407, 2004.
[31] Aris, R., Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Dover, 1962.
[32] Li, X. and Wang, T., “Effects of Various Modeling Schemes on Mist Film Cooling
Simulation,” ASME Paper No. IMECE2005-81780.
[33] Strutt, H. and Lightstone, M., “Analysis of Tracer Particle Migration in Inhomogeneous
Turbulence,” International Journal of Heat and Mass Transfer, 49, pp. 2557-2566, 2006.
[33] Keirsbulck, L., Labraga, L., and Haddad, M., “Influence of Blowing on the Anisotropy of the
Reynolds Stress Tensor in a Turbulent Channel Flow,” Experiments in Fluids, 40, pp. 654-
662, 2006.
7. Figures
Mixture Inlet
Gas Exit
Interaction With
Counter Flow
Stagnation
Mixture Inlet
Gas Exit
Interaction With
Counter Flow
Stagnation
0
1000
2000
3000
4000
5000
6000
0 200 400 600 800 1000
Random DRW Steady State Samples
Carryover [PPMW Liquid Feed]
Figures 1 and 2: Figure 1 contains contours of velocity along axial slices for the preliminary 3-D gas-
only studies. Red indicates gas that has a velocity magnitude greater than the particle slip velocity.
Figure 2 shows typical carryover samples for coupled "base" particle size. The mean is ~ 900 PPMW.
14
Uncoupled CoupledUncoupled Coupled
Uncoupled CoupledUncoupled Coupled
Uncoupled CoupledUncoupled Coupled
Uncoupled CoupledUncoupled Coupled
Figures 3, 4, 5, and 6: Typical instantaneous contours of particle concentration, turbulence viscosity
ratio, production:dissipation ratio, and turbulence intensity, respectively with and without feedback
from particles of base diameter.
0
5
10
15
20
25
0 2 4 6 8 10 12
Jet Diameters From Cone Exit
TVR Coupled : TVR Uncoupled
Axisymmetric, Base d Axisymmetric, Small d 3D, Small d
0
1
2
3
4
5
0 2 4 6 8 10 12
Jet Diameters From Cone Exit
PD Coupled : PD Uncoupled
Axisymmetric, Base d Axisymmetric, Small d 3D, Small d
Figures 7 and 8: Planar average coupled:uncoupled ratios on axial slices showing turbulence viscosity
ratio and turbulence production:dissipation for three pairs of cases.
0
2
4
6
8
10
12
14
16
0 2 4 6 8 10 1 2
Jet Diameters From Cone Exit
I Coupled : I Uncoupled
Axisymmetric, Base d Axisymmetric, Small d 3D, Small d
Figure 9: Planar average coupled:uncoupled ratios on axial slices showing turbulence intensity
(normalized by superficial velocity) for three pairs of cases.
15
Uncoupled CoupledUncoupled Coupled
Uncoupled CoupledUncoupled Coupled
Uncoupled CoupledUncoupled Coupled
Uncoupled CoupledUncoupled Coupled
Figures 10, 11, 12, and 13: Typical instantaneous contours of the first and second Reynolds stress
anisotropy tensor parameters, determinant, and normalized length, respectively.
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
0 2 4 6 8 10 12
Jet Diameters From Cone Exit
Lnorm Coupled :Lnorm Uncoup led
Axisymmetric, Base d Axisym metric, Small d 3D, Small d
0.80
0.84
0.88
0.92
0.96
1.00
1.04
1.08
0 2 4 6 8 10 12
Jet Diameters From Cone Exit
F Coupled : F Uncoupled
Axisymmetric, Base d Axisym metric, Small d 3D, Small d
Figures 14 and 15: Planar average coupled:uncoupled ratios on axial slices showing the normalized
length of departure from isotropy (Eq. 11) and the NRAT determinant (Eq. 10), respectively, for three
pairs of cases.
0.7
0.8
0.9
1.0
1.1
1.2
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Dimensionless Vessel Radius
Coupled : Uncoupled
Lnorm Intensity Axial Vel
Jet Interface
0.7
0.8
0.9
1.0
1.1
1.2
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Dimensionless Vessel Radius
Coupled : Uncoupled
Lnorm Intensity Axial Vel
Jet Interface
0
1/6
- 1/6 0 1/6
Uncoupled C oupled Axi pos Axi neg
Figures 16 and 17: Figure 16 shows azimuthally-averaged coupled:uncoupled ratios on an axial slice
at a distance of 1 jet diameter from the cone exit showing the normalized length of departure from
isotropy, local turbulence intensity, and axial velocity. Figure 17 depicts a Lumley plot at the same
location.
16
Uncoupled CoupledUncoupled Coupled
Uncoupled CoupledUncoupled Coupled
Uncoupled CoupledUncoupled Coupled
Figures 18, 19 and 20: Figures 18 and 19 include typical instantaneous contours of Q and ,
respectively. Figure 20 includes typical instantaneous contours of on a plane at a distance of 1 jet
diameter from the cone exit
-500
-400
-300
-200
-100
0
100
200
300
400
0 2 4 6 8 10 12
Jet Diameters From Cone Exit
Q Coupled : Q Uncoupled
Axisymmetric, Base d Axisymmetric, Small d 3D, Small d
1
Pos.
Neg.
-500
-400
-300
-200
-100
0
100
200
300
400
0 2 4 6 8 10 12
Jet Diameters From Cone Exit
Q Coupled : Q Uncoupled
Axisymmetric, Base d Axisymmetric, Small d 3D, Small d
1
Pos.
Neg.
-1000000
0
1000000
2000000
3000000
4000000
5000000
6000000
7000000
8000000
9000000
10000000
0 2 4 6 8 10 12
Jet Diameters From Cone Exit
Coupled : Uncoupled
Axisymmetric, Base d Axisymmetric, Small d 3D, Small d
1
Pos.
Neg.
-1000000
0
1000000
2000000
3000000
4000000
5000000
6000000
7000000
8000000
9000000
10000000
0 2 4 6 8 10 12
Jet Diameters From Cone Exit
Coupled : Uncoupled
Axisymmetric, Base d Axisymmetric, Small d 3D, Small d
1
Pos.
Neg.
Figures 21 and 22: Planar average coupled:uncoupled ratios on axial slices showing Q and ,
respectively, for three pairs of cases.
