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DRAFT – V9
CFD INVESTIGATION OF GEAR PUMP MIXING USING
DEFORMING/AGGLOMERATING MESH
Wayne Strasser
Eastman Chemical Company, Kingsport, TN, 37660
strasser@voridian.com
ABSTRACT
A moving-deforming grid study was carried out using a
commercial CFD solver, Fluent® 6.2.16. The goal was to
quantify the level of mixing of a lower viscosity additive (at a
mass concentration below 10%) into a higher viscosity process
fluid for a large-scale metering gear pump configuration typical
in plastics manufacturing. Second order upwinding and
bounded central differencing schemes were used to reduce
numerical diffusion. A maximum solver progression rate of
0.0003 revolutions per timestep was required for an accurate
solution. Fluid properties, additive feed arrangement, pump
scale, and pump speed were systematically studied for their
effects on mixing. For each additive feed arrangement studied,
the additive was fed in individual stream(s) into the pump
intake. Pump intake additive variability, in terms of coefficient
of spatial variation (COV), was > 300% for all cases. The
model indicated that the pump discharge additive COV ranged
from 45% for a single centerline additive feed stream to 5.5%
for multiple additive feed streams. It was found that viscous
heating and thermal/shear-thinning characteristics in the process
fluid slightly improved mixing, reducing the outlet COV to
3.2% for the multiple feed stream case. The outlet COV fell to
2.0% for a half-scale arrangement with similar physics. Lastly,
it was found that if the smaller unit’s speed were halved, the
outlet COV was reduced to 1.5%.
INTRODUCTION
Positive displacement gear pumps are commonly used for
precise viscous fluid metering on an industrial scale in both
plastics and food manufacturing. One would likely not propose
installing a new gear pump simply for any potential mixing
capabilities, because of its high level of energy consumption. If
a metering gear pump is already in use, however, the ability to
also utilize this gear pump for additive blending offers certain
advantages. The user could not only avoid both the purchase
and installation costs of new mixing equipment, but it could
also alleviate the need to purchase and install more gear
pump(s) to overcome the additional pressure drop of said
mixing equipment. The goal of this work is, therefore, to
investigate whether or not additives can be blended to a
satisfactory level using an existing in-service industrial-scale
gear pump.
One may browse a mixing equipment manufacture’s
website and find common examples of static mixers for viscous
liquid applications, the use of COV (local standard deviation
divided by the local mean) for a mixedness measure, and how
the COV at the outlet of such devices depends on the COV at
the inlet. There appears to have been few studies of mixing
within metering (positive displacement) gear pumps in the open
literature and patent databases. Patents 6,313,200 and
6,601,987, for example, involve planetary gear arrangements.
In fact, no computational metering gear pump mixing
investigation could be found, nor could concrete data from an
experimental study be located. Kramer [1] stated in his
industrial gear pump performance study, “The mechanism is
basically a straight-through device, though, and does no
mixing” (Underline included). In an agricultural engineering
study by Bouse et al. [2], the improved performance of certain
chemical sprays was noted as a result of multiple passes through
a gear pump. The authors, however, did not make a
quantifiable distinction between shear history effects and
improved mixedness resulting from multiple gear pump passes.
Studies of mixing within axial (primarily) flow units, like
extruders, are more common. An interesting experimental study
is that of Valsamis and Pereira [3] in which enhancements to the
Farrel Continuous Mixer concept are explored. Localized
circulatory cells which promote backmixing and lateral motion
of material are shown to improve blending. In a metering gear
pump, there is little to no flow in the direction normal to the
primary flow axis; Therefore, it would typically not have this
sort of lateral mixing. Also, “inserts/dams” which convert axial
flow to radial flow, were shown to reduce melt defects in these
extruders. It must be noted, however, that the unit in the study
of [3] is non-intermeshing. In other words, the swept
areas/volumes of the rotating members do not interfere with one
another. The difficulties associated with CFD modeling of
intermeshing units (like gear pumps) would not be present in
the study of said device.
2 Copyright © 2005 by ASME
An example of the study of flow in an intermeshing co-
rotating extruder is the computational work of Bruce et al. [4].
The authors discuss forward mixing, back mixing, and generally
swirling compartments within various extruder sections. They
employ a mixing efficiency parameter, also called an
extensional efficiency as in Heniche et al. [5]. The parameter is
defined as the ratio of the symmetric rate of deformation tensor
magnitude to the sum of this magnitude and the vorticity tensor
magnitude. While the tensors, themselves, are not frame
invariant, their magnitudes (scalars) are indeed frame invariant
[6]. As a result, fair comparisons can be made between various
studies. An efficiency value of 0 would represent purely
rotational flow, while a value of unity would correspond to
purely elongational flow. In general, a higher value indicates
better mixing as long as the areas of higher efficiency are not
segregated / isolated. The volume-averaged value of the mixing
parameter ranged from approximately 0.52 to 0.55 in the
extruder studies of [4]. Typical values for a laminar static mixer
from [5] ranged from 0.4 to 0.8 axially, but averaged 0.6.
Bulk FlowBulk Flow
Figure 1: Computational grid from a Polyflow® study
A common technique employed in CFD investigations of
intermeshing systems, as in the study by Bruce et al. [4], is the
process of superposition. Volumes of stationary and rotational
components are constructed and gridded separately. The grid
nodes need not conform to one another at any instant in model
time over the course of the simulation. Grid cells are allowed to
overlap freely, and information is interpolated accordingly.
Figure 1 from a website example by Fluent, Inc. shows the
instantaneous computational grid of a gear pump study using
Polyflow® (finite element formulation). Notice the dramatic
shifts in grid cell size/centroid that occur throughout the domain
during the gear rotational sequence. As shown in the circle,
grid cells are approximately 1/20
th
the area of a neighboring
grid cell at some locations in the computational domain. Sharp
grid cell size or centroid shifts are known to produce substantial
numerical diffusion, making the calculated mixing rate appear
to be better than in reality. Even if the grid were more uniform,
this method may work for pressure profiles and mass flow rate,
but not be ideally suited for mixing studies. Although the
author is not aware of any studies of numerical diffusion of
superposition nodal interpolations, one would expect the
diffusive nature of this approach to be similar to that of non-
conformal interfacial interpolations.
An alternative to superposition is moving-deforming grid.
This concept is relatively new and appears to be somewhat
unexplored. At each computational time step, the model
geometry is advanced a predetermined amount. The
computational grid is then altered/recreated to accommodate
new shapes before the flow or thermal solution is sought
through iterative means. One would expect this to be more
computationally expensive than superposition. Not only are the
grid cell counts likely to be larger, but the grid reconstruction
time is added to each time step. Improved accuracy and
reduced numerical diffusion are the rewards. Fluent, Inc.
offered this method commercially beginning in February of
2003 in their finite volume code. Another commercial CFD
solver, CFX® (mixed finite volume/finite element), began
showcasing this capability around a similar timeframe, but there
is a distinct difference between the two codes’ capabilities. The
mesh within CFX® will only deform to accommodate changes
in shape, while Fluent® will create/agglomerate cells to
maintain grid quality. One can imagine that cell aspect ratio or
skewness can become a significant problem for dramatic
changes in geometry shape with only a deformable grid.
NOMENCLATURE
B Generic constant
C
m
Mixture heat capacity =
k
k
C
k
/
k
k
k
m
Mixture thermal conductivity =
k
k
k
p Static pressure
S Source associated with pump movement
T Mixture static temperature
u
i
Velocity component of the mixture
ij
Stress tensor of the mixture
m
Mixture density =
k
k
k
Volume fraction of phase k
m
Mixture apparent viscosity =
k
k
P0
Zero-shear viscosity of process fluid
Relaxation time of process fluid
Mixture strain rate magnitude
Subscripts
A Additive phase, or fluid
k Generic phase, or fluid
m Mixture property
P Process phase, or fluid
3 Copyright © 2005 by ASME
MODEL
The present study involves unsteady, laminar, multiphase
flow of a mixture of viscous materials through an intermeshing
industrial-scale metering gear pump. The two materials
involved are different "phases" only in the sense that they have
significantly differing properties. The process fluid has a
viscosity which is 1 - 2 orders of magnitude larger than that of
the additive. The additive is fed at relatively low mass
concentrations (<10%) in the pump intake as individual
stream(s) of essentially pure additive. Fluent® 6.2.16, the latest
release commercial finite volume solver from Fluent, Inc., is
used to evaluate the mixing of these streams inside the pump.
The problem involves all three spatial dimensions, because
there is a helical nature to the gears in the pump. The axial flow
components are expected to comparatively small, and the axial
mixing should be minimal. As a result, the problem was
condensed to two dimensions. This makes the study more
conservative in that any 3-D flow would only increase phase
blending.
Physics
The Eulerian mixture model is a simplification of the full
Eulerian-Eulerian multiphase approach. Each liquid is treated
as a separate interpenetrating phase. Mass transfer between
phases is ignored, as well as momentum exchange through
processes such as drag, lift, etc. Properties are volume fraction-
weighted (except for heat capacity, which is mass fraction-
weighted) as shown in the NOMENCLATURE section and are
consistent with the “mixture” model concept. There is a single
pressure and a single velocity field shared by both phases. In
other words, it is assumed that, on a subgrid scale, the phases
have reached local equilibrium. In the mixture approach, the
effects of two-way phase coupling are included in order to be
able to address systems in which the fluids have very different
viscosity ratios. In general, it is expected that this volume-
weighted approach would be more diffusive than the full
Eulerian-Eulerian approach, but this effect will be quantified in
the Numerical Diffusion Tests section. Linear momentum of
the mixture is conserved via Equation 1 (shown in Cartesian
coordinates).
S
xx
uu
t
u
j
ij
j
ji
i
m
+
∂
∂
=
∂
∂
+
∂
∂
τ
ρ
(1)
The materials have similar densities so the effects of natural
convection could be ignored. The mixture stress tensor is
calculated:
∂
∂
+
∂
∂
+−=
i
j
j
i
mij
x
u
x
u
p
ητ
(2)
For the computational cases in which the primary fluid is non-
Newtonian, the deviation from Newtonian behavior is
incorporated into
P
(subset of
m
). The author is aware that
viscous fluids, such as polymers, often exhibit normal stress
differences upon shearing [7]. That is, a viscous tension is
developed along a streamline. In addition, there is sometimes
“memory” associated with these stresses. A thorough discussion
of interesting secondary flows that accompany these types of
fluids is given by Siginer and Letelier [8]. For this particular
process fluid, however, it is known through testing within
Eastman Chemical Company that Carreau-Yasuda relations can
predict the correlation between apparent viscosity and shear rate
throughout the conditions modeled (Equations 3-5). The
concept of having
P
be a scalar function of deformation is
discussed in Bird et al. [9] and is not new.
−⋅=
3
210
11
exp BT
BB
P
η
(3)
−⋅=
3
54
11
exp BT
BB
θ
(4)
[
]
76
)(1
0BB
PP
γθηη
⋅+=
(5)
The additive phase volume fraction is conserved using phase
continuity:
(
)
(
)
0=
∂
∂
+
∂
∂
i
iAA
AA
x
u
t
ρ
α
ρ
α
(6)
Energy of the mixture is conserved via:
j
i
ij
i
m
i
i
mm
x
u
x
T
k
x
Tu
t
T
C∂
∂
−
∂
∂
=
∂
∂
+
∂
∂
τρ
2
2
(7)
Of course, the kinetic and potential energy terms have been
neglected as they have little to no impact in the present
simulation. Viscous heating effects are included for some of the
cases using the definition of the irreversible dissipation of
energy through viscous losses as shown in the last term of
Equation 7. The pressure-work term
i
i
x
u
T
p
T∂
∂
∂
∂
has been
neglected from the RHS of Equation 7 as the models were run
at low pressure differential (< 1 Bar), and the fluid is effectively
incompressible. The walls and gear surfaces were treated as
adiabatic, so there was no net thermal exchange within the
pump or externally. The rotational speed of the gear set
matched the typical for the process fluid at plant throughput.
4 Copyright © 2005 by ASME
Computational Grid
Numerical diffusion is an important consideration in any
mixing study. Artificial smearing of gradients would produce a
computational mixing level higher than reality. (This will be
discussed in further detail in a later section). Solution
boundedness is an equally important consideration in order to
prevent the opposite problem from occurring. The
computational grid was built using lessons learned from
transonic gas turbine passage gridding discussed in Strasser et
al. [10] to minimize the numerical error associated with these
two issues.
Figure 2 displays a typical computational grid (nominally
65,000 grid cells in a pump cross-section) in the present study.
Quadrilateral cells make up the inlet (top of figure) as well as
the outlet (not shown). The “deforming” zone filling the
spaces around the gears is made up of triangular cells. As
shown in Figure 2, grid cell area/centroid shifts are handled
gradually, especially the quadrilateral-triangular cell transition
that is often overlooked in computational studies. Exceptional
Cavity
Tip
Inlet Zone
Deforming
Zone
Floor
Bulk Flow
Cavity
Tip
Inlet Zone
Deforming
Zone
Floor
Cavity
Tip
Inlet Zone
Deforming
Zone
Floor
Bulk Flow
Figure 2: Typical computational grid used in present study
resolution is provided across the inlet zone (nearly 400 grid
points) and outlet zone to minimize cross-diffusion. Grid cell
aspect ratios (in the directions normal to the flow) were
minimized. As the gears rotate, triangular cells in the
deforming zone are agglomerated and/or created (using spring-
based smoothing) to maintain grid quality. Cells are created on
the trailing tooth edge, while cells are destroyed on the leading
tooth edge. In the gear intermeshing zone, where the teeth pass
very near each other, the cell creation/destruction process must
proceed aggressively but smoothly. In the real world gear pump
the teeth actually touch, but computationally they are kept apart
by a small cushion about the size of the smallest cell in the
domain. Area-averaged grid cell equivolume skewness values
for the deforming zone are typically 0.15 (0 being an equilateral
triangle and 1 being a sliver cell) over the course of the run and
never exceed 0.16. Minimizing grid cell skewness is another
factor which is important for minimizing numerical diffusion.
In addition to being diffusive, highly skewed cells (skewness >
~0.8) have been found to cause the local computation of
artificially high strain rates in other internal Eastman Chemical
Company CFD models. Gridding parameters in the code are
utilized such that the computational cell sizes in the deforming
zone are maintained to be nearly the same size as the boundary
cells (smallest in the figure), i.e. cell size growth is minimized
in the boundary-normal direction in order to improve accuracy.
Other methods and measures of controlling deforming grid
quality are available, such as that in [11] in which smaller cells
are adjusted relatively less than larger cells from time step to
time step, but cell skewness was the primary concern in this
study. During cell agglomeration and creation mass and
momentum are conserved, and gradients are interpolated based
on the numerical grid at the previous time step.
Discretization
Care was taken to utilize discretization schemes within the
commercially released Fluent® 6.2.16 that were a blend of
diffusion and boundedness considerations. For example,
switching from first order upwinding to second order upwinding
has been found to have dramatic effects on multiphase
calculations [12]. For the momentum balance, a bounded
central differencing scheme was used for spatial interpolation in
both quadrilateral and triangular cells. This is the most accurate
stencil that is offered in this particular code release for grids
that are not always aligned with the flow. A third-order MUSCL
scheme is available, but does not contain flux limitations and
could cause overshoot for non-aligned flows [13]. Since grid
cell peclet numbers were everywhere less than unity, it was
concluded that central differencing would not experience
dispersion problems. All other variables (except pressure) were
spatially interpolated using a second-order upwind/central
differencing blended scheme. For quadrilateral cells, the
blending constant was solution dependent and varied locally to
stay as close to central as possible without the introduction of
new maxima in the flow field. For triangular cells, the blending
constant was zero, meaning that second-order upwinding was
used. Pressure interpolation was handled using a purely
second-order upwinding scheme for all grid cells. In all cases,
first-order interpolation was used for the transient terms as this
was the only option available for multiphase models in release
6.2.16. Derivatives were discretized using the nodal method
(weighted by nodal values on surrounding faces instead of
simple arithmetical grid cell center averages), which is more
accurate especially for triangular cells. The settings discussed
here are consistent with the best practices proposed by Fluent,
Inc.
5 Copyright © 2005 by ASME
Convergence
The SIMPLE algorithm was used for pressure-velocity
coupling via the segregated implicit solver. Although it is
known for some problems coupling can be more efficiently
handled by SIMPLEC or other methods, numerical tests
confirmed that the flux corrections within SIMPLEC were not
worth the additional CPU time in residual reduction. An
advanced multidimensional slope limiting scheme (total
variation diminishing) was utilized to keep cell gradients under
control. Pressure checker-boarding was prevented using a
second order Rhie-Chow method. The single precision solver
version was used in that it was found that the use of double
precision did not change the results to three significant figures.
Algebraic multigrid was used to reduce large wave error
propagation, and the multigrid termination criteria were lowered
an order of magnitude for pressure and volume fraction to
ensure deep mass convergence. Time stepping was fixed at
approximately 0.0003 revolutions per time step. Time step
increments above this caused the grid deformation algorithm to
fail. Twenty-five sub-iterations were required per time step for
the “flattening” of residuals. Certain underrelaxation factors
were increased to values higher than the default in order to
promote residual reduction. Residual RMS values fell typically
3 orders of magnitude each time step, with the maximum RMS
value at the end of a time step being of order E-5. Additive
mass imbalance, in a time-averaged sense, was maintained to
near-zero. It did not typically exceed +/- 0.5% on an
instantaneous basis. The model ran at a rate of nominally 0.6
computational revolutions per day of CPU time on a Dell
Precision® 450 Workstation. Three to four weeks (>15
revolutions) of run time was normally required to reach a quasi-
steady state. Quasi-steady was defined as the point at which the
outlet additive statistics became stationary in time. Once this
occurred, time-averaging data collection began.
RESULTS
Numerical Diffusion Tests
As would be expected with any CFD “experiment”, a
sanity check should be performed to test the method.
Numerical diffusion cannot be completely avoided although
every reasonable attempt was made to do so. A steady-state
CFD test case was built which somewhat mimicked the grid for
the real gear pump CFD case. It contained a single additive
feed stream (concentrated strand) at the pump intake centerline
and a Newtonian process fluid. The inlet and outlet grid
topology, shape, and size matched the real pump CFD cases
exactly. In the triangular zone, the grid was expanded and
contracted (similar to Figure 2) in and around the outline of
fictitious gear teeth so that opportunities for numerical diffusion
could be brought to the forefront. Of course, there were no
moving parts in the test case, and the flow proceeds straight
through each cell zone. Based on discussions in the classic
Patankar text [14] and shockwave-capturing findings of [15],
the triangular zone is expected to provide the most numerical
mixing. Besides the fact that the test case was run in steady
state mode, all other solver settings were exactly the same as
those discussed in the previous sections. The computational
domain for this straight-through test case is shown in Figure 3.
Bulk FlowBulk Flow
Figure 3: Computational domain for the straight-through
numerical diffusion test case
0
2
4
6
8
10
12
14
0.40 0.45 0.50 0.55 0.60
Normalized Distance Across Boundary
Normalized Concentration
Inlet
End of Inlet Zone
Start of Outlet Zone
Outlet
Figure 4: Normalized additive mass concentration profiles
for the straight-through numerical diffusion test case
Figure 4 shows additive concentration profiles normalized
by the mass flow-weighted area-averaged (MFWAA) additive
concentration of the feed. Notice the additive concentration
profile changes from a square wave to a Gaussian-like
distribution simply as a result of numerical mixing. Again,
there are no moving gears or other blending stimuli in this case.
Specifically, all of the diffusion occurred in the triangular zone.
6 Copyright © 2005 by ASME
The end of the quadrilateral-celled inlet zone is marked with
triangular symbols, while the beginning of the quadrilateral-
celled outlet zone is marked with x-shaped symbols. The
concentration distribution is not changed as the fluid moves
through these zones. The overall COV was reduced through the
triangular zone from 350 (equal to the inlet) to 290.
The author was not satisfied with this result, specifically
the distance traveled by the fluid mixture through the diffusive
triangular zone, so another test case was built. In the new test
case, the Newtonian mixture of materials (single additive feed
stream) had to flow around a perimeter of similar distance
magnitude experienced in the gear pump simulations. The
triangular zone was made up of larger triangles than in the first
test case in order to attempt to maximize the effects of
numerical blending. Figure 5 shows the second test case
(perimeter-flow) computational grid.
Bulk FlowBulk Flow
Figure 5: Computational domain for the perimeter-flow
numerical diffusion test case
The outlet spatial concentration profile was similar in
shape to that in Figure 4 and has not been repeated here. The
profile, however, was more spread out, indicating a higher level
of numerical blending (outlet COV=240). Another mixedness
measure could be the width about the outlet centerline of a
given % accumulated area under the spatial concentration
curve. The resulting width measure depends on the particular
% threshold chosen, so a number of thresholds were tested. In
general the distribution width was approximately doubled
across the triangular zone in the first numerical test case and
nearly tripled in the second numerical test case.
Still not satisfied with these results, in that they did not
take into account gradient smearing through cell
creation/agglomeration (deforming mesh) and first order
upwinding in time, a third test case was constructed. The grid
was similar to that in Figure 5, except that it involved the
Newtonian mixture (single additive feed stream) flowing around
a single center cylinder anchored in the triangular cell zone.
The cylinder's angular velocity and the solution time step were
typical of gear pump simulations discussed in the upcoming
sections. The rotation of the cylinder forced the triangular mesh
to deform (squeeze, create, and agglomerate) just as it would in
a gear pump simulation. The wall of the cylinder was set as a
slip boundary condition so that no vorticity would be created
and no flow disturbance would occur. The time-averaged (TA)
outlet COV was reduced to 180 through the triangular zone in
this test. Because this third test took into account all
foreseeable numerical mixing caused by the mixture approach,
deforming mesh, and spatial/temporal discretization, all
upcoming COV results will be scaled by a similar ratio (~0.5) in
order to isolate real gear pump mixing effects. Incidentally, the
area-averaged value of the mixing parameter for the test cases
were 0.5, indicating simple shear flow.
Single Additive Injection Stream (Case 1)
Figure 6 shows a typical cross-sectional instantaneous
concentration contour plot for the first gear pump case in which
a single additive inlet feed was utilized. Viscous heating and
thermal/shear-thinning (VHTST) were ignored for this case.
The white shade represents anything 60% higher than the
outlet MFWAA mass concentration. Black, of course,
represents zero additive concentration. The contour lines are
scaled in between these two values. The single “strand” of
additive coming down from the inlet is 100% additive, and the
white area enclosed by the gear outline is simply unoccupied
space. The inlet additive COV is >300%.
2
1
3
4
Flow
2
1
3
4
2
1
3
4
2
1
3
4
Flow
Figure 6: Typical instantaneous normalized additive mass
concentration contours for a single additive injection stream
and a Newtonian process fluid; White represents material
having a mass concentration 60% higher than the outlet
instantaneous MFWAA concentration
7 Copyright © 2005 by ASME
Notice how the additive-rich strand is pulled into encircled
region “1” just above the gear meshing zone (encircled region
“4”). For the most part, it remains bound near the cavity floor
throughout the gear movement cycle from the intake to the
discharge. Encircled region “2” marks the near-wall additive-
depleted region that is created by the inlet-fed process fluid
being cut off from the inlet by the sweeping gear teeth. As
consecutive cavity-bound fluid zones merge above the pump
outlet at encircled region “3”, additive-rich material gets split
into two main regions. Some of the cavity-bound rich material
gets pushed towards the outlet walls, while the majority of the
rich material moves towards the outflow core. Gear-induced
fluid motion appears to create interfacial surface area for phasic
mixing. The mechanisms proposed to be responsible for this
are discussed ahead. The TA outlet additive COV (already
adjusted for expected numerical diffusion) is 45% for this case.
Figure 7, showing normalized velocity vectors for a
typical instantaneous flow field, helps to explain further what is
causing the mass concentration trends. Here, the white shade
represents any velocity 10% higher than the tip rotational
velocity magnitude, and black corresponds to zero velocity.
Any vector passing across a surface boundary is simply an
artifact of increasing the length scale of the post-processed
vectors to make them visually insightful. During the rotational
cycle, the peak local velocity magnitude exceeds 15 times the
gear angular velocity in the gear meshing zone (encircled region
“4” in Figure 6). Also in region “4”, peak local strain rate
magnitude exceeds 40 times the gear angular velocity divided
by the diametrical gear-housing clearance. The area-averaged
value of the mixing parameter (used in [4] and [5]) in the
present case is 0.71, indicating that the flow is more
elongational than simple shear flow.
Bulk Flow
Stagnant
region
Mixing
cell
Radially
outward
flow
Radially
inward
flow
Tip
leakage
Bulk Flow
Stagnant
region
Mixing
cell
Radially
outward
flow
Radially
inward
flow
Tip
leakage
Figure 7: Typical instantaneous fluid vectors downstream
of gear meshing zone; White represents a velocity 10%
higher than the gear rotational velocity magnitude
Small mixing cells are created by fluid moving radially
inward on backward-facing teeth faces and fluid moving
radially outward on forward-facing teeth faces. The mixing
cells remain near the cavity floor and mostly unchanged in each
tooth cavity throughout the revolution process, except where the
teeth mesh. In the meshing zone, the mixing cells are mashed
and distorted, but never fully extinguished since there is always
sufficient radial spacing. Tooth-housing diametrical clearance
allows a small amount of each cavity’s fluid to leak (“tip
leakage”) into each subsequent gear swept area. It is difficult to
tell in any of the instantaneous contours shown in this paper, but
the coupling of the mixing cell and the tip leakage creates a
counter-rotating motion in each cavity, providing increased
interfacial area between the phases. The additive-depleted
regions near the walls and the rich regions near the floor tend to
mix through this motion, but the exchange is weak. A wider
radius of vortex rotation in the floor would probably help
exchange these two materials more efficiently. A design change
in the contour of the gear teeth may allow the vortex
entrainment to be increased.
Stagnant regions are created at the outlet side of the gear
meshing zone in an alternating fashion as the fluids moving
radially outward from alternating gear teeth meet and come to
rest. Similar stagnant regions occur on the inlet side (not
shown) of the meshing zone. It must be noted that where the
fluids meet head-on in the stagnant zones, there will likely be a
three-dimensional component to the flow. That has been
ignored in the present 2-D study.
Multiple Additive Injection Streams (Case 2)
The TA outlet COV is dramatically lowered from 45% to
5.5% by the introduction of four other inlet streams of pure
additive across the inlet, resulting in five equally spaced
concentrated feed strands. This is near the 5% COV typically
sought in static mixer applications. Here, again, VHTST effects
are ignored. The total additive feed rate is held constant, and
the inlet COV again is >300%. Figure 8, same grey scale as
Figure 6, provides instantaneous normalized mass concentration
contours on a pump cross-section for this case. The weaving
and breaking up of the strands at the intake side prevent the
building up of relatively high mass concentration additive
regions in the cavity floors as was the tendency of the single-
injection case. Since these strands seem to get pushed radially
outward near the housing wall, the higher mass concentration
regions for this model stayed nearer the walls. The area-
averaged value of the mixing parameter is 0.71 also for Case 2.
Viscous Heating, Thermal/Shear-Thinning (Case 3)
Further reduction in outlet COV, down to 3.2%, was found
by the incorporation of VHTST considerations into the five
stream injection case. The cross-sectional mass concentration
contours look extremely similar to those in Figure 8, so they are
not included here. The area-averaged value of the mixing
8 Copyright © 2005 by ASME
FlowFlow
Figure 8: Typical instantaneous normalized additive mass
concentration contours for five injection streams and a
Newtonian process fluid; White represents material having
a mass concentration 60% higher than the instantaneous
outlet MFWAA concentration
parameter is, again, 0.71 for Case 3. Figure 9, however, shows
new information regarding normalized static temperature. The
white shade represents anything 10°C higher than the inlet
temperature, while black corresponds to the inlet temperature.
FlowFlow
Figure 9: Typical instantaneous normalized temperature
contours for five injection streams in which VHTST effects
are taken into account; White represents material having a
temperature 10° higher than the inlet temperature
There are obvious similarities between the temperature
contours here and the mass concentration contours in Figure 6.
The temperature is relatively higher in the near-floor cavity-
bound material. At the outlet there is a split between core high-
temperature flow and near-wall high-temperature flow as in
Figure 6. The difference, of course, in the two figures is that
the additive source comes from the intake, while the thermal
source lies in the near-tooth regions (especially in the meshing
zone). Moving from the top of the figure to the bottom around
the gear circumferential direction, notice the fact that the high-
temperature material in the cavity floor somewhat disperses into
the fresh feed material captured by the gears. Then, as the
material is exposed to strain, it heats up moving towards the
outlet.
The viscosity of the process fluid varies across the outlet
instantaneously (time-averaged results not available) as can be
seen in Figure 10. The process fluid viscosity is scaled by the
MFWAA process fluid viscosity across the outlet. Mixture
strain rate magnitude, shaped like the typical laminar strain rate
profile, and mixture static temperature are scaled in a similar
fashion. The x-axis is the normalized distance across the outlet
from left to right. Velocity magnitude (not shown here) is
laminar-parabolic just as one might assume.
As expected, the viscosity profile is a mirror image of the
temperature profile. The interesting feature, however, is how
low the viscosity drops near the walls. The temperature doesn’t
rise any more here than it does in the core, yet the viscosity is
much lower than in the core. This lower viscosity occurs
because of the increased strain rate at the wall.
0.88
0.90
0.92
0.94
0.96
0.98
1.00
1.02
1.04
1.06
0.0 0.2 0.4 0.6 0.8 1.0
Normalized Distance Across Outlet
Normalized Viscosity & Temperature
0.0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
3.6
Normalized Strain Rate
Process Fluid Viscosity Temperature Strain Rate
Figure 10: Typical instantaneous normalized outlet process
fluid viscosity, mixture temperature, and mixture strain rate
magnitude for five injection streams and VHTST effects
Bulk fluid viscosity reduction is believed to be the source
of the mixing improvement achieved by the VHTST case. This
is expected to improve stirring action within the cavities
throughout the gear rotational cycle. Although there is variation
in the localized viscosity across the outlet, the overall process
fluid viscosity at the outlet is 22% less than that of the inlet.
This is a result of the overall increase in bulk temperature and
9 Copyright © 2005 by ASME
the instantaneous shear values at the outlet. Again, shear-
history effects have been excluded from this study.
Figure 11 gives insight into the outlet additive mass spatial
concentration profiles for all three full-scale cases. The TA
local concentration is normalized by the TA MFWAA
concentration. Time did not permit running the cases long
enough to developed perfectly symmetrical TA profiles. There
was a dramatic reduction in variability when the additive
injection stream count was increased from 1 to 5. There was
further reduction in variability when VHTST was incorporated.
Both of these effects are consistent with the previously-
mentioned COV values. Another notable feature of this plot is
the inversion of the outlet profile when the additive injection
stream count is increased from 1 to 5. Instead of the highest
concentration being at the outlet centerline (single injection), a
below-average value is seen at the centerline for multiple
injection streams. VHTST effects keep the profile similar, but
the peaks are moved slightly closer to the centerline. The two
multiple additive inlet concentration profiles are similar in
shape to the process fluid viscosity profile in Figure 10.
0.6
0.8
1.0
1.2
1.4
0.0 0.2 0.4 0.6 0.8 1.0
Normalized Distance Across Outlet
Normalized Concentration
Single Injection Five Injections Five Injections With VHTST
Figure 11: Time-averaged outlet additive mass
concentration profiles for the three full-scale cases
normalized by the time-averaged MFWAA outlet
concentration
Scale and Speed Effects (Cases 4 and 5)
Two more runs were carried out to attempt to isolate the
effects of pump scale and pump speed, respectively, in the
presence of multiple additive inlet streams and VHTST effects.
Each of these two new runs was initialized with the resulting
flow field from the most recently discussed case (case 3) and
then run in tandem. For case 4, case 3 was scaled down by a
factor of 0.48. This scaling applies linearly to everything
geometrically in the setup. The relative inlet additive
concentration and pump rotational speed were kept nearly the
same. Since the fluid properties remained the same, the system
Reynolds number fell by the scale factor. The outlet profiles
(like those in Figures 10 and 11) look extremely similar to those
of case 3 and have not been included here. The difference,
however, is that the TA outlet additive COV is even lower than
the full-size pump at a value of 2.0%. It seems counterintuitive
that a lower Reynolds number system would mix more
efficiently. To help relate case numbers, Table 1 tabulates all
CFD cases in this paper.
Table 1: CFD results summary; The shaded cells show the
modeling aspect that was changed from the previous run.
#
TA
Additive
Process
Outlet
Case
Feed
Fluid
Geometric
Speed
Additive
#
Streams
Physics
Scale
Scale
COV
1 1 Newtonian Full Full 45%
2 5 Newtonian Full Full 5.5%
3 5 VHTST Full Full 3.2%
4 5 VHTST Half Full 2.0%
5
5
VHTST
Half
Half
1.5%
Case 5 involved the scaled-down pump (case 4) being
slowed to 55% of the typical speed (Reynolds number lowered
accordingly). Again the outlet profile plots have been omitted
due to their similarity with those already shown in Figures 10
and 11. The TA outlet additive COV fell to 1.5%. Once again,
a system with an even lower Reynolds number shows a better
mixedness at the outlet. The bulk viscosity reduction for the
scaled-down pump (case 4) was 19%, while that of the scaled-
down/slowed pump (case 5) was 14%. Both of these values are
less than that of the full-scale case (case 3 at 22%), due mainly
to less viscous heating for the scaled-down cases. This further
magnifies the inverse Reynolds number – mixing rate trend.
One does not expect, however, that Reynolds number would
play an important part in laminar flow.
It is not clear why the smaller cases are more mixed.
There must be some subtle differences in the counter-rotating
mixing cells inside the cavities and/or the tip leakage rate
between cavities that allow the level of additive blending to be
higher in scaled-down cases 4 and 5. These differences could
not be detected from reviewing transient contour plots and
moving videos from the runs. The area-averaged mixing
parameter remained 0.71 in both scaled-down cases, so it
appears that scale and/or speed do not have a major impact on
the mixing efficiency. The fact that the COV was nearly halved,
while the mixing parameter did not change supports what is
discussed in [5] that a single mixing measure should not be used
to judge improved mixing in all cases. The mixing parameter
value must be unique to this particular pump design.
Experimental Validation of Case 4
A short experimental test of the gear pump additive mixing
concept has been carried out on half scale at full speed. The
setup employed a distributor that was designed to produce
multiple (not necessarily 5), separate feed streams. Of course
10 Copyright © 2005 by ASME
the experimental system involved VHTST effects, since any
omission of these real-world effects is purely a computational
choice. The experimental pump outlet COV varied in the range
15-30%, instead of the 2.0% CFD prediction. There are at least
three explanations for the fact that the CFD variability is 8 – 15
times lower than that of the experimental run. Firstly,
investigations into the performance of the distribution system
revealed that there was a strong potential for the additive feed
strands to become merged together at the pump intake (like
CFD Case 1). There is no way of knowing if this occurred
during the experimental test or to what extent. There was no
CFD run for a strand count between 1 and 5 including VHTST
effects at this scale and speed; However, one can conclude from
the CFD results shown in this paper that if most of the additive
strands were joined together, the COV could easily get pushed
upwards in the range of 15–30%. Secondly, the additive feed
system had stability problems during the test. A consistent
additive feed rate was not attainable. Thirdly, there may be
numerical diffusion that is somehow unaccounted for in the
three numerical diffusion tests discussed earlier. It is known
that any numerical diffusion will artificially bias the CFD COV
results downward; However, it is not likely that whatever has
not been accounted for is causing all the differences between
the CFD and experimental results.
CONCLUSIONS
A high-resolution computational study was undergone to
investigate additive mixing (mass concentrations < 10%) within
a viscous process fluid inside an industrial-scale intermeshing
metering gear pump. Additive feed arrangement, fluid
properties, pump scale, and pump speed effects on mixing were
evaluated. A relatively new moving/deforming grid approach
within a commercial CFD solver was used to resolve transient
additive concentrations within the pump. Careful consideration
was given to numerical discretization, computational grid, and
convergence (building blocks for any CFD study) to ensure
sound results. Typically >15 full computational gear pump
revolutions were required to reach quasi-steady state.
Numerical diffusion is known to play a role in any CFD
simulation, even when the computational grid and numerical
recipes seek to minimize said effects. The COV results are
adjusted for such effects based on three numerical diffusion
tests employed in the present study.
For the case in which a single additive injection stream is
utilized at the intake of the pump, the model predicted an outlet
additive TA COV of 45%. Mixing cells in the gear teeth
cavities and tip leakage work to increase phasic interfacial area
and improve mixing but is relatively small in extent. A
modification to the gear tooth profile may allow the local
cavity-bound circulation to increase. The TA COV is lowered
to 5.5% by the inclusion of five injection streams at the pump
intake. The waving and breaking of these high-concentration
streams helps to prevent the build-up of high-concentration
zones in the teeth cavity floors. This is near the 5% COV value
typically sought in static mixer applications. The inclusion of
viscous heating and thermal/shear-thinning effects in the
multiple inlet stream CFD model shows an even further
reduction in the outlet COV to 3.2%. It is proposed that most
of this improvement came from bulk process fluid viscosity
reduction.
Intriguing results were found by first scaling the geometry
of the pump to nearly half size and then slowing it down to
nearly half speed. Both cases involved multiple inlet additive
streams and VHTST physics. The half-scale pump at full speed
had a TA outlet COV of 2.0%, but the same pump slowed down
had a TA outlet COV of 1.5%. There does not appear to be an
obvious answer for why lower Reynolds number systems would
produce less additive variability at the pump outlet. The value
of the mixing efficiency parameter remained 0.71, in an area-
averaged sense, for all cases studied here.
An experimental study revealed a much higher outlet COV
for a scaled-down pump with multiple feed streams than CFD
predicted. It was concluded after further study of the additive
distribution system that there was no way of ensuring isolated
separate feed streams during the experimental run. Nor was
there a way to ensure a consistently smooth additive feed rate.
It is proposed that this explains the majority of the differences
between the CFD and experimental results. Since it was always
possible, computationally, to get a COV near or less than 5%
with multiple feed streams, the blending was considered
satisfactory.
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11 Copyright © 2005 by ASME
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