Report

1. Department of Mechanical Engineering
FACULTY OF ENGINEERING AND DESIGN
ME40054 Computational Fluid Dynamics
INTERNAL FLOW OF A STATIC MIXER
Candidate Number: 12305
27th November 2019

2. Summary
A CFD study was conducted on a static mixer. Three phases of work were completed. The first
involved creating a baseline mesh and investigated the effect of modelling the boundary layer as an
inflation layer. It was found that velocity wall gradients were more gradual when modelled with no
inflation layer as a turbulent wall profile was not used which can exhibit a jump in velocity wall
gradient from the viscous sub layer. A mixing variation study was completed, showing that no mixing
occurs near the core of the mixing chamber while most mixing occurs in the outer ring of the
chamber, where there are high velocities and temperature gradients. A mesh convergence study
using RMS residuals and pressure at a chosen test point was completed with four groups of
parameter values, with increasing refinement. The first three exhibited convergence while the fourth
did not. It was deduced that unstable oscillations in RMS residuals for the fourth group, showing
nonconvergence, was due to the problem being inherently transient. The refined mesh of the fourth
group was able to capture a higher degree of unsteady physics. Nonetheless, the problem was
assumed to be steady for the last two phases of work using the second parameter group values,
which required half the run time of group 3 and was only worse in accuracy by 2.5%.
The second phase involved using the Design of Experiments tool to optimise outlet temperature
range as a cost function for mixing quality. Two parameters were varied: inlet angle of second inlet
and inlet diameter (both inlets). These were chosen due to their non-effect on fluid flow properties,
e.g. mass flow rate. For a range of inlet diameters of 0.5 m to 1.5 m and range of pipe 2 inlet angle of
0 to -45 degrees, it was found that outlet temperature range increased mostly with inlet diameter.
Outlet temperature range also increased at lower second inlet angle, only at high inlet diameters.
The optimisation tool was used to give the optimised parameters of second inlet angle 0.47987˚ and
inlet diameter 0.50032m. These were used for phase 3.
The simulation was run using different turbulence models in phase 3. These were: K-ε, K-ω, Shear
stress transport (SST) and none (laminar). The model performed well using K-ε and K-ω, exhibiting
convergence. Use of K-ε showed better flow symmetry, attributing to the merit of K-ε having good
accuracy for free stream flows. K-ω showed less flow symmetry but could be modelled up to the wall
without a scalable wall function, allowing wall flows to be modelled more accurately. SST did not
converge due to numerical instabilities as a result of different regional turbulence viscosities. Finally,
to study whether the case might be laminar, no turbulence model was used to run the simulation.
Differences in velocity distribution in comparison to the results from K-ω model indicated that the
flow is highly turbulent and thus, K-ε model should be used with a sufficient scalable wall function.
These studies show the importance of different CFD tools and how they can be used for the users’
advantage to improve accuracy, run time and variable optimisation. A static mixer was chosen due
to its prevalence in industry and lack of open literature on turbulent flows in them.
1 Introduction
Static mixers are motionless mixing devices that allow blending of fluids [1]. Motionless parts
mounted in the mixer utilize energy from the flow stream to create flow patterns, causing the fluids
to mix as they are pumped through the pipeline [2].
In both laminar and turbulent multi-phase flow, static mixers are well established in meeting
industrial requirements for absorption, reaction, extraction and heat transfer. The other option in
mixing turbulent flow is the use of a valve. However, static mixers provide good design performance
and minimal moving parts in comparison. In addition, designs can be engineered to achieve specific
results at low cost and energy expenditure. Thus, static mixers are used extensively in industry. They

3. are applicable to various installations, from refineries and treatment plants to manufacturing
facilities [3].
In comparison to laminar flows in static mixers, experimental and computational open literature on
turbulent flows in static mixers are rarer [4]. Thus, in addition with the importance of static mixers, a
turbulent study on a static mixer was justified and conducted. This involved solving for a mixer with
generic geometry, undertaking a parametric investigation and using different turbulence models to
study flow variable outputs.
2 Methodology: Phase One
2.1 Geometry and Boundary Conditions
Firstly, ANSYS CFX tutorial 5.1 to 5.5 was followed, which involved using Design Modeller, ANSYS
CFX’s integrated CAD programme to create the static mixer geometry:
Figure 1 – Mixer Geometry
The following features were applied:
Table 1 - Model and solver features
Analysis Type Steady State
Turbulence Model k-epsilon
Boundary Conditions Inlet (Subsonic)
Outlet (Subsonic)
Wall: No-slip
Wall: Adiabatic
The two inlets had the following fluid properties:
Table 2 - Inlet fluid properties
Inlet 1 Inlet 2
Mass flow rate 1500 kg/s 1500 kg/s
Temperature 315K 285K

4. 2.2 Baseline Mesh and Boundary Layer Investigation
A baseline mesh was then created using the following options:
• Element Size: 0.2m
• Max size: 0.2m
• Curvature min size: 0.015m
• Curvature normal angle: 18 degrees
• Proximity min size: 0.015m
This generated a mesh containing 8000 nodes and 41,000 elements.
To account for boundary layer effects, an inflation layer was added. A maximum thickness of 0.2m
and 5 layers were applied. This increased refined the mesh to 15,000 nodes and 49,000 elements.
Max iterations were increased to 2000 and residual target reduced to 1.5e-7 to ensure stable
solutions, given by the convergence of residuals in solving.
To investigate how modelling the boundary layer can affect fluid variable results, various output
variables, such as pressure and velocity of fluid in the mixer were explored and plotted in both cases
of omitting and including an inflation layer. Variation of mixing was also explored.
2.3 Mesh Sensitivity Study
Next, a mesh sensitivity study was conducted, to explore how changes in element geometry
parameters vary the solution. Prior to the mesh sensitivity study, the convergence of residuals for
each parameter group was explored, in order to study the accuracy of their solution. One parameter
group was chosen going forward for completing Phases 2 and 3. Four parameter groups were
applied:
Table 3 - Parameter groups
Parameter group 1 2 3 4
Element size (m) 0.4 0.2 0.15 0.12
Max size (m) 0.4 0.2 0.15 0.12
Curvature min size (m) 0.03 0.015 0.011 0.009
Proximity min size (m) 0.03 0.015 0.011 0.009
Inflation thickness (m) 0.3 0.2 0.15 0.12
3 Methodology: Phase Two
In this phase, a parameter study was completed to optimise the amount of flow mixing. The first
stage involved using the Design of Experiments tool to vary the values of chosen parameters. Two
parameters were chosen: input angle of one input pipe (this allowed the angle between the two
inputs to be studied) and the diameter of the input pipes (they were both the same). These variables
were chosen as varying parameters, as they would not fundamentally affect particular fluid
properties during service, such as mass flow rate (fluid speed can be increased to compensate for
reduced cross section area). Rather, they only affect the design and geometry of the mixer. Inlet
diameter was varied between 0.5m and 1.5m and inlet angle for pipe 2 was varied between -45˚ and
0˚. These range of values were most appropriate for the mixing chamber size and produced nine
permutations of cases. The amount of mixing is represented by temperature variation at the outlet.
After the nine cases were solved, a response surface was viewed, showing how outlet temperature
variation varied with inlet angle and inlet diameter. The geometry was then optimised based on the

5. response surface, with the top candidate point chosen and the model updated with its parameter
values.
4 Methodology: Phase Three
As justified in Section 1, open literature on turbulent flows in static mixers are rare. CFX also does
not give a definitive answer as to whether the flow is laminar or turbulent. Thus, a range of turbulent
models as well as using no model (laminar case) were tested on the mixer and results for each model
were correlated, to conclude whether the flow is laminar or turbulent. In addition, the most suitable
model was identified and justified.
The following turbulence models were tested:
1. K-𝜀
2. K-𝜔
3. Shear stress transport (SST)
4. None (laminar)
K-𝜀 , K-𝜔 and SST were chosen as they are 2 equation models, which have been extensively validated
and all three have pros and cons. These are discussed in Section 7. For all models, pressure at point
(0, 0, -1) for convergence testing and velocity contour in the X-Y plane at y = 0, for model comparison
were plotted, if the data was deemed valid.
5 Results and Discussion: Phase One
Unless specified, all plane contour plots are modelled in the X-Z plane at y = 0, intersecting the
mixing channel along the x axis. This allowed the fluid from both inputs to be modelled as well as the
output centreline. See Figure 2. Views from +y axis are shown. However, there is negligible
difference compared to views from -y axis as the mixer is symmetric.
Figure 2 – X-Z plane at y = 0

6. Figure 3 - Velocity contour, no inflation layer
Figure 4 - Velocity contour, with inflation layer
Figure 5 - Velocity Streamlines with inflation layer

7. Figure 8 - Convergence of Momentum and Mass Flux Residuals, top left: Parameter group 1, top right: Parameter group
2, bottom left: Parameter group 3, Bottom right: Parameter group 4
Figure 6 - Temperature contour taken in X-Y plane at z = 1 Figure 7 - Temperature contour with inflation layer

8. 5.1 Boundary Layer Investigation
Figure 3 and Figure 4 indicate the difference an inflation layer makes to solution output variables.
Velocity contour plots in the X-Z plane at y=0 of the mixer are shown, with a significant difference at
the output boundary. Figure 3 shows the velocity (4.397 m/s) being at the maximum value across
much of the outlet cross section, while max velocity is confined to local regions (4.616 m/s) at the
Figure 9 - Convergence of pressure at (0, 0, -1)
Figure 10 - Velocity Streamline variation with parameter groups, top left: group 1, top right: group 2, bottom left: group 3, bottom right:
group

9. beginning of the outlet in Figure 4. Values of max velocity are also different by 0.219 m/s, a 5%
increase on the max velocity given without an inflation layer.
Wall velocities in Figure 3 are more “gradual” while they “jump” to higher values in Figure 4. This is
because the inflation layer considers the use of a turbulent boundary layer, which has a sharp wall
velocity gradient after the viscous sub-layer.
The inflation layer has refined the mesh locally around the boundaries. A finer mesh gives a more
accurate solution, as there would be many more elements, reducing discretization errors. The
programme uses a finite difference method to compute solutions [5]. As each element is dependent
on the neighbouring solution in the finite difference method, this gives rise to more accurate
solutions. Thus, it can be assumed that the solution given by the model with inflation layer added is
more accurate.
5.2 Mixing Variation
Mixing variation was studied for the inflation layer added case.
One way to study variation of mixing is to explore the velocity variation. Areas of high velocity will
accommodate evenly distributed mixing. Figure 4 and Figure 5 show the variation of fluid velocity in
the mixer. In both figures, the fluid near the centreline z axis exhibits very poor mixing, with zero
velocity near centre. The velocity increases with x distance from the centre, due to direct input of
energy and fluid from the two input channels. The velocity then reduces again, with a ring of almost
zero velocity near the wall. This is due to the inlet channels and the mixing chamber not flushed with
each other (seen in Figure 5). As the fluid approaches the outlet, the velocity increases rapidly, with
a ring of high velocity at the start of the outlet, giving rise to a maximum velocity of 4.616 m/s in the
mixer.
Given that the two inlet pipes input fluid at two different temperatures, mixing of the two fluids can
also be studied through temperature. shows that as the fluids enter the mixer, their temperatures
are equalised by the counterpart input. This shows that the ‘outer ring’ accommodates good mixing.
The fluids then reach an equilibrium temperature of about 297K at the centre of the mixer. The fluid
mostly stays at this temperature as the combined fluids leave the mixer through the outlet, as
shown in .
Thus, from the velocity and temperature studies, it can be deduced that much of the mixing occurs
in the outer ring of the mixer.
5.3 RMS Residual Convergence Study
Before conducting post-processing, residuals for each parameter group were investigated. Residuals
are errors in the solution. Various residuals can be studied. For example, a root mean squared (RMS)
residual gives a root mean average error over all elements, for one type of variable – thus are ideal
in gaining an idea of error over the entire mesh [6]. In this study, RMS residuals for momentum in all
three directions and mass flux were studied.
Figure shows the residuals for all four parameter groups. The magnitudes are not as important in a
residual stability study; the stability of these errors is the pivotal point. Time taken for convergence
is shown in Table 4.

10. Table 4 - Time taken for residuals to converge
Parameter Group Run time
1 4 minutes 44 seconds
2 13 minutes 0.4 seconds
3 24 minutes 46 seconds
4 N/A
Parameter group 2 has a finer mesh than group 1, group 3 has a finer mesh than group 2 etc. Thus,
the plots show that as the meshes become more refined, the longer it takes for the residuals to
converge. This is true for groups 1, 2 and 3. This occurs, as finer meshes have more elements. Thus,
the program must solve for more elements and will therefore take longer.
It is clear from the plot for group 4 that the residuals do not converge but instead, oscillate to
instability. Oscillations can also be seen in the residuals for group 2 and 3, albeit with much smaller
amplitudes. These oscillations occur due to the problem being inherently transient; a characteristic
of transient problems are oscillations in output variables. Thus, the simulation would have to be run
as an unsteady simulation, rather than using a steady-state solver when using a more refined mesh
[7]. The results for group 4 may be unstable due to the refined mesh capturing a higher degree of
physics, unsteady in nature [8].
Figure 9 shows the pressure solution given for all four parameter groups at coordinate (0, 0, -1). This
coordinate point was chosen as it is in the middle of the mixing chamber, where velocity is zero and
pressure is highest. It is clear from the trend that the pressure tends to a value as the parameter
group number increases and mesh becomes more refined. This is because as mentioned before, the
residual values become smaller, while the residual never goes to zero due to the nature of iterative
solutions [9]. However, this plot also shows the instability of results from parameter group 4, which
has not been included for the curve of best fit.
5.4 Mesh Convergence Study
Using the parameter groups, velocity streamlines were plotted (Figure 10). The results show that as
the mesh becomes more refined, variables tend to their true values as the residuals become smaller.
As discussed before, only results for groups 1, 2 and 3 are valid with group 3 results being the most
accurate. Thus, it can be said that the max velocity tends towards 11.13 m/s. The results for group 4
must be discarded.
Although group 3 yields the most accurate results out of all four groups, it also requires the greatest
amount of computation effort, almost twice the amount of run time compared to group 2. Group 3
also only improves accuracy by 2.5% (by percentage calculation from data in Figure 9). Thus, by
weighing up computation effort with accuracy, it has been decided to use Parameter Group 2 for
phases 2 and 3.
6 Results and Discussion: Phase Two
The following results were obtained from the parameter study (Table 5):
Table 5 - Parameter Study Results
Inlet Diameter (m) Pipe 2 Inlet Angle (degree) Outlet Temperature Range (K)
0.5 0 0.00040
0.5 -22.5 0.00073

11. 0.5 -45 0.00061
1 0 0.0032
1 -22.5 0.0033
1 -45 0.0038
1.5 0 0.0091
1.5 -22.5 0.0098
1.5 -45 0.0140
These values were plotted on a 3D graph:
Figure 11 - Outlet temperature range vs pipe2 inlet angle vs inlet diameters
Figure 11 shows that the effect of inlet angle is minimal in comparison to inlet diameter. However, at
high values of inlet diameter, the inlet angle becomes more significant, with a difference of 0.005K
at 1.5m inlet diameter. Table 5 shows that the lowest temperature range of 0.0004K occurs with an
inlet angle of 0 degrees and inlet diameter of 0.5m. Other values of inlet angle also yield
comparatively low temperature range results. Thus, any inlet angle is acceptable for an inlet
diameter of 0.5m. The optimisation tool concluded that an inlet angle of -0.47987˚ and inlet
diameter of 0.50032m produced the best outlet temperature range and was thus used for Phase 3.
7 Results and Discussion: Phase Three
Table 6 - Convergence Times for Turbulence Models
Turbulence Model Time taken for convergence (no. of timesteps)
K-ε 1000
K-ω 500
SST N/A
None (laminar) N/A

12. 7.1 K-𝜀 Model
The solution was recalculated with updated
geometry from Section 6 using the K-ε model.
This model was chosen due to its wide usage
and extensive validation. It is successful for a
wide range of flows, such as confined flows
(notes page 155). However, the K-ε Model is
only suitable for flows which have turbulence
with high Reynolds number. Thus, it can only be
used where flow is fully turbulent (notes page
150). CFX uses a scalable wall function to
compute solutions at the inflation layer. Figure
12 shows pressure at (0, 0, -1) converging to a
value after around 1000 time steps, offering
good convergence. Therefore, the results for all
other elements can be assumed to be valid.
A near symmetrical velocity
distribution is shown in Figure 13.
This may be attributed to K-ε
Model’s good accuracy, especially
for free stream flows. The
robustness of K-ε may also
contribute to the solution’s
symmetry, as changes in flow do
not seem to be sensitive to any
small changes. Finally, K-ε is not
memory intensive, as the mesh
near walls are coarse. A scalable
wall function is used at the
inflation layer as the ε equation
contains a term which cannot be calculated at
the wall [10]. However, this may produce
inaccurate calculations. Although K-ε also
models complex flows involving adverse
pressure gradients, separation and strong
streamline curvature poorly, these are
irrelevant as there are only negligible pockets
of separation near the inlets.
7.2 K-ω Model
Figure 14 shows that pressure at the test
point converges but oscillates about a value.
This agrees with the limitation that K-ω has
more difficulty with convergence compared to
K-ε. In addition, Figure 15 shows the
axisymmetric nature of the velocity solution.
This is because the model does not predict free
Figure 12 - Pressure at (0, 0, -1), K-ε Model
Figure 13 - Velocity contour, K-ε Model
Figure 14 - Pressure at (0, 0, -1), K- ω Model

13. stream flows accurately,
especially ones with low-
Reynolds numbers. Due to
the flow axisymmetry, it can
be deduced that the flow is
mostly turbulent [11]. K-ω is
also sensitive to initial
conditions, shown by
velocity contour differences
as a result of slight
difference in inlet angle with
respect to the plane in
Figure 15. However, near
wall interactions are
thought to be modelled more accurately compared to K-ε
as a scalable wall function is not required.
7.3 Shear Stress Transport (SST)
Model
SST is a combination of K-ε and K-ω. K-ω is
used in the near wall region while K-ε is
used in the fully turbulent region away
from the wall (notes page 159). The
models are activated through a blending
function, triggering K-ε near the wall and
K-ω in free stream.
Both options of no transitional turbulence
and fully turbulent transitional turbulence
yielded pressure nonconvergence for the
test point (Figure 16). These numerical
instabilities occurred due to the difference
in turbulence viscosity between the two
regions. As the pressure did not
converge for the test point, the results
were invalid for SST.
7.4 Laminar (No Model)
Finally, using no turbulence model was also tested, thereby assuming that fluid flow is laminar. The
results of using no model was then compared with the results yielded from using the
aforementioned turbulence models.
Figure 17 shows the nonconvergence of the laminar case. This unsteady nature which seems to
repeat cyclically with timestep, as well as that shown in Figure 16 can be due to the nature of the
flow being transient. Thus, it is worth experimenting by modelling the simulation as a transient case.
As the K-ε model is based on flows with high Reynolds number, the results for a laminar case would
not match K-ε results. Rather, results for laminar flow would be similar to those obtained by using K-
Figure 15 - Velocity contour, K- ω Model
Figure 16 - Nonconvergence of pressure at (0, 0, -1), SST Model

14. ω as K-ω assumes low Reynolds numbers. The velocity contour for the laminar case is shown in
Figure 18.
Comparison of Figure 15 and Figure
18 show vastly different velocity
contour patterns. This is especially
true at the core of the mixing
chamber, where K- ω shows a
column of low velocity, high pressure
fluid while laminar shows two
localised spots. Variation of velocity
at the high velocity regions are also
different, with K- ω exhibiting almost
maximum velocity in this plane while
there are three localised spots of
intermediate velocity for the laminar
model. Thus, it can be concluded
that the fluid, given the parameters
and boundary conditions, is highly
turbulent. For this reason, K-ε should
be used to model the mixer.
Conclusion
Using a baseline mesh, investigation on the boundary layer found that the velocity “jumps” towards
larger values near the boundary when an inflation layer is unused, in comparison to a gradual
change in velocity for an inflation layer in use. A study on mixing variation using the same baseline
mesh showed that fluid near the centreline z axis exhibits very poor mixing, with zero velocity near
the centre. Due to input of kinetic energy from the two inputs, velocity increases with radial distance
from the centre of the mixing chamber. Thus, much of the mixing occurs in the outer radial areas of
the chamber. The parameter study for four different parameter groups showed convergence for the
Figure 17 - Pressure at (0, 0, -1), No Model
(laminar)
Figure 18 - Velocity contour, No Model (laminar)

15. first three groups. It was concluded that the fourth group did not exhibit convergence due to a more
refined mesh capturing transient fluid nature. The convergence natures of these groups were also
exhibited in their RMS momentum and mass flux errors over solution timesteps. Despite group 3
yielding the most accurate results, parameter group 2 was used for phase 2 and 3 as this group
required half the run time of group 3 and group 3 only improved accuracy in comparison to group 2
by 2.5%.
The parameter study in phase 2 concluded that outlet temperature range increases with inlet
diameter. At high inlet diameter, a greater negative second inlet angle also yields higher outlet
temperature range. Using the optimisation tool, a second inlet angle of -0.47987˚ and inlet diameter
of 0.50032m produced the best outlet temperature range and these parameters were used for
phase 3. K-ε Model was found to be the most appropriate for the mixer, after the K- ω Model and
laminar case results were compared, concluding that the mixer exhibits mostly high turbulent flow.
Undertaking this CFD study on a static mixer has highlighted the significance of various CFD tools.
These include the use of an inflation layer, design of experiments, optimisation and different
turbulence models. Choosing a mesh refinement size after conducting a mesh convergence study
and ensuring convergence in results at each stage was vital.
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