Supplemental Information
Analysis of Particle Transport and Deposition of Micron-sized Particles in a 90˚
Bend Using a Two-fluid Eulerian-Eulerian Approach
Erick S. Vasqueza, Keisha B. Waltersa and D. Keith Waltersb
aDave C. Swalm School of Chemical Engineering
bDepartment of Mechanical Engineering
Mississippi State University, Mississippi State, MS 39762
S1. Comparison of Boundary Conditions on the Eulerian-Eulerian Model
The Lagrangian based model with near-wall velocity corrections of Longest and Oldham (2008)
was also implemented in the developed Eulerian-Eulerian model. However, no significant
changes in cumulative particle deposition percentage were observed in comparison to the
Eulerian-Eulerian model developed in this study. In Table S1 total particle deposition results are
shown for three particle diameters. Using the lowest level of mesh refinement for case I (Re =
1000), no significant change in the particle deposition percentage was observed from applying
the near-wall correction boundary condition (Longest and Oldham, 2008). We attribute this result
to the highly refined near-wall mesh, which confirms that the Eulerian-Eulerian model established
in this work does not require a special boundary condition to accurately predict total particle
deposition. A full comparison of these data with previous models and the experimental data is
provided in Section 3.2.1.
Table S1. Boundary condition (BC) effects for total deposition percentage using
the Eulerian-Eulerian method with a near-wall velocity correction approximation
(BC on) and the Eulerian-Eulerian method (BC off)
Case I: Re 1000 non-refined mesh
Deposition
Particle
%
St
diameter
BC
BC
(micron)
on
off
0.28
7.3
32.66 33.16
0.37
8.39
67.81 68.91
0.9
13.09
94.64 95.59
Table S2. Particle sizes (and corresponding Stokes number) examined on refined
and coarse grids at different Re numbers
Re = 100
Particle diameter (m)
Stokes #
5.39E-06
5.91E-06
6.87E-06
7.65E-06
8.83E-06
9.23E-06
9.77E-06
0.28
0.34
0.46
0.57
0.76
0.83
0.93
1.15E-05
1.22E-05
1.29
1.46
Re= 1000
Particle diameter (m)
5.69E-06
7.30E-06
7.92E-06
8.39E-06
9.15E-06
1.04E-05
1.15E-05
1.31E-05
1.46E-05
1.54E-05
Stokes #
0.17
0.28
0.33
0.37
0.44
0.57
0.70
0.90
1.12
1.24
S2. Airflow velocity profiles
Airflow velocity profiles are shown in Figure S1 for Reynolds number (Re) of 100 and
1000 along two different symmetry axes (z=0; x=0) and three elbow locations (0°; 45°;
90°). At both Re numbers, higher velocity magnitudes were observed towards the outer
elbow walls due to inertial fluid forces along the symmetry axis (z = 0) (Figures S1A and
B). Conversely, on the perpendicular axis (x = 0), velocity profiles were symmetric for
both cases (Re = 100 and Re = 1000) as shown in Figures S1C and S1D, respectively.
These results agree with numerical calculations by Tsai and Pui (1990) and simulation
results of Pilou et al. (2011) for an elbow geometry under the same flow conditions (Re =
100, De = 38; Re = 1000, De = 419). Using these confirmed airflow velocity profiles,
Eulerian simulations were performed for the particle phase, as discussed in Section 3.2.
Figure S1. Axial velocity profiles at z = 0 (symmetry plane) for (A) Re = 100 and (B) Re
= 1000 at three different elbow locations (left to right): 0° (inlet), 45°, and 90° (outlet).
Velocity profiles are shown for the perpendicular symmetry plane (x = 0) for (C) Re =
100 and (D) Re = 1000 at the same elbow locations (0°; 45°; 90°).
S3. Secondary Airflow Streamlines
Secondary flow streamlines of the air flow velocity profiles were also examined (Figure
S2). Formation of counter-rotating vortices was observed for both the Re = 100 and Re =
1000 cases, as seen in the 45° (elbow curvature) and 90° (geometry exit) cross-sections
(Figure S2). Specifically, two counter-rotating vortices were observed at each condition.
For the highest Re case (Re = 1000), this pair of vortices is closer to the inner wall of the
bend. Conversely, at Re 1000 the maximum axial velocities move towards the outer wall
of the bend (outer wall is opposite the inner wall, as depicted in Fig. S2). These results
are in agreement with previous studies showing increased centrifugal forces and the
resultant effect on vortices position at higher Re and De number airflow in a 90° bend
(Breuer et al., 2006; Pilou et al., 2011).
Figure S2. Secondary airflow velocity streamlines and axial velocity profiles at 45° (top)
and 90° (bottom) for Re = 100 (left) and Re = 1000 (right) showing the formation of
counter-rotating vortices and the effect of higher centrifugal forces on the vortex
positions and maximum velocity positions.
S4. Complementary Equations
Reynolds, Dean, and Stokes numbers were calculated for a parabolic inlet velocity
profile using the fluid and particle properties listed in Table S3. Equations for these
dimensionless numbers are as follows:
𝑅𝑒 =
𝐷𝑒 =
𝑆𝑡 =
𝜌𝐷𝑈𝑚𝑎𝑥
(S1)
2µ
𝑅𝑒
(S2)
√𝑅𝑜
2𝑈
𝜌𝑝 𝑑𝑝
𝑚𝑎𝑥
(S3)
18µ𝐷
The radius of curvature of the bend (R) was calculated based on the dimensionless
curvature radius (Ro) as described by Pui et al. (1987) and shown in Equation S4.
𝑅𝑜 =
2𝑅
(S4)
𝐷
Table S3. Fluid properties and case descriptions
Fluid and Particle Properties
Temperature
273 K
fluid-phase density, ρ
1.18 kg/m3
fluid dynamic viscosity, µ
1.85E-5 kg/(m s)
Particle density, ρp
895 kg/m3
Case I
Re = 1000 (De = 419)
Tube diameter, D
Maximum fluid velocity, Umax
Ro, curvature radius
R, radius of curvature of the bend
Case II
Re = 100 (De = 38)
Tube diameter, D
Maximum fluid velocity, Umax
Ro, curvature radius
R, radius of curvature of the bend
0.00395 m
7.72 m/s
5.7
0.01126 m
0.00093 m
3.371 m/s
7
0.00326 m
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