Direct Numerical Simulation (DLM) of 1204 Spheres in a
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Transcript Direct Numerical Simulation (DLM) of 1204 Spheres in a
Direct Numerical Simulation (DLM)
of 1204 Spheres in a Slit Bed
0.275 in.
Sphere Diameter 0.25”
Compare computed
bed expansion with the
observed
58.5 in.
8 in.
Pan, Sarin, Joseph, Glowinski & Bai 1999
©2002 Regents of University of Minnesota • Fluidization
of 1204 Spheres
1
1204 Particles
0
2
4
6
8
• Crystal configuration
25
20
15
10
5
0
Simulation
Experiment
©2002 Regents of University of Minnesota • Fluidization
of 1204 Spheres
2
1204 Particles
0
2
4
6
8
• V = 2.0 : Particle
position at t = 5.625.
25
•The maximal particle
Reynolds number is
1383.
20
15
•The maximal
averaged particle
Reynolds number is
130.
10
5
0
Simulation
Experiment
©2002 Regents of University of Minnesota • Fluidization
of 1204 Spheres
3
1204 Particles
0
2
4
6
8
• V = 3.0 : Particle
position at t = 20
25
•The maximal particle
Reynolds number is
1142.
20
15
•The maximal
averaged particle
Reynolds number is
131.
10
5
0
Simulation
Experiment
©2002 Regents of University of Minnesota • Fluidization
of 1204 Spheres
4
1204 Particles
0
2
4
6
8
• V = 3.5 : Particle
position at t = 17.238
25
•The maximal particle
Reynolds number is
1671.
20
15
•The maximal
averaged particle
Reynolds number is
236.
10
5
0
Simulation
Experiment
©2002 Regents of University of Minnesota • Fluidization
of 1204 Spheres
5
1204 Particles
0
2
4
6
8
• V = 4.0 : Particle
position at t = 32
25
•The maximal particle
Reynolds number is
1965.
20
15
•The maximal
averaged particle
Reynolds number is
276.
10
5
0
Simulation
Experiment
©2002 Regents of University of Minnesota • Fluidization
of 1204 Spheres
6
1204 Particles
0
2
4
6
8
• V = 4.5 : Particle
position at t = 31
25
•The maximal particle
Reynolds number is
1859.
20
15
•The maximal
averaged particle
Reynolds number is
292.
10
5
0
Simulation
Experiment
©2002 Regents of University of Minnesota • Fluidization
of 1204 Spheres
7
Bed Expansion
volumeof solids
totalvolume
d 3
1204
6
4.437/ H
A
d = 1/4”
HA
Richardson-Zaki
V V 0 1
V V 0 " Bl owou t" velocit y
n (Re)
Superficial
Inlet
V()
Velocity
Vi m V V 0
©2002 Regents of University of Minnesota • Fluidization
of 1204 Spheres
8
Blow Out Velocity
©2002 Regents of University of Minnesota • Fluidization
of 1204 Spheres
9
Bed Height vs. Fluidizing Velocity for
Both Experiment and Simulation
For the monodispersed case studied
in simulation (d = 0.635cm)
Hs = 4.564/(1-e)
The mean sphere size for the
polydisperse case studied in the
experiments is slightly larger
(d = 0.6398cm) and
He = 4.636/(1-e)
©2002 Regents of University of Minnesota • Fluidization
of 1204 Spheres
10
Data from Previous Slide
Plotted in a log-log Plot
The Richardson-Zaki correlation is given by
V() = V(0)e n(Re)
where V(0) is V when e = 1
d
n 4.65 19.5 when Re V (0)d / v 0.2,
D
d
n 4.36 17.6 when 0.2 Re 1,
D
n 4.45 Re1
when 1 Re 500,
n 2.39 when 500 Re 7000
and D is the tube radius. In our experiments and
simulations Re is confined to the range for which n = 2.39.
The slopes of the straight line are given by the RichardsonZaki n = 2.39. The blow-out velocities Vs(0) and Ve(0) are
defined as the intercepts at e = 1.
Vs() = 8.131e 2.39 cm/s
and
Ve() = 10.8e 2.39 cm/s .
©2002 Regents of University of Minnesota • Fluidization
of 1204 Spheres
11
Bed Height vs. Fluidizing Velocity After
Shifting by the Ratio of Blow-out Velocities
d1 = 0.635cm simulation
d2 = 0.6398cm (average d for experiments)
Walls will increase the drag more in the
experiments than in the simulations. Wall
correction of Francis [1933]
s f 2 d 2.25
V (0)
d 1
18 f
D
Vs (0) d1
Ve (0) d 2
2
D d1
D d2
2
2.25
6.35 51
6.398 46.2
2.25
1.233
The value 1.233 is very close to the shift ratio
10.8
1.248
8.131
©2002 Regents of University of Minnesota • Fluidization
of 1204 Spheres
12
Slip Velocity is Computed
on Data Strings at Nodes
U1
U3
U2
The slip velocity for
U1 - U2 is zero + noise
©2002 Regents of University of Minnesota • Fluidization
of 1204 Spheres
13
Transitions Between Power Laws
Logistic dose curve
This curve is fitted to
data and is
convenient to use
with a spread sheet
©2002 Regents of University of Minnesota • Fluidization
of 1204 Spheres
14
Transitions Between Power Laws
Example: Richardson-Zaki Correlation
d
n 4.65 19.5
when R0 0.2;
D
d
n 4.35 17.5 R00.03 when 0.2 R0 1;
D
Power law
V e
e n ( Ro )
V 1
Transition
d
n 4.45 18 R00.1 when 1 R0 200
D
n 4.45R00.1
when 200 R0 500;
n 2.39
when 500 R0
Power law
2.26 19.5 d
D
n 2.39
T 1
Re 0.7
1
T
1.1
12.0
d
D
1 0.1
©2002 Regents of University of Minnesota • Fluidization
of 1204 Spheres
15