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
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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 Re1
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
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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
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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  R00.03 when 0.2  R0  1;
D

Power law
V e 
 e n ( Ro )
V 1
Transition
d

n   4.45  18  R00.1 when 1  R0  200
D

n  4.45R00.1
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
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