Transcript Folie 1 - University of California, San Diego

Numerical Studies of a Fluidized Bed
for IFE Target Layering
Presented by Kurt J. Boehm1,2
N.B. Alexander2, D.T. Goodin2 , D.T. Frey2, R. Raffray1, et alt.
HAPL Project Review
Santa Fe, NM
April 8-9, 2008
1- University of California, San Diego
2- General Atomics, San Diego
Overview
• Cryogenic fluidized bed is under investigation for
IFE target mass production
• Experimental setup is being built at General Atomics
in San Diego
• Numerical model of fluidized bed is being developed
under guidance of R. Raffray at UCSD
–
–
–
–
Improvements to the granular part
The gas – solid flow model
Stepwise validation and verification of the proposed model
Computing the heat and mass transfer
• Future Plans – Research Path
A Fluidized Bed is being Investigated for
Mass Production of IFE Fuel Pellets
• Filled particles (targets)
are levitated by a gas
stream
LAYERING
Fluidized Bed
• Target motion in the
cryogenic fluidized bed
provides a time-averaged
isothermal environment
• Volumetric heating causes
fuel redistribution to form
uniform layer
SPIN
Frit
CIRCULATION
Gas Flow
Unknowns in Bed Behavior call for Numerical
Analysis
• RT - Experimental observations (presented at last meeting by N.
Alexander) are restricted to the particles close to the wall
• The behavior of unlayered shells is unknown (unbalanced
spheres)
• Tests on the cryogenic apparatus are time consuming
• Results might be hard to interpret
 Optimize operating conditions:
Define narrow window of operation for successful deuterium
layering prior to completion of entire setup
– gas pressure, flow speed, bed dimensions, additional heating, frit
design, …
Numerical Model consists of three Parts
• Fluidized Bed Model
– Granular model
– Fluid-Solid interaction
• Layering Model
– Quantification of mass transfer
Fluidized Bed Model
Part I: Granular Model
• Discrete Particle Method
(DPM)
• Motion of individual particles is
tracked by computing the
forces acting on the particles at
each time step
• Apply Newton’s second law of
motion
• Traditionally a spring –
dashpot and/or friction slider
model is applied for particle
collisions
• Limitations: Not developed for
unbalanced spheres
Forces are computed based on relative
velocity at contact point
Cundall and Struck,
Geotechnique, Vol.29, No.1, 1979
When Modeling Unbalanced Spheres the Forces depend on
Particle’s Orientation
Orientation defined by Euler angles
Center of mass
Geometrical Center
wall
Equations for the
force computation
need to be adjusted
to account for the
Torque around
different contact
center of mass
geometry
Contact forces are a
function of relative
velocity at contact
point
 depends on the
orientation of the
particle
wall
displacement
Normal
Force
Tangential Force
Overview Fluidized Bed Model
Initialize position, velocity and quaternion vectors
Particles need to be
spaced apart
Start time stepping
Predictor step
Compute forces based on predicted positions
Particle – wall collisions
Loop over
all
particles
Compute force due to particle – particle collisions
Add gravitational Force
Correct positions, velocities and
accelerations based on the updated forces
Create time averaged statistics
Write Output every
1000 time steps
Fluidized Bed Model
Part II: Fluid- Particle Interaction
Example:
2-D numerical simulation using MFIX
Common Approach for Numerical
Fluidized Bed Model:
Control Volume Method
Void Fraction is determined from
number of grains in each fluid cell
Granular Continuity Equation:
Time:
0.00 s
Time:
0.04 s
Time:
0.08 s
Time:
0.12 s


 g  g    g  gU g ,i   0
t
xi
Granular Navier Stokes Equation:


 g  gU g ,i 
 g  gU g ,iU g , j 
t
x j

g

Pg
xi


 ij
x j


M
I
m 1
gmi
  g  g gi
Particle void fraction = 0.42
Particle void fraction = 0.00
*MFIX – Multiphase Flow with Interphase eXchanges
Developed by National Energy Technology Laboratory
-- http://mfix.org
The Traditional Approach for the Fluid
Model Fails in this Case
Problem with fluid cell sizes:
•
Minimum of seven pellets per fluid cell
for cell average to work in control
volume method
•
Not useful to solve fluid equation for
3x3x4 grid
The Traditional Approach for the Fluid
Model Fails in this Case
• DNS model to resolve flow around each
sphere computationally VERY expensive
Problem with fluid cell sizes:
•
Minimum of seven pellets per fluid cell
for cell average to work in Control
Volume Method
•
Not useful to solve fluid equation for
3x4 grid
The Traditional Approach for the Fluid
Model Fails in this Case
• DNS model to resolve flow around each
sphere computationally VERY expensive
• Choosing a grid size of the same order than the
shells leads to complication determining the
“average void fraction” around a sphere
Problem with fluid cell sizes:
•
Minimum of seven pellets per fluid cell
for cell average to work in Control
Volume Method
•
Not useful to solve fluid equation for
3x4 grid
The most important information we are trying to get is
the particle spin and circulation rate
The most important information we are trying to get is
the particle spin and circulation rate
Experimental observations indicate, that the spin
of the particles is dominantly induced by
collisions, not by fluid interaction
Application of 1-D Lagrangian Model to
Determine Void Fraction
The most important information we are trying to get is
the particle spin and circulation rate
Experimental observations indicate, that the spin
of the particles is dominantly induced by
collisions, not by fluid interaction
Compute the void fraction for each slice
of the fluidized bed, bounded by one
radius in each direction of the center of
each sphere.
Application of 1-D Lagrangian Model to
Determine Void Fraction
The most important information we are trying to get is
the particle spin and circulation rate
Experimental observations indicate, that the spin
of the particles is dominantly induced by
collisions, not by fluid interaction
Compute the void fraction for each slice
of the fluidized bed, bounded by one
radius in each direction of the center of
each sphere.
This “region of interest” moves
with each particle from time step
to time step
Application of 1-D Lagrangian Model to
Determine Void Fraction
The most important information we are trying to get is
the particle spin and circulation rate
Experimental observations indicate, that the spin
of the particles is dominantly induced by
collisions, not by fluid interaction
Compute the void fraction for each slice
of the fluidized bed, bounded by one
radius in each direction of the center of
each sphere.
This “region of interest” moves
with each particle from time step
to time step
Once the void fraction is known, the
drag force can be computed
Knowing Void Fraction, Richardson-Zaki
Drag model is applied
Richardson-Zaki Drag Force for homogeneous fluidized beds:
 dp
U
 p   f g 
fd 
6
 ut
4.8
n

  3.8

Dellavalle Drag Model:

Re t   3.809  3.809  1.832 Ar
Archimedes Number:
2
Ar 
Void Fraction is known
based on 1-D Lagrangian
Model


2
0.5 0.5
gd 3p  f  p   f 
f 2
Terminal Free Fall Velocity is a constant system parameter:
Re t 
dp f
f
ut
Drag force is added to the total force on the particle at each time step
Overview Fluidized Bed Model
Initialize position, velocity and quaternion vectors
Particles need to be
spaced apart
Start time stepping
Predictor step
Compute forces based on predicted positions
Particle – wall collisions
Loop over
all
particles
Compute force due to particle – particle collisions
Compute void fraction
Compute drag force
Compute the resulting
pressure drop
Compute effective weight
Correct positions, velocities and
accelerations based on the updated forces
Create time averaged statistics
Determine bed expansion
Write Output every 1000 time
steps
Preliminary Results from Fluidized Bed Model
indicate Model’s Validity quantitatively
Bubbling behavior can be
predicted theoretically,
seen in the experiment,
and are modeled
numerically
Visualization of the output:
Merrit and Bacon, Meth.
Enzymol. 277, pp 505-524,
1997
Exact System parameters need to be determined
Stability and convergence can be shown
modeling granular collapse (Kinetic Eng)
Total Kinetic Energy in System during Granular Collapse for
decreasing time step size
(J)
6e-05
5e-05
4e-05
3e-05
2e-05
1e-05
n
t 
2N
 n  2
m
k
0.1
Time (s)
200 particles
M = 2E-6 Kg
Diameter = 4 mm
K_eff = 1000 N/m
C_eff = 0.004 N s/m
g = 0.0125 N s/m
 = 0.4
I = 5E-12 Kg s^2
0.2
Stability and convergence can be shown
modeling granular collapse (Rotational Eng)
(J)
Total Rotational Energy in System during Granular
Collapse for decreasing time step size
4e-06
3e-06
200 particles
M = 2E-6 Kg
Diameter = 4 mm
K_eff = 1000 N/m
C_eff = 0.004 N s/m
g = 0.0125 N s/m
 = 0.4
I = 5E-12 Kg s^2
2e-06
1e-06
Time (s)
0.1
 n  2
m 0.2
k
t 
n
2N
Validation of the Flow Model in Packed Beds
•
Pressure Drop (inches
of water)
Packed bed pressure drop
3.175 mm diameter
void fraction ~ 0.40
•
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Experiment
Ergun
Numerical
0
0.2
•
0.4
0.6
0.8
1
•
Flow Speed (m/s)
•
Pressure drop (inches of
water)
Constant void fraction pressure drop
3.96875 mm diameter
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Experiment
Ergun
Numerical
0
0.2
0.4
0.6
Flow speed (m/s)
0.8
1
1.2
Compare the numerical output against
experiment and theory for non-fluidizing
conditions
Experiment: room temperature loop with two
different set of delrin spheres
Established empirical relation: Ergun’s
Equation
 18
 L f U 2 1   
P  
 0.33 
 Re
 d
 4.8
p
 p

Model: Use Richardson-Zaki drag relation, add
drag forces for overall pressure drop
Model, theory and experiment have good
agreement
Homogeneous Fluidization for Validation
Purposes
The Model Prediction Compare with
Theory and Experiments
Homogenious Fluidization of 350 spheres in water
0.12
•
Experiment
Flow speed (m/s)
0.10
0.08
0.06
Numerical Solution
Richardson Zaki
Relation
•
•
0.04
0.02
0.00
50.00%
60.00%
70.00%
Void Fraction
80.00%
90.00%
Experiment: room
temperature setup using two
different sets of shells
Theory: Apply Richardson
Zaki Relation
Model: Use the parameters
describing the system
System Parameters for PAMS shells are
found by analyzing simple Cases
Angled contact
of shell with
table at
1,000 frames
per second
Normal contact
of shell with
table at
10,000 frames
per second
Measurement
Variable to be Determined
Value
Unit
Scale
Mass
1.89 – 0.677
Volume of X spheres
Radius
1.183 and 1.97
Contact Time
k-value
(Stiffness of collision contact)
672-1865
N/m
Energy out vs. Energy in
c-value
(Damping coefficient)
0.01-0.001
(N*s) / m
Transfer from Kinetic to
Rotational Energy
- value
(Coefficient of tangential friction)
tbd
-
From Model
g- value
(Coefficient of dynamic friction)
0.0125 - 0.025
(N*s) / m
Kg
mm
 10 6
The Model Prediction Compare with
Theory and Experiments
Homogenious Fluidization of 350 spheres in water
•
0.12
Experiment
Flow speed (m/s)
0.10
Numerical Solution
0.08
•
Richardson Zaki
Relation
0.06
•
0.04
Experiment: room
temperature setup using two
different sets of shells
Theory: Apply Richardson
Zaki Relation
Model: Use the parameters
determined earlier as input
0.02
Bubbling fluidization 200 PAMS shells ~4mm diameter in N2
0.00
50%
60%
0.9
70%
80%
90%
0.8
Void Fraction
•
•
Large error bars due
to the uncertainty in
pellet radius
Richardson Zaki is
not applicable in
bubbling beds as a
whole
Flow Speed (m/s)
0.7
Experiment
0.6
Numerical Analysis
0.5
0.4
0.3
0.2
0.1
0
30%
40%
50%
60%
70%
Void Fraction
80%
90%
100%
Validation of the Unbalanced Contact is
considered crucial!!!
Validation of the model for off centered particle
collisions is considered very important…
However, has not been done yet.
Layering Model
• Compute the redistribution of fuel based on the fluidized bed
behavior
• Solve 1-D equations simultaneously:
 
C diff  P2 P1 
  
R l ICE  T2 T1 
 1 t ICE 
 h

T1  T2  2   

q


ICE
f

h
k
ICE 

Mass Transfer Equation

Heat Transfer Equation
Marin et alt., J.Vac.Sci.Technol.A. Vol.6, Issue 3, 1988
•
hf
t

This leads to a layering time constant of e
q 
Latent heat
Volumetic heating
• Time step: ~1E-5 s Fluidized Bed vs. ~30-60s Layering
• Based on the time averaged motion and preferential position, we can
compute the average temperature/ temperature difference between
the thick and the thin side of the shell

Summary
STARTING POINT:
A fluidized bed is under investigation for mass production layering of IFE
targets
•
Room temperature fluidized bed experiments
(Presented at the past meeting)
– Promising, but unable to deliver enough
information
•
Numerical model is proposed
– Existing fluidized bed models
– Development of new model
– Validation through theory and experiments
•
Experimental surrogate layering
– Validate layering model
– Show proof of principle
•
Find optimized parameters for
D2 Layering prior to experiment
 Guidelines for Successful Target Layering
Equations to Compute Contact Forces
Distance between two sphere centers
s  s
s


i  sj
sis, j     ri
si  s j
Normal and Tangential Force Component
s
 
 s
Fn  si  s j  ri  r j   k eff  vcp , n  ceff


 
 s
s
v

s
cp ,t
Ft   min   Fns , g  vcp ,t   s
vcp ,t



 i s  s is, j  Ft s
Apply Forces to:


F  m a
Compute contact point velocity

 s


vcp i , j  i , j  sis, j  vis, j
 s  s  s
vcp  vcp i  vcp j
s
    I
s
Orientation of the Particle cannot be determined
Equations to Compute Contact Forces
Distance between two sphere centers
s  s
s
 s

i  sj
scs i     ri
si  s j
Distance between two mass centers
 s

T  b
scg i , j  sis, j  Ai , j  oi , j
Normal and Tangential Force Component
s
 
 s
Fn  si  s j  ri  r j   k eff  vcp , n  ceff





 s
s
v

s
cp ,t
Ft   min   Fns , g  vcp ,t   s
vcp ,t



 i s  scgs , i  Ftots
 b  A  s
Convert spin into space fixed coordinates

  Ai , j  i , j b
s
i, j
T
Compute contact point velocity

 s
 s 
vcp i , j  i , j  scg i , j  vis, j
 s  s  s
vcp  vcp i  vcp j


F  m a
Apply Forces to:
 
b
x
 
b
y
 
b
z
 xb
I xx
 yb
I yy
 zb
I zz
 I yy  I zz  b b
 y  z
 
I
xx


 I  I xx  b b
 z  x
  zz
 I

yy


 I xx  I yy  b b
 x  y
 
I
zz


Quaternion Description allows Following
Orientation of Particles
•
Rotational equations require body fixed and
the space fixed coordinate systems
•
Matrix of rotation is applied to switch between
the two
•
Unlike the translational motion (keeping track
of x-y-z coordinates) the rotational motion
cannot be tracked simply recording pitchyaw-roll angles
•
This rotation matrix depends on the order by
which the rotations are applied
•
Solution: Quaternions
Quaternion representation describes the
orientation of a body by a vector and a scalar
Simple description of
rotational motion
Equations to Compute Contact Forces
Distance between two sphere centers
s  s
s
 s

i  sj
scs i     ri
si  s j
Distance between two mass centers
 s

T  b
scg i , j  sis, j  Ai , j  oi , j
Convert spin into space fixed coordinates

  Ai , j  i , j
s
i, j
T
b
Compute contact point velocity

 s
 s 
vcp i , j  i , j  scg i , j  vis, j
 s  s  s
vcp  vcp i  vcp j
Normal and Tangential Force Component
s
 
 s
Fn  si  s j  ri  r j   k eff  vcp , n  ceff


 
 s
s
v

s
cp ,t
Ft   min   Fns , g  vcp ,t   s

vcp ,t
F  m a



 i s  scgs , i  Ftots  b  A  s
 xb 
Apply Forces to:
 yb 


F  m a
 q 0 
 q0
 


q
 1  1  q1
 q   2  q
 2
 2
 q 
q
 3
 3
 
b
z
 xb
I xx
 yb
I yy
 zb
I zz
 q1
q0
 q2
 q3
q3
 q2
q0
q1
 I yy  I zz  b b
 y  z
 
I
xx


 I  I xx  b b
 z  x
  zz
 I

yy


 I xx  I yy  b b
 x  y
 
 I zz 
 q3  0 
 
q 2   xb 
 q1   yb 
 b 
q 0   z 