Title: Stratification, Fluctuations and Stability in Monodisperse Fluidized Beds' low Re, noninertial : hig
1 Stratification, Fluctuations and Stability in
Monodisperse Fluidized Beds.( low Re,
non-inertial high Pe, No Brownian Motion) Phil
Segre Physics Dept., Emory Univ., Atlanta, Ga.
Jim McClymer - Physics Dept., Univ. of Maine.
Outline - Sedimentation Stability Problem -
Fluidized beds
P.N. Segre and J.P. McClymer, J. Phys. Cond.
Matt. 16, S4219 (2004) P.N. Segre, Phys. Rev.
Lett. 89, 254503 (2002).
2Sedimentation
- Very long range H.I. 1/r
- Sedimentation rate depends upon
- local density.
Chaos !
3Large Collection of Spheres f
Average Settling Velocity vsed(f)/v0
Q. What are the Particle Dynamics during settling
at a concentration f ??
4(No Transcript)
5Prediction for Velocity Fluctuations
Assume a Random Particle Distribution ? Caflisch
and Luke (1985)
v0
x L
Blob of Mass
Size L
v0Dv
Gravitational
Viscous
Blob Model Predicts Diverging Fluctuations with
Cell Size L
6Velocity Fluctuations in a dilute Sample
Fluctuation Divergence with Cell Size
- Theory
- Caflisch, Luke Divergence
- Koch, Shaqfeh x1/f No Divergence
- Experiment
- Koglin Divergence
- Nicolai, Guazzelli No Divergence
- Simulation
- Ladd Divergence
Need to do experiments.x L??
7Sedimentation Experiments - Review
P.N. Segre, E. Herbolzheimer and P.M. Chaikin,
Phys. Rev. Lett. 79 2574 (1997).
x
Data from Middle of Settling Particle Columns
x
P.I.V.
Correlation Lengths
8Evaluate the Blob Model with new x length, not L
Assume a Random Particle Distribution
v0
x
v0Dv
Blob of Mass Size x
Gravitational
Viscous
Blob Model Predicts
Use from Experiments.
9Velocity Fluctuations in Sedimentation.
Summary of picture in Sedimentation.
- With sample of a, f, uniform concentration. ?
Large fluctuations, uniform in height.? No
time-dependent behavior.Lots of theoretical
models, but little concensus
The blob model for fluctuations, which
successfully connects density and velocity
fluctuations ? Big Conceptual Problem
10Velocity Fluctuations sv Result from f
Fluctuations sf
Particle Flux
- For f fluctuations use N1/2 Model
- The flux gradient is then
Dense regions Fall, Dilute Regions Rise ? Net
Particle Flux Downward This cant be stable in
time !
11Time Dependent Velocity Fluctuations
S. Tee et al, Phys. Rev. Lett. 89, 054501 (2002)
Time varying concentration profile
Time varying velocity fluctuations
- Time Varying Properties.
- Can be serious problem if not a stationary,
time-independent state. - Theories for fluctuations have no time
dependence. - To get a Stable, Stationary State Fluidized
Bed.
12Fluidized Bed Perpetual Sedimentation
- Particles of radius104 mm, Polydispersity 1.5
- Stokes Velocity, vst1.21 mm/s, Pump fluid
upwards at vpump0.727 mm/s to expand the
particles to about 18.5 cm, so ltfgt 0.10. - Stable particle column for hours.
z
x
y
Pump
Big Question What are properties of Stationary
State Sedimentation ?
13Were potentially looking for small affects, so
we need to use very, very, Monodisperse Particles
ltagt104 mm
sa / ltagt1.5 !!
14Fluidized Bed
Top
Middle
Pumped Fluid
15Stability Tests in Fluidized Bed view near
mid-height
Fluctuations similar in Fluidized and
Sedimentation modes.
Stable Fluctuations
Mix Sample
16Method to Measure f(z) Q. Is f uniform ??
I(f)
I(z)
I(z)
I(f) calibration
f(z)
17Q. Is f uniform ?? A. No.
f010.0
- The volume fraction f is not uniform.
- But if then how could the bed be balanced
? - Do the local sedimentation velocities vary with
height ? - Where does the pump rate vpump fit in ?
vsed (f) vo(1-f)5.5
18Calculated Sedimentation Velocities
RZ equation
f010.0
v(f0)/v00.56
ftop6.0
v(ftop)/v00.71
Measure (vsed /v0)top0.686
Measure vpump /v00.727
New Variable, the Excess Velocity
vex(z)vpump-vsed(z)
19Particle Fluxes
- Zeroth order term Mean Velocities
So that
This is UNSTABLE, there must be another source of
particle FLUX Lets look at the FLUCTUATIONS
20Velocity Fluctuations in a Fluidized Bed
Sedimentation Scalings
- The average velocities ltVxgt, ltVzgt 0 , Stable
Bed. - The velocity fluctuations sv and correlation
lengths x are not uniform with height !
21Velocity Fluctuations sv Result from f
Fluctuations sf
Particle Flux
- For f fluctuations use N1/2 Model
- The flux gradient is then
Dense regions Fall, Dilute Regions Rise ? Net
Particle Flux Downward !
22Condition for Stability
Integrate in z from TOP for comparison to
experiments
23Stability Test Particle Fluxes in a Fluidized
Bed
The fluxes sum to zero, i.e. STABLE Fluidized Bed
24Summary - Single Fluidized Bed
- Stability found, and modeled in a fluidized bed.
- All properties are height
dependent. - ? Tee et. Al, Start with uniform ?
develops in time
- Stratification is directly related to
Fluctuations - Still no way a priori to close equation, x.
- Q. How do the stratification and
fluctuations vary with Cell Height H ??
25H
Push the System to shorter and taller cellsALL
ltfgt0.10
H18.5 cm
a55 mm H/a3364 1779 1194
755 446 TALL
MED SHORT N1.8 e7
2.6 e6 7.7 e5 1.9 e5
4.0 e4
104 mm
155 mm
245 mm
415 mm
How does Stratification and Fluctuations vary
with H, N?
Simulations Nlt3.5e4 particles NO
stratification 1 A.J.C. Ladd, Phys. Rev. Lett.
88, 48301 (2002) 2 N. Nguyen and A.J.C. Ladd,
J. Fluid Mech. 525, 73 (2005).
26Volume Fraction Profiles f(z)
The stratification increases with increasing
height H
27Stratification vs. cell height H
ftop /f0 e-H/3711
Dftopf0-ftop
Dftop/f0 1- e-H/3711 H/3711 short cell limit
Dftop /f0 ? 0 H? 0 Homogenous bed in limit
of zero height!
28RZ calculated Sedimentation Velocities
SHORT
TALL
vpump matches vsed(ftop), and Excess Velocity
Increases with Taller cells
29Stratification vs. cell height H
(vsed /v0)top
RZ v(f)/vo(1-f)5.5
vpump /v0
ftop /f0 e-H/3711
- The pump velocity depends upon cell height H
The upward pumped flow velocity matches well the
sedimentation velocity of top interface.
vpump (vsed)top 0 No Flux at top-stable
position
30Velocity Fluctuations vs. Bed Height H
Tall
Fluctuations at mid-height
Short
sv / v0 ? 0 H? 0 Homogenous bed in limit
of zero height!
31Correlation Lengths x(z) vs. Particle size a
32Test of Flux Balance (integral of flux deriv.)
The fluxes are well balanced in all beds
33Fluidized Beds
- Stability found, and modeled in all fluidized
beds. - Concentration Gradients increase in Taller
Cells. - Velocity Fluctuations also increase in Taller
Cells. - Properties are never uniform as envisioned in
Sedimentation.
Homogenous beds in limit of zero height!
ftop /f0 e-H/3711
- Stability relation is 1 Equation, 3 Unknowns !
- Still no way a priori to close equation for x
34(No Transcript)
35(No Transcript)
36(No Transcript)