Stratification, Fluctuations and Stability in Monodisperse Fluidized Beds' low Re, noninertial : hig

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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).
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Sedimentation

• Very long range H.I. 1/r
• Sedimentation rate depends upon
• local density.

Chaos !
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Large Collection of Spheres f
Average Settling Velocity vsed(f)/v0
Q. What are the Particle Dynamics during settling
at a concentration f ??
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Prediction 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
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Velocity 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??
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Sedimentation 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
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Evaluate 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.
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Velocity 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
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Velocity 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 !
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Time 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.

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Fluidized 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 ?
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Were potentially looking for small affects, so
we need to use very, very, Monodisperse Particles
ltagt104 mm
sa / ltagt1.5 !!
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Fluidized Bed
Top
Middle
Pumped Fluid
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Stability Tests in Fluidized Bed view near
mid-height
Fluctuations similar in Fluidized and
Sedimentation modes.
Stable Fluctuations
Mix Sample
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Method to Measure f(z) Q. Is f uniform ??
I(f)
I(z)
I(z)
I(f) calibration
f(z)
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Q. 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
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Calculated 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)
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Particle Fluxes

• Zeroth order term Mean Velocities

So that
This is UNSTABLE, there must be another source of
particle FLUX Lets look at the FLUCTUATIONS
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Velocity 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 !

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Velocity 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 !
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Condition for Stability
Integrate in z from TOP for comparison to
experiments
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Stability Test Particle Fluxes in a Fluidized
Bed
The fluxes sum to zero, i.e. STABLE Fluidized Bed
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Summary - 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 ??

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H
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).
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Volume Fraction Profiles f(z)
The stratification increases with increasing
height H
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Stratification 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!
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RZ calculated Sedimentation Velocities
SHORT
TALL
vpump matches vsed(ftop), and Excess Velocity
Increases with Taller cells
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Stratification 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
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Velocity Fluctuations vs. Bed Height H
Tall
Fluctuations at mid-height
Short
sv / v0 ? 0 H? 0 Homogenous bed in limit
of zero height!
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Correlation Lengths x(z) vs. Particle size a
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Test of Flux Balance (integral of flux deriv.)
The fluxes are well balanced in all beds
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Fluidized 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

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