Behavior of Powders - Outline
• Interparticle Forces
–
–
–
–
Van der Waals Forces
Adsorbed Liquid Layers & Liquid bridges
Electrostatic
Solid Forces
• General Classifications for Fluidized Beds
van der Waals
• Weakest force exists between solids; is of
molecular origin
• For the case of a sphere near a wall
R
y
FVW
KH R

6 y2
KH: Hamaker constant
(varies with material)
Between two flat surfaces
FVW
1
 6
y
y
Particles & Liquids
• If particles are present with a condensable vapor,
the surface may have a layer of condensed vapor on it
This bond may be stronger than
bare surface van der Waals forces
• Adsorbed liquid can smooth over defects increasing
contact area
• More liquid leads to liquid bridges
Types of Liquid Bonding
a) Pendular-looks like bridge, but
particles not immersed in liquid
b) Funicular-thicker bridges but not
completely filled
c) Capillary-particles at edge of
cluster not completely wetted by
liquid
d) Droplet-all particles completely wet
Pendular- a closer look
Pc: pressure inside
capillary liquid
1 1 
PC     
 r1 r2 
•
•
•
•
When Pc<PA, particles will want to come together
Surface tension forces always pull particles together
This arrangement creates strongest interparticle bond
With more liquid, particles can move more freely
Electrostatic & solid Bridges
• Same as for aerosols, charged powders can repel each
other
• Solid bridges-imagine liquid above was NaCl/water
• If powder in dried crystallites of salt would remain
holding particles together
• Other compounds called binders (liq. or solid form)
can be used by dissolving in liquid & drying
• Solid binders –another type, dry powders that react
with liquid to form solid bridges
Interparticle Forces are functions of:
•
•
•
•
•
Particle size
Liquid concentration
Humidity
Temperature
Interrelationship of above variables
Behavior of Particles in Fluidized Beds
• Depending on particle characteristics and interparticle forces, fluidization behavior differs
• Group A- can be fluidized by air at ambient conditions(least cohesiveness) over a range of fluidization velocity
• Group B- powders that bubble under some conditions where Group A would not bubble (more
cohesive)
• Group C- powders that can not be fluidized
without bubbling(even more cohesive)
• Group D- large powders that form spouting
beds(coarse powders, may have low cohesivity)
Flow in Packed Beds (not fluidized)
• Darcy’s rule for laminar flow
 P
u
H
u: superficial velocity through bed
H: bed thickness
P: pressure drop
• More exactly for case of randomly packed bed of
monosized particles (diameter=x) , where =void fraction,
=fluid viscosity
(P)
u (1   )
 180 2
3
H
x

• For turbulent flow (f=fluid density)
2
(P)
u (1   )
 1.75  f
3
H
x 
Criteria & overall expression
• Packed Bed Reynolds #
– Laminar Re*<10
– Turbulent Re*>2000
Re 
*
xu f
 (1   )
• General eq’n.=Ergun eq’n
 P 150u (1   ) 2
u 2 (1   )

 1.75 f
2 3
H
x 
x 3
Pressure drop for non spherical Particles
• For laminar flow (xsv=surface-volume mean
diameter)
(P)
u (1   ) 2
 180 2
3
H

x sv
– xsv=sphere having same surface to volume ratio as
particles need mean if particles are not uniform
• For entire range of Re*
 P 150u(1   ) 2
u 2 (1   )

 1.75 f
2
3
3
H
x

sv
x sv 
Friction Factors-Packed Beds
3
 P x

• f*=friction factor=
2
H

u
(
1


)
f
*
• In terms of Re
laminar
f*=150/Re*+1.75
• Three regimes
Laminar f*=150/Re*
Turbulent f*=1.75
turbulent
logf*
f* constant!
Log Re
Fluidization: backwards packed bed
• When upwards drag exceeds apparent
weight of particles bed becomes
fluidized
pressure drop 
gravity
u
force balanceon particles
unit area
• F=gravity-upthrust
Upwards
drag
P  HA
(1   )( p   f )
A
 H (1   )( p   f ) g
g
• This eq’n ignores interparticle forces
Fluidization-Relationship between P & u
Pip
Fluidized bed region
Minimum fluidizatio nvelocity
• Pip=related to extra forces needed to overcome
interparticle forces
Dimensionless numbers
• Ar=Archimedes #
 f (  p   f ) gx

2
3
sv
• Gravity & buoyancy vs. viscous forces
• Remf=Reynolds# at incipient fluidization

umf xsv  f

Fluidized Bed vocabulary
• Mass of particles in bed=MB=(1-)PAH
A:area (cross section) of bed
H: bed height
P:particle density
:void fraction
mass of particle
Absolute density=
volume of solidsmaterial making up the particle
mass of particlein a bed
Bed density= volume of particles& voids between them
mass of particle
Bulk density=
volume of particles& voids between them
What gas velocities are required?
• For particles larger than 100m
– Wen&Yu correlation
• Remf=33.7[(1+3.59*10-5Ar)0.5-1]
– Valid for spheres in the range 0.01< Remf1000
• For particles less than 100 m(xP=particle diameter)
umf 
(  p   f ) 0.934 g 0.934 xP1.8
1110 0.87  g0.066
• For fluidized beds-harmonic mean of mass distribution
used as mean
Bubbles vs. No Bubbles
• umb=superficial velocity at which bubbles
first appear
• umb(Abrahamsen &Fieldart,1980) for
 xP  g0.06 
 2.07Exp(0.716F ) 0.347 
 

• For groups B&D powders, they only bubble,
umf= umb
• For group C, bubbles never form (cohesive
force too high) & channeling occurs
Slugging
• When size of bubbles is greater than 1/3 of diam.
of bed, rise velocity is controlled by equipment
• Slugging leads to large pressure fluctuations &
vibrations
• Don’t want slugging!
• Yagi&Muchi(1952) criteria to avoid slugging
(Hmf:bed height at onset of fluidization, D:diameter
of bed)
 H mf 
1.9


0.3
D
(

x
)


P P
Expansion of a fluidized bed
• For non bubbling, there’s a region where u increases,
particle separation increases but P/H remains
constant
• u is related to uT –single particle terminal velocity in
general u= uTn, =voidage of the bed
  P x P uT 
u= uT4.65
  0.3
Re P  
  
ReP > 500
u= uT2.4
• Between - Khan & Richardson, 1989
0.27 

4.8  n
x
0.57
 0.043Ar 1  2.4  
n  2.4
 D  

More Bed Stuff
Expansion for bubbling beds
• Simple theory-any gas excess of that needed for fluidization
could form bubbles (not perfect since for low cohesive
powders, much increase in gas velocity can occur before
bubbling & increase leads to lower density,bigger bed
volume)
• Relationship between gas as bubbles & gas doing
fluidization depends on type of powder
Entrainment
• Removal of particles from bed by fluidizing gas
• Rate of entrainment & size distribution of entrained particles
will depend on particle size & density, gas density &
viscosity, gas velocity & fluctuations, gas flow regime, radial
position, vessel diameter
Entrainment
All particles are carried up & particle
flux+suspension concentration are
constant with height
Disengagement zone-upward flux and
suspension concentration of fine
particles decreases with increasing
height
Coarse particles fall back down
Applications for fluidized beds
• Drying – minerals, sand, polymers, pharmaceuticals, fertilizers
• Mixing – all kinds of materials
• Granulation – process of making particles
cluster by adding a binder
• Coating
• Heating/cooling – provides uniform temperature and good heat transport
Issues to consider
• Gas distribution
screen
• Erosion – solid, hard particles may cause wear in bed
• Loss of fines- reduces quality of fluidization lowers g
as-solid contact area, reduces catalytic activity
• Cyclones – can be used to separate entrained fines for
recycle
Feeding the bed
• May need to feed fluidized bed
• Important for drying, granulation,
recycle of fines
• Methods of solids feeding
– Screw conveyors
– Pneumatic conveying