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)
b)
c)
d)
Pendular-looks like bridge, but
particles not immersed in liquid
Funicular-thicker bridges but not
completely filled
Capillary-particles at edge of
cluster
not completely wetted by liquid
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- fine powders that cannot 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 for case of randomly packed bed of monosized particles
More exactly
u: superficial velocity through bed
H: bed thickness
P: pressure drop
•
•
(diameter=x) , where =void fraction, =fluid viscosity
(P)
u (1   )
 180
For turbulent flow ( =fluid
density)
2
3
H
x

2
f
(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
force balanceon particles
unit area
• F=gravity-upthrust
pressure drop 
gravity
P  HA
Upwards
drag
u
•
(1   )( p   f )
A
 H (1   )( p forces
 f )g
This eq’n ignores interparticle
g
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
Absolute density=
mass of particle
volume of solidsmaterial making up the particle
mass of particles in a bed
volumeof particles& voids bet ween them
Bed density=
Bulk density=
mass of particles
volumeof particles& voids bet ween them