The role of interparticle forces in thefluidization of micro and nanoparticles
A. CastellanosA. Castellanos
POWDER FLOW 2009London, December 16
Acknowledgements
The experimental work presented here has been done in the laboratory of the group of Electrohydrodynamics and Cohesive Granular Materials of the Seville University in collaboration with my colleagues Miguel Angel Sanchez Quintanilla, Jose Manuel Valverde Millan.
Key idea of the talk
Starting point: in the fluidized state fine cohesive powders form aggregates due to their strong interparticle attractive forces.
Aggregates stop growing when their Cohesive Bond number (ratio of attractive forces to weight) is of order one.
Aggregates may be considered as effective non-cohesive particles, and using the well known empirical relations for non-cohesive powders we extend the Geldart diagram to fine cohesive powders.
Plan of the talk• Introduction to the fluidized state: Geldart map• Important forces: boundaries B-A, C-A • Our materials: toners (microparticles) and silica (nanoparticles)• Fluidization in fine powders: Solid-like and fluid-like regimes• The role of van der Waals forces: aggregation and transitions between fluidization regimes• Prediction of a new regime and experimental confirmation.• Final result: diagram of fluidization regimes for fine cohesive powders•Conclusion
The fluidized state
The gas-fluidized state is a concentrated suspension of solid particles in the upward gas flow.1.- The distance between particles is of the order of the size of the particles2.- The free surface is horizontal (it looks like a liquid but it is not a liquid).
The Fluidized state
Group A powder: Xerographic toner
Bpowder
Cpowder
A powder
0.5
0.1
1.0
5.0
10 50 100 500 1000
C AB
D
m]
(ρp–
ρ g)×
10–3
[kg/
m3 ]
10.0
Cohesive
Bed expansionbefore bubbling
Bubbling at umf
Spouting
Micro and Nano-aggregatesdp [μ
Geldart’s classification
D powder
Forces acting upon grains
• Gravity force (weight) mg (or the force due to a pressure of confinement).
• Attraction force between particles (force of van der Waalsfor dry neutral powders, capillar forces, electrostatic forces, magnetic forces, forces due to solid bridges (sintering) and other chemical bonds)
• Hydrodynamic resistance force (force acting upon particles due to friction with the ambient fluid) and pressure and inertial forces (not considered here). The relations between these three forces give the two important non-dimensional parameters of granular materials:Bond Cohesive number: attractive force / weightNumber C: attractive force / viscous drag
Interparticle and external forces change the type of fluidization
• Decreasing interparticles forces: C A B
• Increasing interparticles forces: B A C
Varying the external force may also induce transitions between the fluidization types. This have been shown for vibrations, sound, and electric and magnetic forces.
Drag (viscous friction) force
For small Reynolds number Re (ratio of inertial to viscous forces) and spherical particles
6πηRvη the viscosity of the fluidR the radius of the particlev the velocity of the particle relative to the fluid
Dimensionless numbers in fluidization types A, B
• In the fluidized state:
Drag (viscous friction) / weight ~ 1
• Re <<1 near onset of fluidization for type B • Re<<1 for micron and nanoparticles (type A if fluidizable)
The boundary A-BMolerus in 1982 postuled, and experiments confirmed, that number C (attractive force/drag) of order 1 define the boundary A-B. Therefore Bond number (attractive force/weight) is of order 1. Here, we will show the physical mechanism responsible for this transition in fine powders.
The boundary C-A• We need to break the primary aggregates into the primary
particles or smaller aggregates so that the powder becomesfluidizable
• For the fluidization to be stable the subsequent growth by collisions in larger aggregates must be limited in size
• The first step is process dependent (history) and thereforethere is not a clear-cut criterion to define this boundary.
Gas flow
CollisionsHydrodynamic
interactions
Interparticle contacts
Different mechanism of stress transmission in the discrete phase:
Gas flow
Stress transmitted byinterparticle contacts
Stress transmitted by collisions and hydrodinamic
interactions
Solid-like behaviour
Fluid-like behaviourσc > 0
σc = 0
Stresses in fluidization
In previous studies the fluidized region before bubbling was very small (powdersabove 70 microns in size). This made very difficult to resolve directly this controversy.
We have solved this controversy using micro size fine powders
Controversy: has type A a solid-like or fluid-like behaviour?
Our materials: micron size particles
Our materials: nano size particles• Aerosil R974
Primary SiO2 nanoparticles. 12 nm diameter.
Surface treated to render it hydrophobic
Sub-agglomerates due to fusing of contacts in the fabrication process at high temperatures, of size < 1 μm
Sub-agglomerates agglomerate into simple-agglomerates, due to van der Waals forces, of size of order 10-100 μm.
Method of fluidization
Both solid-like and fluid-like regimesexist for fine powders
Diffusion in the fluidized bed (type A)
Porous filter
Gas flow
Removable slide
Toner CLC700
•The bed is set to bubble and then thenflow is reduced to the desired value
•The slide is removed and the toners start mixing
•White paper strips are immersed in the yellow side of the cell at regular intervals
Example: gas velocity U0 = 1.8 mm/s
Δt = 14 s
Δt = 51 s
Δt = 96 s
Δt = 188 s
The fluid-like region shrinks to zero as the particle size increasesand type A changes to type B (particles above 75 microns)
Fine powders: solid fraction-gas velocity
Can we predict where are the boundaries solid-like to fluid-like
and fluid-like to bubbling?
• Yes, but for that we need to understand the behavior of particles in the fluidized state.
• The crucial factor is: cohesive particles may aggregate, and the aggregates behave as cohesionlessparticles (key idea)
For our materials the dominant interparticleforces are van der Waals forces, and we need first to determine the size of these aggregates.
Attractive forces: dry uncharged grains
Aggregation of fine particles in fluidized bedsAggregation of fine particles in fluidized beds
Distribution of adhered particles in a paper strip immersed in the bed
Distribution of adhered particles in a paper strip immersed in the bed
a) 100 mμ 19.1 m particle sizeμ b) 100 mμ 7.8 m particle sizeμ200 mμ 40 m particle sizeμ
Bo ~ 10Bo ~ 10 Bo ~ 1000Bo ~ 1000Bo ~ 100Bo ~ 100
pwF
weightparticleforceattractivecleInterpartiBo 0==
The boundary between solid-like and fluid-like regimes is given by the jamming transition (when the aggregates start to be in permanent contact).
We assume that the aggregates sediment like effective cohesionless spherical particles (key idea). Then, in the boundary solid-like/fluid-like their effective volume fraction should be 0.56 (the random loose packing volume fraction of spheres).
If we are able to estimate the particle solid fraction (total mass/volume) as a function of the effective volume fraction of aggregates in the fluid-like region, then we can estimate the particle solid fraction at the solid-like/fluid-like boundary.
The last step is comparison of the model with experiments.
How to predict the solid-like/fluid-like boundary? Our plan
Settling experiments to characterize aggregatesSettling experiments to characterize aggregates
Toner
2 0 0
2 0 0 01 . 5 0 0
Flow controllerManometer
Ultrasound sensor
Valve
Flow controllerOpen end
The valve shuts the gas flow and the bed collapses. The valve shuts the gas flow and the bed collapses.
The ultrasound sensor measures the height of the bed to determine the settling velocity (Acquisition rate 10 - 40 Hz)
The ultrasound sensor measures the height of the bed to determine the settling velocity (Acquisition rate 10 - 40 Hz)
3
4
5
6
7
8
9
0 10 20 30 40 50 60
time (s)
h (c
m)
V = dh/dts
Toner (Seville)Toner (Seville)
Modified Richardson-Zaki law for aggregatesModified Richardson-Zaki law for aggregates
vp0: terminal velocity of an isolated particle
N: number of particles in aggregate
φ* :volume fraction occupied by aggregates
d*: size of aggregate
•Assuming hydrodynamic and geometric radius are equal:•Assuming hydrodynamic and geometric radius are equal:
•For aggregated particles, the R-Z law must be modified:•For aggregated particles, the R-Z law must be modified:
where n = 5.6 if Ret(v*) < 0.1
( )0
1 n
p
vv
φ= −
0* pNv vκ
=
*
p
dd
κ =3
*Nκφ φ=
3
0
1n
ag
p
v N kv N
φκ
⎛ ⎞= −⎜ ⎟
⎝ ⎠
2
01
18p p
pgd
vρ
μ=
Velocity of an aggregate
( )1 **
nsvv
φ= −
Effect of particle size on aggregationEffect of particle size on aggregation
dp(μm) N k D 7.8 63 5.218 2.523 11.8 23.7 3.549 2.512 15.4 12.4 2.724 2.499 19.1 9.6 2.448 2.508
SAC=32% (constant F0)
dp(μm) N k D 7.8 63 5.218 2.523 11.8 23.7 3.549 2.512 15.4 12.4 2.724 2.499 19.1 9.6 2.448 2.508
SAC=32% (constant F0)
* 40 45pd d mκ μ= ≈ −
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
particle volume fraction
v s (c
m/s
) 15.411.87.8
19.1
dp ( m)μ
Theoretical estimation of the maximum size of stable aggregates
Drag acts mainly at the surface of the aggregate whereas gravity is a body force acting uniformly through the aggregate. This results in shear forces distributed across the aggregate limiting its size.
Drag acts mainly at the surface of the aggregate whereas gravity is a body force acting uniformly through the aggregate. This results in shear forces distributed across the aggregate limiting its size.
Spring model derived by Kantor and Witten (1984) and used by Manley et al. (2004) to study the limits to gelation in colloidal suspensions
Spring model derived by Kantor and Witten (1984) and used by Manley et al. (2004) to study the limits to gelation in colloidal suspensions
Theoretical model
Interparticle attractive forceparticle weight
size of aggregateparticle size
fractal dimension of aggregate
g
D
Bo
k
D N κ
≡
≡
≡ =
2DgBo k +≈
pd
1
10
100
1 10 100 1000Bo
N
N ~ Bo
D =~ 2.5 (DLA)
0.7
Theoretical prediction:
Size and number of particles in the aggregateSize and number of particles in the aggregate
0.62D
D Dg gN k Bo Bo+= ≈ ≅
Solid-like Fluid-like boundarySolid-like Fluid-like boundary
33* * 2
DD
J J J gBoNκφ φ φ
−+= ≈ 2D
gBo κ +≈
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 10 100 1000 10000
Predicted (D=2.5)
glass beads
xerographic toners
Cornstarch samples
gBo
φ φ∗J/ J φJ is measured at the jamming transition
k and N are obtained from settling experiments
φ∗J must be close to the RLP limit of noncohesive spheres (0.56) ( aggregates behave as low cohesive effective particles).
φJ is measured at the jamming transition
k and N are obtained from settling experiments
φ∗J must be close to the RLP limit of noncohesive spheres (0.56) ( aggregates behave as low cohesive effective particles).
At the fluid-to-solid transition aggregates jam in a expanded solidlikestructure with a particle volume fraction φJ.
At the fluid-to-solid transition aggregates jam in a expanded solidlikestructure with a particle volume fraction φJ.
Transition A-B
If φb = φJ the bed will transit directly from solid to bubbling (Geldart B)
How to predict the fluid-like/bubbling boundary? Our plan
We estimate this boundary via Wallis criterion for aggregates as effective non-cohesive particles (key idea).
Macroscopic bubbling is a nonlinear process that has been related to the formation of solids concentration shocks when the propagation velocity of a voidage disturbance (uφ ) surpasses the elastic wave velocity (ue) of the fluidized bed (Wallis 1969)
Macroscopic bubbling is a nonlinear process that has been related to the formation of solids concentration shocks when the propagation velocity of a voidage disturbance (uφ ) surpasses the elastic wave velocity (ue) of the fluidized bed (Wallis 1969)
{ }
{ }
0
1 2
2
; : (1 ) ;
at bubbling onset1 ;
g ng p
e
e p pp
dvu R Z v v
du u
pu p gd
φ
φ
φ φφ
ρ φρ φ
/
⎫=− − = − ⎪
⎪ ≈⎬⎡ ⎤⎛ ⎞∂ ⎪=⎢ ⎥⎜ ⎟ ⎪∂⎝ ⎠⎢ ⎥⎣ ⎦ ⎭
Bubbling boundary
Modified Wallis criterion for aggregatesModified Wallis criterion for aggregates0
*
3*
*0
*
fractal dimension of aggregate
volume fraction of aggregates
settling velocity of individual agregate
aggr
g
p
D
p
Interparticle attractive force FBoparticle weight
size of aggregate dkparticle size d
D N
NNv v
κ
κφ φ
κρ
≡
≡
≡ =
= ≡
= ≡
≡
{ }
{ }
* * * **
* *1 2*
* * * * *2* *
egate density
; : (1 ) ;
at bubbling onset1 ;
g ng
e
e
dvu R Z v v
du u
pu p gd
φ
φ
φ φφ
ρ φρ φ
/
⎫= − − = − ⎪
⎪ ≈⎬⎡ ⎤⎛ ⎞∂ ⎪= ⎢ ⎥⎜ ⎟ ⎪∂⎝ ⎠⎣ ⎦ ⎭
2DgBo κ +≈
dp N k φb exp. φb pred.
7.8μm 63 5.22 0.089 0.087
11.8 μm 23.7 3.55 0.140 0.146
15.4 μm 12.4 2.72 0.177 0.188 19.1 μm 9.6 2.45 0.228
0.229
dp N k φb exp. φb pred.
7.8μm 63 5.22 0.089 0.087
11.8 μm 23.7 3.55 0.140 0.146
15.4 μm 12.4 2.72 0.177 0.188 19.1 μm 9.6 2.45 0.228
0.229
Comparison with experimental resultsComparison with experimental results
Fluidization with Nitrogen of tonersFluidization with Nitrogen of toners Fluidization with other gasesFluidization with other gases
CH4
φb(exper.)
φb(pred.)
0
0.1
0.2
0.3
0.4
0 0.1 0.2 0.3 0.4 0.5
0.5
FCC 63 m, Rietema 1991μCanon CLC700 toner 8.5 m μ
N2
N2H2
Ne
Ne HeAr
The maximum stable size Db of bubbles (Harrison et al. 1961)The maximum stable size Db of bubbles (Harrison et al. 1961)
Single, isolated bubble in gas-fluidized bed showing a cloud (Davidson, 1977)
Single, isolated bubble in gas-fluidized bed showing a cloud (Davidson, 1977)
To have an estimation of Db/dp, Harrison et al. hypothesized that bubbles are no longer stable if their rising velocity Ub exceeds the settling velocity of the individual particles.
To have an estimation of Db/dp, Harrison et al. hypothesized that bubbles are no longer stable if their rising velocity Ub exceeds the settling velocity of the individual particles.
( ) ( )2
2 30
2 2 2
1118
18 0.70.7
pp f pp p f b
pb b
g dg dv D
dU g D
ρ ρρ ρμ
μ−−
≈
≈
Criterion for existence of homogeneousfluidization before bubbling
Implications of the Harrison and Wallis criteriaImplications of the Harrison and Wallis criteria
( )
* * * 1/ 2
1/ 2* ** * * * 1 1/ 2* * 1
* *
*
*
* *
*
( ) ;
(1 ) 1 0.7 (1 )
0.7 max 0.1950.098
min( ) 0
1.72
en e nb
e
b
e
b
u g du u Du v n n
u dg D v
at
u u
Dford
φφ
φ
φ
φ φ φ φ
φ
− −
⎫=⎪ −⎪ ⎛ ⎞= − = − −⎬ ⎜ ⎟
⎝ ⎠⎪= ⎪⎭
=
− >
<
14444244443
1442443
* * **0 1.72b
eDThus u u fordφ φ> ∀ > <
2 3(2 3) ( 2)
2 2 2
1 1.7218 0 7
determines the SFE-SFB boundary
p p D Dbg
g dD Bod
ρμ
− / +∗ ≈ =
.
1.5 2 2.5 3 3.5 4
2
4
6
8*bD
d
μ (x 10 ) Pa s5
H2
CΗ2
CO2
N2
Kr Ne
APF-B
APF-E
0
3
27.8
1135 /
2.5
p
p
F nNd m
kg m
D
μ
ρ
≈
≈
≈
0
0.05
0.01
NeonNitrogen
0.2
0.25solidlike
0.1
0.15
0.1 1v (cm/s)g
elutriation (Ne)
Nonbubbling fluidlike
bubbling (N )2
φ
Fluidization of 7.8μm toner with Nitrogen and NeonFluidization of 7.8μm toner with Nitrogen and Neon
Direct transition to elutriation is observed when Neon is used (as predicted)Direct transition to elutriation is observed when Neon is used (as predicted)
Comparison with experimental resultsComparison with experimental results
Nitrogen Neon
Final agglomerates in unsievednano-silica powders
This powder is non-fluidizable
Agglomerate size for sieved nanosilica with a 500 mm grid (Bed expansion)
Using the agglomerate diameter da and fractal dimension D in the RZ equation to fit the expansion of the bed in the uniform fluidlike regime, we obtain:
da: 226 μm
D: 2.588
( )nDp
Dg kvkv φ−− −= 31 1
These are the values we have used in the calculations.
Types of fluidization of cohesive particles
The A-B boundary, identified by φJ=φbcoincides with Bog=1 (limit for aggregation).
Thus the existence of an expanded nonbubblingregime is directly related to aggregation of the cohesive particles
ρp = 1135kg/m3, ρf = 1kg/m3, μ = 1.79x10-5Pa s, F0 = 2nN, g =9.81m/s2, D = 2.5 (typical values).
Types of Fluidization (in other variables)
Conclusion
Using the well known empirical relations for the behaviour of fluidized beds of noncohesivepowders and applying them to aggregates:
1.- We have estimated the boundaries between the different types of fluidization, and 2.- We have predicted a new type of fluidization (solid-like to fluid-like to elutriation).
Thank you very much for your attention