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