Aggregation and Breakage of Nanoparticle Dispersions in Heterogeneous Turbulent Flows

CFD in Chemical Reaction Engineering CFD in Chemical Reaction Engineering BargaBarga, Italy, June 2005, Italy, June 2005

M. SoosM. Soos, D. Marchisio*, J. Sefcik, M. Morbidelli, D. Marchisio*, J. Sefcik, M. MorbidelliSwiss Federal Institute of Technology ZSwiss Federal Institute of Technology Züürich, CHrich, CH

**PolitecnicoPolitecnico didi TorinoTorino , IT, IT

Coagulator types:Coagulator types:Stirred tankStirred tank

Couette deviceCouette devicePipePipe

Problem definition - coagulation process

AggregatesAggregates / Granules/ GranulesDispersion of stable Dispersion of stable primary particlesprimary particles

destabilizationdestabilization

10-7 10-6 10-5 10-4 10-3

[m]

Problem definition - coagulation process

Turbulent flowTo have good product quality:To have good product quality:•• appropriate morphologyappropriate morphology•• effective mixingeffective mixing

Stirred tank Taylor-Couette device

V = 1.3 L

V = 2.5 L

Particle/aggregate characterization technique

1

23

1 – Stirred vessel2 – Light scattering

instrument3 – Syringe pump

Focused Beam Reflectance Method

(LASENTEC)

Light Scattering(Malvern Mastersizer)

Probe of FBRM

Breakage

Aggregation

k-j

j

Bk kK N

Aggregation Rate

,Ai j i jK N N

Breakage Rate

j

i

k

k=i+j

Aggregation and Breakage kinetics

Assumption about daughter distribution function

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

, ,1 1

; , ; , ; ,

1 ; , ; , d ; , ; , d2

; , d ; ,

i tti i i

A A

B B

n t n tu n t D

t x x x

K n t n t n t K n t

K b n t K n t

ξ

ξ ξ ξ ξ ξ

ξ ξξ

ξ ξξ

ξ ξ ξ ξ ξ ξ ξ

ξ ξ ξ ξ ξ

∞

′ ′ ′−

∞

′

∂ ∂⎡ ⎤∂ ∂⎡ ⎤+ − =⎢ ⎥⎣ ⎦∂ ∂ ∂ ∂⎣ ⎦

′ ′ ′ ′ ′− −

′ ′ ′+ −

∫ ∫

∫

x xx

x x x x

x x

Reynolds-averaged mass balance (“CFD & PBE”): (PBE mass based)

Aggregation source term

Breakage source term

There are several numerical approaches to solve this equation:

• Classes method• Method of moments*• Monte Carlo method…

Population Balance Equation

Only unknown are the values of aggregation and breakage kernels

b(ξ |ξ’) - Daughter distribution function of produced fragments

( )3 1/ 1/3, A = f fd dSA

pK G Rξ ξ α ξ ξ′ ′+

( )( ) 3 1/1 = f

PdBK P Gξ ξ

Aggregation and Breakage rate expressions (kernels)

1/ fd

p

RRξ ξ∝

Fractal scaling:

( )1OCC n Vξφ ξ∞

= ∫

Occupied volumefraction:

Breakage kernel:Aggregation kernel:

( )( )1 1 1 1,

2 13

f f f fd d d dBA Bk TKWξ ξ ξ ξ ξ ξ

µ− −

′ ′ ′= + +

, , ,A BA SAK K Kξ ξ ξ ξ ξ ξ′ ′ ′= +

( )2 1 1

12

1

fN dg i ii

Np i ii

R wR w

ξ

ξ

+

=

=

= ∑∑

wi – weights of quadrature app.ξi – abscissas of quadrature app.

Connection to light scattering

αA, P1, P3, df – from experiment or assume

G – either from experiment (difficult)or from CFD

12

G εν⎛ ⎞= ⎜ ⎟⎝ ⎠

0.5occφ ≅

gelation occurs

0 15 30 45 60

3

6

9

12

Fixed pivot QMOM N = 2 QMOM N = 3 QMOM N = 4 QMOM N = 5

τ = t G φ

⟨Rg⟩

/ R

p

100

104

108

1012

100 101 102 103100

104

108

1012

100 101 102 103

m(i) / m(1)

Num

ber

dens

ity (#

/ m

3 )

7.5 min 15 min

25 min 45 min

• Steady state is accurately modeled with two nodes (N = 2)

• For lower fractal dimensions (df ≤ 2) larger number of nodes (N = 4 - 5) needed to be used

Cluster mass distribution (CMD)

Solution of PBE - QMOM vs. fixed pivot method

quadrature approximation

( )0

1d

Nk k

k i ii

m n wξ ξ ξ ξ∞

=

= ≈∑∫

( ) ( ) ( )

( ) ( ),

1 1 =1 1

, , ,

1 2

k ki k t

i i i

N N N Nk kA k k B B ki j i j i j i j i i i i i i

i j i i

m t m tu m t D

t x x x

K w w K b w K wξ ξ ξ ξ ξ= = =

∂ ∂⎡ ⎤∂ ∂⎡ ⎤+ − =⎢ ⎥⎣ ⎦∂ ∂ ∂ ∂⎣ ⎦

⎡ ⎤+ − − + −⎢ ⎥⎣ ⎦∑∑ ∑ ∑

x xx

Weights wi and abscissas ξi calculated by PD algorithm

Quadrature Method Of Moments (QMOM)

0 5 10 15100

101

102

200 rpm 400 rpm 600 rpm 800 rpm

τ = t G φ

⟨Rg⟩

/ Rg,

p

Breakage is nonlinearly dependent on G

Estimation of model parameters from experiment

( )( ) 3 1/1 = f

PdBK P Gξ ξ

( )3 1/ 1/, A = f fd dAK Gξ ξ α ξ ξ′ ′+

Aggregation is linearly dependent on G

αA, df

P1 (G), P3

RP = 1.085 µmφ0 = 5 × 10-5

αA = 0.2df = 2.1

P1 = 1.551 × 1012 G 2.82

P3 = 4

Dimensionless form of PBE

10 0

1 0 0

( ) 1 ( ) ( ) ( ) ( )2

+ ( ) ( )

A Aijk ik

i j k ii j k i

B Bmk m k

m km k

Kd x Kx x x xd V G V G

K Kx xG G

τ τ τ τ ττ

τ τφ φ

∞

+ = =

∞

= +

= −

Γ−

∑ ∑

∑

0t Gτ φ=0

0

( )( ) ii

N Vx ττφ

=

Dynamics of the system – full CFD (TC)

G1 = G2 = Gi = ⟨G⟩

G1

G2

G1 ≠ G2 ≠ Gi ≠ ⟨G⟩

G1

G2

CFD model:• particle distribution heterogeneous• shear rate distribution heterogeneous

Homogeneous model:Homogeneous model:• particle distribution homogeneous• shear rate distribution heterogeneous

breakage kernels properly averaged over volume

Lumped model (often used in literature):Lumped model (often used in literature):• particle distribution homogeneous• shear rate is everywhere equal to the ⟨G⟩

2 2P PG G≠

Is the effect of spatial shear rate heterogeneity significant ?

Innerwall

Outerwall

Taylor-Couette apparatus, 2D simulation, RSM of turbulence

12

G εν⎛ ⎞= ⎜ ⎟⎝ ⎠

Dynamics of the system – full CFD (TC)

0 5 10 15 200

3

6

9

12

Lumped model CFD Homogeneous model

⟨Rg⟩

/ R

p

τ

φ0 = 5 × 10-5

φ0 = 5 × 10-5

⟨Rg⟩ / Rp

Assumption about particle distribution homogeneity is valid

Dynamics of the system – full CFD (TC)

0 5 10 15 200

3

6

9

12

Lumped model CFD Homogeneous model

⟨Rg⟩

/ R

p

τ

φ0 = 5 × 10-5

0 5 10 15 200

10

20

30

⟨Rg⟩

/ R

p

τ

φ0 = 1 × 10-3

φ0 = 5 × 10-5 φ0 = 1 × 10-3

⟨Rg⟩ / Rp

Assumption about particle distribution homogeneity is NOT

valid

Fluid element trajectories

Two examples of fluid element trajectories corresponding to the starting point from low and high

shear rate value8 12 16 200

200

400

600

3 6 9 12 15

G, s

-1

Time, s

Spatial shear rate distribution in the gap of Taylor-Couette

Innerwall

Outerwall

G = 110 s-1

Shear rate [1/s]

Fluid element tracking – Process timescales

Macro-mixing:

Aggregation / breakageat steady state:

( )1

; ,A A

g gK G R R Nτ =

( )1;

B BgK G R

τ =

τ M =

kε

12

G εν⎛ ⎞= ⎜ ⎟⎝ ⎠

k, εCFD

τA, τB >> τM

Homogeneous model:• particle distribution homogeneous• aggregation / breakage kernels properly

averaged over volume

( ), ,1 dVA A

V

K K GVξ ξ ξ ξ′ ′= ∫ ( )1 dVB B

V

K K GVξ ξ= ∫

8 12 16 200

15

30

45

3 6 9 12 15

⟨Rg⟩

/ R

p

Time, s

8 12 16 200

200

400

600

3 6 9 12 15

G, s

-1

Time, s

Shear rate history

Calculation starts from the steady state distribution

corresponding to the starting point of shear rate

φS = 5 x 10-5

Fluid element tracking – Process timescales

τA, τB << τM

Equilibrium model:• particle distribution relaxes

immediately to steady state according to local shear rate

8 12 16 200

100

200

300

3 6 9 12 15Time, s

⟨Rg⟩

/ R

p

8 12 16 200

200

400

600

3 6 9 12 15

G, s

-1

Time, s

Shear rate history

Calculation starts from the steady state distribution

corresponding to the starting point of shear rate

Macro-mixing:

Aggregation / breakageat steady state:

( )1

; ,A A

g gK G R R Nτ =

( )1;

B BgK G R

τ =

τ M =

kε

12

G εν⎛ ⎞= ⎜ ⎟⎝ ⎠

k, εCFD

φS = 1 x 10-2

Analysis of timescales – Conclusion

0

20

40

60

80

100 Lumped model Homogeneous model CFD model Equilibrium model

⟨Rg⟩

/ R

p

max

min

1.1g

g

R

R≈

0.1OCCφ ≈

10-4 10-3 10-210-3

10-2

10-1

100

101

102

103

times

cale

s, s

φS

τM

τA,B

Hom

ogen

eous

mod

elH

omog

eneo

us m

odel

Cou

pled

CFD

& P

BE

Cou

pled

CFD

& P

BE

mod

el a

ppro

pria

tem

odel

app

ropr

iate

Nee

d fu

ll C

FD &

PB

EN

eed

full

CFD

& P

BE

w/ m

icro

mix

ing

mod

el

w/ m

icro

mix

ing

mod

el

& m

ultip

hase

flow

& m

ultip

hase

flow

(τA < τB for this system)

Note: region of validity of homogeneous model in stirred tank is shifted to left compare to TC

• Based on timescales analysis it is possible to decide which model is appropriate for certain conditions

• At low volume fraction (< 4 × 10-4) CFD model can by efficiently replaced by homogeneous model

• Kinetic parameters obtained from lumped model are not applicable for different vessel geometry – significant effect of shear rate heterogeneity

• CFD + PBE need to be used already for rather mild solid volume fractions

Note: occupied volume fraction can be used to check significance of viscosity on flow field