Chapter 7. Transport Phenomena of Nanoparticles

7.1 Drag force

(1) Stokes' law

* Drag force, FD:

- net force exerted by the fluid on the spherical particle in the direction of flow

where U : relative velocity between particle and fluid

CD: drag coefficient

cf. For pipe flow

where f: Fanning friction factor

Area exerted by friction

Kinetic energy of unit-mass fluid

24

2

2U

dCFf

pDD

ρπ

=

( )22

22 UDLfF

Uf

f

w

f

w

ρπ

ρτ =→=

* CD vs. Rep

where

, : density and viscosity of fluid

- For Rep < 1 (creeping flow region)

Stokes' law

cf. for pipe flow

- For 500<Rep<200,000 CD=~0.44

For 1<Rep<500

ρ µµ

ρ fp

p

Ud=Re

UdF pD µπ3=

24Re

242

2U

dFf

p

p

D

ρπ

=

Re

16=f

( )687.0Re15.01Re

24p

p

DC +=

p

DCRe

24=∴

(2) Non-continuum Effect

* Mean-free path of fluid

where nm : number concentration of molecules

dm: diameter of molecules

For air at 1 atm and 25oC, = 65.1nm

For water at 25oC, = ?

* Particle-fluid interaction

λ

λ

Continuum regime transition regime free-molecule regime

22

1

mm dn πλ =

* Knudsen number

-Continuum regime (<0.1)

-Transition regime (0.1~10)

-Free-molecular regime (>10 )

- Particles in water is always in continuum regime…

* Corrected drag force

where Cc: Cunnigham correction factor

- Particles in water do not need noncontinuum correction…

(3) Nonspherical particles

* Shape correction factor,

In air at 1atm and 25oC

22.75.062.911.1681.017

0.010.050.11.010

χ

Dynamic shape factors of powders

1.001.08

1.321.07

1.05-1.111.361.572.04

spherecubeCylinder (L/D=4)

axis horizontalaxis vertical

bituminous coalquartzsandtalc

Dynamic shape factorPowders

p

nd

Kλ

=

0~nK

1~nK

∞~nK

C

p

dC

UdF

µπ3=

( )[ ]nnC KKC /55.0exp8.0205141 −++=

Ud

FD

µπχ

υ3=

7.2 Migration in Gravitational Force Field

For the particle suspending in the fluid

- Force balance

Terminal settling velocity

- For Stokes’ law regime

8.8×10-5

1.0×10-3

3.5×10-3

7.8×10-2

0.31

, cm/sa

0.10.51.05.010.0

aFor unit density particle in air at 1atm and 25oC

BgD FFF −=

( ) gdU

dC pfp

fpD

3

2

2

624ρρ

πρπ−=

∴

21

3

4

−=∴

f

fp

D

p

TC

gdU

ρ

ρρ

( ) gdC

Udpfp

c

p 3

6

3ρρ

πµπ−=

( )µ

ρρ

18

2

cpfpTCgd

U−

=∴

* Dynamic equivalent diameters – calculated in air

In general

* Migration velocity

where Um: migration or drift velocity in the fields

Note gravitational migration velocity: terminal settling velocity, UT

* Number flux by migration

where n: particle number concentration

Irregular particles and its equivalent spheres

Stokes’diameter

Aerodynamic diameter

== m

c

p

Dext UC

dFF

πµ3

mUnJrr

=

Centrifugal migration

Electrical Migration

where q: charge of particles

E : strength of electric field

e: charge of electron (elementary unit of charge)

ne : number of the units

* Charging of particles

- Applied to electrostatic precipitation

Acceleration of centrifugation>>g

22

11 ωρ

ρ

ρ

ρrm

r

UmF

p

f

p

t

p

f

pc

−=

−=

( )r

CUdU

ctpfp

cf µ

ρρ

18

22−=∴

eEnEqF e==rr

p

cee

d

eECnU

πµ3=∴

7.3 Electrical Migration

where q: charge of particles

E : strength of electric field

e: charge of electron (elementary unit of charge)

ne : number of the units

- Electrical mobility

- Applied to electrostatic precipitation in gas

* Charging of particles in gas (Later for the case of liquid)

- Direct ionization

- Static electrification: electrolyte, contact, spray, tribo, flame)

- Collision with ions or ion cluster

eEnEqF e==rr

p

cee

d

eECnU

πµ3=∴ Electrical migration velocity

πµ3

0 cei

eCn

E

UZ ==

* Diffusion charging

- Collision by ions and charged particles in Brrownian motion

Where : mean thermal speed of the ions(=240m/s at SC)

Ni: ion concentration

KE: proportionality factor depending on unit used…

* Field charging

- Charging by unipolar ions in the presence of a strong electric field

Where : relative permissibility of the particle

Zi: mobility of ions(=0.00015m2/V s

- Saturated after sufficient time…

+=

kT

tNecdK

eK

kTdtn

iipE

E

p

21ln

2)(

2

2

π

ic

+

+

=tNeZK

tNeZK

eK

Edtn

iiE

iiE

E

p

ππ

εε

142

3)(

2

rε

04

1

πε=EK

* Charge limit

- By electron ejection from mutual repulsion on the surface

where EL: surface field strength required for spontaneous emission of

electrons(=9.0 108 V/m)

- Rayleigh limit

If mutual repulsion > surface tension force for liquid droplets

* bipolar charging: Kr85

eK

Edn

E

Lp

4

2

max =

2/1

2

3

max

2

=

eK

dn

E

pπσ

×

Electrophoresis

- Movement of nanoparticles in liquid medium

* Zeta potential : potential at the slip plane*

- Plane that separates the tightly bound liquid layer from the rest of liquid

- ~Stern layer

- determines the stability of colloidal dispersion or a sol

- requires >25mV for the stability

* Electrical migration velocity

* Electrical mobility

)2/1(2 0 ppr dd

q

κεπεζ

+=

πµζεε

3

2 0 EU r

E =

πµζεε

3

2 00 rE

E

UB ==

7.3 Migration by Interaction with Fluids

(1) Diffusion

* Brownian motion

: Random wiggling motion of particles by collision of

fluid molecules on them

* Brownian Diffusion :

Particle migration due to concentration gradient by Brownian motion

Fick's law

where Dp: diffusion coefficient of particles, cm2/s

n : particle concentration by number

cf. Diffusion of molecules

nDJ p∇−=rr

* Coefficient of Diffusion

cf. Liquid diffusivity 10-5cm2/s

p

c

pd

kTCD

πµ3=

Diffusion Coefficient of Unit-density sphere at 20oC in air

0.195.2 10-4

6.7 10-6

2.7 10-7

2.4 10-8

0.00037(air molecule)0.010.11.010

Diffusion coefficient,cm2/s

Particle diameter,

0,, →∆∆∆ zyx

Jz

J

y

J

x

J

t

n zyxrr⋅∇−=

∂

∂+

∂

∂+

∂

∂−=

∂∂

∴

( ) nDnDt

npp

2∇=∇⋅∇=∂∂ rr

0=n

and

- Introducing Fick's law

B.C.

(x,y,z)

x∆y∆z∆

yxJzxJzyJzyxt

nzyx ∆∆∆−∆∆∆−∆∆∆−=∆∆∆

∂∂

∴

zyJJJzyxt

nxxx ∆∆∆+−=∆∆∆

∂∂

)]([

zyx ∆∆∆/

zxJJJ yyy ∆∆∆+−+ )]([

yxJJJ zzz ∆∆∆+−+ )]([

Net input in x direction

Net input in y direction

Net input in z direction

Constant Dp

- Mass balance for the cube in the fluid

- Integration (or solution) gives: n=n (x,y,z,t),

or particle number concentration at (x,y,z) at t

x

y

z

),,( zyxn

At given time t

* One-dimensional diffusion from the origin

At t=0 n=0 for all x except x=0

At x=0, n=n0 for all t and

The solution is :

Where

Differentiating with respect to x

Normal distribution with respect to x-axis

( ) nDnDt

npp

2∇=∇⋅∇=∂∂ rr

2

2

x

nD

t

np ∂∂

=∂∂

∴

00

=

∂∂

=xx

n

)()(12

1)(0

2

ηηηπ

ηη

η erfcerfden ≡−≡−= ∫ −

tD

z

p2≡η

dxtD

x

tDn

txdntxn

pp

−=

4exp

)4(

1),(),(

2

2/1

0 π

- Mean displacement:

- Standard deviation or root-mean square displacement

- represent particle movement (displacement) by diffusion

0=x

tDx prms 2==σ

Net displacement in 1s due to Brownian motion and gravity for standard-density spheres at standard conditions

7.1 10-43.1 10-32.2 10-610

0.213.5 10-67.4 10-61.0

428.8 10-73.7 10-50.1

48006.9 10-83.3 10-40.01

xrms /xsettSettling in

1s(m)xrms in 1s(m)

Particle diameter, m

(2) Thermophoresis

- Discovered by Tyndall in 1870

- Examples of thermophoresis

- Dust free surface on radiator or wall near it

- Movement cigarette smoke to cold wall or window

- Spoiling of the surface of the cold wall

- Scale formation on the cold side in the heat exchanger

* In free molecular regime

Waldmann and Schmidt(1966)

- independent of dp

T

TdpF pth

∇−=

rr

2λ

T

T

T

TU th

∇=

+

∇−=∴

rrr

νπα

ν55.0~

814

3

* Correction for continuum fluid-particle interaction

Brock(1962)

where

Terminal settling and thermophoretic velocities in a temperaturegradient of 1oC/cm at 293K

2.8 10-6

2.0 10-6

1.3 10-6

7.8 10-7

6.7 10-8

8.6 10-7

3.5 10-5

3.1 10-3

0.010.11.010.0

Thermophoretic velocities in a temperature gradient

of 1oC/cm at 293Ka

Terminal settling velocity (m/s)

Particle diameter( )

T

THdF

G

p

th ρ

πµ

2

9 2 ∇−=

rr

++

+

+Kn

k

k

Knk

k

KnH

p

G

p

G

8.821

4.4

61

1~

T

THCU

G

cth ρ

µ2

3 ∇−=∴

rr

ap

a kk 10=

mµ

(3) Phoresis by Light

Photophoresis

- Where to be heated depends on the refractive index of the particle

e.g. submicron particles in the upper atmosphere

Radiation pressure

e.g. tails of comet, laser-lift of particles

(4) Diffusion of medium

Diffusiophoresis

Stefan flow

- For evaporating surface

- For condensing surface

e.g. Venturi scrubber

7.4 Inertial Motion and Impact of Particles

(1) Inertial motion

- For Stokesian particles

Momentum (force) balance for a single sphere

Neglecting buoyancy force

Integrating once, defining relation time as

Integrating twice

As ,

stop distance

Dp Fdt

dUm −=

c

p

pp

C

Ud

dt

dUd

πµρ

π 3

6

3 −=∴

µ

ρτ

18

2

cpp Cd=

τt

eUU−

= 0

−= − ττ

t

eUx 10

∞→τt

sCUd

Uxcpp ≡=

µ

ρτ

18~

0

2

0

0.065*127*66100

8.5 10-4*2.3*6.610

1.1 10-50.0350.661.0

2.7 10-79.0 10-40.0660.1

2.0 10-87.0 10-50.00660.01

time to travel 95%

of S

S at U0=10m/

sRe0

Particle diameter,

m

Net displacement in 1s due to Brownian motion and gravity for

standard-density spheres at standard conditions

* out of Stokes' range

(2) Simiulitude Law for Impaction : Stokesian Particles

* Impaction: deposition by inertia

- For Re < 1

Force balance around a particle

Defining dimensionless variables

where U , L : characteristic velocity and length of the system

Define Stokes number

( )fpp

p

p UUddt

Udm −−= πµ3

L

UU ≡1

L

UU

f

f ≡1L

tU≡θand ,

( )111

fUUd

UdSt −−=∴

θ

obstacle of size

epersistenc particle

18

2

≡=≡L

U

L

UdSt

pp τµ

ρ

,

(subscript p: omitted)( )1113

3)/(6

fpp ULULddUL

ULdd −−= πµ

θπ

( )111

2

18f

ppUU

d

Ud

L

Ud−−=

θµ

ρ

Or in terms of displacement

↑

where , : displacement vector

-

* Particle trajectory

↑

where

- Two particle impaction regimes are similar

when the geometric, hydrodynamic and particle trajectories are the same…

- Applications

- Cyclone, particle impactor, filter

11

2

1

2

fUd

rd

d

rdSt =+

θθ

rr

L

rr

rr≡1

( )RStfr Re,,)(1 =∴ θ

Geometric similarity

L

dR

p=

The solution gives , or where the particle is at time t…)(tr

Re

)(1 θr

x

y

z

)(tr