3. The Motion of ParticlesDrag force

d particle diameterV flow velocity

Spherical particle, Re < 13DF dV

Drag coefficient

212

24

ReD

Da

FC

V A

A projected area24d

Case 1: With slip3

Dc

dVF

C

Ccis Cunningham correction factor

For d > 0.1 m

2.521cC d

For d > 0.01 m

0.551 2.514 0.8expcC d d

Case 2: High Re, Re > 1

3Case : Nonspherical particle3 V

Dc

d VF

C

is shape factorVdis equivalent volume diameter

Shape/type

spherical fiber (L/d 4= )

quartz dust fused alumina

talcum (platelet)

11.32 (axis perpendicular to flow)1.07 (axis parallel to flow) 1.361.04-1.492.04

Motion under gravity3 V

Dc

d VF

C

Equation of motion

0y ydV Vg

dt

Particle relaxation time or time constant2

3 18pdm

d

Terminal settling velocity

2

18p

TS

d gV g

Mechanical mobility2

18pTSdV

Bmg m

Terminal settling velocity with slip, shape factor2

18pc

TS

d gCV

Motion under electrical forces

EF Eq neE

q neq particle chargen number of chargee electron charge = 1.6x10-19 CE electric field

In equilibrium

E DF F

3

c

dVneE

C

Terminal electrical velocity

3c

TE

neECV

d

Electrical mobility

3TE cV neC

ZE d

Relation between VTE and E for two particle sizes

Motion under thermal gradients

1T k T V

Thermophoretic force - > Temperature gradient

Thermophoretic velocity

Motion under no external force

Equation of motion

3dV

m dVdt

Velocity

0 expV V t

Traveling distance

0

0 0

0

expt t

t

x t V dt V t dt

V e

Stopping distance, t >>

0

0

20

18p

S V

BmV

d V

Similarity in particle motion

1. Reynolds number (Re) must be equal

2

stopping distancecharacteristic lengthStk

18p

S

D

d U

D

With slip

2 . Stokes number (Stk) must be equal

2

Stk18p cC d U

D

Particle motion for several values of Stokes number

3. When gravity is important, gravitational parameter (G) must be equal

G TSV

U

To determine if inertia or gravity is more important, use Froude number (Fr)

2StkFr =

G

V

gD

Aerodynamic diameter

Aerodynamic diameter (da ) is the diameter of a spherical particle of density0

= 1 g/cm3 which has the same terminal settling velocity in air as the particle of interest. 1 2

0

pa pd d

Stokes diameter (ds ) is the diameter of a spherical particle that has the same density and terminal settling velocity in air as the particle of interest.

1 2

0

ba sd d

bis the bulk density

Comparison of equivalent volume diameter, Stokes diameter, and aerodynamic diameter.

IIIIIIII II IIIIIII

Stokes number

2

Stk2 9

p p c

j j

d U CU

D D

jDis the jet diameter

Collection efficiency characteristics of an impactor

- - Collection efficiency characteristics of an impactor: Ideal v real

Diffusion (Brownian motion)

Random motion of an aerosol particle in still air

Jis the particle flux (# particles per unit area per unit time)is the diffusion coefficientis the number of particlesis the direction of motion

Fick’s first law

B

dnJ D

dx

BD

n

x

3c

B

kT CD kT B

d

Stokes-Einstein derivation

RMS and average velocity1 2

1 2

3

3

18

rms

p

kTc

m

kT

d

1 2

1 2

2 3

8

48

p

kTc

m

kT

d

dp(micron)

DB

(m2/sec)B

(m/N sec)c

(m/sec)

0.00037*

0.01

0.1

1.0

10.0

2.0x10-5

5.4x10-8

6.9x10-10

2.7x10-11

2.4x10-12

-

1.3x1013

1.7x1011

6.8x109

6.0x108

460

4.4

0.14

0.0044

0.00014

* diameter of air molecule

Diffusion-related properties of standard-density spheres at 293 K

Deposition by diffusion

2

2B

dn d nD

dt dx

Fick’s second law

Aerosol particle collides and sticks to the surface

Boundary and initial conditions

0,0 , 0n x n x

0, 0 , 0n t t

Solution ,

2 B

xn x t erf

D t

Concentration profile for a stagnant aerosol of 0.05-mm particles near a wall

General form of the concentration profile near a wall

Cumulative number of particle deposited per unit area during time t

1 2

02 BD tN t n

Deposition velocity: velocity that particles move to a surface and is analogous to the terminal settling velocity due to gravity.

0dep

JV

n

Cumulative deposition of particles on a horizontal surface during 100 sec.

Cumulative deposition

dp

(micron)Diffusion(#/m2)

Settling(#/m2)

RatioDiffusion/settling

0.0010.010.11.010100

2.6x104

2.6x103

3.0x102

59175.5

0.686.988

35003.1x105

2.5x107

3.8x104

3803.4

1.7x10-2

5.5x10-5

2.2x10-7

Diffusion of aerosol particles on the tube wall

Penetration for circular tube

2 3 , 0.0091 5.50 3.77

, 0.0090.819exp 11.5 0.0975exp 70.1out

in

nP

n

Deposition parameter

2

4 B B

t

D L D L

d U Q

Lis the length of the tubeis the diameter of the tubeis the average velocityis the flow rate

td

UQ

Peclet number : another dimensionless parameter used in diffusion motion

PeB

UD

D

Dis the characteristic length

Penetration for rectangular tube

2 3 , 0.0051 2.96 0.4

, 0.0050.910exp 7.54 0.0531exp 85.7P

Penetration of aerosol particles in a tube.

Fractional loss to the walls by diffusion for an aerosol flowing through a 1-m-long tube

dp Flow rate (L/min)

(micron) 0.1 1.0 10

0.0010.010.11.0

1.0000.4280.0290.003

0.9780.1080.0060.0008

0.4220.0250.0010.0002