04/22/2023

Aerosol & Particulate Research Lab 1

Motion of Aerosol

• Newton’s Resistance Law and Stokes’ Law• Cunningham Slip Correction Factor• Settling Velocity, Mechanical Mobility• Particle Acceleration• Aerodynamic Diameter• Settling Chamber• Brownian Motion & Diffusion

Reading: Chap. 3.3-3.4

http://aerosol.ees.ufl.edu/aerosol_trans/section01.html

04/22/2023

Aerosol & Particulate Research Lab 2

History

History: Galileo’s (1564-1642) experiment in Pisa tower

Newton’s Resistance Law: The force is proportional to the gas pushed away and the relative velocity between the sphere and the gas (negligible viscous force)

)2

(8

222pg

pDppgDD V

ACVdCF

Vp

CD = 0.44 for Rep > 1000

Tim Tebow can be a good air pollution engineer because he knows how to control the movement of particles in the air …..

04/22/2023

Aerosol & Particulate Research Lab 3

ppD dVF 3

pppgD dV

CRe2424

Stokes Law: negligible inertial force compared to viscous force (Rep < 1)

Reynolds Number: inertial force/frictional force

ppgp

dVRe

Two major parameters: V & dp

Plot CD as a function of Rep for Newton’s LawPlot CD as a function of Rep for Stokes’ Law

Transition Regime

6Re

1Re24 3/2

p

pDC

04/22/2023

Aerosol & Particulate Research Lab 4

What is the max velocity of a particle to be in the Stokes regime? 1 m, 10 m, 100 m (g = 1.2 kg/m3; = 1.81×10-5 Pa•s)

04/22/2023

Aerosol & Particulate Research Lab 5

Categorize aerosol movement based on interaction between the particle and gas molecules

Mean free path of gas (l): average distance traveled by a gas molecule between successive collisions; 0.066 m for air at STP

Ruler of aerosol movement:

)1.0( 52.21

3

mdd

C

CVd

F

pp

c

c

pD

l

(derivation for dp < 1 m @ STP)

Handout Appendix A11

1 mm 0.1 m

gas velocity at the surface of small particles is not zero --> slipCunningham Slip Correction Factor:

http://aerosol.ees.ufl.edu/aerosol_trans/section06.html

04/22/2023

Aerosol & Particulate Research Lab 6

Settling Velocity• When the drag force is equal and opposite to the

gravitational force

• Particle Mechanical Mobility

6 )( 3

3 gd

CdV

mgFF pgp

C

ppGD

1Refor 18

2

pcpp

TS

gCdV

p

C

D dC

FVB

3

)regimelaminar Law, sStoke'(

~ 0

What is the impact of considering Cc on a particle’s settling velocity? Why?

What is the physical meaning of B?Does a smaller or a larger particle

have larger mobility?http://worldphotocollections.blogspot.com

http://onprogram.blogspot.com

04/22/2023

Aerosol & Particulate Research Lab 7

Relaxation Time: indication of the time required for a particle to adjust/relax its velocity to a new condition of force

Terminal/Settling Velocity

18

2Cpp Cd

mB

)considered ison acceleratiother if (or agBFV GTS

(Appendix A11 or Figure 5-8 on p.9 or Figure 3.8 in text)(remember B is an intrinsic property of a given aerosol)

Q: What other acceleration?

Transition Regime Turbulent Regime

29.043.0

71.014.171.0153.0

air

ppTS

dgV

74.1air

ppTS

gdV

04/22/2023

Aerosol & Particulate Research Lab 8

If flow regime is unknown because VTS is unknown --> K factor33.0

2

gp

p

gdK

Laminar K< 3.3Turbulent K > 43.6

Can we clean the dust in this classroom by waiting them to settle down? How long will it take? Assume the particle size is: 1 m, 10 m, 100 m. The room is 3 m high.

Fig 5-8 on P.9

For a 100 µm unit-density particle, which flow regime is applicable?

04/22/2023

Aerosol & Particulate Research Lab 9

04/22/2023

Aerosol & Particulate Research Lab 10

Particle Acceleration

)1()( )()(

)()(3

/

tTS

cpDG

eVtVdt

tdVtVg

dttdVmCdtVmgFF

FG=mg

t=0V(t)=0

FG=mg

t=V(t)=?

FG=mg

t>3V(t)=VTS

FD=3V(t)dp

FD=3VTSdp

• Newton’s law

http://aerosol.ees.ufl.edu/aerosol_trans/section04.html

04/22/2023

Aerosol & Particulate Research Lab 11

Non-zero Initial velocityV t V V V ef f

t( ) ( ) / 0

Stopping DistanceS V BmV 0 0

)1()()( /0

tff eVVtVtx For Re0 < 1

Small aerosols adapt to the new environment (i.e. following the flow well) in a very short time, almost instantly!!!

dp (m) S (mm) 3 (ms) Re0.01 7.0×10-5 2.0×10-5 0.00660.1 9.0×10-4 2.7×10-4 0.0661.0 0.035 1.1×10-2 0.6610 2.3 0.85 6.6

100 127 65 66

Stopping Distance and Time to Travel 95% of the Stopping Distance for Standard Density Spheres with an Initial Velocity of 10 m/s

04/22/2023

Aerosol & Particulate Research Lab 12

Aerodynamic Diameter• The Stokes diameter, ds, is the diameter of the sphere that has the

same density and settling velocity as the particle.• The aerodynamic diameter, da, is the diameter of the unit density

(0=1 g/cm3) sphere that has the same settling velocity as the particle.

1818

20

2cacsp

TSgCdgCd

V

0 p

sa dd

PM10 and PM2.5 are aerodynamic diameters. Why?Is optical diameter or aerodynamic diameter more relevant?

04/22/2023

Aerosol & Particulate Research Lab 13

LHVV

x

TS

Horizontal Elutriator/Settling Chamber(Plug flow model: no radial or axial mixing)

If monodisperse aerosols are uniformly distributed at the entrance, what is the collection efficiency as a function of VTS (dp)?

04/22/2023

Aerosol & Particulate Research Lab 14

Brownian Motion & Diffusion

• The primary transport mechanism for small particles (< 0.1 m); Important when transport distance is small: e.g. filter, airway in human lung

• Definition:– Brownian motion: irregular wiggling motion of a

particle caused by random bombardment of gas molecules against the particle

– Diffusion: the net transport of the particles from a region of higher concentration to a region of lower concentration

http://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/brownian/brownian.html

04/22/2023

Aerosol & Particulate Research Lab 15

Fick’s First Law of Diffusion The net flux of aerosols (the net number of particles

traveling through a unit area per unit time) is proportional to the concentration gradient

J: flux (#/area/time)D: diffusion coefficient (area/time)n: particle number concentration (#/cm3)

dxdnDJ x

Fick’s Second Law of Diffusion

xnDJ

tn

2

2

The rate of loss of particles is proportional to the gradient of the flux.

driving force

04/22/2023

Aerosol & Particulate Research Lab 16

• Solve 1-D equationSpread of particles over time and space

nt

Dnx

2

2

1

21

161

Numbers on curves are values of Dt

04/22/2023

Aerosol & Particulate Research Lab 17

SolutionDtx

DtNtxn o

4exp

2),(

2

Mean Square Displacement of particles

Result:

Dt

x

2

2

xN

x n x t dxo

2 21

( , )

Stokes-Einstein Equation for Diffusivity

p

c

dkTCD3

How to have a larger diffusivity? Why? Air, water, universe?

04/22/2023

Aerosol & Particulate Research Lab 18

Quick Reflection