Introduction to Aerosols Drag Forces Cunningham ... · Introduction to Aerosols ! Drag Forces! Cunningham Corrections! Lift Forces! Brownian Motion! Particle Deposition Mechanisms!

1

G. AhmadiME 437/537-Particle G. AhmadiME 437/537-Particle

!! Introduction to Aerosols Introduction to Aerosols !! Drag ForcesDrag Forces!! Cunningham CorrectionsCunningham Corrections!! Lift ForcesLift Forces!! Brownian MotionBrownian Motion!! Particle Deposition MechanismsParticle Deposition Mechanisms!! Gravitational SedimentationGravitational Sedimentation!! Aerosol CoagulationAerosol Coagulation

G. AhmadiME 437/537-Particle

Aerosols are suspension of Aerosols are suspension of solid or solid or liquid particles in a gas.liquid particles in a gas.Dust, smoke, mists, fog, haze, andDust, smoke, mists, fog, haze, andsmog are common aerosolssmog are common aerosols..Aerosol particles are foundAerosol particles are foundin different shapesin different shapes..


Equivalent area diametersEquivalent area diametersFeretFeret’’ss diameter diameter (maximum distance (maximum distance edge to edge) edge to edge) StokesStokes’’ diameter diameter (diameter of a (diameter of a sphere with the same density and the sphere with the same density and the same velocity as the particle)same velocity as the particle)Aerodynamic diameter Aerodynamic diameter (diameter of(diameter ofa sphere with the density of water a sphere with the density of water and the same velocity as the particle)and the same velocity as the particle)

2


Aerosols Air

Number Density (Number/cm)

100-105 1019

Mean Temperature(K)

240 – 310 240 – 310

Mean Free Path Greater than 1 m 0.06 µm

Particle Radius 0.01 – 10 µm 2 10-4 µm

Particle Mass (g) 10-18 - 10 -9 4.6 10-23

Particle Charge (Elementary Charge Units)

0 – 100 Weakly Ionized Single Charge


d2Kn λ

=

fc||M

fp vv −=

4dn

DSc

2fλ=

ν=

|'v||'v|)

vv(Br

f

p2/1

2,f

2,p

==

nKM4d||Re =

ν−

=fp vv

Knudsen NumberKnudsen Number

Mach NumberMach Number

Schmidt NumberSchmidt Number

Brown NumberBrown Number

Reynolds NumberReynolds Number


p

f

f

λλ = Mean Free Path= Mean Free Path νν = = KinematicKinematic ViscosityViscosity

d = Particle Diameterd = Particle Diameter D = DiffusivityD = Diffusivity

v = Particle Velocityv = Particle Velocity vv’’ = Thermal Velocity= Thermal Velocity

v = Fluid (Air) Velocityv = Fluid (Air) Velocity n = Number Densityn = Number Density

c = Speed of Soundc = Speed of Sound


p

f

f

Pd2kT

nd21

2m

2m π=

π=λ

PT1.23)m( =µλ

K/J101.38k -23×= Molecular Molecular DiameterDiameter

AirAir

3


410310210110010110−210−310−Particle Diameter,

410−

Electro.Wave X-Ray

µm

UV Vis Infrared Microwaves

FumeDefinition DustMist Spray

Solid

Liquid

Soil

Atmospheric

Typical Particles

Size Analysis methods

Clay Silt Sand Gravel

Smog Cloud/Fog Mist Rain

Viruses Bacteria HairSmoke Coal Dust Beach Sand

Microscopy

Electron Microscopy Sieving

Ultra Centrifuge Sedimentation


410310210110010110−210−310−

Particle Diameter, 410−

Gas CleaningMethod

µm

Ultrasonic Settling Chamber

Centrifuge

Air Filter

Diffusion Coeff. cm2/s

Impact Separator

Electrostatic Separator

HE Air Filter

Thermal Separator

Terminal Velocity cm/sS=2

AirWater

AirWater

2105 −× 510− 9102 −× 11102 −×12105 −×10105 −×8105 −×6105 −×

610− 4102 −×7106 −× 3106 −×1010−

6.0 600

12


µUd3 = F πStokesStokes

Re24

AU21

FC2

DD =

ρ=Drag Drag

CoefficientCoefficient

µρ

=UdRe

Reynolds Reynolds NumberNumber


DC

Re

4


OseenOseen Re]16Re/31[24CD

+=

Re]Re15.01[24

C687.0

D

+=

4.0CD =

1000 Re 1 <<

53 105.2Re10 ×<<

NewtonNewton


Turbulent Turbulent Boundary LayerBoundary Layer

Laminar Laminar Boundary LayerBoundary Layer


0

1

10

100

1000

CD

0 1 10 100 1000 10000 Re

Stokes Eq. (5)

Oseen

Newton

Experiment

Predictions of various models for drag coefficient for a sphericPredictions of various models for drag coefficient for a spherical particle.al particle.


U

d

h

Normal to the WallNormal to the Wall

((BernnerBernner, 1961), 1961)

)h2

d1(Re24CD +=

5


Ud

h

1543D ])

h2d(

161)

h2d(

25645)

h2d(

81)

h2d(

1691[

Re24C −−−+−=

Normal to the WallNormal to the Wall

((FaxonFaxon, 1923) , 1923)


For 1000 > For 1000 > KnKn > 0> 0

cD C

Ud3F πµ=

]e4.0257.1[d

21C 2/d1.1c

λ−+λ

+=

StokesStokes--Cunningham Cunningham DragDrag

Cunningham Cunningham CorrectionCorrection


1

10

100

1000

Cc

0.001 0.01 0.1 1 10 100 Kn

Variation of Cunningham correction with Knudsen number.Variation of Cunningham correction with Knudsen number.G. AhmadiME 437/537-Particle

c

cDiameter, µm C

10 µm 1.018

1 µm 1.176

0.1 µm 3.015

0.01 µm 23.775

0.001 µm 232.54

Variations of CVariations of Ccc with d for with d for λλ = 0.07 = 0.07 µµmm

6


( )

S6.0ReMexp1

M2.0M1.0Re48.0Re03.01

Re48.0Re03.038.05.4ReM5.0exp

SRe247.0exp567.133.4SRe24C

82

1

D

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−−+

+⎥⎦

⎤⎢⎣

⎡++

++++

⎟⎠⎞

⎜⎝⎛−+

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛−×++=

−

Henderson (1976)Henderson (1976) M < 1M < 1


Henderson (1976)Henderson (1976) M > 1M > 1

21

42

21

2

D

ReM

86.11

S1

S058.1

S22

ReM86.1

M34.09.0

C

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎥⎥⎦

⎤

⎢⎢⎣

⎡−++⎟

⎠⎞

⎜⎝⎛++

=


Carlson and Carlson and HoglundHoglund (1964)(1964)

)MRe25.1exp(28.182.3

ReM1

Re

3M

427.0exp(1

Re24C

88.063.4

D

−++

−−+=


pf

pff

D /13/21Ud3Fµµ+µµ+

πµ=

Ud2F fD πµ=For BubblesFor Bubbles

7


K=Correction FactorK=Correction Factor

KUd3F eD πµ=

3/1e )Volume6(d

π=


Cluster Shape

Correction Cluster Shape

Correction Cluster Shape

Correction

oo K = 1.12 oooo K = 1.32 oooo

K = 1.17

ooo K = 1.27 ooooo K = 1.45 o oo

o o

K = 1.19

oo o

K = 1.16 oooooo K = 1.57 oooooo

K = 1.17

oooooooo

K = 1.64 ooooooo K = 1.73


µUaK'6 = FD π

b

a

b

a

β−−β+β−β−β

−β=

])1(ln[)1(

)12(

)1(34

K'2/12

2/12

2

2

β+−β+β−β−β

−β=

])1(ln[)1(

)32(

)1(38

K'2/12

2/12

2

2

ab

=β


ab

ab

β+−β−β−ββ

−β=

− ])1(tan)1(

)2(

)1(34

K'2/121

2/12

2

2

β−−β−β−ββ

−β=

− ])1(tan)1(

)23(

)1(38

K'2/121

2/12

2

2

ba

=β

8


a

aaU16FD µ=

3/aU32FD µ=


b

b

βπµ

=2lnUb4FD

βπµ

=2lnUb8FD


)Rln002.2(U4F

eD −

πµ=

ν=

aU2R e


Drag

Gravity

guuu pfp

m)(C

d3dt

dmc

+−πµ

=

Equation of Motion Equation of Motion

9


guuu pfp

τ+−=τ )(dt

d

ν=

µρ

=πµ

=τ18

CSd18

Cdd3

mC c2

cp2

c

Relaxation Time Relaxation Time

f

p

Sρρ

=

)m(d103)s( 26 µ×≈τ −


)e1)(( /tfp τ−−τ+= guu

µρ

=τ=18

gCdgu c2p

t

Terminal Velocity = Equilibrium Velocity after Large TimeTerminal Velocity = Equilibrium Velocity after Large Time

)m(d30)s/m(u 2t µ≈µ


Stopping Distance = Penetration distance for Stopping Distance = Penetration distance for an initial velocity of an initial velocity of uuoo

)e1( /t τ−−τ= po

p uxτ−= /teop uu

τ= po

p ux)m(d3)m(x 2p µ≈µ


76.2 mm7.62 mm7.6×10-37.47 cm/s503.09 mm309 µm3.1×10-43.03 mm/s10

0.786 mm78.6 µm7.9×10-50.77 mm/s50.0357 mm3.6 µm3.6×10-635 µm/s10.0103 mm1.03 µm1×10-610.1 µm/s0.50.0009 mm0.092 µm9.1×10-80.93 µm/s0.10.0004 mm0.04 µm4×10-80.39 µm/s0.05

Stopping Distance u= 10 m/s

Stopping Distance u= 1 m/s

τ sec Terminal Velocity

Diameter, µm

10


)]e1(t)[(

)e1(/tf

/t

τ−

τ−

−τ−τ++

−τ+=

gu

uxx po

po

p

)]e1(/t[u/x /tfp τ−−−τ=τ

)]e1(/t[u/y /tfp τ−−−τα−=ττ

τ=α fu

g

Components Components


-12

-10

-8

-6

-4

-2

0

y/ut

au

0 1 2 3 4 5 6 t/tau

α =0.1

α =1

α =2

Variations of the particle vertical position with time.Variations of the particle vertical position with time.


-12

-10

-8

-6

-4

-2

0

y/ut

au

0 1 2 3 4 5 6 x/utau

α =0.1

α =1

α =2

Sample particle trajectories.Sample particle trajectories.


guuu pfp

)mm()(C

d3dt

d)mm( f

c

a −+−πµ

=+

6dm

f3f ρπ=

.m21m fa =

Fluid MassFluid Mass

Apparent MassApparent Mass

11


)S11()(

dtd)

S211( −τ+−=τ+ guuu pf

p

)1(18

gCd)S11(gu p

fc

2pt

ρρ

−µ

ρ=−τ=

Terminal VelocityTerminal Velocity


Lift

ufup

)dydusgn(|

dydu|)uu(d615.1F

f2/1

fpf22/1

)Saff(L −ρν=

SaffmanSaffman (1965, 1968)(1965, 1968)


1d|uu|Rpf

es <<ν−

=

1dR2

eG <<νγ

=&

1dR2

e <<ν

Ω=Ω

1RRε

es

2/1eG >>=

McLaughlin (1991)McLaughlin (1991)


⎩⎨⎧

>α≤α+−α−

=40 Rfor )R(0524.0

40 Rfor 3314.0)10/Rexp()3314.01(F

F

es2/1

es

es2/1

es2/1

)Saff(L

L

es

eG2

espf R2

R2

R|uu|2

d=

ε=

−γ

=α&

)]32.0(6tan[667.0)]191.0(log5.2tanh[13.0F

F10

)Saff(L

L −ε++ε+=

20 0.1 ≤ε≤

Mai (1992)Mai (1992)

12


⎩⎨⎧

<<εεε−>>εε−

=−

−

1 for )ln(1401 for 287.01

FF

25

2

)Saff(L

L

McLaughlin (1991)McLaughlin (1991)


CherukatCherukat and McLaughlin (1994)and McLaughlin (1994)

4/IdVF L22

)LC(Lρ=

−

luuuV pfp γ−=−= &

V2d

Gγ

=∧&

l2dK =

)K,(II GLL Λ=

Lift

d l


Lift

24)AL(L d576.0F γρ=− &

2/332/1)Saff(L d807.0F γρν= &

Leighton and Leighton and AcrivosAcrivos (1985) (1985)

SaffmanSaffman


Velocity Field in the Inertial Velocity Field in the Inertial SublayerSublayer

Byln1u +κ

= ++

ν=+ yuy

*

5B ≈ 300y30 ≤< +

Wall UnitsWall Units *uuu =+

13


dydU

0 µ=τ

dydUu 2* ν=

1dydU

=+

+ ++ = yu 5y0 ≤< +

Turbulent stress is negligibleTurbulent stress is negligible


+y

+u

5.5yln5.2u += ++

++ = yu

3012 300

10

20

30


ν=γ

2*u

4)AL(L d576.0F ++

− = 3)Saff(L d807.0F ++ =

2L

LFFρν

=+

ν=+

*dud

31.2)Hall(L d21.4F ++ = 87.1

)MN(L d57.15F ++ =

Hall (1988)Hall (1988) MollingerMollinger and and NieuwstadtNieuwstadt (1996) (1996)


1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

Fl+

0.1 1 10 d+

Mollinger

Hall

Saffman

Leighton

Experiment

Documents

Introduction to Aerosols Drag Forces Cunningham ... · Introduction to Aerosols ! Drag Forces! Cunningham Corrections! Lift Forces! Brownian Motion! Particle Deposition Mechanisms!