me 437/537-particle g. ahmadi€¦ · me 437/537-particle g. ahmadi ¾equivalent area diameters...

G. AhmadiME 437/537-Particle


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


Equivalent area diametersFeret’s diameter (maximum distance edge to edge) Stokes’ diameter (diameter of a sphere with the same density and the same velocity as the particle)Aerodynamic diameter (diameter ofa sphere with the density of water and the same velocity as the particle)


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

2f λ=ν=

|'v||'v|)

vv(Br

f

p2/1

2,f

2,p

==

nKM4d||Re =

ν−=

fp vv

Knudsen Number

Mach Number

Schmidt Number

Brown Number

Reynolds Number


p

f

f

λ = Mean Free Path ν = Kinematic Viscosity

d = Particle Diameter D = Diffusivity

v = Particle Velocity v’ = Thermal Velocity

v = Fluid (Air) Velocity n = Number Density

c = Speed of Sound


p

f

f

Pd2kT

nd21

2m

2m π

=π

=λ

PT1.23)m( =µλ

K/J101.38k -23×=Molecular DiameterMolecular Diameter

AirAir


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

CoefficientDrag Coefficient

µρ= UdRe

Reynolds NumberReynolds Number


DC

Re


OseenOseen Re]16Re/31[24CD

+=

Re]Re15.01[24

C687.0

D

+=

4.0CD =

1000 Re 1 <<

53 105.2Re10 ×<<

NewtonNewton


Turbulent Boundary LayerTurbulent Boundary Layer

Laminar Boundary LayerLaminar Boundary Layer


0

1

10

100

1000 C

D

0 1 10 100 1000 10000 Re

Stokes Eq. (5)

Oseen

Newton

Experiment

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


U

d

h

Normal to the Wall

(Bernner, 1961)Normal to the Wall

(Bernner, 1961)

)h2

d1(Re24CD +=


Ud

h

1543D ])

h2d(

161)

h2d(

25645)

h2d(

81)

h2d(

1691[

Re24C −−−+−=

Normal to the Wall

(Faxon, 1923) Normal to the Wall

(Faxon, 1923)


For 1000 > Kn > 0For 1000 > Kn > 0

cD C

Ud3F πµ=

]e4.0257.1[d

21C 2/d1.1c

λ−+λ+=

Stokes-Cunningham DragStokes-Cunningham Drag

Cunningham CorrectionCunningham Correction


1

10

100

1000 C

c

0.001 0.01 0.1 1 10 100 Kn

Variation of Cunningham correction with Knudsen number.


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 Cc with d for λ = 0.07 µm


( )

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 Hoglund (1964)Carlson and Hoglund (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


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


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


guuu pfp

τ+−=τ )(dt

d

ν=

µρ

=πµ

=τ18

CSd18

Cdd3

mC c2

cp2

c

Relaxation Time

f

p

Sρρ=

)m(d103)s( 26 µ×≈τ −


)e1)(( /tfp τ−−τ+= guu

µρ=τ=

18gCdgu c

2pt

Terminal Velocity = Equilibrium Velocity after Large Time

)m(d30)s/m(u 2t µ≈µ


Stopping Distance = Penetration distance for an initial velocity of uo

)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


)]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 fug

Components


-12

-10

-8

-6

-4

-2

0 y/

utau

0 1 2 3 4 5 6 t/tau

α =0.1

α =1

α =2

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.


guuu pfp

)mm()(C

d3dt

d)mm( f

c

a −+−πµ=+

6dm

f3f ρπ=

.m21m fa =

Fluid Mass

Apparent Mass


)S11()(

dtd)

S211( −τ+−=τ+ guuu pf

p

)1(18

gCd)S11(gu p

fc

2pt

ρρ−

µρ=−τ=

Terminal Velocity


Lift

ufup

)dydusgn(|

dydu|)uu(d615.1F

f2/1

fpf22/1

)Saff(L −ρν=

Saffman (1965, 1968)


1d|uu|Rpf

es <<ν

−=

1dR2

eG <<ν

γ=&

1dR2

e <<ν

Ω=Ω

1RRε

es

2/1eG >>=

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)


<<εεε−>>εε−

=−

−

1 for )ln(1401 for 287.01

FF

25

2

)Saff(L

L

McLaughlin (1991)


Cherukat 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 andAcrivos (1985)

Saffman


Velocity Field in the Inertial Sublayer

Byln1u +κ

= ++

ν=+ yuy

*

5B ≈ 300y30 ≤< +

Wall Units *uuu =+


dydU

0 µ=τ

dydUu 2* ν=

1dydU =+

+ ++ = yu 5y0 ≤< +

Turbulent 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

LFF

ρν=+

ν=+

*dud

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

)MN(L d57.15F ++ =

Hall (1988) Mollinger and Nieuwstadt (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


! Introduction to Aerosols! Drag Forces ! Cunningham Corrections! Lift Forces! Brownian Motion! Particle Deposition Mechanisms! Gravitational Sedimentation! Aerosol Coagulation

me 437/537-particle g. ahmadi€¦ · me 437/537-particle g. ahmadi ¾equivalent area diameters...

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