1 G. Ahmadi ME 437/537-Particle G. Ahmadi ME 437/537-Particle ! ! Introduction to Aerosols Introduction to Aerosols ! ! Drag Forces Drag Forces ! ! Cunningham Corrections Cunningham Corrections ! ! Lift Forces Lift Forces ! ! Brownian Motion Brownian Motion ! ! Particle Deposition Mechanisms Particle Deposition Mechanisms ! ! Gravitational Sedimentation Gravitational Sedimentation ! ! Aerosol Coagulation Aerosol Coagulation G. Ahmadi ME 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, and Dust, smoke, mists, fog, haze, and smog are common aerosols smog are common aerosols . . Aerosol particles are found Aerosol particles are found in different shapes in different shapes . . G. Ahmadi ME 437/537-Particle Equivalent area diameters Equivalent area diameters Feret Feret ’ ’ s s diameter diameter (maximum distance (maximum distance edge to edge) edge to edge) Stokes Stokes ’ ’ 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 of a 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)
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1
G. AhmadiME 437/537-Particle 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..
G. AhmadiME 437/537-Particle
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)
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G. AhmadiME 437/537-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
G. AhmadiME 437/537-Particle
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
G. AhmadiME 437/537-Particle
p
f
f
λλ = Mean Free Path= Mean Free Path νν = = KinematicKinematic ViscosityViscosity
d = Particle Diameterd = Particle Diameter D = DiffusivityD = Diffusivity
Predictions of various models for drag coefficient for a sphericPredictions of various models for drag coefficient for a spherical particle.al particle.
G. AhmadiME 437/537-Particle
U
d
h
Normal to the WallNormal to the Wall
((BernnerBernner, 1961), 1961)
)h2
d1(Re24CD +=
5
G. AhmadiME 437/537-Particle
Ud
h
1543D ])
h2d(
161)
h2d(
25645)
h2d(
81)
h2d(
1691[
Re24C −−−+−=
Normal to the WallNormal to the Wall
((FaxonFaxon, 1923) , 1923)
G. AhmadiME 437/537-Particle
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
G. AhmadiME 437/537-Particle
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
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G. AhmadiME 437/537-Particle
( )
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
G. AhmadiME 437/537-Particle
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
⎟⎟⎠
⎞⎜⎜⎝
⎛+
⎥⎥⎦
⎤
⎢⎢⎣
⎡−++⎟
⎠⎞
⎜⎝⎛++
=
G. AhmadiME 437/537-Particle
Carlson and Carlson and HoglundHoglund (1964)(1964)
)MRe25.1exp(28.182.3
ReM1
Re
3M
427.0exp(1
Re24C
88.063.4
D
−++
−−+=
G. AhmadiME 437/537-Particle
pf
pff
D /13/21Ud3Fµµ+µµ+
πµ=
Ud2F fD πµ=For BubblesFor Bubbles
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G. AhmadiME 437/537-Particle
K=Correction FactorK=Correction Factor
KUd3F eD πµ=
3/1e )Volume6(d
π=
G. AhmadiME 437/537-Particle
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
G. AhmadiME 437/537-Particle
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
G. AhmadiME 437/537-Particle
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
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G. AhmadiME 437/537-Particle
a
aaU16FD µ=
3/aU32FD µ=
G. AhmadiME 437/537-Particle
b
b
βπµ
=2lnUb4FD
βπµ
=2lnUb8FD
G. AhmadiME 437/537-Particle
)Rln002.2(U4F
eD −
πµ=
ν=
aU2R e
G. AhmadiME 437/537-Particle
Drag
Gravity
guuu pfp
m)(C
d3dt
dmc
+−πµ
=
Equation of Motion Equation of Motion
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G. AhmadiME 437/537-Particle
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 −
G. AhmadiME 437/537-Particle
)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
G. AhmadiME 437/537-Particle
Stopping Distance = Penetration distance for Stopping Distance = Penetration distance for an initial velocity of an initial velocity of uuoo