me 437/537-particle g. ahmadi€¦ · me 437/537-particle g. ahmadi ¾equivalent area diameters...
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G. AhmadiME 437/537-Particle
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.
G. AhmadiME 437/537-Particle
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)
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
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
G. AhmadiME 437/537-Particle
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
G. AhmadiME 437/537-Particle
p
f
f
Pd2kT
nd21
2m
2m π
=π
=λ
PT1.23)m( =µλ
K/J101.38k -23×=Molecular DiameterMolecular Diameter
AirAir
G. AhmadiME 437/537-Particle
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
G. AhmadiME 437/537-Particle
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
G. AhmadiME 437/537-Particle
µUd3 = F πStokesStokes
Re24
AU21
FC2
DD =
ρ=Drag
CoefficientDrag Coefficient
µρ= UdRe
Reynolds NumberReynolds Number
G. AhmadiME 437/537-Particle
DC
Re
G. AhmadiME 437/537-Particle
OseenOseen Re]16Re/31[24CD
+=
Re]Re15.01[24
C687.0
D
+=
4.0CD =
1000 Re 1 <<
53 105.2Re10 ×<<
NewtonNewton
G. AhmadiME 437/537-Particle
Turbulent Boundary LayerTurbulent Boundary Layer
Laminar Boundary LayerLaminar Boundary Layer
G. AhmadiME 437/537-Particle
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.
G. AhmadiME 437/537-Particle
U
d
h
Normal to the Wall
(Bernner, 1961)Normal to the Wall
(Bernner, 1961)
)h2
d1(Re24CD +=
G. AhmadiME 437/537-Particle
Ud
h
1543D ])
h2d(
161)
h2d(
25645)
h2d(
81)
h2d(
1691[
Re24C −−−+−=
Normal to the Wall
(Faxon, 1923) Normal to the Wall
(Faxon, 1923)
G. AhmadiME 437/537-Particle
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
G. AhmadiME 437/537-Particle
1
10
100
1000 C
c
0.001 0.01 0.1 1 10 100 Kn
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 Cc with d for λ = 0.07 µm
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 Hoglund (1964)Carlson and Hoglund (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
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
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
G. AhmadiME 437/537-Particle
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 −
G. AhmadiME 437/537-Particle
)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
G. AhmadiME 437/537-Particle
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
G. AhmadiME 437/537-Particle
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
G. AhmadiME 437/537-Particle
)]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
G. AhmadiME 437/537-Particle
-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.
G. AhmadiME 437/537-Particle
-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.
G. AhmadiME 437/537-Particle
guuu pfp
)mm()(C
d3dt
d)mm( f
c
a −+−πµ=+
6dm
f3f ρπ=
.m21m fa =
Fluid Mass
Apparent Mass
G. AhmadiME 437/537-Particle
)S11()(
dtd)
S211( −τ+−=τ+ guuu pf
p
)1(18
gCd)S11(gu p
fc
2pt
ρρ−
µρ=−τ=
Terminal Velocity
G. AhmadiME 437/537-Particle
Lift
ufup
)dydusgn(|
dydu|)uu(d615.1F
f2/1
fpf22/1
)Saff(L −ρν=
Saffman (1965, 1968)
G. AhmadiME 437/537-Particle
1d|uu|Rpf
es <<ν
−=
1dR2
eG <<ν
γ=&
1dR2
e <<ν
Ω=Ω
1RRε
es
2/1eG >>=
McLaughlin (1991)
G. AhmadiME 437/537-Particle
>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)
G. AhmadiME 437/537-Particle
<<εεε−>>εε−
=−
−
1 for )ln(1401 for 287.01
FF
25
2
)Saff(L
L
McLaughlin (1991)
G. AhmadiME 437/537-Particle
Cherukat and McLaughlin (1994)
4/IdVF L22
)LC(Lρ=
−
luuuV pfp γ−=−= &
V2d
Gγ=∧&
l2dK =
)K,(II GLL Λ=
Lift
d l
G. AhmadiME 437/537-Particle
Lift
24)AL(L d576.0F γρ=− &
2/332/1)Saff(L d807.0F γρν= &
Leighton andAcrivos (1985)
Saffman
G. AhmadiME 437/537-Particle
Velocity Field in the Inertial Sublayer
Byln1u +κ
= ++
ν=+ yuy
*
5B ≈ 300y30 ≤< +
Wall Units *uuu =+
G. AhmadiME 437/537-Particle
dydU
0 µ=τ
dydUu 2* ν=
1dydU =+
+ ++ = yu 5y0 ≤< +
Turbulent stress is negligible
G. AhmadiME 437/537-Particle
+y
+u
5.5yln5.2u += ++
++ = yu
3012 300
10
20
30
G. AhmadiME 437/537-Particle
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)
G. AhmadiME 437/537-Particle
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
G. AhmadiME 437/537-Particle
! Introduction to Aerosols! Drag Forces ! Cunningham Corrections! Lift Forces! Brownian Motion! Particle Deposition Mechanisms! Gravitational Sedimentation! Aerosol Coagulation
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