1
G. AhmadiME 637-Particle-II G. AhmadiME 637-Particle-II
OutlineOutline44ElectrostaticsElectrostatics44Particle ChargingParticle Charging44Charged Particle KineticCharged Particle Kinetic44ThermophoreticThermophoretic ForceForce44PhotophoreticPhotophoretic ForceForce
G. AhmadiME 637-Particle-II
e4πρ=⋅∇ D eρ=⋅∇ D
tc1
c4
∂∂
+π
=×∇DJH
tc1∂∂
+=×∇DJH
0tc
1=
∂∂
+×∇BE 0
t=
∂∂
+×∇BE
0=⋅∇ B0=⋅∇ B
Coulomb’s Coulomb’s LawLaw
Ampere’s Ampere’s LawLaw
Faraday’s Faraday’s LawLaw
Absence of Free Absence of Free Magnetic PolesMagnetic Poles
Gaussian UnitsGaussian Units MKS UnitsMKS Units
G. AhmadiME 637-Particle-II
0te =
∂ρ∂
+∇JContinuity Continuity Equation Equation
ED 0ε= 10 =ε mVolt/Coul10854.8c4
10 122
7
0 ⋅×=π
=ε −
HB 0µ= 10 =µ 70 104 −×π=µ
EJ σ= 2/100 )(c −µε=
Constitutive Equations, Free space Constitutive Equations, Free space
Ohm’s LawOhm’s Law
2
G. AhmadiME 637-Particle-II
l
9103×
ρ 3103×
9103×
41031 −×
300/1
5103×
statvoltstatvolt1 volt1 voltVVElectric PotentialElectric Potentialstatvoltstatvolt/cm/cm1 volt/m1 volt/mEEElectric FieldElectric Fieldstatamp/cmstatamp/cm221 amp/m1 amp/m22JJCurrent DensityCurrent Densitystatamperestatampere1 1 ampere(coul/sampere(coul/s))IICurrentCurrent
statcoul/cmstatcoul/cm331 coul/m1 coul/m33Charge DensityCharge Densitystatcoulombstatcoulomb1 coulomb (1 coulomb (coulcoul))qqChargeCharge
10107 7 ergs/sergs/s1 watt (W)1 watt (W)PPPowerPower101077 ergs ergs 1 joule (J)1 joule (J)W, UW, UWork, EnergyWork, Energy10105 5 dynes dynes 1 1 newtonnewton (N)(N)FFForceForce1 second (s)1 second (s)1 second (s)1 second (s)ttTimeTime10103 3 gram (gm)gram (gm)1 kilogram (kg)1 kilogram (kg)mmMassMass
10102 2 centimeter (cm)centimeter (cm)1 meter (m)1 meter (m)LengthLengthGaussianGaussianMKSMKSSymbolSymbolPhysical QuantitiesPhysical Quantities
G. AhmadiME 637-Particle-II
For accurate works, all factors of 3 in the coefficients should be replaced by 2.99793.
1 henryLInductance
mag. moment/cm31 amp/mMMagnetic Inductionoersted1 amp-turn/mHMagnetic field
gauss1 weber/m2BMagnetic inductiongauss cm2 (maxwell)1 weberFMagnetic flux
cm1 faradCCapacitances/cm1 ohmRResistance
1/s1 mho/mConductivity
statcoul/cm2
statvolt/cm)1 coul/m2DDisplacement
dipole moment/cm31 coul/m2PPolarizationGaussianMKSSymbolPhysical Quantities
5103×51012 ×π
σ 9109×1110)9/1( −×
11109×810410
3104 ×π310
41 −×π
11109×
G. AhmadiME 637-Particle-II
Most aerosol particles carry some electrical chargesMost aerosol particles carry some electrical charges
EFE q= neq =
coul106.1e 19−×= statcoul108.4e 10−×=
cC/Ud3qE πµ=d3
qCZu cp
πµ==
Coulomb Coulomb ForceForce
Electric ChargeElectric Charge
Particle Particle MobilityMobility
G. AhmadiME 637-Particle-II
BoltzmannBoltzmannEquilibrium Charge Equilibrium Charge
DistributionDistribution ∑∞
−∞=
−
−=
n
22
22
}dkT/enexp{
}dkT/enexp{)n(f
}dkT
enexp{dkT
e)n(f222
−π
= µ02.0>d
}d
n05.0exp{d24.0)n(f
2
−π
= µ02.0>d
3
G. AhmadiME 637-Particle-II
Average Number of Charge Average Number of Charge
µ02.0>d2edkTdn)n(f|n|)n(f|n|nπ
≈≈= ∫∑∞
∞−
∞
∞−
2r4qEπγ
= επ=γ /4
1=ε π=γ 4εε
=γo
1
Point Point ChargeCharge cgscgs
MKSMKSAirAir
)m(d,d36.2n µ≈=
G. AhmadiME 637-Particle-II
2r4q'qE'qF
πγ
== εε=γ
o
1
metervoltsecamp10859.8 12
o −−
×=ε −
)109(rq'qF 92 ×
ε=
PermittivityPermittivity
Coulomb’s Coulomb’s LawLaw
G. AhmadiME 637-Particle-II
e4Ed)
2)1(2
1](1tneZ
tneZ[n
2
p
p
ii
ii
+ε−ε
++π
π=
∞
∞
∞→+ε−ε
+=∞ tase4
Ed]2
)1(21[n
2
p
p
Quartzfor3.4p =ε
cgscgs
cgscgs
G. AhmadiME 637-Particle-II
]tden)kTm
2(1ln[e2
dkTn 2i
2/1
i2 ∞
π+=
38i cmsec/ion10~tn ∞
4
G. AhmadiME 637-Particle-II G. AhmadiME 637-Particle-II
-
Gas+-
Gas, U=1-3 m/s+
-
b=15-40 cm
G. AhmadiME 637-Particle-II
CuyC)D(J e
T −∂∂
ν+−=
d3EqCu c
e πµ= ∫
∫∞
∞
ν+−−
ν+−−
=
0T
e
y
0T
e
}D
dyuexp{1
}D
dyuexp{1
CC
}uuexp{1
Cu
}D
dyuexp{1
Cu)x(J
D
e
e
0Te
e
−−−=
ν+−−
−= ∞
∞
∞
∫ ∫∞
ν+
=
0T
D
Ddy1u
G. AhmadiME 637-Particle-II
De uu << ∞= Cu|J| D
eD uu <<∞= Cu|J| e
dxCu2Jdx2CbUd e ∞∞ −==
}bU
xu2exp{CC e0 −= ∞∞ }
bULu2exp{
CC e
0
L −=∞
∞
5
G. AhmadiME 637-Particle-II
0.1 1 1030
60
90
99
99.9
d (µm)
η
G. AhmadiME 637-Particle-II
EGD FFFu++=
dtdm
p
mq
dtd τ
−τ+−=τ Eguuu pfp
mq
dtd τ
−=+τ Euuuo
pp
guuo τ+= f
Equation Equation of Motionof Motion
G. AhmadiME 637-Particle-II
Γ−=+τ 1udtud p^
p^
0
pp^
uuu =
0muEqτ
=Γ
inertianeglecting,1||For >>Γ
Γ−=p^
U mEqup τ
−=
Euo ||For
G. AhmadiME 637-Particle-II
}2
dexp{T|'v|
d158F
^
f
f2
t λθ
−∇κ
−=∞<< nK25.0
1<<Μ
)2
1(21.012.09.0 p
ft
mm
^
κκα
−α+α+=θ2/1f
ff)
mkT8(|'v|cπ
==
ationmodAccomMomentumm =α
ationmodAccomThermalt =α
6
G. AhmadiME 637-Particle-II
)KC221)(C31(
T]C33.1)C33.11)(KC[(KCd3
ntp
f
nm
nmnmntp
f
nmt2
+κκ
+κ+
∇κ−κ++κκ
πµ−=tF
2.0K0 n <<
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
κκ
+∇
πµν−=
f
p0 2
1TT
d9tF
1n >κ
Continuum RegimeContinuum Regime
G. AhmadiME 637-Particle-II
)TR'v2
1(48
pd
p
2ff
3
p
κ+ρ
π−=
IF∞→nK
DiffusiophoreticDiffusiophoretic ForceForce
G. AhmadiME 637-Particle-II
4
62o
3
3
2o
2
e
y128dE3
y16qEd
y16qqEF
πε−
+πε
−=
Coulomb Coulomb ForceForce
Image Image ForceForce
Polarization Polarization ForceForce
Dipole Dipole InteractionsInteractions G. AhmadiME 637-Particle-II
Particles with Particles with Single ChargeSingle Charge
7
G. AhmadiME 637-Particle-II
BoltzmannBoltzmannChargeCharge
Saturation Saturation ChargeCharge
G. AhmadiME 637-Particle-II
44ElectrostaticsElectrostatics44Particle ChargingParticle Charging44Charged Particle KineticCharged Particle Kinetic44ThermophoreticThermophoretic ForceForce44PhotophoreticPhotophoretic ForceForce
G. AhmadiME 637-Particle-II