ME 637-Particle-II G. Ahmadi ME 637-Particle-II G. Ahmadi · PDF file1/300 3×105...
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1 G. Ahmadi ME 637-Particle-II G. Ahmadi ME 637-Particle-II Outline Outline 4 4 Electrostatics Electrostatics 4 4 Particle Charging Particle Charging 4 4 Charged Particle Kinetic Charged Particle Kinetic 4 4 Thermophoretic Thermophoretic Force Force 4 4 Photophoretic Photophoretic Force Force G. Ahmadi ME 637-Particle-II e 4πρ = ⋅ ∇ D e ρ = ⋅ ∇ D t c 1 c 4 ∂ ∂ + π = × ∇ D J H t c 1 ∂ ∂ + = × ∇ D J H 0 t c 1 = ∂ ∂ + × ∇ B E 0 t = ∂ ∂ + × ∇ B E 0 = ⋅ ∇ B 0 = ⋅ ∇ B Coulomb’s Coulomb’s Law Law Ampere’s Ampere’s Law Law Faraday’s Faraday’s Law Law Absence of Free Absence of Free Magnetic Poles Magnetic Poles Gaussian Units Gaussian Units MKS Units MKS Units G. Ahmadi ME 637-Particle-II 0 t e = ∂ ρ ∂ + ∇J Continuity Continuity Equation Equation E D 0 ε = 1 0 = ε m Volt / Coul 10 854 . 8 c 4 10 12 2 7 0 ⋅ × = π = ε − H B 0 µ = 1 0 = µ 7 0 10 4 − × π = µ E J σ = 2 / 1 0 0 ) ( c − µ ε = Constitutive Equations, Free space Constitutive Equations, Free space Ohm’s Law Ohm’s Law
Transcript of ME 637-Particle-II G. Ahmadi ME 637-Particle-II G. Ahmadi · PDF file1/300 3×105...
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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
