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