Outline Outline 4 Electrostatics 4 Electrostatics 4 Particle - PowerPoint PPT Presentation

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

Physical Quantities Physical Quantities Symbol Symbol MKS MKS Gaussian Gaussian Physical Quantities Symbol MKS Gaussian 3 × 5 10 Length l 1 meter (m) 10 2 2 centimeter (cm) centimeter (cm) Polarization P 1 coul/m 2 dipole moment/cm 3 Length 1 meter (m) 10 12 × π 1 coul/m 2 5 statcoul/cm 2 Displacement D 10 Mass Mass m m 1 kilogram (kg) 1 kilogram (kg) 10 3 10 3 gram (gm) gram (gm) statvolt/cm) Time t 1 second (s) 1 second (s) Time t 1 second (s) 1 second (s) σ 9 × 9 Conductivity 1 mho/m 10 1/s 5 dynes Force Force F F 1 newton 1 newton (N) (N) 10 5 10 dynes × − 11 ( 1 / 9 ) 10 Resistance R 1 ohm s/cm 7 ergs Work, Energy Work, Energy W, U W, U 1 joule (J) 1 joule (J) 10 7 10 ergs 9 × 11 10 Capacitance C 1 farad cm 10 7 7 ergs/s Power Power P P 1 watt (W) 1 watt (W) 10 ergs/s gauss cm 2 (maxwell) Magnetic flux F 1 weber 8 10 Charge Charge q q 1 coulomb (coul 1 coulomb ( coul) ) 3 × 9 statcoulomb statcoulomb 10 ρ 1 weber/m 2 4 Magnetic induction B 10 gauss Charge Density 1 coul/m 3 3 3 × 3 statcoul/cm 3 3 Charge Density 1 coul/m statcoul/cm 10 4 × π 3 Magnetic field H 1 amp-turn/m 10 oersted 3 × 9 10 Current Current I I 1 ampere(coul/s 1 ampere(coul/s) ) statampere statampere 1 mag. moment/cm 3 Magnetic Induction M 1 amp/m × − 3 × 3 5 10 10 Current Density Current Density J J 1 amp/m 2 1 amp/m 2 statamp/cm 2 statamp/cm 2 π 4 1 9 × 11 × − 4 Inductance L 1 henry 10 Electric Field Electric Field E E 1 volt/m 1 volt/m 10 statvolt/cm statvolt /cm 3 1 / 300 Electric Potential Electric Potential V V 1 volt 1 volt statvolt statvolt For accurate works, all factors of 3 in the coefficients should be replaced by 2.99793. ME 637-Particle-II G. Ahmadi ME 637-Particle-II G. Ahmadi Most aerosol particles carry some electrical charges Most aerosol particles carry some electrical charges − 2 2 Boltzmann exp{ n e / dkT } Boltzmann = f ( n ) ∞ Equilibrium Charge Equilibrium Charge ∑ − Coulomb = q = 2 2 Coulomb exp{ n e / dkT } ne F E q E Distribution Distribution Force Force = −∞ n Electric Charge Electric Charge 2 2 2 > µ e n e 0 . 02 = − d = × − − 19 = × f ( n ) exp{ } 10 e 1 . 6 10 coul e 4 . 8 10 statcoul π dkT dkT qC 2 = πµ = = 0 . 24 0 . 05 n Particle Particle p qE 3 Ud / C c u Z = − > µ f ( n ) exp{ } d 0 . 02 c πµ Mobility Mobility 3 d π d d ME 637-Particle-II G. Ahmadi ME 637-Particle-II G. Ahmadi 2

γ Average Number of Charge Average Number of Charge q ' q 1 = = γ = F q ' E ε ε π 2 4 r ∞ ∞ o dkT ∑ ∫ = ≈ ≈ > µ n | n | f ( n ) | n | f ( n ) dn 0 . 02 d π 2 e − ∞ − ∞ = ≈ µ n 2 . 36 d , d ( m ) − amp sec ε = × − γ Permittivity 12 Permittivity 8 . 859 10 q Point Point γ = π ε o − = 4 / volt meter E cgs cgs Charge Charge π 2 4 r q ' q 1 Coulomb’s Coulomb’s γ = π = × ε = γ = 9 4 F ( 9 10 ) Air Air 1 MKS MKS ε 2 ε ε Law Law r o ME 637-Particle-II G. Ahmadi ME 637-Particle-II G. Ahmadi π ε − eZ n t 2 ( 1 ) 2 Ed π ∞ = + i i p dkT 2 n [ ]( 1 ) cgs cgs π + ε + = + 1 / 2 2 eZ n t 1 2 4 e n ln[ 1 ( ) n de t ] ∞ i i p ∞ i 2 2 e m kT i ε − 2 2 ( 1 ) Ed = + → ∞ p n [ 1 ] as t ∞ ε + 2 4 e cgs cgs p 8 3 n ∞ t ~ 10 ion sec/ cm i ε p = 4 . 3 for Quartz ME 637-Particle-II G. Ahmadi ME 637-Particle-II G. Ahmadi 3

- + Gas - b=15-40 cm + Gas, U=1-3 m/s - ME 637-Particle-II G. Ahmadi ME 637-Particle-II G. Ahmadi ∂ u << C = u = − + ν − | J | u C T y J ( D ) u C ∞ u dy e D ∫ − − D ∂ e e y 1 exp{ } + ν T D C = 0 << ∞ u u C u dy = EqC ∞ ∫ = − − | J | u C c e ∞ u 1 exp{ } D e e πµ + ν e T 3 d D 0 = = − bUd C 2 Jdx 2 u C dx ∞ ∞ e 1 − u C u C = ∞ ∞ = = − u e e J ( x ) ∞ D ∞ u dy dy − − ∫ ∫ − − e 1 exp{ } 1 exp{ u } 2 u x C 2 u L e + ν u ∞ T + ν = − D T L = − D e C C exp{ } e D exp{ } 0 ∞ ∞ 0 0 bU C bU ∞ 0 ME 637-Particle-II G. Ahmadi ME 637-Particle-II G. Ahmadi 4

p d Equation Equation u 99.9 = + + m F F F of Motion of Motion D G E dt 99 η τ p d u 90 τ = − + τ − f p q u u g E dt m 60 τ p d u = + τ τ + = − f p u o u g q 30 u u E 0.1 1 10 o dt m d ( µ m) ME 637-Particle-II G. Ahmadi ME 637-Particle-II G. Ahmadi p < < ∞ ^ ^ 0 . 25 K κ θ f p 8 d d u ^ n = − ∇ − τ + = − Γ 2 F d T exp{ } u 1 t λ For u o || Μ << E 15 f 2 dt | v ' | 1 Eq τ p p u ^ Γ = u = α κ mu 8 kT f ^ u f = = f 1 / 2 θ = + α + α − c | v ' | ( ) 0 . 9 0 . 12 0 . 21 ( 1 t ) 0 0 π f κ m m m p 2 Γ >> τ For | | 1 , α m = Momentum Accom mod ation = − p u p Eq ^ = − Γ U m neglecting inertia α t = Thermal Accom mod ation ME 637-Particle-II G. Ahmadi ME 637-Particle-II G. Ahmadi 5

− π 3 d p I Continuum Regime Continuum Regime < n < 0 K 0 . 2 = F p 1 → ∞ K 48 ( ) κ f n − πµ + + κ − κ ∇ 2 3 d C K [( C K )( 1 1 . 33 C ) 1 . 33 C ] T 2 κ ρ + κ t m n p t n m n m n f f 2 v ' R T = F κ p t f + κ + + ( 1 3 C )( 1 2 2 C K ) κ m n t n p Diffusiophoretic Force Force Diffusiophoretic ⎡ ⎤ ⎢ ⎥ πµν 9 d 1 = − ∇ ⎢ ⎥ T F κ n > t κ p 1 T ⎢ ⎥ + 0 2 ⎢ ⎥ ⎣ κ ⎦ f ME 637-Particle-II G. Ahmadi ME 637-Particle-II G. Ahmadi Coulomb Coulomb Image Image Polarization Polarization Force Force Force Force Force Force Particles with Particles with Single Charge Single Charge 2 3 q qEd = − + F qE e πε 2 3 16 y 16 y o πε 2 6 3 E d − o 4 128 y Dipole Dipole Interactions Interactions ME 637-Particle-II G. Ahmadi ME 637-Particle-II G. Ahmadi 6

4 Electrostatics 4 Electrostatics Saturation Saturation 4 Particle Charging 4 Particle Charging Charge Charge 4 Charged Particle Kinetic 4 Charged Particle Kinetic 4 Thermophoretic 4 Thermophoretic Force Force Boltzmann Boltzmann 4 Photophoretic Charge 4 Charge Photophoretic Force Force ME 637-Particle-II G. Ahmadi ME 637-Particle-II G. Ahmadi ME 637-Particle-II G. Ahmadi 7