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

SLIDE 1

1

• G. Ahmadi

ME 637-Particle-II

• G. Ahmadi

ME 637-Particle-II

Outline Outline

4 4Electrostatics Electrostatics 4 4Particle Charging Particle Charging 4 4Charged Particle Kinetic Charged Particle Kinetic 4 4Thermophoretic Thermophoretic Force Force 4 4Photophoretic 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 t c 1 = ∂ ∂ + × ∇ B E t = ∂ ∂ + × ∇ B E = ⋅ ∇ B = ⋅ ∇ 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

t

e =

∂ ρ ∂ + ∇J

Continuity Continuity Equation Equation

E D ε = 1

0 =

ε

m Volt / Coul 10 854 . 8 c 4 10

12 2 7

⋅ × = π = ε

−

H B µ = 1

0 =

µ

7

10 4

−

× π = µ E J σ =

2 / 1

) ( c

−

µ ε =

Constitutive Equations, Free space Constitutive Equations, Free space Ohm’s Law Ohm’s Law

SLIDE 2

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• G. Ahmadi

ME 637-Particle-II

l

9

10 3×

ρ

3

10 3×

9

10 3×

4

10 3 1

−

×

300 / 1

5

10 3× statvolt statvolt 1 volt 1 volt V V Electric Potential Electric Potential statvolt statvolt/cm /cm 1 volt/m 1 volt/m E E Electric Field Electric Field statamp/cm statamp/cm2

2

1 amp/m 1 amp/m2

2

J J Current Density Current Density statampere statampere 1 1 ampere(coul/s ampere(coul/s) ) I I Current Current statcoul/cm statcoul/cm3

3

1 coul/m 1 coul/m3

3

Charge Density Charge Density statcoulomb statcoulomb 1 coulomb ( 1 coulomb (coul coul) ) q q Charge Charge 10 107

7 ergs/s

ergs/s 1 watt (W) 1 watt (W) P P Power Power 10 107

7 ergs

ergs 1 joule (J) 1 joule (J) W, U W, U Work, Energy Work, Energy 10 105

5 dynes

dynes 1 1 newton newton (N) (N) F F Force Force 1 second (s) 1 second (s) 1 second (s) 1 second (s) t t Time Time 10 103

3 gram (gm)

gram (gm) 1 kilogram (kg) 1 kilogram (kg) m m Mass Mass 10 102

2 centimeter (cm)

centimeter (cm) 1 meter (m) 1 meter (m) Length Length Gaussian Gaussian MKS MKS Symbol Symbol Physical Quantities Physical Quantities

• G. Ahmadi

ME 637-Particle-II For accurate works, all factors of 3 in the coefficients should be replaced by 2.99793.

1 henry L Inductance

• mag. moment/cm3

1 amp/m M Magnetic Induction

• ersted

1 amp-turn/m H Magnetic field gauss 1 weber/m2 B Magnetic induction gauss cm2 (maxwell) 1 weber F Magnetic flux cm 1 farad C Capacitance s/cm 1 ohm R Resistance 1/s 1 mho/m Conductivity statcoul/cm2 statvolt/cm) 1 coul/m2 D Displacement dipole moment/cm3 1 coul/m2 P Polarization Gaussian MKS Symbol Physical Quantities

5

10 3×

5

10 12 × π

σ

9

10 9 ×

11

10 ) 9 / 1 (

−

×

11

10 9 ×

8

10

4

10

3

10 4 × π

3

10 4 1

−

× π

11

10 9 ×

• G. Ahmadi

ME 637-Particle-II

Most aerosol particles carry some electrical charges Most aerosol particles carry some electrical charges

E FE q =

ne q = coul 10 6 . 1 e

19 −

× = statcoul 10 8 . 4 e

10 −

× =

c

C / Ud 3 qE πµ = d 3 qC Z u

c p

πµ = =

Coulomb Coulomb Force Force Electric Charge Electric Charge Particle Particle Mobility Mobility

• G. Ahmadi

ME 637-Particle-II

Boltzmann Boltzmann Equilibrium Charge Equilibrium Charge Distribution Distribution

∑

∞ −∞ =

− − =

n 2 2 2 2

} dkT / e n exp{ } dkT / e n exp{ ) n ( f } dkT e n exp{ dkT e ) n ( f

2 2 2

− π = µ 02 . > d } d n 05 . exp{ d 24 . ) n ( f

2

− π = µ 02 . > d

SLIDE 3

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• G. Ahmadi

ME 637-Particle-II

Average Number of Charge Average Number of Charge µ 02 . > d

2

e dkT dn ) n ( f | n | ) n ( f | n | n π ≈ ≈ =

∫ ∑

∞ ∞ − ∞ ∞ −

2

r 4 q E π γ = ε π = γ / 4 1 = ε π = γ 4 ε ε = γ

• 1

Point Point Charge Charge cgs cgs MKS MKS

Air Air

) m ( d , d 36 . 2 n µ ≈ =

• G. Ahmadi

ME 637-Particle-II

2

r 4 q ' q E ' q F π γ = =

ε ε = γ

• 1

meter volt sec amp 10 859 . 8

12

• −

− × = ε

−

) 10 9 ( r q ' q F

9 2

× ε =

Permittivity Permittivity Coulomb’s Coulomb’s Law Law

• G. Ahmadi

ME 637-Particle-II

e 4 Ed ) 2 ) 1 ( 2 1 ]( 1 t n eZ t n eZ [ n

2 p p i i i i

+ ε − ε + + π π =

∞ ∞

∞ → + ε − ε + =

∞

t as e 4 Ed ] 2 ) 1 ( 2 1 [ n

2 p p

Quartz for 3 . 4

p =

ε

cgs cgs cgs cgs

• G. Ahmadi

ME 637-Particle-II

] t de n ) kT m 2 ( 1 ln[ e 2 dkT n

2 i 2 / 1 i 2 ∞

π + =

3 8 i

cm sec/ ion 10 ~ t n ∞

SLIDE 4

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• G. Ahmadi

ME 637-Particle-II

• G. Ahmadi

ME 637-Particle-II

• Gas

+

• Gas, U=1-3 m/s

+

• b=15-40 cm
• G. Ahmadi

ME 637-Particle-II

C u y C ) D ( J

e T

− ∂ ∂ ν + − = d 3 EqC u

c e

πµ =

∫ ∫

∞ ∞

ν + − − ν + − − =

T e y T e

} D dy u exp{ 1 } D dy u exp{ 1 C C

} u u exp{ 1 C u } D dy u exp{ 1 C u ) x ( J

D e e T e e

− − − = ν + − − − =

∞ ∞ ∞

∫

∫

∞

ν + =

T D

D dy 1 u

• G. Ahmadi

ME 637-Particle-II

D e

u u <<

∞

= C u | J |

D e D

u u <<

∞

= C u | J |

e

dx C u 2 Jdx 2 C bUd

e ∞ ∞

− = = } bU x u 2 exp{ C C

e

− =

∞ ∞

} bU L u 2 exp{ C C

e L

− =

∞ ∞

SLIDE 5

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• G. Ahmadi

ME 637-Particle-II

0.1 1 10 30 60 90 99 99.9

d (µm) η

• G. Ahmadi

ME 637-Particle-II

E G D

F F F u + + = dt d m

p

m q dt d τ − τ + − = τ E g u u u

p f p

m q dt d τ − = + τ E u u u

• p

p

g u uo τ + =

f

Equation Equation

• f Motion
• f Motion
• G. Ahmadi

ME 637-Particle-II

Γ − = + τ 1 u dt u d

p ^ p ^ p p ^

u u u = mu Eqτ = Γ inertia neglecting , 1 | | For >> Γ Γ − =

p ^

U m Eq up τ − = E uo || For

• G. Ahmadi

ME 637-Particle-II

} 2 d exp{ T | ' v | d 15 8 F

^ f f 2 t

λ θ − ∇ κ − =

∞ < <

n

K 25 .

1 << Μ

) 2 1 ( 21 . 12 . 9 .

p f t m m ^

κ κ α − α + α + = θ

2 / 1 f f f

) m kT 8 ( | ' v | c π = = ation mod Accom Momentum

m =

α ation mod Accom Thermal

t =

α

SLIDE 6

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• G. Ahmadi

ME 637-Particle-II

) K C 2 2 1 )( C 3 1 ( T ] C 33 . 1 ) C 33 . 1 1 )( K C [( K C d 3

n t p f n m n m n m n t p f n m t 2

+ κ κ + κ + ∇ κ − κ + + κ κ πµ − =

t

F

2 . K

n <

< ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ κ κ + ∇ πµν − =

f p

2 1 T T d 9

t

F 1

n >

κ

Continuum Regime Continuum Regime

• G. Ahmadi

ME 637-Particle-II

) T R ' v 2 1 ( 48 p d

p 2 f f 3 p

κ + ρ π − = I F

∞ →

n

K

Diffusiophoretic Diffusiophoretic Force Force

• G. Ahmadi

ME 637-Particle-II

4 6 2

• 3

3 2

• 2

e

y 128 d E 3 y 16 qEd y 16 q qE F πε − + πε − =

Coulomb Coulomb Force Force Image Image Force Force Polarization Polarization Force Force Dipole Dipole Interactions Interactions

• G. Ahmadi

ME 637-Particle-II

Particles with Particles with Single Charge Single Charge

SLIDE 7

7

• G. Ahmadi

ME 637-Particle-II

Boltzmann Boltzmann Charge Charge Saturation Saturation Charge Charge

• G. Ahmadi

ME 637-Particle-II

4 4Electrostatics Electrostatics 4 4Particle Charging Particle Charging 4 4Charged Particle Kinetic Charged Particle Kinetic 4 4Thermophoretic Thermophoretic Force Force 4 4Photophoretic Photophoretic Force Force

• G. Ahmadi

ME 637-Particle-II