Electrodynamics

Most natural and manmade aerosols carry charges, and their behavior is strongly affected by their charge. In this section the consequences of charges on dynamic of aerosols are discussed. A brief review of electrodynamics is first presented. The Maxell equations governing electrodynamics are listed in Table 1.

Table 1. Electrodynamics equations. Maxwell’s Equations MKS Units

Coulomb’s Law ∇ ⋅ D = 4πρe ∇⋅ D = ρe Ampere’s Law 4π 1 ∂D 1 ∂D ∇ × H = J + ∇ × H = J + c c ∂t c ∂t Faraday’s Law 1 ∂B ∂B ∇ × E + = 0 ∇ × E + = 0 c ∂t ∂t Absence of Free Magnetic Poles ∇ ⋅ B = 0 ∇ ⋅ B = 0 Continuity Equation ∂ρ ∇J + e = 0 ∂t

Constitutive equations for free space are given as:

107 D = ε E ε = 1 ε = = 8.854 ×10−12 Coul / Volt ⋅ m (1) 0 0 0 4πc2

−7 B = µ0H µ0 = 1 µ0 = 4π ×10 (2)

Ohm’s Law is given by

−1/ 2 J = σE c = (ε0µ0 ) (3)

In these equations:

D = Electric Displacement c = Speed of Light

B = Magnetic Induction ε = Dielectric Constant

E = Electric Field µ = Permeability

H = σ = Conductivity

J = Current ρe = Charge Density

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Note that 1 electronic unit of charge = 4.8×10−10 = 1.59 ×10−19 Coulombs (MKS). A conversion table for different physical quantities are given in Table 1.

Table 1. Conversion table. Physical Quantities Symbol MKS Gaussian

2 Length l 1 meter (m) 10 centimeter (cm) Mass m 1 kilogram (kg) 103 (gm) Time t 1 (s) 1 second (s) Force F 1 newton (N) 105 Work, Energy W, U 1 joule (J) 107 Power P 1 watt (W) 107 ergs/s Charge q 1 coulomb 3×109 (coul) 3 Charge Density ρ 1 coul/m 3×103 statcoul/cm3 Current I 1 ampere 3×109 statampere (coul/s) 2 Current Density J 1 amp/m 3×105 statamp/cm2 Electric Field E 1 volt/m 1 ×10−4 /cm 3 Electric Potential V 1 volt 1 statvolt 300 2 Polarization P 1 coul/m 3×105 dipole moment/cm3 2 Displacement D 1 coul/m 12π ×105 statcoul/cm2 (statvolt/cm) Conductivity σ 1 mho/m 9 ×109 1/s Resistance R 1 ohm 1 ×10−11 s/cm 9 Capacitance C 1 farad 9 ×1011 cm F 1 108 cm2 (maxwell) 2 Magnetic induction B 1 weber/m 104 gauss Magnetic field H 1 amp-turn/m 4π ×103 Magnetic Induction M 1 amp/m 1 ×10−3 magnetic moment/ cm3 4π Inductance L 1 henry 1 ×10−11 9 For accurate works, all factors of 3 in the coefficients should be replaced by 2.99793.

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Aerosols Charging and Their Kinetics

Most aerosol particles carry some electrical charges. In an electric field of strength E, a charged particle experiences a force, which is given by

FE = qE . (4)

Here

q = ne (5) where e is the elementary unit of charge and n is the number of elementary units carried by the particle. (e = 1.6 ×10−19 coulombs (MKS)= 4.8×10−10 Statcoulombs (electrostatic cgs unit).)

Particle Mobility

Particle mobility is defined as the velocity that it will acquire when subjected to an electric field of unit strength. Equating the drag and Columbic forces, it follows that

qE = 3πµUd / Cc (6)

For E = 1, the mobility becomes

qC u = Zp = c (7) 3πµd

Particle Charging

Aerosol particle can be electrified by a number of different processes. Among the important ones are: Direct ionization, static electrification, electrolytic effects (liquids of high dielectric constant exchange ions with metals), contact electrification, spray electrification, frictional electrification, collisions with ion, diffusion, and field charging.

Boltzmann Equilibrium Charge Distribution

Under equilibrium conditions, the aerosols approach the Boltzmann charge distribution. Accordingly, the fraction of particles having n unites of charge of one sign, f(n) , is given by

exp{−n2e2 / dkT} f(n) = ∞ . (8) ∑exp{−n2e2 / dkT} n=−∞

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For d > 0.02µm , the approximate expression for the charge distribution becomes

e2 n 2e2 f (n) = exp{− } . (9) dkTπ dkT

For a room temperature of 20o C , (9) reduces to

0.24 0.05n2 f (n) = exp{− } d in µm for d > 0.02µm . (10) dπ d

The average number of charge per particle is given by

∞ ∞ dkT n = | n | f (n) ≈ | n | f (n)dn ≈ for d > 0.02µ m . (11) ∑ ∫ 2 −∞ −∞ πe

Electric Field of a Point Charge

The electric field strength of a point charge at a distance r is given by

γ q E = , (12) 4πr 2 where γ = 4π / ε for cgs or electrostatic unit with ε being the dielectric constant of the 1 medium. For air, ε = 1, γ = 4π , and γ = (for MKS). εoε

Coulomb’s Law

Combining (5) and (12), we find

γ q'q F = q' E = , (cgs) (13) 4πr 2 which is Coulomb’s law for forces between two charge particles. Note that in the MKS 1 units, γ = , and the permittivity (dielectic constant) of free space, εoε amp − sec ε = 8.859 ×10−12 . o volt − meter

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q'q F = (9 ×109 ) , (MKS). (14) ε r 2

Field Charging

When a particle is in an electric field, the particle acquires charges due to collisions with ions, which are moving along the lines of force that intersect the particle surface. This process is known as field charging. The number of charges accumulated by the particle is given by

πeZ n t 2(ε −1) Ed2 n = [ i i∞ ](1+ p ) , (cgs), (15) πeZini∞ t +1 εp + 2 4e

where t is time, Zi is the mobility of ions, ni∞ is the ions concentration far from the particle, e is the electronic charge, εp is the dielectric constant of the particle, and E is the electric field intensity.

For sufficiently large time,

2 2(εp −1) Ed n∞ = [1+ ] as t → ∞ . (16) εp + 2 4e

2(ε −1) The factor [1+ p ] is a measure of distortion of the electrostatic field by the εp + 2 particle. The factor varies between 1 and 3 for εp = 1 to ∞ . For dielectric materials, the values of εp usually are less than 10. εp is 4.3 for quartz and 2.3 for benzene.

Diffusion Charging

Even in the absence of an external electric field, particles exposed to an ion cloud become charged. Ions will collide with the particle due to their thermal motions. As the particle becomes charged, it will repel ions of the same sign and leads to a non- homogenous distribution of ions in its neighborhood. After time t , the number of charge acquired by the particle is given by

dkT 2π 1/ 2 2 n = 2 ln[1+ ( ) ni∞de t] , (cgs) (17) 2e mikT

kT 1/ 2 where mi is the ionic mass and ( ) is the mean velocity with which the ions strike 2πmi the particle surface.

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As t→ ∞ , the number of charge (according to the formula) also approaches 8 3 infinity, which is not correct. However, for typical values of ni∞t ~10 ion sec/ cm , the equation gives reasonable results, especially for particles, which are smaller than the mean free path.

Electrical Precipitation

Electrical precipitation is used widely in power plants for removing particles from discharging smokestacks. In the most common type, the dusty gas flows between parallel plate electrodes. The particles are charged by ions generated in a corona discharge surrounding rods or wires suspended between the plates. Figure 1 shows schematics of electrostatic precipitators.

-

Gas +

-

-

+ b=15-40 Gas, U=1-3 m/s

-

Figure 1. Schematics of electrostatics precipitators.

In the precipitators the cloud of electrons and negative ions moves towards the collecting electrodes. Particles are charged by field or diffusion charging, depending on their size. The Reynolds number is of the order of 104 or greater and hence the flow is turbulent. The flux normal to the collecting plate is given by

∂C J = −(D + νT ) − u C , (18) ∂y e

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where Ue is the particle migration velocity towards the plate given by

EqC u = c . (19) e 3πµd

Here, q is the charge on the particle and E is the intensity of the electric field.

For ue being a constant, assuming J remains constant, it follows that

y u dy 1− exp{− e } ∫ T C 0 D + ν = ∞ (20) C∞ u dy 1− exp{− e } ∫ T 0 D + ν and

− ue C∞ ue C∞ J(x) = ∞ = − (21) dy ue 1− exp{−u } 1− exp{− } e ∫ T u 0 D + ν D where

1 u= , (22) D ∞ dy ∫ T 0 D + ν is the deposition velocity due to combined Brownian and turbulent diffusion.

For ue << u D , it follows that | J |= u D C∞ and deposition is controlled by diffusion. When u D << ue as is the case for electrical precipitators, | J |= ue C∞ . For a lengthdx length,

bUdC∞ = 2Jdx = −2ue C∞dx . (23)

Hence,

2uex C∞ = C∞0 exp{− } . (24) bU

For a length L ,

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C∞ L 2u L = exp{− e }. (25) C∞0 bU

Figure 2 shows typical collection efficiencies of electrostatic precipitators.

99.9

99

η 90

60

30 0.1 1 10 d (µm)

Figure 2. Variation of collection efficiency of an electrostatic precipitator.

Field Strength

The electrostatic potential V satisfies a Poisson equation given as

2 ∇ V = −γρs , (26)

where ρs is the space charge per unit volume. Then electric field is given as

E = −∇V (27)

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Particle Drift in an Electric Field

The equation of motion of a particle in an electric field is given by

dup m = F + F + F , (28) dt D G E where the forces are due to drag, gravity, and electric field. Equation (28) may be restated as

dup τ τ = uf − up + τg − Eq . (29) dt m

Equation (29) may be restated as

dup τ τ + up = u − Eq , (30) dt o m

f where uo = u + τg . Suppose E is in the same direction as uo , non-dimensionalizing with respect to uo , we find

^ p d u ^ p τ + u = 1− Γ , (31) dt where

^ p u p Eqτ u = , Γ = (32) u0 mu0

If | Γ |>> 1, then the electrical force dominates. In this case, if the inertia is also neglected, we find

^ p τ U= −Γ or up = −Eq . (33) m

In addition to the gravitational and electrical forces, particles could experience forces induced by temperature gradient, concentration gradient, and electromagnetic radiation. These are discussed in the next section.

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