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 Gaussian Units 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 = Magnetic Field σ = Conductivity J = Current ρe = Charge Density ME637 1 G. Ahmadi Note that 1 electronic unit of charge = 4.8×10−10 Statcoulombs = 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 gram (gm) Time t 1 second (s) 1 second (s) Force F 1 newton (N) 105 dynes Work, Energy W, U 1 joule (J) 107 ergs Power P 1 watt (W) 107 ergs/s Charge q 1 coulomb 3×109 statcoulomb (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 statvolt/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 Magnetic flux F 1 weber 108 gauss cm2 (maxwell) 2 Magnetic induction B 1 weber/m 104 gauss Magnetic field H 1 amp-turn/m 4π ×103 oersted 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. ME637 2 G. Ahmadi 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=−∞ ME637 3 G. Ahmadi 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 > 0.02µm . (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 ME637 4 G. Ahmadi 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. ME637 5 G. Ahmadi 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 ME637 6 G. Ahmadi 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 , ME637 7 G. Ahmadi 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) ME637 8 G. Ahmadi 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.