• Correction to Phys Phenom I slides

• Movement in electric fields

• Movement in thermal fields

Physical phenomena II

Corrections to drag coeff. slide

For settling under gravity

Regime

Continuum Cd Rep2 = 4/3 dp

3 f(p - f) g / 2

Here, this expression is valid from Stokes through Newtons

Continuum, Stokes Cd = 24/Rep

Continuum, Intermediate Cd ~ (24/Rep) (1+0.15Rep0.687) to 7%

Continuum, Newton’s Cd ~ 0.44

Movement of particles in external fields

• Previously - particles diffusing down a concentration gradient by Brownian motion

• Particles can also move under the influence of external fields in addition to gravity.

• Examples: electric fields, magnetic fields, and temperature gradients

New Term - Particle Mobility

• Mobility = velocity given to a particle by a constant unit of driving force, “B”

• Driving forces could be gravity, electric fields, thermophoretic forces, magnetic forces etc…

€

B =C

3πμdp

What units does this have???

Mobility con’t

€

B =τ

m

In terms of relaxation time:

where p dp2/18 and m is particle mass

Electrostatics - definitions

Fundamental equation: Coulomb’s law

F Kqq'

RE E 2=FE = electrostatic repulsive forceq, q’ = point charges of like signR = separation distanceKE = constant of proportionality

cgs units = 1SI = 9.0 x 109 N m2 C-2

Ampere -- current required to produce a specified force between two parallel wires 1 m apartCoulomb -- amount of charge transported in 1 second by a current of 1 AVolt -- potential difference between two points along a wire carrying 1 A of current, and dissipating 1 watt of power between the points.

Electric fields: • An electric field exists in the space around a

charged object• Charged particles in this space are acted upon by

the electrostatic repulsive force, FE

• Field strength is given by: E = FE/q• Charge, q, normally expressed as n multiples of

the smallest unit of charge, the charge on an electron (e = 1.6 x 10-19 C) q = ne

• So: force on a particle with n elementary units of charge, in a field of strength E: FE = neE

E-fields: simple geometries

• Field around a single point charge, q

• Field strength between two oppositely charged, closely spaced parallel plates (neglecting edge effects)

EKE=q

R2

EW

x

W - difference in voltage between two plates

x - separation distance

=Δ

Δ

Particles in the E-Field

• If a charged particle is placed in an electric field, it will move, and the resulting velocity can be found by a force balance (Electrostatic force = Stoke’s drag)

For no net force:

• This we will call terminal electrical velocity, UTE

€

neE =3πUd

C

UTE =neEC

3πμd

Electrical mobility

• Ability of a particle to move in an electric field usually expressed as electrical mobility, Z, given by:

• For Re < 1, UTE = ZE• Z has units of m2 V-1 s-1 and is related to

mechanical mobility, B, as Z = neB€

Z =UTE

E=

neC

3πμd for particle Reynolds numbers < 1

Example problem: What is the electrical mobility of a) a 1 micron particle, carrying 40excess electrons and b) a 0.01 micron particle carrying 4 excess electrons?

If we place these particles between two charged plates, charged at +1000 V, and -1000 V, separated by 1 cm, what are their terminal electrostatic velocities?

+ -

Particle charging mechanisms:

• Static electrification - particles are charged by mechanical action– Electrolytic charging = liquids with high-dielectric

constant are separated from solid surfaces. Can happen in atomization, where liquids strip charge off atomizer surface. Results in slightly to moderately charged droplets.

– Spray electrification = results from disruption of charged liquid surfaces. Principle can be used as aerosol generator.

More static charging

• Static electrification– Contact charging - also known as triboelectrification =

occurs during separation of dry, non-metallic particles from solid surfaces. Friction increases the amount of charge acquired, and since most methods of resuspending dry powders involve friction, these methods produce charged particles. Ineffective charging mechanism at relative humidities above 65%.

More charging mechanisms: • Diffusion charging- when ions are present, collisions

between particles and ions occur. The ions stick, and the particle becomes charged.

• If particles are mixed with unipolar ions, over time, as charge accumulates, a field is produced around the particle, repelling additional ions, so charging rate approaches zero.

• Never exactly reaches zero because no upper limit of Boltzman distribution of ion velocities. (always probability that some ions have sufficient momentum to overcome repulsive force).

• This charging mechanism does NOT require an external electric field.

Charging mechanisms:

• Field charging - charging by unipolar ions in the presence of a strong electric field.

• Motion of ions in electric field along field lines results in frequent collisions between particle and ions.

• As particles become charged, field strength decreases, and rate of ions reaching particle decreases.

• At saturation charge, no ions reach particle.

How to charge aerosols?• Why? Electrostatic precipitators for particle collection

(powders or pollution) also, electrostatic samplers. • Need source of unipolar ions• Best source is corona discharge• Created when there is a strong nonuniform electric

field between– needle - plate – wire-tube

• Want: electrical breakdown occurring near needle or wire, but not arcing across whole separation distance.

Corona discharge

• In region near wire, E >Eb, and electrons are accelerated to velocity sufficient to knock electron from air molecules, creating a postive ion and a free electron.

• If wire is positively charged, electrons move to wire, but positive ions stream away.

• If wire is negatively charged, positive ions go to it, and electrons go towards tube, attaching to air molecules creating negative ions.

• Either way, ions produced in high concentrations.

• Aerosols entering - leave with same charge as wire.

wire corona

tube wall

Charge limits• Maximum amount of charge that can be acquired by a

negatively charged particle

• where EL is surface field strength required for spontaneous emissions of electrons (9.0 x 108 V/m)

• For positively charged particles, same equation, but EL = surface field strength for emission of positive ions (2.1 x 1010 V/m)

• For liquid drops: called ‘Rayleigh limit’ is liquid surface tension

nL =d E

K ep L

E

2

4

nd

K eL

p

E

=⎛

⎝⎜

⎞

⎠⎟

2 3

2

1 2πγ

/

How to neutralize aerosols?

• Why? Want to have particles with known charge distribution for sampling.

• Can aerosols have zero charge? Yes, but air has 103 bipolar ions/cm3, so the equilibrium charge state is a distribution, called the Boltzmann equilibrium charge distribution.

• Highly charged particles loose charge by collision with oppositely charged ions, leading to predictable (!) distribution, shaped like normal distribution for particles > 0.5 microns.

• empirical approximation for the average number of charges is:

( )n dp= 2 371 2

./

>-3 -3 -2 -1 0 1 2 3 >3

10

50

2000

10

20

30

40

50

60

70

80

90

100

Percent particles

charged as indicated

Number and sign of charge per particle

Particle diameter, nm

Boltzmann Distribution of Charge

10 20 50 100 200 500

Source of bipolar ions

• Common approach is to use radioactive source (usually polonium-210 or krypton-85) to ionize air molecules inside a chamber through which aerosol flows.

• To compare, neutralization of highly charged particles takes 2 s in commercial radioactive neutralizers, but would take 100 minutes in air.

Electrostatic CollectionMigration of Charged Particle in Electric Field

ve

+

+

Particles and thermal fields

• In addition to electric fields, particles also move in presence of temperature gradients

• Movement called thermophoresis

• Thermal force and aerosol particle motion always in direction of decreasing temperature

Thermophoresis

Drift of aerosol particle from hot to cold caused by collision with more energetic gas molecules on the hot side

Hot

Cold

• thermal force on a particle given by:

• thermophoretic velocity given by: • independentof particle size!

T = assume particle has same T as surrounding gas at that location

Thermophoresis - free molecular

direction of thermophoretic force

hot side cold side

€

FTH =−pλdp

2∇T

T

€

UTH =−0.55μ∇T

ρ gT

Thermophoresis - more

• Continuum - more complicated since a temperature gradient is established in particle, which affects gas surrounding particle, UTH not independent of size

• Comparison of terminal settling velocities, temp gradient = 1 K/cm, T = 300 K

dp

micronsterminal settlingvelocity, m/s

thermophoreticvelocity, m/s

0.01 6.7 x 10-8 2.8 x 10-6

0.1 8.6 x 10-7 2.0 x 10-6

1.0 3.5 x 10-5 1.3 x 10-6

10.0 3.1 x 10-3 7.8 x 10-5

Thermophoresis- implications

• for small particles, temperature gradients used to sample with no size bias

• in clean rooms, heated surfaces used to keep particles from depositing

• can use thermophoresis for collecting powders