Physical phenomena II
• Correction to Phys Phenom I slides
• Movement in electric fields
• Movement in thermal fields
Corrections to drag coeff. slide
For settling under gravity
Regime
Continuum Cd Rep2 = 4/3 dp3 rf(rp - rf) g / m2
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
3md p
What units does
this have???
Mobility con’t
In terms of relaxation time:
B

m
where  rp dp2/18 m,and m is particle mass
Electrostatics - definitions
Ampere -- current required to produce a specified force between
two parallel wires 1 m apart
Coulomb -- amount of charge transported in 1 second by a
current of 1 A
Volt -- potential difference between two points along a wire
carrying 1 A of current, and dissipating 1 watt of power between
the points.
Fundamental equation: Coulomb’s law
qq'
FE = electrostatic repulsive force
FE  K E 2
q, q’ = point charges of like sign
R
R = separation distance
KE = constant of proportionality
cgs units = 1
SI = 9.0 x 109 N m2 C-2
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
KEq
E 2
R
• Field strength between two oppositely
charged, closely spaced parallel plates
(neglecting edge effects)
W
E
x
W - difference in voltage between two plates
x - separation distance
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)
3Ud
For no net force: neE 
UTE
C
neEC

3md
• This we will call terminal electrical velocity, UTE


Electrical mobility
• Ability of a particle to move in an electric
field usually expressed as electrical
mobility, Z, given by:
UTE neC
Z

for particle Reynolds numbers
E
3md
• For Re < 1, UTE = ZE
• Z has units of m2 V-1 s-1 and is related to
mechanical mobility, B, as Z = neB
<1
Example problem:
What is the electrical mobility of a) a 1 micron particle, carrying 40
excess 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
tube wall
wire
corona
•
•
•
•
•
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.
Charge limits
• Maximum amount of charge that can be acquired by a
negatively charged particle
dp2EL
nL 
4K E e
• 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:
3 1/ 2
called ‘Rayleigh limit’
 2 gdp 

nL  
g is liquid surface tension
2
 KE e 
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  2.37 dp
1/ 2
Boltzmann Distribution of Charge
100
90
80
70
Percent
particles 60
charged as 50
indicated 40
30
20
10
0
200
>-3 -3
50
-2
-1
0
1
10
2
3
>3
Number and sign of charge per particle
10
20
50
100
200
500
Particle diameter,
nm
Source of bipolar ions
• Common approach is to use radioactive
source (usually polonium-210 or krypton85) 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 Collection
Migration 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
Thermophoresis - free molecular
hot side
cold side
direction of
thermophoretic force
• thermal force on a particle given by:
pd p T
2
FTH 
T
• thermophoretic velocity given
by:
• independent
0.55mT
of particle size!
U 

TH
rgT
T = assume particle has
same T as surrounding gas at that location
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
microns
0.01
terminal settling
velocity, m/s
6.7 x 10 -8
thermophoretic
velocity, m/s
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