Particle Aerodynamics
S+P Chap 9.
Need to consider two types of motion
•
Brownian diffusion – thermal motion of particle, similar to gas motions.
– Direction is random, leads to diffusion
– Directly involves the molecular nature of the gas
•
Forces on the particle
– Body forces: Gravity, electrostatic
• Direction follows force field
– Surface forces: Pressure, friction
• Direction opposes motion of particle with respect to the fluid
– Treats the gas as a continuous fluid
Relevant Scales
•
•
Diameter of particle vs. mean free path in the gas – Knudsen #
Inertial “forces” vs. viscous forces – Reynolds #
Brownian Motion Review
• We have looked at Brownian motion from both
Einstein’s and Langevin’s points of view
• Einstein came up with
N
 D 2 N
t
Where D = BkT
And B is particle mobility
Multiply times x2 and do an average <>, and you get
and then convert to 3-D space, noting x2 = x2 + y2 + z2
• Langevin looked at the problem stochastically and
came up with
N
x
 Dx2 2 N
t
2
 x2
t
 x2
t
 x2
t
 2D
 6D
 6kTB
Using Brownian Diffusion
Problem:
– Consider the DMA. If it takes 20 seconds for
air to flow from the top of the column to the
bottom, what is the rms lateral distance an
uncharged particle of mobility B will travel
during transit. Hint This is diffusion in 1
direction
Gaussian Plumes
• Random walk  binomial distribution
• Binomial distribution at large N  Gaussian
• We have seen for diffusion that <x> = 0 and <x2> = 2Dt
in 1 dimension.
• Thus our Gaussian plume looks like
2

N0
1  x  x0  
N ( x, t ) 
exp 
 
4Dt
 2  2 Dt  
• You can show that this satisfies the continuous diffusion
equation
N
2
 D N
t
Drag and Mobility
• 3 “drag force” results so far
– vt = BFG
– FD = -v / B
– <v> = <v0>exp(-t/mB) – If no external force
• t = mB  “relaxation time” for a particle
• 1st two are equilibrium results
• Third is the approach towards equilibrium for an arbitrary initial
velocity.
• All three presume that the air is not moving, or rather that the
particle velocity is expressed in the reference frame of the air.
• The next step is to link the particle mobility to the properties of
the particle and the air.
Stokes Drag
• This is the simplest case for drag.
• Think of the air flowing around the particle at speed u
• There are two drag stresses that must be integrated over
surface:
– Pressure drag
– Friction drag stress GF = n(du/dr)
• In the “Stokes” regime, these drag effects are equal in
magnitude, producing a Stokes Drag Force
– FS = 6nRu
Mobility Vs. Diameter
• Using Stokes drag with our definition of mobility, we have
– FS = 6nRu
– FD = -v / B = u/B
– BS = 1/(6nR)
• As Particle radius goes up, mobility goes down.
• This is how the DMA works – it finds the mobility of a
particle based on its drift velocity, and then infers the
diameter from the assumed drag.
• Stokes drag doesn’t work at all sizes and velocities,
however. Other factors are at play
Relaxation Time and Stop Distance
• <v> = <v0>exp(-t/mB) –No external force
• t = mB  “relaxation time” for a particle
• We can integrate this to see how far a particle
gets that starts at x = 0 with v = v0
v  v0 exp(t / t )
v0t
t
x  x0   vdt
x
0
 x0  tv0 (1  exp(t / t ) 
“Stop Distance” ~
R2
t
for stokes flow – increases w/ size
Flow Around A Corner
• Key to this problem is variable
gas velocity and variable
particle velocity
• Drag force on particle takes
more general form:
– ] FD = -(v-u)/ B
• Solve for particle trajectory in x
and y directions.
Knudsen #
2
Kn 
Dp
 = mean free path of air molecule
Dp = particle diameter

Gas molecule selfcollision cross-section
1
2  B N B
Gas # concentration
Quantifies how much an aerosol particle influences its
immediate environment
• Kn Small – Particle is big, and “drags” the air nearby along with it
• Kn Large – Particle is small, and air near particle has properties about
the same as the gas far from the particle
Kn
Free Molecular
Regime
Transition
Regime
Continuum
Regime
DP
Slip Correction for low Kn
Relaxation Time and Start Distance
• Now consider a particle starting at rest, but
suddenly feeling an external force Fe
• We can integrate this to see how far a particle
gets that starts at x = 0 with v = v0
v  v0 exp(t / t )
v0t
t
x  x0   vdt
x
0
 x0  tv0 (1  exp(t / t ) 
“Stop Distance” ~
R2
t
for stokes flow – increases w/ size