Particle Aerodynamics S P Chap 9.

1
Particle AerodynamicsSP 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

2
Brownian Motion Review

• We have looked at Brownian motion from both
  Einsteins and Langevins points of view
• Einstein came up with
• Where D BkT
• And B is particle mobility
• Multiply times x2 and do an average ltgt, 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

3
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

4
Gaussian Plumes

• Random walk ? binomial distribution
• Binomial distribution at large N ? Gaussian
• We have seen for diffusion that ltxgt 0 and ltx2gt
  2Dt in 1 dimension.
• Thus our Gaussian plume looks like

• You can show that this satisfies the continuous
  diffusion equation

5
Drag and Mobility

• 3 drag force results so far
• vt BFG
• FD -v / B
• ltvgt ltv0gtexp(-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.

6
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 6pnRu

7
Mobility Vs. Diameter

• Using Stokes drag with our definition of
  mobility, we have
• FS 6pnRu
• FD -v / B u/B
• BS 1/(6pnR)
• 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 doesnt work at all sizes and
  velocities, however. Other factors are at play

8
Relaxation Time and Stop Distance

• ltvgt ltv0gtexp(-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

9
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.

10
Knudsen
l mean free path of air molecule Dp particle
diameter
Gas molecule self-collision cross-section
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
DP
Continuum Regime
Transition Regime
11
Slip Correction for low Kn
12
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