Brownian Motion

1
Brownian Motion Diffusion
Reading Chap7
HISTORY In 1827 the English botanist Robert Brown
noticed that pollen grains suspended in water
jiggled about under the lens of the microscope,
following a zigzag path. Even more remarkable was
the fact that pollen grains that had been stored
for a century moved in the same way. Magic?
http//galileo.phys.virginia.edu/classes/109N/more
_stuff/Applets/brownian/brownian.html
http//www.geocities.com/piratord/browni/Difus.htm
l
Q Observation?
2

• Definition
• Brownian motion irregular wiggling motion of a
  particle caused by random bombardment of gas
  molecules against the particle
• Diffusion the net transport of particles from a
  region of higher concentration to a region of
  lower concentration
• The primary transport mechanism for small
  particles (lt 0.1 ?m) Important when transport
  distance is small e.g. filter, airway in human
  lung, loss in sampling path

3
Ficks First Law of Diffusion

• The net flux of aerosols (the net number of
  particle traveling through a unit area per unit
  time) is proportional to the concentration
  gradientJ flux (/area/time)D diffusion
  coefficient (area/time)n particle number
  concentration (/cm3)

4
Ficks Second Law of Diffusion
A
A
B
B
D
D
C
C
5
Spread of particles over time and space
1/16
Numbers on curves are values of Dt
1/2
1
6

• Solve 1-D equation

Solution
(normal distribution!)
Mean Square Displacement of particles
Recall Mean Surface Area
Q Why not mean displacement?
7
Stokes-Einstein Equation for Diffusion
Coefficient

• Assumptions
• the Brownian motion of an aerosol is equivalent
  to that of a giant gas molecule the kinetic
  energy for aerosol Brownian motion is the same as
  the gas molecules (KE3/2KT)
• the diffusion force on a particle is equal to the
  friction force

HISTORY The first mathematical theory of Brownian
motion was developed by Einstein in 1905. For
this work he received the Nobel prize
for particles greater than the mean free path of
air
Q What are the parameters that affect an
aerosols diffusivity?
8
Particle Mean Free Path
Mean thermal velocity
(Chap. 2.2.2-2.3)
Particle mean free path
aerosol
Gas molecule
9
Diffusional Deposition

• Aerosols adhere when they hit a surface.
• Ex. A vertical surface in a infinitely large
  volume of aerosol

Solution
Q What if dp is larger?
10
Deposition Rate

• Concentration gradient at the surface
• Using Ficks First Law
• Integrating over time
• Deposition velocity

General form of the concentration profile near a
wall
11
Q Which mechanism (diffusion and gravitation
settling) is more important to deposit
particles?
12
Penetration

• Dimensionless deposition parameter
• Penetration

Q How does P change if the tube diameter doubles
while the volume flow rate remains the same? Why?
Tube
Rectangular channel
tube
rectangular channel
13
Penetration versus deposition parameter for
circular tubes and rectangular cross-section
channels
14

• Diffusion Battery tube bundle parallel plate.
  Same equation for a single long tube or by
  cutting it into n parallel tubes each carrying
  1/n of the total flow

Penetration for an aerosol flowing through a
tube (1m length) at 3 laminar flow rates
15
Screen-type Diffusion Battery/Classifier
Aerosol Measurement, Ed. Willeke, K. Baron, P.
A., 1993.
16
Aerosol Measurement, Ed. Willeke, K. Baron, P.
A., 1993.
17
Diffusion Denuder
Aerosol Measurement, Ed. Willeke, K. Baron, P.
A., 1993.

• Diffusion Dryer

18
Reflection