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Single Particle Motion. Reading: Chaps 3 & 5. Want to be great athletes? Study aerosol science & engineering because you surely need to know how to control particle movement in the air!. V.

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single particle motion
Single Particle Motion

Reading: Chaps 3 & 5

Want to be great athletes? Study aerosol science & engineering because you surely need to know how to control particle movement in the air!



  • Newton’s Resistance Law The force is proportional to the gas pushed away and the relative velocity between the sphere and the gas (negligible viscous force)

CD = 0.44 (sphere) for 103 < Rep < 2×105

Delivery van 0.04

Sports car 1.0

Airplane 0.25


  • Reynolds Number: ratio of inertial forces to frictional forces (Chap.2.5)

For a spherical particle, L = dp,

Q: Choose a combination of dp and V when the Newton’s Law can be applied.


Stokes’ Law: negligible inertial force compared to viscous force (Rep < 1); in a laminar flow for a spherical particle

  • Assumptions used in solving Navier Stokes’ equation:
  • Incompressible flow (g = cons)
  • Constant motion (V = const)
  • Rigid sphere
  • Fluid velocity on the particle surface

Q: How does FD change wrt V and dp in the laminar regime?

Q: Under what scenario will you expect to apply Newton’s law instead of Stokes’ Law?

Q: How does CD change wrt Re from 10-3 to 105?

settling velocity
Settling Velocity
  • When the drag force is equal and opposite to the gravitational force
  • Particle Mechanical Mobility
  • Settling Velocity


Q: What is the physical meaning of B?

Q: Does a smaller or a larger particle

have larger mobility?

terminal settling velocity of unit density spheres

1 mm

0.1 mm

Terminal Settling Velocity of unit-density spheres

: mean free path (Chap 2.3)

0.066 mm for air @STP

  • Cunningham Slip Correction Factor: gas velocity at the surface of small particles is not zero --> slip

(Re < 1)

Q: So is the corrected VTS larger or smaller?

Appendix A11

Q: How long do you have to wait for 0.1 mm particles to completely settle in this classroom (4 m high)?


Knudson Number

Kn = 2/dp

  • A dimensionless number for the relative size of a particle in the system
  • Continuum regime: Kn << 1. Gas molecules striking the particle surface are strongly affected by those leaving.
  • Free Molecular regime: Kn >> 1. The gas molecule movement is rarely affected by the presence of the particle.

Q: Kn for a 10 mm particle? In which regime?

nonspherical particles
Nonspherical Particles
  • Dynamic shape factor: the ratio of the actual resistance force of the nonspherical particle to the resistance force of a sphere having the same volume and velocity
  • Drag force
  • Settling velocity

de: equivalent volume diameter

Q: Does an irregular particle experience a larger or smaller force (compared to a particle with the same volume)?

aerodynamic diameter
Aerodynamic Diameter
  • The Stokes diameter, ds, is the diameter of the sphere that has the same density and settling velocity as the particle.
  • The aerodynamic diameter, da, is the diameter of the unit density (0 = 1 g/cm3) sphere that has the same settling velocity as the particle.

Cunningham factor should be included if dp < 1 mm

Q: Can we design an instrument by applying settling velocity?

horizontal elutriator settling chamber spectrometer
Horizontal Elutriator (settling chamber, spectrometer)

Q: Can it be vertical?

Q: Can we make the acceleration > g?

vertical elutriator
Vertical Elutriator

centrifuge separator
Centrifuge Separator

Cheng et al., 1988

particle acceleration



Particle Acceleration
  • Newton’s law

 = mB : relaxation time

Q: What is the physical meaning of relaxation time?











Non-zero Initial velocity


Stopping Distance

Time for unit density particles to reach their terminal velocity

  • An aerosol can adjust itself very quickly to a new environment!!!
inertial impaction
Inertial Impaction
  • Stokes number: the ratio of the stopping distance of a particle to a characteristic dimension of the obstacle
  • For an impactor

Q: Stk << 1? Stk >> 1?


Assumptions of Simplified Theory:

Uniform streamlines in the jet

Streamlines are arcs of a circle with

the centers at A



Determine dp50

Q: How to collect smaller particles

using an impactor?

Downstream pressure

(Chap 3.4, Pdin kPa, dp in um)

Q: (low pressure impactor) Advantages? Disadvantages?

Q: Again, how to collect smaller particles using an impactor?


Cascade Impactor

Aerosol flow In

Clean air out


Q: Advantages?



Virtual Impactor

Collection efficiency & internal loss

curves for a virtual impactor

Aerosol concentrator?















Time of Flight

Q: Comparison with an impactor?

aerodynamic focusing lens
Aerodynamic Focusing Lens

Time of Flight (TOF) vs Time in Beam (TIB)

TSI Operating Manual, 2004