Mechanisms of Particle Deposition - Aerosols: Measurement, Dosimetry, and Health Effects - page 49

By recognizing that F g = mg and substituting in Equation 3.1, this relationship may also be
expressed as

3πμ d V

p = m

g (3.3)

p

The equation for terminal settling velocity, V TS , is derived by expressing the particle mass in terms
of ρ p and solving for V p :

2

= ρ

d g

p p

V (3.4)

TS

18 μ

The time it takes for a particle starting at rest to reach its terminal settling velocity is the relaxation
time, τ, expressed as

2

d ρ

μ

d

2

ρ
μ

p p

ae o

(3.5)

τ = =

18 18

Note that τ is dependent upon both particle properties (e.g., particle size and density) and luid vis-
cosity. When we substitute Equation 3.5 into Equation 3.4, we obtain V TS expressed in terms of τ:

V TS = τ g

(3.6)

3.2.3 a erodynaMic d iaMeter
The aerodynamic diameter of a particle ( d ae ) is an important parameter used to relate particles
of differing shapes and densities. The d ae of a particle is deined as the diameter of a unit density
sphere that has the same V TS as the particle in question. The d ae is commonly used to characterize
the kinetic behavior of larger (>1 μm) aerosol particles, and can be calculated as

d ae = d g ρ

p

(3.7)

where d g is the geometric diameter of the particle.

3.2.4 M odiFications to tHe a erodynaMic e quations
Modiications may be made to adapt Stokes's Law for use in nonidealized situations. For example,
Equation 3.1 assumes that a particle is spherical in shape, an assumption that may not always be
appropriate (e.g., ibers, spheroids, cubes). Without appropriate correction, calculations of aerody-
namic properties and deposition probabilities for inhaled particles may be inaccurate. 6

3.2.4.1 Correction for Nonspherical Particles
When particles are nonspherical, a dynamic shape correction factor, Π, may be applied to Stokes's
Law. This quantity is deined as the ratio of the drag force of the nonspherical particle to the drag
force of a sphere having the same volume and velocity. Stokes's Law, corrected for shape, may then
be expressed as

F = 3πμ d V χ (3.8)

D p