Mechanisms of Particle Deposition - Aerosols: Measurement, Dosimetry, and Health Effects - page 51

Kn is the ratio of the mean free path of the gas molecules (λ) to the particle diameter. It may be
expressed as

= 2λ

Kn (3.12)

d p

The mean free path may be calculated using

1
n d m

λ = (3.13)

2

2 π

If the Kn is very small ( Kn <<1), a particle will decelerate due to gas molecule collisions upon the
particle's surface. Conversely, if the Kn is large ( Kn >> 1), a particle will not be affected by gas
molecules.
When the Cunningham slip correction factor is included, the Stokes drag force on a particle
becomes

D = 3πμ

Vd
C

χ

p

F (3.14)

c

The terminal settling velocity, corrected for shape and slip, is given by

d 2
18

= ρ

gC

p p c

V (3.15)

TS

μχ

Table 3.2 demonstrates the effect of slip correction on V TS for particles of various sizes. Note that
there are signiicant differences between the corrected and uncorrected V TS values for submicron
particles.

TABLE 3.2
Terminal Settling Velocity Values Calculated Using the
Cunningham Slip Correction Factor

Particle Diameter ( μ m) C c V TS a , w/o C c (cm/s) V TS a , with C c (cm/s)

0.001 228.234 3.02E-09 6.61E-07

0.01 23.3443 3.02E-07 6.77E-06

0.1 2.97393 3.02E-05 8.71E-05

0.5 1.34743 7.54E-04 1.005E-03

1 1.17273 3.02E-03 3.516E-03

3 1.05757 2.71E-02 2.864E-02

5 1.03454 7.54E-02 7.790E-02

10 1.01727 3.02E-01 3.066E-01

Source: Crowder, T.M. et al., Pharm. Res ., 19, 239, 2002.
a Values given are calculated for a unit density sphere of a given size at 20°C and 101 kPa
using Davies's constants. 13