Fundamentals of Particle Dynamics - Computational Fluid and Particle Dynamics in the Human Respiratory System - page 152

at time t 3 τ a particle will reach 95 % of its stopping distance. This measure is

important when considering particle flows in 90 ◦ turns, such as those found in the
nostril-to-nasal passage and the nasopharynx region. In these regions, particles with
short stopping distances will be able to follow the fluid streamlines, while those with
longer stopping distances will not adapt to the change in the flow direction, but will
continue their paths until their stopping distances are reached. Thus, if these particles
are at distances from the wall that are shorter than their stopping distances, then they
will deposit on the airway bends.
As was noted before, the particle relaxation time characterises the particle re-
sponse time, but the particle behaviour is also affected by the airflow conditions. For
example, consider a particle with a high relaxation time, τ , moving through a narrow
90 ◦ pipe bend with velocity u . We are interested to know if the particle is likely to
impact the pipe walls. The likelihood of particle impaction depends, in addition to
the relaxation time, on the velocity u and pipe diameter. If the pipe has a wider di-
ameter, the likelihood for impaction decreases. To account for the flow environment
surrounding the particle, the ratio of the particle relaxation time to a characteristic
fluid flow time scale is defined as the particle Stokes number. That is,

=

τ u f
d c

St p

=

(6.27)

where u f is the fluid low velocity and d c is the characteristic dimension of the geom-
etry. For small Stokes numbers ( St 1), the particle relaxation time is sufficiently

small compared with the time scale of the flow, and therefore the particles act as gas
tracers in the flow field. For larger Stokes numbers ( St 1), particles will separate

from any curved streamlines and continue in their original direction.

6.4.4 Non-spherical Particles

For non-spherical particles such as fibres, typically a shape factor correction is used
to modify the Stokes drag law as a first approximation. Thus,

3 πμ ( u f u p ) d ve k F D = − (6.28)

where d ve is the diameter of a sphere having the same volume as the non-spherical
particle. That is,

6

π Volume 1 / 3

d ve = (6.29)

and k is a correction factor (Fig. 6.10 ).
For a cluster of n spheres, d e n 1 / 3 d . For tightly packed clusters, k 1.25.

For a chain of equal size spheres moving perpendicularly to the line of the chain, k
depends on the number of spheres in the chain. For example chains of two, four and

= ≤