Fundamentals of Particle Dynamics - Computational Fluid and Particle Dynamics in the Human Respiratory System - page 158

Fig. 6.15 Variations of RMS
particle vertical displacement
with time for a density ratio
of ρ p /ρ f 2000 and

different diameters

=

6.5.2 Eulerian Diffusion Models

The Eulerian diffusion model assumes a dilute suspension of nanoparticles that is
treated as a chemical component or species with the use of a diffusion-convection
equation. The diffusion of nanoparticles is governed by Fick's law:

dC
dx

=−

J D (6.40)

where C is the concentration, J is the flux, and D is the diffusion coefficient given
by

τk b T
m

k b TC c
3 πμd

D = = (6.41)

where m is the mass of the spherical particle. Equation (6.41) was first obtained by
Einstein (1905) from a thermodynamics analysis. It is seen that the particle diffusivity
decreases as particle diameter increases. Furthermore, the diffusivity is independent
of particle density. Typical values of diffusivity for particles in the size range of
10 nm to 10 m under normal temperature conditions in air are listed in Table 6.5 .

It is seen that the diffusivity decreases sharply as particle diameter increases.

μ

Table 6.5 Particle mass
diffusivity for different
particle diameters

d ( μ m) D (cm 2 /s)

10 − 2 5 . 24 × 10 − 4

10 − 1 6 . 82 × 10 − 6

2 . 74 × 10 − 7

1

2 . 38 × 10 − 8

10