Page 66 - Air pollution and greenhouse gases from basic concepts to engineering applications for air emission control

40 2 Basic Properties of Gases

Example 2.6: Air mean free path
For air, the molecule diameter is approximately 3:7 10 10 m. Determine the mean
free path of air at sea level at 293 K. What is the value for air mean free path at an
elevation of 2,000 m by assuming same temperature?
Solution
3
Air density at the sea level is q ¼ 1:21 kg/m . At the elevation of 2,000 m, air
0
3
density becomes q 2000 ¼ 0:95 kg/m . The corresponding mean free paths are cal-
culated using Eq. (2.51).
At sea level,
0:02884 1
k 0 ¼ p
ffiffiffi 23 10 2
2p 6:0221 10 3:7 10ð Þ 1:21
¼ 0:066 10 6 m ¼ 0:066 lm

At 2,000 m above the sea level,

0:02884 1
k 2000 ¼ p ffiffiffi
23 10 2 0:95
2p 6:0221 10 3:7 10ð Þ
¼ 0:083 10 6 m ¼ 0:083 lm

Conversion between the mean free paths under different conditions was given by
Allen and Raabe [1], Cited by [14]

2
k p 0 T 0 þ T s
T
¼ ð2:52Þ
k 0 p T 0 T þ T s
where the parameters with subscript of 0 is for the standard condition and those
without subscripts are for any other conditions. Examples of values of mean free
path for air are given in Table 2.1.

2.1.8 Number of Collisions with Wall/Surface

By applying the kinetic theory, one can also calculate the number of collisions on
the walls of a container. The quantitative analysis of the collision on the wall of the
container is important for the study of the kinetic molecular theory of transport
properties such as diffusion and viscosity.
Again, consider the scenario in Fig. 2.1, a cubic container with a wall area, A,
and assume elastic impact between the molecules and the wall. During a small time
interval of Dt; the distance that the molecule with the moving speed of c x travel is
Dx ¼ c x Dt in the +x-direction if they do not collide with the wall. In another word,
the molecules will collide with the wall if they are within a distance Dx ¼ c x Dt from
the wall.