Page 68 - Air pollution and greenhouse gases from basic concepts to engineering applications for air emission control

42 2 Basic Properties of Gases

Substitute Eq. (2.7) into this equation and we can get

r ffiffiffiffiffiffiffiffiffi
kT
J x ¼ C N ð2:60Þ
2pm
Considering the relationship between gas density and the molecule number
concentration described in Eq. (2.50), the total number of collisions per unit time
per unit area can also be expressed in terms of gas density as
r ffiffiffiffiffiffiffiffiffiffiffi
kT
J x ¼ q : ð2:61Þ
2pm 2
2.1.9 Diffusivity of Gases

We can derive the diffusivity of a single gas by applying the preceding analysis of
molecule collision on a surface to an imaginary cubic container that is formed by a
distance of mean free path, 2k. We can apply finite element analysis from x kÞ to
ð
ð x þ kÞ. The concentration at x is C N . Assuming a constant gas concentration
gradient of dC N from x kÞ to x þ kÞ; the concentrations at x þ kÞ and x kÞ
ð
ð
ð
ð
dx

dC N dC N
are C N þ k and C N k , respectively.
dx dx
Considerations of symmetry lead us to assert that the average number of particles
traveling in a given direction ð x; y or zÞ will be one-sixth of the total, and
thus the mean rate at which molecules crosses a plane is N c=6 per unit area in unit
time. N is the total number of molecules in the container. This differs slightly from
the exact result, although it is suitable for some simplified argument.
Statistically, 1/6 of the molecules at x þ kÞ will move along x direction.
ð
According to the definition of mean free path, these molecules leaving the plane
ð x þ kÞ along x direction will reach plane x. Therefore, the number of molecules
leaving plane x þ kÞ per second per unit area is
ð
1 dC N
J x ¼ C N þ k c
6 dx
Similarly, the number of molecules per second per unit area leaving plane
ð x kÞ to plane x is

1 dC N
J !x ¼ C N k c
6 dx