Environmental Engineering Reference
In-Depth Information
where
r
2
is the mean square of the distance from the origin. Integrating twice by
parts and using the normalization condition (4.30), we transform the right-hand
side of the equation into
Z
1
Z
1
2
r
4
dr
1
r
@
r
2
dr
@
@
D
4
π
r
2
(
rW
)
D
3
D
4
π
r
(
rW
)
@
0
0
Z
1
r
2
dr
D
6
D
W
4
π
D
6
D
.
0
The resulting equation is
d r
2
6
Ddt
. Since at zero time the particle is located at
the origin, the solution of this equation has the form
D
r
2
D
6
Dt
.
(4.32)
Because the motion in different directions is independent and has a random char-
acter, it follows from this that
x
2
D
y
2
D
z
2
D
2
Dt
.
(4.33)
The solution of (4.30) can be obtained from the normal distribution (1.77), which
is appropriate for this process. Diffusion consists of random displacements of a
particle, and the result of many collisions of this particle with its neighbors fits the
general concept of the normal distribution. In the spherically symmetric case we
have
W
(
r
,
t
)
D
w
(
x
,
t
)
w
(
y
,
t
)
w
(
z
,
t
),
Δ
D
˝
x
2
˛
D
and substituting
2
Dt
into (1.77), we obtain
Dt
)
1/2
exp
x
2
4
Dt
w
(
x
,
t
)
D
(4
π
for each
w
function. This yields
Dt
)
3/2
exp
.
r
2
4
Dt
W
(
r
,
t
)
D
(4
π
(4.34)
4.2.3
The Einstein Relation
If a particle is subjected to an external field while traveling in a vacuum, it is uni-
formly accelerated. If this particle travels in a gas, collisions with gas particles create
a frictional force, and the mean velocity of the particle in the gas is established both
by the external field and by the collisions with other particles. The proportionality
coefficient for the relationship between the mean velocity
w
of a particle and the
