Environmental Engineering Reference
In-Depth Information
which maintain a permanent out- and inflow. Still the concentration gradient is
accompanied by diffusive flux.
The empirical (first) Fick's Law
1
is a quantification of diffusive flux (physical
unit: mass/(area
time)), here stated for the fluid phase:
¼Drc
j
(3.5)
In words: the diffusive flux is proportional to the negative concentration gradi-
ent. The minus sign guarantees that the direction of the net flux is from high
concentrations to low concentrations. The factor of proportionality is the diffusion
constant or diffusivity
D
with the physical unit [area/time]. Note that here too the
∇
-operator is used. Here it is not working in connection with a vector product,
because the following variable is a scalar.
2
The result of such an operation is
a vector. In three dimensions it can be written as:
0
@
1
A
@c=@x
@c=@y
@c=@z
rc ¼
(3.6)
In general, the diffusivity
D
depends on the fluid and on the transported compo-
nent; it depends on temperature and pressure and on the geochemical environment.
For all substances there is a diffusivity in gases, which differs from the diffusivity in
liquids, and it even depends on the type of gas or liquid. The diffusivity in saltwater
usually is different from the diffusivity in freshwater.
The diffusivity, as defined by (
3.5
), is defined in single phase systems, i.e. in
liquids or in gases, and is a characteristic of the molecules involved, i.e. of the
component and of the medium. For that reason it is common to speak of
D
as
molecular diffusivity. In the following this will be taken into account by writing
D
mol
, while
D
remains the notation for diffusivity in general.
The order of magnitude of
D
mol
in water at common temperatures of 20
Cfor
most chemical components is around 10
9
m
2
/s. In air it is in the range of 10
5
m
2
/s.
For ammonium gas NH
3
for example,
D
mol
10
9
m
2
/s at 20
Cand
in water is 1.46
1
Adolf Eugen Fick (1829-1901), German physiologist; Fick's second Law is valid for mass
conservation in a single phase environment:
2
@x
2
c
@t
c ¼ D
@
@
The formula is obtained when Fick's Law, as given in (
3.5
), is used as replacement for the flux in
(
3.4
).
2
In contrast to a vector a scalar has a single value only. A scalar function has a single value,
depending on space and/or time. Velocity is a typical vector variable, which in a one-dimensional
case is reduced to a scalar variable.
