Environmental Engineering Reference
In-Depth Information
Using the notation of the 2D case in (
3.2
), the corresponding equation without
sources or sinks is simply
ð Þ
D
y
D
t þ j
yþ
j
y
D
x
D
t
. Equation
3.2
is simplified in two steps. First one divides through the product of all spatial
extensions and the finite time step
D
V ¼ j
xþ
j
x
D
x
D
y
D
z
D
t
and obtains:
j
Ax
ðx þ
D
2
; y; z; tÞj
Ax
ðx
D
2
A
ð
x
;
y
;
z
;
t
þ
D
t
Þ
A
ð
x
;
y
;
z
;
t
Þ
D
t
; y; z; tÞ
¼
D
x
j
Ay
ðx; y þ
D
2
; z; tÞj
Ay
ðx; y
D
2
; z; tÞ
þ
D
y
j
Az
ðx; y; z þ
D
2
; tÞj
Az
ðx; y; z
D
2
; tÞ
þ
þ Q
D
z
(3.3)
The second step is the transition from finite steps to infinitesimal steps,
D
x ! @x;
D
y ! @y;
D
z ! @z;
D
t ! @t
, according to the differential calculus in
order to get the continuity equation in the formulation given in (
3.1
). Using the
vector notation, the same equation can be expressed briefly as:
@A
@t
¼r
j
A
þ Q
(3.4)
Thus the aim to express the flux as a function of the concentration is simple for
advective transport. In order to achieve this for diffusive/dispersive flux, an empiri-
cal relationship has to be introduced, e.g. Fick's Law.
3.2 Fick's Law and Generalizations
3.2.1 Diffusion
There is a natural tendency in natural systems to level out concentration differences.
The process which causes this tendency is called diffusion. When in a system there
is a high concentration at one place and a smaller concentration at another place,
there will be a net diffusive flux of the component from the location with higher
concentration to the one with lower concentration. In the molecular scale, diffusion
is a random motion of molecules in all directions. In systems without concentration
differences all random walks together maintain the same concentration level. But if
concentration is not constant, there is a net flux in one direction, from the high to the
low concentrations.
A system with initial concentration differences will finally reach a constant
concentration level if no other processes are present. Other processes can stabilise
the concentration gradient. Then the diffusive flux may be balanced by processes
