Environmental Engineering Reference
In-Depth Information
Fig. 3.4 Illustration for the
derivation of 1D transport
equation in streams
i-1
deposition
inflow
i+1
tributaries
pollution
outflow
This is the so-called
mass transport equation
, which is valid for biogeochemical
species of all kinds. The mathematical characterization is as follows: it is of second
order in space, as there appear second derivatives in
x
,
y
and
z
but not third derivatives.
It is of first order in time. It is parabolic concerning the mathematical classification of
partial differential equations. In case of constant coefficients it is a linear equation.
The simplifications, performed for the 1D equation, can be made for the multi-
dimensional situation as well. For the equation
y
@
c
@t
¼r
ð
y
D
rc
Þ y
v
rc þ q
(3.20)
the generalized condition is that the flow field is divergence-free, or in mathemati-
cal formulation:
0.
4
Formulation (
3.20
) is then obtained from formulation
r
v
¼
(
3.19
) due to:
r
v
c ¼
v
rc þ c r
v
ð
Þ¼
v
rc
4
For an incompressible fluid the condition
r
v
¼
0means that there are no internal sources or sinks
for the fluid (see chap. 12).
