Environmental Engineering Reference
In-Depth Information
Equality of the inside and outside terms for any weighting function
W
ensures
that the local balance equations are respected throughout the area, therefore also
ensuring the global balance. If equality is checked for a limited number of functions,
the average of local balances is also ensured.
Several types of boundary conditions were implicitly introduced in the equations
above. In pollutant transport, we mainly meet the following conditions:
-
waterproof condition
that results in a boundary without any impositions
(neither in flows nor in concentration);
-
imposed concentration
, especially in the case of a boundary in connection with
a river or lake at constant concentration (or considered as such);
- the case of
imposed flow
may be more difficult to determine because the flow
can come from either advective or convective phenomena, or from the combination
of both. Imposed flow analysis is complex because the effects of advective and
convective phenomena are difficult to separate.
13.5. Numerical modeling of transport by advection
The first transport mechanism of miscible pollutant in an aquifer is generally
advection. Equations governing advection are reduced to a law of particle flow:
vCu
=
[13.27]
and to an equation of mass conservation involving the storage law. If degradation
and storage in other phases are neglected, whichever they are, the equation of
conservation becomes:
T
(
)
∇
Cu
.
+ =
C
0
[13.28]
where:
T
T
Cu
(.
∇+∇ +=
)( ).
Cu
C
0
[13.29]
For an established flow of an incompressible fluid (usually the case of saturated
underground flows) the first term disappears. Thus, a hyperbolic differential
equation is obtained:
T
Cu C
(
∇+ =
).
0
[13.30]
