Environmental Engineering Reference
In-Depth Information
To simplify things, let us assume the convective velocity is constant on an element.
This hypothesis may subsequently be waived without major difficulty. Let us introduce
discretization [13.32] into weighted residue equations [13.25] and [13.26]. In a steady
state, we get the following pollutant nodal flows:
∫
T
δ
= -
∇
(
) dV
δ
=
δ
[13.34]
v
W
N
C
F
C
I
L
L
L
L
V
with:
diffus.+dispers.
= - D
'
∇
[13.35]
v
NC
L
L
v
advec.
=
NC
u
[13.36]
L
L
The iteration matrix is calculated by deriving nodal flows. We obtain:
∫
T
∫
T
=
(
∇
)
D(
'
∇
)dV
−
u
(
∇
)dV
[13.37]
K
N
N
N
N
K
LK
K
L
L
V
V
This matrix allows us to obtain the concentrations at any node of the mesh using
the following equation:
KC f
+=
0
[13.38]
LK
L
K
The first term in [13.37] is a standard orthotropic diffusion matrix. The second
term is new. It comes from advection and
is not symmetrical.
13.6.2.
One-dimensional case
Let us consider the solution given by this model (see Figure 13.4 - Galerkin's
standard solution) to an academic one-dimensional problem in which the advection
velocity is constant and the concentration imposed is equal to one at one end and
zero at the other end.
