Environmental Engineering Reference
In-Depth Information
power law of the length travelled, i.e.,
D L x
L
ʸ
D
(
x
) =
.
(6)
In the above equation D L is the dispersion coefficient at the extraction length ( x
=
L )
and
is the connectivity index associated to the hydrodynamic dispersion.
There are several parameters involved in the model, specifically in Eqs. ( 4 ) and
( 6 ) appear
ʸ
ˆ
, A cs , D L and
ʸ
. The tracer flux for constant fluid density is given by
M F
A cs ˆˁ 0 C
D L L ʸ x ʸ
C
(
x
,
t
)
J
(
x
,
t
) =
(
x
,
t
)
,
(7)
x
where the first term describes advection, and the second one depicts dispersion.
Accordingly, the tracer conservation equation for uniformflow is (Herrera-Hernández
et al. 2013 )
C
(
x
,
t
)
+ ʓ(α)
x α 1
J
(
x
,
t
)
=
S T .
(8)
t
x
=
/
After substituting Eq. ( 7 ) into Eq. ( 8 ), using dimensionless variables, x D
x
L ,
C D =
/
C Max and t D =
C
t
t , and assuming that no source is present, we obtain
x D
M F ʔ
C D
t D + ʓ(α)
t
A cs ˆˁ 0
C D
x D ʓ(α)
D L ʔ
t
C D
x 1 α
D
x 1 α
D
=
0
.
(9)
L α
L α 1
L 2
x D
x D
ʔ
t is an arbitrary reference time and C Max is a reference tracer concentration.
The mathematical model given by Eq. ( 9 ) contains various parameters that can be
grouped as
M F ʔ
U ad = ʓ(α)
L α
t
A cs ˆˁ 0 ,
(10)
D ad = ʓ(α)
D L ʔ
t
.
(11)
L α 1
L 2
Rewriting Eq. ( 9 ) in terms of these parameters, it becomes
x D
C D
C D
C D
U ad x 1 α
D ad x 1 α
t D +
x D
=
0
.
(12)
D
D
x D
x D
 
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