Agriculture Reference
In-Depth Information
where
L
,
T
, and
c
0
are the same as in Equations 4.10 through 4.12. The third-
type boundary condition is applied to the upper boundary. However, because
the dispersion coefficient vanishes at
x
=
, the third-type boundary condi-
tion formally reduces to the first-type boundary condition. The governing
equation (4.16) subject to initial and boundary conditions (Equations 4.17
through 4.20) were solved numerically (see Appendix 4B for the detailed
finite-difference scheme).
4.2.1.3 CDE with a Nonlinearly Time-Dependent Dispersivity
Zhou and Selim (2002) developed a fractal model
t
o describe a time-depen-
dent dispersivity in terms of mean travel distance
x
. The fractal model reads:
α=
−
xax
D
fr
1
()
(4.21)
3
2−
, and
D
fr
is the fractal
dimension of the tortuous stream tubes in the media.
D
fr
varies from 1 to 2.
If
D
fr
=
, we recover the time-invariant constant dispersivity. Similarly, if
D
fr
=
, Equation 4.21 reduces to Equation 4.5. Again, we assume molecular
diffusion can be ignored and dispersion coefficient for a nonlinear disper-
sion function is given by
D
fr
where now
a
3
is a constant with dimension
L
D
−
1
D
D
−
1
Dt
()
=α
()
xv
=
ax
vavt
=
(4.22)
fr
fr
fr
3
3
Substituting Equation 4.22 into Equation 4.7 and rearranging yields the
following governing equation:
2
∂
∂
=−
∂
c
t
c
x
∂
∂
c
DD
fr
−
1
v
+
av
t
fr
(4.23)
3
2
∂
x
Equation 4.8 is recovered if we let
D
fr
=
in the above equation. The upper
boundary conditions are
∂
∂
c
x
DD
fr
−
1
vc xt
(,)
=−
vc
av
t
,
x
=
0,
0
< ≤
t
T
(4.24)
fr
0
3
∂
∂
c
x
DD
fr
−
1
vc xt
(,)
=−
av
t
,
x
=
0,
t
>
T
fr
(4.25)
3
The remaining initial and lower boundary conditions are the same as those
for linear dispersivity model (Equations 4.9 and 4.12). The above system was
also solved using the finite-difference method (see Appendix 4C).
Search WWH ::
Custom Search