Environmental Engineering Reference
In-Depth Information
the total concentration of
i
per m
3
. If transport is in all three directions, we can
generalize the equation to the following:
¯
∇·
D
∇[
]
−
U
·
∇
[A]
−
∂
[A]
tot
∂t
A
r
A
=
,
(6.203)
where
∇=
∂/∂x
+
∂/∂y
+
∂/∂z
is called the “del operator.” Using the
local equilib-
rium assumption
(LEA) between the solid- and liquid-phase concentrations,
W
A
=
K
sw
[A], and if
r
A
=
0, we obtain
D
∂
2
[
A
]
U
∂
]
∂x
=
[
A
∂
]
∂t
,
[
A
−
R
F
(6.204)
∂x
2
where
R
F
= ε + ρ
b
K
SW
is the
retardation factor
as defined earlier.
ρ
b
=
(
1
− ε
)
ρ
s
is
the soil bulk density.
In a plug-flow reactor, if the initial concentration in the fluid is [A]
0
and at time
=
t
0, a step increase in concentration [A]
s
is applied at the inlet, the following
initial and boundary conditions apply to the advection-dispersion equation in the
x
-direction:
[
A
]
(x
,0
)
=[
A
]
0
for
x
≥
0,
[
A
]
(
0,
t)
=[
A
]
0
for
t
≥
0,
(6.205)
[
A
]
(
∞
,0
)
=[
A
]
0
for
t
≥
0.
The solution is
erfc
x
exp
Ux
D
erfc
x
,
[
A
]−[
A
]
0
1
2
−
(
U/R
F
)
t
2
√
Dt/R
F
(
U/R
F
)
t
2
√
Dt/R
F
+
A
0
]
=
+
[
A
]
s
−[
(6.206)
where erfc is the complementary error function (see Appendix 8).
It is instructive to consider the simplifications of the advective-dispersion equation
for specific cases:
(i) If dispersion (diffusion) in much larger than advection, that is,
D
∂
2
[
A
]
∂x
2
U
∂
[A]
∂x
and
r
A
=
0, we have
D
∂
2
[
A
]
R
F
∂
[
]
∂t
,
A
=
(6.207)
∂x
2
which is the well-known Fick's equation, for which, with the boundary
conditions given earlier, the solution is
erf
x
.
[
A
]−[
A
]
0
A
0
]
=
1
−
√
4
Dt
(6.208)
[
A
]
s
−[
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