Environmental Engineering Reference
In-Depth Information
Fick's first law describes the diffusive mass flux of chemical species in a
saturated soil medium under chemical concentration gradients as:
d
*
)
(5.5)
J
=
D C
∇
(
−
i
i
i
where,
J
i
d
is the diffusive mass flux of the ith chemical species per unit cross
sectional area of the porous medium (ML-2T-1),
-2
T
-1
),
C
i
is the molar concentra-
tion of the ith chemical species (ML
-3
), and
D
i
*
is the effective diffusion
coefficient of the ith chemical species (L2T-1).
2
T
-1
). The migrational mass flux
of the free ionic species in the soil pore fluid due to the applied electrical
gradient is given by:
(
)
e
(5.6)
J
=∇−
uC E
*
i
i
i
where,
J
i
e
is the migrational mass flux of the ith species (ML-2T-1),
-2
T
-1
), and
u
i
*
is
the effective ionic mobility of the ith species (L
2
T
-1
V
-1
) defined as:
DZF
RT
*
(5.7)
u
=
i
i
*
i
where,
Z
i
is the charge of the ith species (Coulomb), F is Faraday's constant
(96,485 C/mo1 electrons), R is the universal gas constant (8.3144 J/Kmol),
and T is the absolute temperature (K). The other contributing mechanism
to the flux of species is advection by the soil pore fluid. The advective mass
flux of species i relative to the soil particles is:
(
)
+∇−
(
)
J
=
C
J
=
Ck
∇−
hk
E
(5.8)
⎣
⎦
i
i
w
i
h
eo
where,
C
i
is the molar concentration of pore fluid. The total species mass
flux (diffusion, migration, and advection) is given as:
(
)
+∇−
(
)
+
(
)
+∇−
(
)
(5.9)
J
=∇−
D
*
C uC
ECk
⎣
∇−
hK
E
⎦
*
i
i
i
i
i
i
h
eo
Figure 5.1 shows a schematic of mass transport profile of cationic and
anionic species based on the assumption that water advection components
(EO and hydraulic) act from anode to cathode. As observed in figure 5.1,
the advective EO flow enhances transport of cationic species migrating from
anode toward cathode, and retards transport of anionic species migrating
from cathode to anode (Shapiro and Probstein, 1993; Acar et al., 1997).
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