Agriculture Reference
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where solute distribution in the sphere is not considered uniform (Rao et al.,
1980a). Moreover, solute diffusion into the aggregates can be governed by
Fick's second law, which may be expressed in spherical coordinates as:
im
2
im
im
∂
C t
t
()
∂
∂
C
r
2
∂
C
r
=
D
+
(8.5)
e
∂
2
r
∂
where
D
e
(=
D
o
τ
2
) is an effective molecular diffusion coefficient,
D
o
is molecu-
lar diffusion in water, τ
2
is a tortuosity factor (<1), and
r
is the radial coordi-
nate in a sphere of diameter
a
. Here,
D
e
was assumed to be independent of
concentration within the aggregate
C
im
. Average concentration in the sphere
can be calculated using:
a
3
__
im
∫
3
im
Ct
()
=
rC rtdr
(,)
(8.6)
a
0
As a result, Rao et al. (1980a, 1980b) derived an approximate expression for
α assuming spherical aggregates as:
im
α
D
Θ
e
α=
*( )
t
(8.7)
2
a
where the parameter α* is estimated based on the aggregate size
a
,
D
o
,
t
, and
F
, the fraction of the mobile to total water content (
F
=
Θ
m
/
Θ). As a result, this
α or α(
t
) in Equation 8.7 is time dependent and approximates the diffusion
process in a sphere. In an earlier attempt to arrive at an expression for an
overall dispersion
D
for nonreactive solute transport in spherical aggregates,
Passioura (1971) approximated the overall
D
for soils composed of spherical
aggregates as:
22
(1
−
Fav
D
)
m
DDF
=
+
(8.8)
15
e
with the constraint
(1
−
FDL
av
)
e
>
0.3
(8.9)
22
where
L
is solute transport length. As evidenced by Equation 8.8, when phys-
ical nonequilibrium is dominant, the overall
D
increases with increasing
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