Agriculture Reference
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versus distance from which the diffusion coefficient may be estimated.
Several investigators have employed this method in the laboratory to study
the effect of diffusion on solute transport in different soils (Kemper, 1986;
Oscarson et al., 1992). To estimate the diffusion coefficient, the governing
solute diffusion equation for nonreactive solutes must be used:
2
∂
∂
C
t
∂
∂
C
z
(3.18)
=D
2
CC at
=
t
=
0,
z
<
0
(3.19)
o
CC at
=
t
=
0,
z
<
0
(3.20)
i
where
C
o
and
C
i
are the initial concentrations for blocks 1 and 2, respec-
tively. Analytical solution of the diffusion equation is available and can be
expressed as:
CztC
(,)
=+ −
0.5(
CC erfcz
)
(/ 4)
Dt
(3.21)
i
o
i
and
erfc
is the complmentary error function given by:
∞
2
∫
2
−
r
erfc
()
=
e
d
(3.22)
π
Equation 3.18 is result of the incorporation of Fick's law of diffusion with
the continuity equation of Equation 3.12 for the
z
dimension only. Equation
3.18 is commonly known as the diffusion equation in one dimension and is
the subject of several topics and journal articles. Solutions of Equation 3.18
for various initial conditions are available in numerous sources, including
Crank (1956), Carslaw and Jaeger (1959), and Ozisik (1968).
In Figure 3.3, the concentration distributions are shown for different times
after joining the two blocks. In the example it is assumed that
C
o
= 1,
C
i
=
0 mg L
-1
, and
D
= 1 cm
2
d
-1
. As expected as the time
t
for diffusion increases
solute spreading from the block of high concentration increases. Similar
results can be obtained for increased values of the diffusion coefficient
D
.
3.2.3 Dispersion
Dispersion is a highly significant transport mechanism that is unique to
porous media. The term dispersion is sometimes referred to as mechanical or
hydrodynamic dispersion and includes all solute-spreading mechanisms that
are not attributed to molecular diffusion. The mechanical or hydrodynamic
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