Agriculture Reference
In-Depth Information
If the movement is solely by diffusion, then from Fick's first law,
D
∂C
∂x
F
=−
(
2
.
3
)
t
where
D
is the solute diffusion coefficient. If the soil solution is also moving,
then the solute will also be carried by mass flow, and
=−
D
∗
∂C
∂x
+
vC
L
F
(
2
.
4
)
where
v
is the water flux in the
x
direction,
C
L
is the concentration of the solute
in the soil solution and
D
∗
is the dispersion coefficient which differs from the
diffusion coefficient because the movement of the solution itself causes some
dispersion of the solute.
The continuity equation for combined diffusion and mass flow is obtained by
combining Equations (2.2) and (2.4):
D
∗
∂C
∂x
−
vC
L
∂C
∂t
=
∂
∂x
(
2
.
5
)
t
This is an expression of Fick's second law.
The concentration of the solute may also change as a result of processes occur-
ring within the volume
δx
. This is allowed for by adding a term
R(C, x, t)
to
Equation (2.2) to give
D
∗
∂C
∂x
−
vC
L
∂C
∂t
=
∂
∂x
t
+
R(C, x, t)
(
2
.
6
)
Note that
R
can be positive or negative. Generally conditions in submerged
soils are strongly affected by the vegetation present, which acts as the main
conduit for gas transfer between the soil and overlying atmosphere. The effects
of vegetation can be allowed for in the
R
term, suitably modified with depth in
the soil and time. Time-dependent reactions, microbially mediated reactions and
other reactions adding or removing the solute can be represented with additional
R
terms.
The equivalent equation for movement normal to a cylinder, such as a plant
root, is
∂r
r
D
∗
∂C
∂r
−
vC
L
∂C
∂t
=
1
r
∂
t
+
R(C, x, t)
(
2
.
7
)
where
r
is the radial distance from the axis of the cylinder.
In simple cases these equations can be solved analytically but more often
numerical solutions are necessary.
