Agriculture Reference
In-Depth Information
solute continuity equation for rectangular coordinates as (omitting the sub-
script i ):
∂χ
∂Θ ρ
(C+S)
t
=
=
div
J
(3.11)
t
The above equation is the general solute transport formulation dealing
with the total amount of solute present in the soil system. Equation 3.11 does
not include rates of production or removal of solutes from the soil, however.
To achieve this, we introduce the term Q to represent a sink or a source term
which accounts for the rate of solute removal (or addition) irreversibly from
a unit volume of a bulk soil (Μg cm -3 h -1 ). Incorporation of Q into Equation
3.11 yields:
J
y
∂Θ ρ
(C+S)
t
J
x
J
z
y
=
x
+
+
z
Q
(3.12)
This irreversible term Q can also be considered as a rate of volatilization or
a root uptake term representing the rate of extraction ( Q positive) of a solute
from the bulk soil or the rate of exudation of a solute ( Q negative). Moreover,
in the following sections, we will restrict our analysis to one-dimensional
flow in the z-direction where the flux J z is dominant.
Equation 3.12 is universally accepted as the continuity equation for solutes
in the x , y , and z or in cartesian coordinates where the concentration C is
presented as C ( x , y , z ). Analogous formulations can be derived for cylindrical
and spherical coordinate systems as shown in Figure  3.2. Here the concen-
tration C is presented as C ( r , ψ, z ) and C (ρ, ψ, Φ) for cylindrical and spherical
Spherical Coordinates
Cylindrical Coordinates
( r , ψ, z )
(ρ, ψ, φ)
r
r
ψ
φ
z
ρ
r
ψ
Angle ψ
FIGURE 3.2
Spherical and cylindrical coordinate systems.
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