Agriculture Reference
In-Depth Information
This term was also expressed in a finite difference (with iteration) as:
b
K
ρ
Θ
f
r
r
R
=
[
]
= +
1
[
b
-1
]
R
Y
(3.66)
ij
,
where
Y
represents the average concentration at time
j
(known) and that at
time
j
+ 1 (unknown) for which solution is being sought, such that
r
[
C
]
2
+
C
ij
,
+
1
ij
,
(3.67)
r
[
Y
]
=
where
r
refers to the iteration step. Iteration is implemented here due to the
nonlinearity of the equilibrium retention reaction (i.e.,
b
≠ 1).
For the kinetic retention equations, the time derivative for
S
1
,
S
2
, and
S
3
were also expressed in their finite-difference forms in a similar manner to
the above equations. Therefore, omitting the error terms and incorporating
iteration, the term associated with
S
1
was expressed as:
n
r
−ρ
[
]
2
+
ρ
∂
∂
C
C
S
t
ij
,
+
1
ij
,
1
r
≅Θ
[(
)]
k
k
S
(3.68)
1
2
1
ij
,
Similarly, the kinetic term associated with
S
2
, when
S
3
is neglected, is
approximated as:
m
r
[
]
2
+
ρ
∂
∂
C
C
S
t
r
ij
,
+
1
ij
,
2
−ρ
≅Θ
(
)
k
k
S
(3.69)
3
4
2
,
ij
Moreover, the irreversible term
Q
was expressed in an implicit-explicit
fashion as:
+
CC
ij
,
+
1
ij
,
Q
≅Θ
(3.70)
k
s
2
The number of iterations for the above calculations must be specified since
no criteria are commonly given for an optimum number of iterations. A con-
venient way is based on mass balance calculations (input versus output) as a
check on the accuracy of the numerical solution.
For each time step (
j
+ 1), the finite difference of the solute transport
equation, after rearrangement and incorporation of the initial and bound-
ary conditions in their finite-difference form, can be represented by a
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