Agriculture Reference
In-Depth Information
soil profile. Solution for a set of linear equations such as the Thomas algo-
rithm for tridiagonal matrix-vector equations can be used (Pinder and Gray,
1977). The newly calculated C values can be used subsequently as input val-
ues in the solution for solute retention. The solution of these equations thus
provides the amount of sorbed phases due to the irreversible and reversible
reactions at the same time ( j + 1) and all incremental distances along the soil
profile.
3.9.2 Transport versus Batch
In most experimental investigations dealing with the fate of chemicals in
soils or porous media transport is absent. For example, methods for quan-
tifying adsorption isotherm for a range of concentrations and over time are
rarely conducted under flow conditions. A list of such methods where net
transport of a solute is absent is discussed in Chapter 1.
When transport is absent, the problem is simplified considerably and
becomes that of an initial-value-problem where space is ignored. Simple ana-
lytical solutions are available for the linear case (Equation 3.56). However,
for Freundlich (Equation), Langmuir (Equation), or other nonlinear formula-
tions to describe the retention reaction processes, iterations must be used
to solve the initial-value-problem. The solution becomes that representing
no-flow batch conditions where the retention is to be described over time.
The major exception between the above formulations and that excluding
transport is in the way the equilibrium sorbed phase concentration ( S e ) is
calculated. For any given time step j , the amount in soil solution C and that in
the sorbed phase S e are in local equilibrium and their amounts are related by
the K f value according to the nonlinear Freundlich equation (3.42). Therefore,
the total amount in the solution and sorbed phases ( S e ) is
HC S
=Θ +ρ =Θ +ρ
C
b ,
t
0
K
C
(3.74)
e
f
As a result, one can calculate, from C and S e , the amount H at any time step
j . Now to estimate these variables at time step j + 1, subsequent to the calcu-
lations of all other variables (i.e., S 1 , S 2 , etc.), one can calculate a new value
for H and partition such a value between C and S e (based on the Freundlich
equation) using the following expression:
H
K
C
= Θ+ρ
(3.75)
b
1
C
f
which is derived directly from Equation 3.74, which is based on the newly calcu-
lated H for the sum of concentration and equilibrium sorbed phases. Equation
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