Agriculture Reference
In-Depth Information
or other nonspecifically adsorbed (e.g., Na, Ca, Cl, and NO
3
) or specifically
adsorbed (e.g., PO
4
, AsO
4
, and transition metals) species. Vacant or unfilled
refers to sites vacant or unfilled by the specific solute species of interest. The
process of occupying a vacant site by a given solute species actually is one of
replacement or exchange of one species for another. However, the simplify-
ing assumption on which this model is based is that the filling of sites by a
particular solute species need not consider the corresponding replacement
of species already occupying the sites. The Langmuir-type approach con-
sidered here (Equations 6.11 to 6.14) is a specialized case of an ion-exchange
formulation (ElPrince and Sposito, 1981). Alternatively, the competitive
Langmuir approach may be used if the identities of the replaced solute spe-
cies are known.
6.1.1 Transport Model
Incorporation of the second-order reactions into the classical (convection-
dispersion) transport equation yields (Brenner, 1962; Nielsen, van Genuchten,
and Biggar, 1986):
2
Θ
∂
∂
C
t
+ρ
∂
∂
+
∂
∂
C
x
∂
∂
C
x
S
t
S
t
=Θ
∂
∂
1
2
D
−
q
−
Q
(6.15)
2
where
D
is the hydrodynamic dispersion coefficient (cm
2
h
-1
),
q
is Darcy's
water flux (cm h
-1
), and
x
is depth (cm). Here the term
Q
is a sink represent-
ing the rate of irreversible reactive chemical reactions by direct removal from
the soil solution (mg h
-1
cm
-3
). In this model, the sink term was expressed in
terms of a first-order irreversible reaction for reductive sorption, precipita-
tion, or internal diffusion:
Q=
Θ
k
C
(6.16)
s
where
k
s
is the rate constant for the irreversible reaction (h
-1
). Equation 6.16 is
similar to that for a diffusion-controlled precipitation reaction if one assumes
that the equilibrium concentration for precipitation is negligible and that
k
s
is
related to the diffusion coefficient.
For convenience, we define the dimensionless variables:
X=
x
L
(6.17)
T=
qt
L
Θ
(6.18)
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