Agriculture Reference
In-Depth Information
8.5 A Modified Two-Region Approach
Due to the uncertainty of obtaining independent measurement for the frac-
tion of sites
f
, Selim and Ma (1995) reexamined the original assumption of
the mobile-immobile approach within the scope of the second-order formu-
lation described above. Figure 8.17 presents a multiple processes based on a
combined physical and chemical nonequilibrium on the concepts of mobile-
immobile model and multireaction models. Because there is no practical
approach for separating the chemical reactions in the dynamic and stagnant
flow regions, it is assumed that the same rate coefficients apply to both soil
regions (Ma and Selim, 1997).
In the modified approach, they considered the dynamic and stagnant
soil regions in the soil as a continuum, and connected to one another
(Figure 8.10). Solute retention may occur concurrently in the dynamic and
stagnant regions until equilibrium conditions are attained or all vacant sites
for a soil aggregate become occupied (filled). They proposed that the rates of
retention reactions in the mobile and immobile phases are a function of the
total vacant sites in the soil. Specifically, this modified approach does not
distinguish between the fraction of sites associated with the dynamic region
and that of the stagnant region. That is, the amount retained from the mobile
phase, for example, affects the total number of vacant sites for retention of
solutes in the immobile water phase, and vice versa. In fact, the fraction of
sites
f
has been shown to be highly affected by experimental conditions,
such as particle size, water flux, solute concentration, and species considered
(van Genuchten and Wierenga, 1977; Nkedi-Kizza et al., 1983). Selim and Ma
(1995) also assumed that the second-order approach accounts for two revers-
ible kinetic reactions and one irreversible reaction. Specifically,
S
e
and
S
k
are
associated with reversible and
S
i
with irreversible reactions. According to
the second-order rate law, the rate of reaction is not only a function of solute
concentration in solution but also of the number of available retention sites
on matrix surfaces. As the sites become filled or occupied by the retained
solute, the number of vacant or unfilled sites, which we denote as φ (μg per
gram soil), approaches zero. In the mean time, the amount of solutes retained
by the soil matrix (
S
) approaches the total capacity or maximum sorption
sites
S
max
.
Incorporating the modified concept into the mobile-immobile approach,
the transport convection-dispersion equation with reactions in the dynamic
soil region can be rewritten as:
∂
m
+ρ
∂
∂
m
2
m
−
νΘ
∂
∂
m
C
t
S
t
∂
∂
C
x
C
x
m
=
Θ
m
D
mm
−α
(
mim
−
)
Θ
CC
(8.57)
∂
2
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