Agriculture Reference
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controlled heterogeneous reaction and the other physically controlled reac-
tion (Rubin, 1983). The chemically controlled heterogeneous reaction was
considered to be governed according to the second-order approach. In the
meantime, the physically controlled reaction is chosen to be described by
diffusion or mass transfer based on the mobile-immobile concept (Coats and
Smith, 1964; van Genuchten and Wierenga, 1976). A comparison between the
mobile-immobile concept and that of the two-site approach indicates that
the dynamic and stagnant regions for solute retention are analogous to sites
1 and 2 of the two-site concept. Nkedi-Kizza et al. (1984) presented a detailed
discussion on the equivalence of the mobile-immobile and the equilibrium-
kinetic two-site models.
An important feature of the second-order approach is that an adsorp-
tion maximum (or capacity) is assumed. This maximum represents the
total number of adsorption sites per unit mass or volume of the soil
matrix. It is also considered an intrinsic property of an individual soil
and is thus assumed constant. In previous two-region models, a finite
number of sites has not been specified and thus an adsorption maximum
is never attained (e.g., van Genuchten and Wierenga, 1976; Rao et al., 1979;
Brusseau et al., 1989; van Genuchten and Wagenet, 1989). Specifically, in
Equations 8.3 and 8.4, the dimensionless term
f
denotes the ratio or frac-
tion of dynamic or active sites to the maximum or total adsorption sites
S
max
(μg/g soil). In addition, the terms (Μ
S
m
/Μ
t
) and (Μ
S
im
/Μ
t
) represent
the rates of reversible heavy metal reactions between C in soil solution
and that present on matrix surfaces in the mobile (or dynamic) and the
immobile regions, respectively. Moreover, irreversible reactions of heavy
metals were incorporated in this model as may be seen by the inclusion
of the sink terms
Q
m
and
Q
im
(μg cm
-3
h
-1
) in Equations 8.3 and 8.4, respec-
tively. It is assumed that irreversible retention or removal from solution
will occur separately in the mobile and immobile water regions. However,
the governing mechanism of retention for each region was assumed to
follow first-order-type reactions. Specifically, it is assumed that irrevers-
ible retention for the mobile and immobile regions are considered in this
transport model as follows:
Q
m
= Θ
m
k
s
C
m
(8.21)
Q
im
= Θ
im
k
s
C
im
(8.22)
where
k
s
is an irreversible rate coefficient (h
-1
) common to both regions. These
formulations for the sink terms do not occur elsewhere in the literature and
were first proposed by Selim and Amacher (1988). Since soil matrix surfaces
may behave in a separate manner to heavy metal retention, it is conceivable
that the rate of irreversible reactions in the mobile region is characteristi-
cally different from that for the mobile region. One way to achieve this is to
distinguish between the rate coefficient (
k
s
) controlling the reaction for the
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