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concentration and tailing response of observed P effluent concentrations.
They further modified their model by considering first-order irreversible
precipitation described by:
Qc
c
=α
and first-order irreversible chemical immobilization via physical adsorption
described by:
QS
S
=α
where α
c
and α
s
(h
-1
) are the rate coefficients for precipitation and chemi-
cal immobilization, respectively, and Q is the rate of solute consumption
(sink). Incorporating an irreversible sink for chemical immobilization or pre-
cipitation into the convective-dispersive transport equation (CDE) provided
significantly better agreement between the observed data and the model-
predicted curve (Figure 5.3).
5.1 Two-Site Models
One of the earliest multireaction models is the two-site model proposed by
Selim, Davidson, and Mansell (1976). This model was developed to describe
observed batch results, which showed rapid initial retention reactions fol-
lowed by slower retention reactions. The model was also developed to
describe the excessive tailing of breakthrough curves (BTCs) obtained from
pulse inputs in miscible displacement experiments. The two-site model is
based on several simplifying assumptions. First, it is assumed that a frac-
tion of the total sites (referred to as type I sites) reacts rapidly with the solute
in soil solution. In contrast, we assume that type II sites are highly kinetic
in nature and react slowly with the soil solution. The retention reactions
for both types of sites were based on the nonlinear (or nth order) reversible
kinetic approach and may be expressed as:
∂
∂
Θ
ρ
S
t
1
=
n
−
k
C
k S
(5.1)
1
2
1
∂
∂
Θ
ρ
S
t
2
=
m
−
k
C
k S
(5.2)
3
4
2
=+
SSS
(5.3)
T
1
2
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