Environmental Engineering Reference
In-Depth Information
9.7.2 Transport Prediction
For contaminants that can be partitioned in the transport process in the subsoil system, it
is not uncommon to use the transport relationship given as Equation 2.2 in Chapter 2. For
some situations, such as equilibrium-partitioning processes, this is an adequate method
for predicting the transport and fate of contaminants that can be partitioned. The assump-
tion is generally made that the rate of reactions is independent of the concentration of con-
taminants, i.e., a zero-rate reaction process. However, for many other situations relating to
contaminants that are partitioned during transport, such as nonequilibrium partitioning,
this relationship needs to be knowledgeably applied and perhaps modiied to meet the
conditions of partitioning. The relationship given as Equation 2.2 in Chapter 2 can be writ-
ten in its original expanded form as:
2
∂
∂
c
t
∂
∂
c
x
∂
∂
c
xn
ρ
ρ
∂
∂
c
t
*
,
=
D
−
v
−
(9.6)
L
2
w
where
c
is the concentration of contaminants of concern,
t
is the time,
D
L
is the diffusion
coeficient,
v
is the advective velocity,
x
is the spatial coordinate, ρ is the bulk density of
soil media, ρ
w
is the density of water,
n
is the porosity of soil media, and
c*
is the concen-
tration of contaminants adsorbed by soil fractions (see ordinate in graph shown in Figure
2.15 in Chapter 2). We recall that the adsorption isotherms portrayed in Figures 2.14 and
2.15 are derived from batch equilibrium tests with soil solutions, and we further recall the
discussion in Section 9.5.1 and Figure 9.15 that the distribution coeficient
k
d
obtained from
the adsorption isotherms refers directly to a maximum reactive surface reaction process.
Equation 2.2 is obtained when
c*
is assumed to be equal to
k
d
c
, i.e., if a linear adsorption
isotherm is assumed. Substituting for
c*
in Equation 9.6 gives us:
2
∂
∂
c
t
∂
∂
x
x
∂
∂
c
xn
ρ
ρ
∂
(
kc
x
=
D
−
v
−
d
.
(9.7)
L
2
∂
w
ρ
ρ
Collecting terms and deining R as the
retardation
=+
1
k
, the equation previously
d
n
seen as Equation 2.3 is obtained:
w
2
∂
∂
c
t
∂
∂
c
x
∂
∂
c
x
R
=
D
−
v
.
(9.8)
L
2
If the partition isotherms obtained from actual laboratory tests do not show linearity,
the obvious solution is to use the proper function that describes
c
*, e.g., the Freundlich
or Langmuir (see Figure 2.15) or some other equivalent relationship. In this instance, the
opportunity to properly relect the partitioning process through a compact soil mass
should be taken. Instead of using the nonlinear adsorption isotherm obtained from batch
equilibrium tests on soil solutions, the sorption relationship obtained from column leach-
ing tests through compact soil should be used. This has been discussed in Section 9.5.1 and
illustrated in terms of adsorption curves' differences in Figure 9.15.
Equations of the type shown as Equation 2.2 or Equation 9.7 have been described as
nonreactive transport relationships. This description has been applied to such equations
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