Environmental Engineering Reference
In-Depth Information
Solution
5.5.3  Combined Sorption and Decay
In some cases, both sorption and first-order decay pro-
cesses occur simultaneously. Assuming that sorption is
described by the linear sorption isotherm, Equation
(5.49), and that the sorbed mass decays as a first-order
process, described by
(a) From the data given, M = 10 kg, H = 1 m, V =
0.5 m/day, n = 0.2, D L = 1 m 2 /day, D T = 0.1 m 2 /day,
and λ = 0.01 day −1 . neglecting decay, the con-
taminant concentration downstream of the spill is
given by
c
t
s
= −
λ
c
(5.63)
M
Hnt D D
(
x Vt
D t
)
2
y
D t
2
c x y t
* ( ,
, )
=
exp
4
4
4
π
L
T
L
T
where c s is the sorbed mass per unit volume of the
porous medium, the mass flux per unit volume of
groundwater into the aqueous phase due to desorption,
S 1
and the maximum concentration, at x = Vt , is
given by
, is given by
M
Hnt D D
*
c
( ) =
t
c
c
λβ
c
n
max
β
λ
c
n
β
aq
s
aq
aq
1 = −
S
=
(5.64)
L
T
m
n
t
n
t
which yields
where c aq is the aqueous concentration of the contami-
nant. In applications to flow in porous media, the con-
taminant concentration, c , used in the advection-diffusion
equation, is equal to the aqueous concentration, c aq ,
hence
10
12 6
.
c
*
( )
t
=
=
kg/m
3
max
t
4
π
( )( . )
1 0 2
t
( )( . )
1 0 1
12 600
,
=
mg/L
t
c
aq =
c
(5.65)
Accounting for first-order decay, the maximum con-
centration, c max ( t ), is given by
and Equation (5.64) can be expressed as
β
c
t
λ β
12 600
,
*
λ
t
0 01
.
t
S
1 = −
n c
(5.66)
c
( )
t
=
c
( )
t e
=
e
m
max
max
t
n
Hence, the maximum concentrations at 1, 10, 100,
and 1000 days are as follows:
In addition to the mass flux into the aqueous phase
due to desorption, there is the additional mass flux, S 2 ,
being removed from the groundwater due to decay of
the dissolved contaminant, where
*
t
c
( )
t
c max ( t )
max
(d)
(mg/L)
(mg/L)
S
2 = −λ
c
(5.67)
1
12,600
12,450
10
1260
1140
The total rate at which mass is added to the ground-
water, S m , is equal to the mass flux into the groundwater
due to desorption plus the mass flux due to first-order
decay of the dissolved contaminant; therefore,
100
126
46.5
1000
12.6
0.0005
(b) As time increases, the decay effect becomes more
pronounced. For example, after 100 days, the
maximum concentration is 126 mg/L without decay
compared with 46.5 mg/L with decay. The aqueous
concentrations calculated should be compared with
the aqueous solubility of the contaminant. If the
concentration calculated exceeds the solubility of
the contaminant in water, not all of the spilled con-
taminant will dissolve, and the initial concentration
at the spill location can be taken as equal to the
solubility.
1
2
S
=
S
+
S
m
m
m
β
c
t
λ β
= −
n c
λ
c
(5.68)
n
β
c
t
β λ
= −
1
+
c
n
n
Substituting this fate model into the advection-
diffusion equation, Equation (5.17), and simplifying
yields
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