Environmental Engineering Reference
In-Depth Information
transport equation (3.19) by restriction to the one-dimensional steady state situa-
tion, when diffusion due to bioturbation, additional source- and sink-terms and
inter-phase exchanges are assumed to be irrelevant. Furthermore, both remaining
relevant terms are divided by bulk density
r
b
, which they have in common. For the
spatial coordinate we use
z
here in order to distinguish the vertical direction.
Concerning sulphate, Berner (
1964
) takes into account the processes of diffu-
sion, fluid flow and degradation. The resulting differential equation for the 1-
dimensional steady state is:
2
D
@
@z
2
c
S
u
@
@z
c
S
l
S
c
org;s
¼
0
(9.15)
where
c
S
denotes sulphate concentration.
D
is the relevant diffusion coefficient,
u
denotes fluid flow and
l
S
the kinetic degradation coefficient. Equation (
9.15
)is
a differential equation for sulphate concentration
c
S
. If there is no compaction and
no connection to the ambient groundwater regime, one can set
v ¼ u ¼ w
. The
general solution of the given system for both components is:
v
z
c
org;s
¼ c
org;s
0
exp
D
z
v
2
c
org;s
0
v
2
v
v
z
c
S
¼ C
0
þC
1
exp
exp
(9.16)
þDl
with integration constants
C
0
and
C
1
. The particular solution given by Berner
(
1964
) results for
C
1
¼
0:
v
z
c
org;s
¼ c
org;s
0
exp
þc
S1
v
z
c
S
¼ c
S
0
c
S1
ð
Þ
exp
(9.17)
where the subscript '0' denotes the concentration at the sediment-water interface,
while the subscript '
' represents the concentration in the deep sediment. Berner
(
1964
) presented the solution (
9.17
) to describe sulphate reduction in maritime
sediments and later used it to describe a part of the nitrogen cycle (Berner
1971
). If
the nitrogen cycle is concerned,
c
S
in the equations given above has to be replaced
by the concentration of total ammonia.
As soon as the sulphate disappears, the solution of (
9.16
) produces negative
concentrations. Boudreau and Westrich (
1984
) suggest the use of the Monod or
Michaelis-Menton kinetics (see Chaps. 7 and 9 above) to describe sulphate reduc-
tion and organic carbon content. In the here used notation one obtains:
1
w
@
kc
org;s
c
S
K
S
þc
S
¼
@z
c
org;s
0
@z
D
@
@
@z
c
S
u
@
fkc
org;s
c
S
K
S
þc
S
¼
@z
c
S
0
(9.18)
