Environmental Engineering Reference
In-Depth Information
Loss to air
Input
Air
Water
uC
w
(
x
)
uC
w
(
x
+
Δ
x
)
Sediment
x
x
+
Δ
x
Loss to sediment
FIGURE 6.16
Material balance on a section of a stream assuming complete mixing across
the width and depth of the stream.
material balance across the volume
WH
Δ
x
:
input
=
output
+
reaction
+
accumulation.
uC
w
(x)WH
=
uC
w
(x
+ Δ
x)WH
+
rWH
Δ
x
+
loss to air
x
d
C
w
+
loss to sediment
+
WH
Δ
d
t
.
At steady state, we have d
C
w
/
d
t
=
0. The rate of loss to air is
K
w
(C
w
−
(C
a
/K
aw
))(W
Δ
x)
. The rate of loss to sediment is
K
s
(C
w
−
(w
i
/K
aw
))(W
Δ
x)
.Ifa
=
first-order rate of reaction is considered,
r
k
r
C
w
. The overall material balance is
C
w
(x
C
w
−
C
w
−
+ Δ
x)
−
C
w
(x)
K
w
H
C
a
K
aw
K
s
H
W
K
sw
u
·
=−
−
−
k
r
C
w
.
(6.77)
Δ
x
Since time,
t
=
x/u
,
Δ
t
= Δ
x/u
, and taking lim
Δ
, we obtain the following
x
→
0
differential equation:
C
w
−
C
w
−
d
C
w
d
t
=−
K
w
H
C
a
K
aw
K
s
H
W
K
sw
−
−
k
r
C
w
.
(6.78)
If both sediment and air concentration remain constant, we can solve the above
equation using the initial condition,
C
w
=
C
0
at
t
=
0, to obtain (Reible, 1998)
+
β
α
C
0
e
−α
t
e
−α
t
)
,
C
w
=
(
1
−
(6.79)
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