Environmental Engineering Reference
In-Depth Information
Therefore, the concentration at the source after 10
minutes is 567 mg/L.
(d) The steady-state concentration at the source is
obtained by taking the limit as
t
→ ∞ in Equation
(3.105), which gives
2
2 10 0 20 1 63
exp
( .
0 20 100
2 10
)(
)
c
(
100 600 =
,
)
(
1 1 63
+
.
)
(
)( .
)( .
)
(
)
100
+
( .
0 20 600 1 63
)(
)( .
)
−
erf
1
2
(
10 600
)(
)
( .
0 20 100
2 10
)(
)
(
)
−
exp
1 1 63
−
.
M
AV
M
AV
(
)
c
(
)
=
( ,
0 ∞ =
)
erf
∞
Γ
Γ
100
−
( .
0 20 600 1 63
)(
)( .
)
−
erf
1
2
10 0 20 1 63
2
(
10 600
)(
)
c
( ,
0
∞ =
)
=
0 612
.
kg/m
3
=
612
mg/L
(
)( .
)( .
)
=
0 234
.
kg/m
3
=
234
mg/L
Therefore, the steady-state concentration at the
source is 612 mg/L.
Therefore, the concentration 100 m downstream of
the source after 10 minutes is 234 mg/L.
3.3.2 Diffusion in Two Dimensions
(b) The steady-state concentration at
x
= 100 m is given
by Equation (3.106) as
In cases where a tracer is completely mixed over one
dimension, further mixing can only occur in the other
two dimensions. Such a case is illustrated in Figure 3.17,
where at any (
x
,
y
) location, complete mixing has already
occurred in the
z
direction, and further mixing can only
occur in the
x
and
y
dimensions. The fundamental dif-
fusion equation describing two-dimensional diffusion is
given by
M
AV
Vx
D
x
c x
( ,
∞ =
)
exp
(
1
∓
Γ
)
Γ
2
2
10 0 20 1 63
( .
0 20 100
2 10
)(
)
c
(
100
,
∞
)
=
)
exp
(
1 1 63
−
.
)
(
)( .
)( .
(
)
3
=
0 325
.
kg/m
=
325
mg/L
∂
∂
c
t
∂
∂
2
c
∂
∂
2
c
Therefore, the steady-state concentration 100 m
downstream of the source is 325 mg/L.
(c) The concentration at the source at
t
= 10 minutes =
600 seconds is given by Equation (3.105) as
=
D
+
D
(3.109)
x
y
x
2
y
2
with initial and boundary conditions given by
M
AV
Vt
D t
x
Γ
M
L
c
( , )
0
t
=
erf
(3.110)
c x y
( ,
, )
0 =
δ
( ,
x y
)
Γ
2
2
10 0 20 1 63
( .
0 20 600 1 63
2
)(
)( .
)
c
(
±∞ ± ∞
,
, )
t
= 0
(3.111)
c
( ,
0 600
)
=
erf
(
)( .
)( .
)
(
10 600
)(
)
where
M
is the mass of tracer injected instantaneously
at
t
= 0,
L
is the length over which the mass is uniformly
=
0 567
.
kg/m
3
=
567
mg/L
typical (
x,y
) location
z
top boundary
L
tracer uniformly mixed over depth
bottom boundary
y
x
Figure 3.17.
Two-dimensional diffusion.
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