Environmental Engineering Reference
In-Depth Information
t = t 1
t = t 2
t = t 3
t = t 4
t = t 5
t = t 6
Figure 3.18. Two-dimensional diffusion from a finite source.
where V is the mean velocity, which occurs in the x
direction, and k is the first-order decay constant. The
steady-state concentration resulting from a constant-
flux line source is obtained by taking the limit of Equa-
tion (3.121) as t → ∞, which gives
τ
( )
m d
τ
t
c x y t
( ,
, )
=
(
)
4
π
t
L D D
x
D t
τ
0
x
y
(3.119)
2
y
D t
2
exp
)
(
(
)
4
τ
4
τ
x
y
where the transient source is located at x = 0, y = 0. In
the case of a distributed transient source, m x y t
M
L D D
Vx
D
(
)
c x y
( ,
)
=
exp
K
2
β
(3.122)
( , , ) , the
resulting concentration distribution, c ( x , y , t ), is given by
0
2
2
2
π
x
x
y
where M is the constant mass flux (MT −1 ), K 0 is the
modified Bessel function of the second kind of order
zero, and β 2 is defined as
(
)
m
ξ η
,
,
t d d dt
ξ η
t
x
y
2
2
c x y t
( ,
, )
=
(
)
4
π
t
τ
L D D
0
x
y
1
1
x
y
(3.120)
2
2
(
)
(
)
x
ξ
y
D t
η
exp
)
(
(
)
4
D t
τ
4
τ
x
y
2
2
2
(
D x D y V D
+
)(
+
4
D D k
)
y
x
y
x
y
(3.123)
β 2
=
4
D D
x
y
3.3.2.2  Continuous  Line  Source.  The concentration
distribution resulting from the continuous release of
nonconservative mass from a line source (in the z direc-
tion) in a flowing environment can be derived from
Equation (3.119) as
The Bessel function, K 0 , is explained in Appendix D.2,
and tabulated values of this function can be found there
also. The Bessel functions are built in to commonly used
electronic spreadsheets.
In cases where longitudinal diffusion can be neglected
(i.e., Pe >> 1), the steady state concentration distribution
given by Equation (3.122) becomes
( )
τ τ
m d
t
c x y t
( ,
, )
=
(
)
4
π
t
τ
L D D
0
x
y
(
)
2
2
x Vt
D t
y
D t
(
)
exp
)
)
k t
τ
(
(
4
τ
4
τ
x
y
M
Vy
D x
2
kx
V
c x y
( ,
)
=
exp
(3.124)
(3.121)
4
L
4
π
xVD
y
y
 
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