Environmental Engineering Reference
In-Depth Information
particularly in the context of heat conduction, and ana-
lytical solutions for a wide variety of initial and bound-
ary conditions are available. using these solutions
together with the transformation given by Equation
(3.29) provides many useful results to describe the
mixing process in water environments when the mean
flow,
V
i
, is steady and spatially uniform.
2
2
2
∂
∂
c
t
∂
∂ ′
c
∂
∂ ′
c
∂
∂ ′
c
=
D
+
D
+
D
(3.40)
x
y
z
x
2
y
2
z
2
where
x
′,
y
′, and
z
′ are the coordinates relative to the
centroidal axis. Multiplying Equation (3.40) by
x
′
2
and
integrating over
x
′ between ±∞ yields
∂
∂
c
t
x dx D
∂
∂
′
2
c
∞
∞
∫
∫
∫
′
2
′ =
x
′
2
dx
′ +
x
x
2
3.2.2.2 Nonconservative Tracers with First-Order
Decay.
In the case of nonconservative tracers undergo-
ing first-order decay, the source flux,
S
m
is given by
−∞
−∞
2
2
∂
∂
′
c
∂
∂
′
c
∞
∞
∫
′
′
+
′
′
D
x
2
dx D
x
2
dx
y
z
2
2
y
z
−∞
−∞
(3.41)
S
m
= −
kc
(3.36)
To evaluate these integrals, assume that the tracer con-
centrations are equal to zero at
x
′ = ±∞, which requires
the following boundary conditions
and hence the the governing advection-diffusion equa-
tion is given by
3
3
2
∂
∂
c
t
∂
∂
c
x
∂
∂
c
x
∂
∂
′
=
c
x
∑
∑
+
V
=
D
−
kc
(3.37)
0
at
x
′ = ±∞
(3.42)
i
i
i
2
i
i
i
i
=
1
i
=
1
c
=
0
at
x
i
′ = ±∞
(3.43)
If the actual concentration,
c
, is expressed in terms of a
modified concentration,
c
′, where
Applying these boundary conditions to Equation (3.41)
and integrating by parts yields
c
′ =
ce
kt
.
(3.38)
∂
∂
∞
∞
∫
∫
then substituting Equation (3.38) into Equation (3.37)
yields the following governing differential equation in
terms of the modified concentration,
x c dx
′
2
′ =
2
D
c dx
′
x
t
−∞
−∞
2
2
∂
∂ ′
∂
∂ ′
∞
∞
∫
∫
+
D
x c dx D
′
2
′ +
x
′
2
c dx
′
y
z
y
2
z
2
−∞
−∞
3
3
2
(3.44)
∂ ′
∂
c
t
∂ ′
∂
c
x
∂ ′
∂
c
x
∑
∑
+
V
=
D
(3.39)
i
i
2
i
i
i
=
1
i
=
1
Integrating Equation (3.44) with respect to
y
′ from −∞
to +∞, applying Equations (3.42) and (3.43), and simpli-
fying yields
The remarkable result here is that the term accounting
for first-order decay has now disappeared, such that
when expressed in terms of a modified concentration,
c
′, the tracer behaves as a conservative substance. In
practical terms, this means that when dealing with a
tracer undergoing first-order decay, the tracer can be
treated as a conservative substance to determine the
concentration distribution,
c
′, and then the calculated
concentration distribution multiplied by
e
−
kt
to deter-
mine the actual concentration distribution.
∂
∂
∞
∞
∞
∞
∫
∫
∫
∫
2
x c dx dy
′
′
′ =
2
D
c dx dy
′
′
x
t
−∞
−∞
−∞
−∞
∂
∂
′
2
∞
∞
∫
∫
2
+
D
x c dx d
′
′
y
′
z
z
2
−∞
−∞
(3.45)
Integrating Equation (3.45) with respect to
z
′ from −∞
to +∞, applying Equations (3.42) and (3.43), and simpli-
fying yields
3.2.3 Moment Property of the Diffusion Equation
∂
∂
∞
∞
∞
∞
∞
∞
∫
∫
∫
∫
∫
∫
2
x c dx dy dz
′
′
′
′ =
2
D
c dx dy dz
′
′
′
x
t
The moment property of the diffusion equation is the
basis for estimating diffusion coefficients from field
measurements using conservative tracers, such as dyes.
The diffusion of a tracer cloud relative to its centroidal
axes is given by Equation (3.35) as
−∞
−∞
−∞
−∞
−∞
−∞
(3.46)
The integral term on the right-hand side of Equation
(3.46) is equal to the total mass,
M
, of the tracer, where
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