Environmental Engineering Reference
In-Depth Information
tion. The utility of these nondimensional parameters
also carry over into interpreting and simplifying analytic
solutions of the advection-diffusion equation.
cases where the tracer distribution is symmetric relative
to the
x
i
-axes. The derivatives in the (
x
i
,
t
) space are
related to the derivatives in the (
′
′
x
i
,
t
) space by the fol-
lowing relations derived from the chain rule,
3.2.2 Transformation to the Diffusion Equation
3
∂ ′
∂
x
x
∂ ⋅
∂
( )
∂ ⋅
∂ ′
( )
∑
j
=
(3.30)
Several practical analytical solutions of the advection-
diffusion equation (Eq. 3.17) can be obtained by trans-
forming this equation into a simple diffusion equation
(without advection) for which many analytical solutions
have been developed in other fields of engineering. The
most commonly used transformations are described
below.
x
x
i
j
i
j
=
1
3
∂ ⋅
∂
( )
∂ ⋅
∂ ′
( )
∂ ′
∂
x
t
+
∂ ⋅
∂
( )
∑
j
=
(3.31)
t
x
t
j
j
1
=
where (·) represents any scalar function of
x
i
and
t
.
Combining Equations (3.29-3.31) yields
3.2.2.1 Conservative Tracers.
In cases where the total
tracer mass remains constant, the contaminant is called
conservative
, and
S
m
= 0 in Equation (3.17). In these
cases, the transport of the contaminant is described by
∂ ⋅
∂
( )
=
∂ ⋅
∂
′
( )
(3.32)
x
x
i
i
3
∂ ⋅
∂
( )
∂ ⋅
∂ ′
+
∂ ⋅
( )
( )
∑
= −
V
(3.33)
j
t
x
∂
t
3
3
2
∂
∂
c
t
∂
∂
c
x
∂
∂
c
j
∑
∑
j
1
=
+
V
=
D
(3.28)
i
i
x
2
i
i
i
=
1
i
=
1
Substituting Equations (3.32) and (3.33) into the
advection-diffusion equation, Equation (3.28), yields
the transformed equation in (
′
When the diffusing tracer is being advected at a con-
stant mean velocity,
V
i
, Equation (3.28) can be simplified
by changing the independent variables from
x
i
and
t
to
′
x
i
,
t
) space,
3
2
∂
∂
c
t
∂
∂ ′
c
∑
x
i
and
t
, where the new variables
x
i
are defined by
′
=
D
(3.34)
i
x
2
j
i
=
1
x
′ =
x V t
−
(3.29)
i
i
i
which is more commonly written in the Cartesian form
′
x
i
coordinate measures locations relative to
the mean position of the tracer particles, given by
V
i
t
.
This transformation of coordinate system is illustrated
in Figure 3.3, where the point
O
is the mean position of
the tracer particles at time zero,
O
' is the mean position
of the tracer particles at time
t
, and P is a fixed point in
space. This coordinate transformation will lead to a sim-
plified description of the tracer distribution in many
where the
∂
∂
c
t
∂
∂ ′
2
c
∂
∂ ′
2
c
∂
∂ ′
2
c
=
D
+
D
+
D
(3.35)
x
y
z
2
2
2
x
y
z
Equation (3.35) is generally referred to as the
diffusion
equation
. This equation has been studied in detail
in many engineering and scientific applications,
x'
3
P
(x
1
,
x
2
,
x
3
)
x'
3
P
O
'
x
3
(x
1
-V
1
t
,
x
2
-V
2
t
,
x
3
-V
3
t)
x'
2
(
V
1
t
,
V
2
t
,
V
3
t
)
O
'
x'
2
x'
1
O
(0,0,0)
x
2
x'
1
(0,0,0)
x
1
Figure 3.3.
Change of coordinate system.
Search WWH ::
Custom Search