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.
 
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