Environmental Engineering Reference
In-Depth Information
This equation can be combined with the expression for
the mass flux given by Equation (3.6) and written in the
expanded form
control surface
c
t
q ( ux vector)
+ ∇⋅
( V
c D c
− ∇ =
)
S
m
(3.12)
dA
control volume
n (unit normal)
which simplifies to
streamline in ambient ow eld
c
t
m
(3.13)
+
V
⋅∇ + ∇⋅
c
c
(
V
)
= ∇ +
D c S
2
Figure 3.2. Control volume in a fluid transporting tracer.
This equation applies to all tracers in all fluids. In the
case of an incompressible fluid, which is typical of the
water environment, conservation of fluid mass requires
that
the control volume (L 2 ), q is the flux vector (Eq. 3.6)
(ML −2 T −1 ), n is the unit outward normal to the control
volume (dimensionless), and S m is the mass flux per unit
volume originating within the control volume (ML−3T−1). −3 T −1 ).
In Equation (3.7), Term A is the rate at which tracer
mass is being added from a source within the control
volume, Term B is the rate of mass accumulation within
the control volume, and Term C is the rate at which
tracer mass is leaving the control volume. Equation
(3.7) can be simplified using the divergence theorem,
which relates a surface integral to a volume integral by
the relation
∇⋅ V 0
(3.14)
and combining Equations (3.13) and (3.14) yields the
following diffusion equation for incompressible fluids
with isotropic diffusion:
c
t
+
V
⋅∇ = ∇ +
c D c S
2
(3.15)
m
In cases where there are no sources or sinks of tracer
mass (i.e., a conservative tracer), S m is zero, and Equa-
tion (3.15) becomes
q n
dA
= ∇⋅
q
dV
(3.8)
A
V
c
t
Combining Equations (3.7) and (3.8) leads to the result
2
+
V
⋅∇ = ∇
c D c
(3.16)
=
(3.9)
S dV
c dV
− ∇⋅ q
dV
m
t
If the diffusion coefficient, D , is anisotropic, the princi-
pal components of the diffusion coefficient can be
written as D i , and the diffusion equation becomes
V
V
V
Since the control volume is fixed in space and time, the
derivative of the volume integral with respect to time is
equal to the volume integral of the derivative with
respect to time, and Equation (3.9) can be written in the
form
3
3
c
t
c
x
2
c
+
V
=
D
+
S
m
(3.17)
i
i
x
2
i
i
i
=
1
i
=
1
c
t
where x i are the principal directions of the diffusion
coefficient tensor. Equation (3.17) is the most com-
monly used relationship describing the mixing of con-
taminants in aquatic environments, and it is known as
the advection-diffusion equation .
+ ∇⋅ −
q
S
dV
=
0
(3.10)
m
V
This equation requires that the integral of the quantity
in parentheses must be zero for any arbitrary control
volume, and this can only be true if the integrand itself
is zero. Following this logic, Equation (3.10) requires
that
3.2.1 Nondimensional Form
A reference concentration, C , such as the background
concentration of the contaminant, can usually be
defined, along with a reference velocity, V , and a
c
t
+ ∇⋅ −
q
S
=
0
(3.11)
m
 
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