Environmental Engineering Reference
In-Depth Information
velocity
∂
∂
c
x
y
q
d
= −
D
(3.3)
plume at
t
=
t
1
i
plume at t = 0
i
Whereas the Fickian relation given by Equation (3.3)
parameterizes the mixing effect of macroscopic velocity
variations with correlation length scales smaller than
the support scale of the Fickian relationship, tracer mol-
ecules are also advected by larger-scale fluid motions.
The mass flux associated with the larger-scale (advec-
tive) fluid motions is given by
x
(a)
y
q
a
=
V c
(3.4)
i
i
x
where
q
i
a
is the tracer mass flux (ML−2T−1)
−2
T
−1
) due to advec-
tion in the
x
i
direction, and
V
i
is the large-scale fluid
velocity (LT
−1
) in the
x
i
direction. Since tracers are trans-
ported simultaneously by both advection and diffusion,
the total flux of a tracer within a fluid is the sum of the
advective and diffusive fluxes given by
(b)
y
∂
∂
c
x
q
=
q
a
+
q
d
=
V c D
−
(3.5)
i
i
i
i
i
x
where
q
i
is the tracer flux in the
x
i
direction. Equation
(3.5) can also be written in vector form as
(c)
q V
=
c D c
− ∇
(3.6)
Figure 3.1.
Advection and diffusion. (a) uniform advec-
tion + molecular diffusion. (b) uniform advection + (molecu-
lar + turbulent diffusion).
where
q
is the flux vector, and
V
is the large-scale fluid
velocity. The expression of the tracer flux in terms of an
advective and diffusive component must generally be
associated with a length scale,
L
, that is a measure of
the averaging volume used to estimate the advective
velocity,
V
, and the diffusion coefficient,
D
. The main
distinction between advection and diffusion is that
advection is associated with a net movement of the
center of mass of a tracer, whereas diffusion is associ-
ated with spreading about the center of mass.
Consider the finite control volume shown in Figure
3.2, where this control volume is fixed in space and is
within the fluid that is transporting the tracer. In accor-
dance with the law of conservation of mass, the net flux
(MT
−1
) of tracer mass into the control volume must be
equal to the rate of change of tracer mass (MT
−1
) within
the control volume. The law of conservation of mass can
be put in the form
(c) Dispersion + (molecular +
turbulent diffusion).
tensor (L
2
T
−1
),
c
is the tracer concentration (ML
−3
), and
x
j
is the coordinate measure in the
x
j
direction (L).
Although Fick's law was originally developed to describe
molecular diffusion, in most environmental applications
it is used to describe contemporaneous molecular and
turbulent diffusion, such that the diffusion coefficient,
D
ij
, is understood to equal the sum of the molecular
diffusion coefficient,
D
m
, and the turbulent diffusion
coefficient, ε
ij
, such that
D D
ij
=
+
ε
(3.2)
m
ij
Typically, ε
ij
>>
D
m
and hence
D
ij
mostly accounts for
turbulent diffusion (i.e.,
D
ij
≈ ε
ij
). In cases where the
(combined) diffusion coefficient,
D
ij
, varies with direc-
tion, the diffusion process is called
anisotropic
, and in
cases where the diffusion coefficient is independent of
direction, the diffusion process is called
isotropic
. Hence,
for isotropic dispersion,
D
ij
=
D
for all values of
i
and
j
,
in which case Fick's law (Eq. 3.1) is given by
=
∂
∫
∫
∫
S dV
∂
+ ⋅
c dV
q n
dA
(3.7)
m
t
V
V
A
Term A
Term C
Term B
where
V
is the volume of the control volume (L
3
),
c
is
the tracer concentration (ML
−3
),
A
is the surface area of
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