Environmental Engineering Reference
In-Depth Information
reference length,
L
, which characterizes the dimension
of the space in which the contaminant is moving. The
concentration,
c
, coordinates,
x
i
, time,
t
, and velocity
components,
V
i
, can be normalized relative to these ref-
erence quantities to yield the following nondimensional
variables:
advective flux
diffusive flux
VC
DC L
VL
D
Pe
=
=
=
(3.24)
i
/
i
i
Large values of Pei
i
indicate that advective flux domi-
nates in the
x
i
direction, and small values of Pei
i
indicates that diffusive flux dominates in the
x
i
direction.
kL/V.
This nondimensional group represents the
ratio of the advection time scale (
L
/
V
) to the
decay time scale (1/
k
) and is called the
Damköhler
number
, denoted by Da. Therefore, the Dam-
köhler number is defined by
c
C
c
*
=
(3.18)
x
L
i
*
=
(3.19)
x
i
t
L V
*
=
(3.20)
t
/
advection time scale
decay time scale
L V
k
/
/
kL
V
Da
=
=
=
(3.25)
1
V
V
*
=
i
(3.21)
V
i
Large values of Da indicate that that advection time
scale is much longer than the decay time scale and
so accounting for decay is important, and con-
versely, small values of Da indicate that the decay
time scale is much longer than the advection time
scale and so decay is relatively unimportant.
where the asterisk indicates that the variable is nondi-
mensional. In cases where the contaminant of interest is
not conservative, the source term,
S
m
, in the advection-
diffusion equation (Eq. 3.17) is nonzero. In the special
case where the nonconservative contaminant exhibits
first-order decay, the source term,
S
m
, can be expressed as
using the definition of Pei
i
(=
VL
/
D
i
) and Da (=
kL
/
V
),
Equation (3.23) can be written as
(3.22)
S
m
= −
kc
3
3
∂
∂
c
t
*
∂
∂
c
x
*
∂
∂
2 *
c
x
∑
∑
(
)
−
1
+
(
)
where
k
is the decay constant. Substituting Equations
(3.18-3.21) into Equation (3.17), taking
S
m
= −
kc
, and
simplifying yields the following nondimensional form of
the advection-diffusion equation
*
*
+
V
=
Pe
Da
c
(3.26)
i
i
*
*
*2
i
i
i
=
1
i
=
1
According to Equation (3.26), when Pei
i
>> 1, the diffu-
sion term can be neglected (i.e., advection dominates),
and when Pe
i
<< 1 diffusion is an influential process that
should be taken into account. Similarly, when Da >> 1,
decay should be taken into account, and when Da << 1,
decay can be neglected. When both diffusion and decay
are present, the ratio of Da to 1/Pei
i
gives a measure of
the relative importance of decay compared with diffu-
sion, and this ratio is given by Da·Pei,
i
, where
3
3
−
1
∂
∂
c
t
*
∂
∂
c
x
*
VL
D
∂
∂
2 *
c
x
kL
V
+
∑
∑
*
c
*
+
V
=
(3.23)
i
*
*
*2
i
i
i
i
=
1
i
=
1
The utility of this nondimensional representation is that
all the terms involving only nondimensional variables
are on the order of unity (i.e., “1”), since each of the non-
dimensional variables have been normalized by a refer-
ence quantity that is characteristic of the ambient
environment. Consequently, the only terms whose mag-
nitudes are not necessarily on the order of unity are the
diffusion terms and the decay term, whose magnitudes
are determined by the magnitudes of the nondimen-
sional groups,
VL
/
D
i
and
kL
/
V
, respectively. The physical
meaning of these nondimensional groups are as follows:
2
kL
V
VL
D
kL
D
Da Pe
⋅
=
⋅
=
(3.27)
i
i
i
Hence, when
kL
2
/
D
i
>> 1, decay is a much more impor-
tant process than diffusion, and conversely. when
kL
2
/
D
i
<< 1, diffusion is a much more important process
than decay. In practical terms, this means that when
kL
2
/
D
i
>> 1, diffusion can be neglected when decay is
taken into account.
It is apparent that the Péclet number (Pe) and the
Damköhler number (Da) provide basic measures of the
relative importance of advection, diffusion, and decay
processes incorporated in the advection-diffusion equa-
VL/D
i
.
This nondimensional group represents the
ratio of the advective flux (
VC
) to the diffusive
flux (
D
i
C
/
L
) and is called the
Péclet number
. The
Péclet number is commonly denoted by Pei,
i
, where
i
represents the coordinate direction of diffusion.
Hence,
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