Environmental Engineering Reference
In-Depth Information
value for
Pe
then represents a reduced importance of diffusion, which in the profiles
is reflected by steeper gradients.
The corresponding M-file
'analtrans_s2'
is included in the accompanying
software.
The lower most curve in Fig.
5.4
shows the profile for advection and decay only,
i.e. for no diffusion, calculated as solution of the simple first order ordinary
differential equation
v
@
c
0or
@c
@x
¼
v
c
@x
l
c ¼
(5.23)
The solution is:
v
l
cðxÞ¼c
in
exp
ð
xÞ
(5.24)
or in dimensionless form (for dimensionless concentration, with dimensionless
parameter
Da
1
and dimensionless independent variable
x
)
c=c
in
¼
exp
ðDa
1
xÞ
(5.25)
. As could be
expected for increasing values of
Pe
, the function of (
5.25
) is approached as
asymptote. But the convergence is quite slow. For the value of
Da
1
¼
Formally one can represent the no-diffusion case by
Pe ¼1
1, the
value of the P
´
clet number should be distinctly above 10 in order to obtain a
good correspondence between the concentration distribution and the asymptote.
For a good correspondence close to the outlet (
x
close to
L
),
Pe
should be 100 or
higher.
Figure
5.5
shows graphs of functions according to (
5.21
) for a fixed value of the
P´clet number
Pe ¼
ohler number
Da
1
. The
constant value of
Pe
guarantees a constant relation between diffusion and advec-
tion, while with changing values of
Da
1
decay or degradation processes change
1 and selected values of the first Damk
€