Biomedical Engineering Reference
In-Depth Information
Substituting
(2.51)
in
(2.54)
results in:
r
0
u
2
48
D
vc
vx
:
J
convection
¼
(2.55)
The area-averaged axial flux now can be expressed as:
D
vc
vx
:
r
0
u
2
48
D
r
0
u
2
48
D
D
vc
vc
vx
¼
J
¼
vx
þ
(2.56)
According to Fick's law, the term in the square bracket of the above equation can be called the effective
diffusion coefficient or, more accurately, the dispersion coefficient:
r
0
u
2
D
¼ D þ
48
D
:
(2.57)
The specie conservation equation can be now formulated as:
vc
vt
þ
v
2
c
vx
2
:
u
vc
D
vx
¼
(2.58)
The above partial differential equation can be solved analyti
ca
lly. For instance, if the initial
condition of the specie concentration is a pulse
cðx; t ¼
, where
d
(
x
) is the Dirac
function, the transient one-dimensional solution of the average concentration is:
0
Þ¼C
0
dðxÞ
exp
"
#
:
2
C
0
4
pD
t
ðx utÞ
c
ð
x
;
t
Þz
p
(2.59)
4
D
t
Figure 2.10
shows the typical concentration distribution of the one-dimensional dispersion at
different time instances.
A few years after Taylor's publication, Aris provided a firmer theoretical framework for this theory
by using a moment analysis
[12]
. He also generalized the problem to handle time-periodic flows
[13]
.
Following the works of Taylor and Aris, there are several other contributions to the theory of deriving
FIGURE 2.10
One-dimensional dispersion, typical concentration distribution at different time instances.
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