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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