Biomedical Engineering Reference
In-Depth Information
Contrary to the diffusion problem, there is no known analytical solution for
the system (7.58). We just mention here that, for very long times, there exists an
approximate solution [13] using the variable
R
= c
A
c
B
.
7.4.1.3 Reaction and Advection-Diffusion in a Microchannel
In biotechnology, chemical reactions are very often performed in microchannels
with parallel flowing buffer fluids (Figure 7.30).
If
y
is the cross direction and
x
the direction of the flow, the two reactants
A
and
B
produce the component
C
by reaction-diffusion in the transverse direction
y
(Figure 7.31).
Suppose a reaction symbolized by
A B C
+ ®
For a steady state flow of reactants, the system of equations for the reaction is
[14, 15]
2
¶
c
¶
c
A
A
U
=
D
-
kc c
A
A B
2
¶
x
¶
y
2
¶
c
¶
c
B
B
U
=
D
-
kc c
(7.60)
B
A B
2
¶
x
¶
y
2
c
c
¶
¶
C
C
U
=
D
+
kc c
C
A B
2
¶
x
¶
y
where, for the diffusion term, the second derivative in
x
has been neglected in front
of that in
y
. We use a change of variables resembling to that of (7.57)
1
2
æ
ö
x
y
D
*
*
x
=
,
y
=
,
D
=
D D
,
L
=
ç
(7.61)
÷
A B
2
L
k c
c
UL
D
è
ø
A
,0
B
,0
Figure 7.30
Reaction of two components flowing in parallel in a microchannel.
