Biomedical Engineering Reference
In-Depth Information
where
L
is the characteristic dimension of the microsystem and
D
the order of
magnitude of the diffusion coefficients of the reactants. After substitution, one
obtains
D
τ
C
Da
=
2
L
If
Da
is large, the reaction time is much larger than the mixing time. The
concentrations in
[A]
and
[B]
can then be considered uniform in the reacting
volume, and system (7.52) collapses to
·
¶
c
A
» -
k c c
A B
¶
t
¶
c
B
(7.54)
» -
nk c c
A B
¶
t
¶
c
C
»
mk c c
A B
¶
t
This system is considerably easier to solve since it does not requires the knowl-
edge of the velocity field and of the diffusion process.
If
Da
is small, the picture is much more complicated. There are reaction
fronts that form and diffuse progressively before obtaining a homogeneous
final state [12]. Numerical treatment is usually required for such systems.
·
7.4.1.2 Reaction-Diffusion at a Front Separating Two Reactants
Start from the same second order reaction
A B C
+ ®
and suppose that it takes place in a volume at rest (no convective transport), as
sketched in Figures 7.28 and 7.29.
In the case of a one-dimensional space (Figure 7.29), we have indicated the
solution for the concentration alone in Chapter 4. This solution was an error (
erf
)
function and the concentration spreads proportionally to the square root of time.
Now we add a second-order reaction. The equations governing this type of reaction
diffusion in one dimension geometry are
2
¶
c
¶
c
A
A
=
D
-
kc c
A
A B
2
¶
t
¶
x
(7.55)
2
¶
c
¶
c
B
B
=
D
-
kc c
B
A B
2
¶
t
¶
x
