Biomedical Engineering Reference
In-Depth Information
The reaction itself may be performed in two different ways: first, in the whole
volume of the reaction chamber; second, on a functionalized surface located on the
wall of the reaction chamber. The first type is called homogeneous reaction, the
second heterogeneous reaction.
Thus, we will consider successively the reactions kinetics coupled with advec-
tion-diffusion phenomena for homogeneous or heterogeneous situations.
7.4.1  Homogeneous Reactions
7.4.1.1 Governing Equations
Let us consider a second-order reaction of the type
A nB mC
+ ®
occurring in a fluid volume where the reactants A and B are transported by a flow of
velocity u . If we recall the advection-diffusion equation (Chapter 5), and notice that
there is now sink-source term for concentration, the governing equations are [11]
c
A
+ Ñ =
u c
D c
D
-
k c c
A
A
A
A B
t
c
B
+ Ñ =
u c
D c
D
-
nk c c
(7.52)
B
B
B
A B
t
c
C
+ Ñ =
u c
D c mk c c
D
+
C
C
C
A B
t
where D A , D B and D C are the diffusion coefficients of species A , B , and C , and k
the reaction rate. In (7.52), we have adopted the concentration notations c A = [ A ],
c B = [ B ], and c C = [ C ]. Remark that the sink-source term has the characteristic form
of a second-order reaction k [ A ][ B ]. The advection-diffusion equations are in this
case nonlinear due to the nature of the sink-source term. Moreover, the two first
equations in [ A ]and [ B ] are strongly coupled via their sink term. The third equation
for [ C ] is only weakly coupled to the two other. The solution of the system is not
easy and requires a numerical approach.
Typically, there are two main cases of problems. Note t C the characteristic time
of the reaction and t M the mixing time—which depends on dynamic fluid motion
or only diffusion. Define a nondimensional number by
τ
τ
C
M
Da
=
(7.53)
Da is called the Dammköhler number.
For a purely diffusive situation, the diffusion mixing time t M is of the order of
2
L
D
τ »
M
 
 
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