Biomedical Engineering Reference
In-Depth Information
or, in a Cartesian coordinate system,
é
2
2
2
ù
¶
c
¶
c
¶
c
¶
c
¶
c
¶
c
¶
c
+
u
+
v
+
w
=
D
+
+
ê
ú
(6.8)
2
2
2
¶
t
¶
x
¶
y
¶
z
¶
x
¶
y
¶
z
ë
û
Suppose for an instant that the diffusion coefficient
D
= 0. In such a case
Dc
Dt
= 0
(6.9)
so that the concentration
c
remains the same along a trajectory (Figure 6.4). This
propriety is valid for short times where diffusion process has not had time to smear
out the concentration. It is well known that very laminar flows, usually associated
with biotechnological devices, are very unfavorable to diffusion, “short” times are
often rather long, and mixing devices promoting mixing have been developed to
enhance diffusion [1].
Now, let us come back to the general case where the diffusion coefficient is not
zero. Let us define a concentration norm in the whole domain occupied by the fluid
by the mathematical function
Q =
ò
c dxdydz
Mathematically, this function represents a measure of the concentration. In the
absence of source or sink of substance, it is possible to derive [2]
¶Q
= -
2
ò
D c dxdydz
(
)
Ñ
¶
t
showing that Q always decreases with time. Thus the concentration smears out with
time (Figure 6.5). If
D
were negative, which does not happen, the particles would
concentrate and there would be antidiffusion, which, of course, does not exist.
6.2.2 Source Terms
If there are concentration source or sink terms, (6.7) becomes
Figure 6.4
Sketches of mass transport in the absence of diffusion. (a) Near a solid wall and
(b) in the bulk of the flow.
