Biomedical Engineering Reference
In-Depth Information
Pi theorem implies that there is a 4 - 3 = 1 nondimensional number that governs the
nondimensional equation and characterizes the phenomenon.
Suppose that we take for references a velocity
U
¥
, a length scale
L
, and a con-
centration
c
0
. Relevant dimensionless variables may be defined as
c
x
y
z
u
v
w
t
*
*
*
*
*
*
*
*
c
=
,
x
=
,
y
=
,
z
=
,
u
=
,
v
=
,
w
=
,
t
=
(6.29)
L
c
L
L
L
U
U
U
0
¥
¥
¥
U
¥
Substitution of (6.29) in (6.28) yields
é
ù
*
*
*
*
2 *
2 *
2 *
¶
c
¶
c
¶
c
¶
c
D
¶
c
¶
c
¶
c
*
*
+
u
+
v
+
w
=
+
+
(6.30)
*
ê
ú
*
*
*
*
*2
*2
*2
U L
¶
t
¶
x
¶
y
¶
z
¶
x
¶
y
¶
z
ë
û
¥
As was expected from Buckingham's theorem, only one dimensionless param-
eter appears in (6.30). This parameter represents the ratio of inertia to diffusion
and is referenced by
=
U L
¥
Pe
(6.31)
D
The Peclet number is a key feature in the problems of dispersion under the
action of diffusion and advection. We shall see in the following sections many ex-
amples where the Peclet number determines the solution of the advection-diffusion
problem.
Note that the Peclet number may be written as a function of the Reynolds
number
U L
ν
¥
P
R S
=
=
(6.32)
e
e
c
ν
D
where
S
c
is the nondimensional Schmitt number.
6.2.7 Concentration Boundary Layer
In Chapter 2, we have seen that the entrance length of microflows in capillary tubes
is very short, because the hydrodynamic boundary layer develops and reaches very
quickly the middle of the tube. It would be wrong to conclude that the same will
happen to mass transfer boundary layer. In fact, the picture looks like that of Figure
6.12 where the hydrodynamic flow is established but not the concentration field.
Figure 6.13 shows the calculated mass transfer boundary layer inside a usual
detection chamber for a buffer fluid carrying DNA strands. The results are obtained
by solving the advection-diffusion equation using a finite difference numerical
scheme. It appears immediately that the vertical distance (1 mm) is too important
because the compounds carried by the buffer flow (DNA strands) are mostly unaf-
fected by the labeled wall and keep flowing through the chamber.
An estimate of the boundary layer thickness may be found by a dimensional
analysis. The starting point is the advection-diffusion equation, assuming a steady
state concentration field, no source terms, and a two-dimensional problem
