Biomedical Engineering Reference
In-Depth Information
that the side exits close to the needle tip should have larger cross sections than
those in the vicinity of the flow inlet. If all the side exits were to have the same
section, then the flow would exit in the first side branches and not reach the
needle tip.
One solution might be to assign initial cross sections to the side exits, run a
calculation with a numerical software, memorize the values of the flow rates at
each exit, calculate a “distance” to the expected uniform flow solution, find an op-
timization algorithm to change the dimensions of the side exits, and run the process
iteratively until convergence. However, this way of proceeding is long and costly.
Optimization over
N
parameters (the
N
widths of the 2
N
side channels [
a
i
,
i
= 1,
N
]) is very complex.
In such a case, it is preferable to search for an inverse algorithm based on a
lumped model formulation. Using a lumped element model to take care of all the
different microfluidic segments in the needle, and imposing the constraint that the
flow rate at all outlets is the same, a recurrence relation for the pressure at the nodes
can be derived and solved to obtain the desired channel widths.
2.2.9.3
Algorithm
The algorithm has three steps: (1) calculation of the velocities in the central channel
by a recurrence relation starting from the needle tip, (2) establishment of a recur-
rence relation for pressure at the nodes starting from the needle tip, and (3) calcula-
tion of the pressure at each node using the first two steps.
Step 1: Velocities in the Axial (Central) Channel
Let the letters
P
,
Q
,
V
stand for, respectively, the pressure, flow rate, and fluid veloc-
ity. The density of the liquid is
ρ
. Because of the process of fabrication of the needle,
the vertical dimension of the microchannels
b
is the same for all the channels. For
simplicity, it is assumed that the spacing
L
(axial distance) between the side chan-
nels is constant and that the width
a
0
of the main channel is a given constant. Con-
sequently, the length of the side channels is also a constant,
L
s
. A schematic view of
the flow channels is given in Figure 2.35.
Because there are 2
N
exits, the total mass conservation equation can be written
as
Q
=
(2
N Q
)
(2.73)
in
exit
By a recurrence approach, starting from the far end of the needle and progress-
ing to the front end, we obtain
Figure 2.35
Schematicviewofthemainandsecondarychannels.
