Biomedical Engineering Reference
In-Depth Information
Equation (2.79) is a recurrence relation for the pressure at the nodes versus
the widths of the side channels; this relation has been obtained by considering
the velocities in the axial main channel only (with the exception of the last side
channel). The pressures at the nodes are now directly calculated using the side
channels.
Step 3: Pressure at the Nodes (From Side Channels)
The pressure at a node is directly related to the outside pressure by the relation
(
)
8
η
L
+
9
a Q
s
i
o
( )
P
=
P
+
g
λ
(2.80)
i
o
i
3
ba
i
and the solution is obtained by equating the values of the pressure at the nodes from
(2.80) and (2.79)
(
)
(
)
L
+
9
a
L
+
9
a
L
s
i
s
N
( )
(
)
(
0
g
λ
=
g
λ
+
g
λ
(
N i N i
-
)(
- +
1)
(2.81)
i
N
3
3
3
0
a
a
a
i
N
For any given value of the width of the last side channel
a
N
, the right-hand side
of the previous relation is known, and we find the implicit relation for the
a
i
of the
type
(
)
L
+
9
a
s
i
( )
g
λ
=
B
(2.82)
i
i
3
a
i
where
B
i
is the value of the right-hand side (which depends only of the width of the
last side channel). The inverse solution is then reduced to an implicit solution of an
analytical function, and this is tractable. Depending on the value of the
λ
i
, one has
to solve either one of these two third-order polynomials.
For
λ
i
> 4.45,
2
3
B a
-
9
a
-
L
=
0
(2.83)
i
i
s
i
3
For
λ
i
< 4.45,
(
)
(
)
(
)
2
3
2
2
B b
-
9
a
-
L
+
18
b a
-
b L
2
+
9
b a
=
L b
(2.84)
i
s
s
i
s
i
i
It can be shown that these two polynomials have one real root and two imagi-
nary roots, so that the real root is the desired solution. It is interesting to note that
the fluid viscosity does not appear in (2.82), and neither does the value of the total
flow rate (
Q
i
), so that the calculated dimensions of the microsystem will satisfy the
constraint of delivering the same flow rate at the outlets for different fluids and dif-
ferent inlet flow rates. That shows a generality in the system. Another advantage
of this method is that it is straightforward to change the angle between the main
and side channels by changing the pressure drop expression at an intersection and
replacing (2.77) by
