Biomedical Engineering Reference
In-Depth Information
For complex hydraulic circuits, including many parts having different functions,
modeling with finite elements numerical software can quickly become impossible
because too many nodes are required and the capacity of the computer is exceeded.
Besides, the description of the geometry can be quite difficult. In such a case, the
circuit may be decomposed in connecting parts, with each part—or branch—
corresponding to a precise function, such as connecting channels, microchambers,
micropumps, and valves. Such a model is called a
lumped model
(Figure 2.31) [34,
35].
The model requires defining nodes {
i
= 1,
N
} and branches {
j
= 1,
M
}, the nodes
being the extremity of the different branches. As we deal with a flow field calcula-
tion, the unknowns are the average velocities
U
j
and the pressure at the nodes
P
i
.
Thus, the vector of unknowns is of the dimension
N
+
M
U
U
ì
ü
1
ï
ï
ï
ï
M
A
=
í
ý
P
P
ï
1
ï
ï
ï
î
þ
N
A first set of equation is constituted by the mass conservation equations at each
node
i
of the net. At a node
i
, the equation for the conservation of the flow rate is
å
U S
=
0
(2.70)
j
j
i
i
j
i
where
j
i
is the index corresponding to all the branches connected to node
i
.
S
j
i
and
U
j
i
are, respectively, the cross section of these branches and the average velocities.
Note that the velocities
U
j
i
are signed.
The second set of equations is constituted pressure drop relations. For the
branch [
i
- 1,
i
], this relation can be cast into the form
(2.71)
If the branch [
i
- 1,
i
] is simply a channel of constant section, or any type of
channel where Bernoulli's equation applies, the preceding relation collapses to
P
-
P
=
f U
(
,
S
)
i
-
1
i
i
-
1,
i
i
-
1,
i
8
η
-
L
U
i
1,
i
i
1,
i
-
P
-
P
=
(2.72)
i
1
i
-
2
R
h i
, 1,
-
i
Figure 2.31
Schematicviewofamicroluidiccircuit.
