Biomedical Engineering Reference
In-Depth Information
Figure 2.30
Bernoulli'slaw:energyconservationalongastreamline.
P
isthepressureand
V
isthe
velocity.
third, friction can be accounted for, by assuming a loss of energy due to friction on
the rigid wall between any two cross sections. Streamlines do not account for wall
friction since they are not bounded by walls, but energy conservation in a cross
section does.
As we have seen before, the pressure drop caused by friction on the wall is usu-
ally expressed by
4
L
æ
1
2
ö
2
D =
P f
ρ
U
ç
÷
D
è
ø
h
where
f
is the friction factor (nondimensional). The value of
f
depends on the par-
ticular geometry of that part of the duct where the conservation equation is applied.
In some cases, such as for laminar flows in circular or rectangular ducts, the coef-
ficient
f
can be calculated; most of the time it is given by algebraic expression using
one or more parameters derived from experiments.
Bernoulli's equation incorporating friction pressure drop is then
2
2
ρ
u
ρ
u
1
2
+
ρ
g z
+
P
=
+
ρ
g z
+
P
+ D
P
(2.69)
1
1
2
2
1,2
2
2
where D
P
12
is the pressure loss by friction on the walls between points 1 and 2.
Written in such a form, Bernoulli's equation is a powerful tool for many applica-
tions in microfluidics, as we will see in the following sections.
2.2.8 Modeling: Lumped Parameters Model
Because it takes into account the complex geometry of the boundaries, the finite
element method is the preferred method of modeling microfluidic flows. We will not
deal here with the finite element method and its application to microfluidics; this
would be a book itself, and some aspects are already detailed in the literature [33].
However, we present the lumped parameters model, which is a simplified calcula-
tion method and gives very interesting and accessible results in some cases.
