Biomedical Engineering Reference
In-Depth Information
For a square duct
14.2
Re
f
=
(2.59)
D
In the case of an equilateral triangular duct
13.3
Re
f
=
(2.60)
D
In the literature, there exist catalogs of friction factors for very different shapes of
ducts, with or without obstacles [29].
2.2.6.4
LaminarPressureDrop
We have just seen that in the laminar case, the pressure drop is proportional to the
mean velocity or to the flow rate. The hyd
ra
ulic resistance of a fluidic channel can
then be defined as D
P
=
R
Q
Q
or
D =
. The two expressions are equivalent if
we remark that
R
V
=
R
V
S
, where
S
is the cross-section area. Next, for simplicity, we
give expressions of
R
V
. For an arbitrary cross section, with
x
- and
y
-dimensions of
the same order, a general approximation of the hydraulic resistance is
P R V
V
2
2
p
R
»
2
µ
L
S
(2.61)
V
where
S
is the cross section area and
p
is the wetted perimeter. Exact or approxi-
mate values have been found for particular shapes. These values are listed in
Table 2.1. Note that the resistances correspond to an interior flow without free
surface (all the walls are taken into account).
2.2.6.5
TheEffectofRoughness
In the preceding sections we have derived expressions for the pressure drop in mi-
crochannels assuming that the walls were perfectly smooth. By “perfectly smooth,”
we assume that the hypothesis for the derivation of the Poiseuille-Hagen flow is
satisfied, that is, the flow has a unique axial component and no transverse compo-
nent. If the walls have roughness of a sufficient size to induce some 3D component
of the flow near the wall, then the Poiseuille-Hagen hypothesis is not met and the
expression for the pressure drop must be revised.
It has been observed that friction factors were usually higher than the values
predicted by the conventional theory for smooth walls. Besides, the theoretical re-
sults are approximately valid for relatively large channels, but deviate significantly
from experimental observations in the case of small channels.
The effect of roughness on pressure drop can be approached by a perturbation
method where the roughness is modeled by a spatial sinusoidal function [30-32].
It is concluded that the additional friction is related first to the roughness ratio
ε
=
h
/
w
, to the value of the Reynolds number and to the wave number of the wall sur-
face (Figure 2.26). A general conclusion is that the roughness can be neglected for
ε
< 1-3%, but increases quickly above this threshold.
