Biomedical Engineering Reference
In-Depth Information
rates scale is
k
2
because the product (
dw
) scales as
k
2
. A natural choice for the flow
rates is then
(2) 2 (1)
Q Q
(4.117)
An interesting consequence is that the velocities of each phase remain constant
in the scaling:
V
(2)
=
V
(1)
. Using the model, we can transpose (4.115) and (4.116) in
terms of pressure
=
κ
P
<
P
=
(
R R f Q
+
)
+
2
R f Q
i
ic
i
s
i
s
ec
c
(4.118)
P
<
P
=
(
R
+
2
R f Q R f Q
)
+
e
ec
e
s
ec
s
ic
As the hydraulic resistances
R
scale like 1/
k
3
and the flow rates
Q
scale as
k
2
,
the critical pressures
P
c
governing the dripping-jetting transition should be of the
ratio 1/
k
. In consequence, we have for the pressure the relation
(2) (1)
(1
P P
(4.119)
When the dimensions are scaled homothetically by
k
, and the driving pressures
are in the ratio 1/
k
. It has been experimentally observed that the droplet emission
frequency
fr
is in the ratio 1/
k
=
κ
1
(2)
(1)
(4.120)
fr
=
fr
κ
and the droplet diameter
f
in the ratio
k
(2)
(1)
φ κφ
=
(4.121)
It is not coincidental that the droplet diameter and the geometrical dimen-
sions scale identically; the same applies for the pressure and droplet frequency.
These experimental scaling rules have been experimentally verified. Figure 4.75
shows the droplet diameter
f
as a function of the dimension of the device. In
Figure 4.76 the inverse of the frequency 1/
fr
has been plotted against the device
depth.
Figure 4.75
Droplet diameter
f
varies linearly with the device dimension. The reference system is
a 50
m
m device functioning with driving pressures
P
i
=
P
e
= 200 mbars.
