Biomedical Engineering Reference
In-Depth Information
depends on the flow rate ratio
q
*
=
cste
=
q
*
2,3
, or, in a pressure actuated system,
P
i
/P
e
=
p
*
=
cste
=
p
*
2,3
. Using (4.107), these transitions can be written in terms of
hydraulic resistance ratios:
1
-
æ
R
ö
e
s
p
*
= -
1
B
=
1
+
1,2
ç
÷
è
2
R f
ø
(4.109)
-
1
æ
ö æ
ö
R
R
i
e
p
*
=
1
+
q
*
1
+
ç
÷ ç
÷
2,3
2,3
2
R f
2
R f
è
ø è
ø
s
s
For rectangular channels, the hydraulic resistances can be expressed under the
form [87]
3
(4.110)
R
»
4
η
L wd
/ (
ζ
)
where
z
is a coefficient depending on the aspect ratio. In order to conserve the limits
of the droplet regime for new channel dimensions, the following conditions must
be respected if the additional friction factor
f
is assumed to be always close to unity
(e.g., if the droplet frequency is not too high),
R
L w
ξ
ξ
e
e
s
s
=
=
cste
R
L w
s
s
e
e
(4.111)
R
η
L w
ξ
i
i
i
s
s
=
=
cste
R
η
L w
ξ
s
e
s
i
i
An immediate solution for this system is the homothetic scaling, where all the
dimensions are scaled by the same constant
k
. Using the superscripts 1 for the origi-
nal geometry and 2 for the scaled up geometry, we have
(2)
(2)
(2)
w
L
d
=
=
=
κ
(1)
(1)
(1)
w
L
d
(4.112)
(2)
(2)
ξ α
ζ α
=
=
1
(1)
(1)
and the ratio of the hydraulic resistances is constant in the scaling-up. This homo-
thetic scaling up based on the analytical model is verified by the experiment. Figure
4.74, corresponding to a pressure plot, shows that the limits of the droplet regimes
respect the similarity rules, even in the nonlinear domain when
f
starts to slightly
depart from unity.
Flow Conditions—Pressure and Flow Rate Scaling
To limit polydispersity, the droplets should be produced in the dripping regime. The
transition from dripping to jetting regime has been subject of many investigations
[73, 74]. It appears that two nondimensional numbers pinpoint this transition: the
critical Weber number
We
c
of the discontinuous phase and the critical capillary
number
Ca
c
of the continuous phase. The first condition to observe a dripping
regime can be expressed as
