Biomedical Engineering Reference
In-Depth Information
where
Q
is the liquid fl ow rate,
K
the electrical conductivity,
γ
the surface tension,
ε
0
the permittiv-
ity of a vacuum,
ρ
the density of the solution, and
Q
0
is the characteristic fl ow rate and is equal to
ρK/ε
0
γ
.
It is observed that the droplet size is dependent on the liquid fl ow rate and the liquid properties
such as electrical conductivity, surface tension, and
density. The experimental results have revealed
that when the electrical conductivity of the liquid is too low, such as in the range of 10
−
8
to 10
−
10
S/m,
the liquid cannot be electrosprayed in a cone-jet mode because of an insuffi cient current of the drop-
lets [41]. However, if the electrical conductivity is too high, an unstable mode of electrospraying in a
polydispersity will occur. A solution with a high surface tension is diffi cult to electrospray in a cone-
jet mode, and this high electrical potential is needed to overcome the surface tension to break up the
droplets. On the other hand, when the fl ow rate of the solution is too low, a stable cone-jet mode cannot
be formed because of the insuffi cient current. However, when the fl ow rate is too high and more liquid
leaves the capillary, the charges and current needed to form a stable cone-jet spraying mode become
insuffi cient. The relationship between the current and the droplet size can be found from Equation
11.7 and is only valid for liquid with a low viscosity and electrical conductivity, which suggests that
the droplet size will decrease as the fl ow rate of the solution increases [2,27]:
(
ρε
0
Q
4
1/6
)
______
d
=
c
I
2
(11.7)
where
d
is the droplet size,
c
a constant,
ρ
the density of the liquid,
Q
the fl ow rate of the liquid,
ε
0
the permittivity of vacuum, and
I
is the current of the droplets and a function of the fl ow rate, sur-
face tension, and electrical conductivity of the liquid.
Gañán-Calvo et al. proposed universal scaling laws of the electrospraying current as well as the
charge and size of the droplets in a cone-jet mode from a theoretical model of the charge transport
and through experimentation [30]. The two scaling laws are applicable to different solutions—those
with relatively high viscosity and electrical conductivity and those with low viscosity and electrical
conductivity. This variation will depend on the dimensionless parameter, which governs the liquid
acceleration process and ultimately the electrospraying current and droplet size. The dimensionless
parameter is defi ned as follows [30]:
γ
3
ε
0
2
1/3
[
]
______
δµ
·
δ
1/3
=
µ
3
K
2
Q
(11.8)
where
Q
is the fl ow rate of liquid,
ε
0
the permittivity of vacuum,
K
the electrical conductivity,
γ
the
surface tension, and
µ
the viscosity of the solution.
If
δµ
∙
δ
1/3
≤
1, the current
I
and size of the droplet
d
in a cone-jet electrospraying mode are as
follows:
I/I
0
=
6.2[
Q
/(
β
−
1)
1/2
·
Q
]
1/2
(11.9)
d
/(
β
−
1)
1/3
d
0
=
1.6[
Q
/(
β
−
1)
1/2
Q
0
]
1/3
−
1.0
(11.10)
If
δµ
∙
δ
1/3
≥
1, the current
I
and size of the droplet
d
in a cone-jet electrospraying mode are as
follows:
I/I
0
=
11.0[
Q
/
Q
0
]
1/4
−
5.0
(11.11)
d
/
d
0
=
1.2[
Q
/
Q
0
]
1/2
−
0.3
(11.12)
where
I
0
=
(
ε
0
γ
2
/
ρ
)
1/2
,
Q
0
=
ε
0
γ
/(
ρK
), and
d
0
=
[
γε
0
2
(
ρK
2
)]
1/3
.
