Environmental Engineering Reference
In-Depth Information
these equations reduce to a simple closed form and the drag force in Eq. (
14.16
)is
given by Stoke
s law:
'
F
drag
¼
6
ˀʷ
m
a
v
m
v
p
ð
14
:
18
Þ
where v
p
is the velocity of the particle and v
m
the velocity of medium. Stoke
s law
has been experimentally verified to be an accurate estimate of the drag force when
Re
'
<
0.5 and deviates by only about 10 % at
Re
1[
16
].
Under the influence of an electric field, heat is generated in the medium,
resulting in local temperature gradients which in turn give rise to gradients in the
conductivity and permittivity of the medium. These gradients can induce fluid
movement, and experiments show that for a given set of parameters (applied
voltage, medium conductivity, frequency, electrode geometry, etc.) the resulting
fluid flow has a reproducible pattern. As a consequence the drag force exerted by
the moving fluid must be considered in the total force F expression.
The other hydrodynamic force exerted on particles manipulated by dielectro-
phoresis is buoyancy [
42
]:
¼
g
F
buoyancy
¼
V
p
ˁ
p
ˁ
m
ð
14
:
19
Þ
where g is the acceleration due to gravity and the
ˁ
p
and
V
p
represent the density and
volume of the particle. Since the volume of a nanoparticle is small, the magnitude
of the buoyancy force is also small. However, the densities may be such that will
have to overcome particles
natural tendency to float or sediment over time.
'
14.2.3.2 Electrothermal Forces
The high intensity electric fields often needed to manipulate particles have been
observed to produce Joule heating inside the fluidic medium [
43
], especially when
dielectrophoresis is occurring at AC frequencies in the MHz range. This ohmic
heating causes a temperature gradient that in turn results in spatial conductivity and
permittivity gradients within the suspending medium. The variation of electrical
properties within the medium results in Coulombic and dielectric body forces that
will induce extra fluid flow. The time-averaged body force on the fluid is given by
[
44
]:
E
Re
˃
m
ε
m
ʱ ʲ
ð
Þ
1
2
E
2
ε
m
ʱ
E
h
F
thermal
i
¼
T
T
ð
14
:
20
Þ
ˉε
m
∇
∇
˃
m
þ
j
where
are the linear and volumetric coefficients of thermal expansion and
T
is the absolute temperature. The additional drag force from the electrothermally
induced flow can be found by solving the Navier-Stokes equations and using
Eq. (
14.18
) as the volume force term.
ʱ
and
ʲ
