Chemistry Reference
In-Depth Information
The basic equation governing bubble behaviour under the action of a fluctuating sound
or pressure field is the Rayleigh-Plesset equation:
þ
2
¼
d
2
R
dt
2
3
2
dR
dt
1
r
l
4
R
dR
dt
2
R
R
p
i
p
1
(7.1)
where R is the instantaneous radius of the cavity,
r
l
is the density of the cavitating medium,
m
is the viscosity and
s
the surface tension of the medium, p
i
is the pressure inside the
bubble and p
1
is the fluctuating pressure field, which is responsible for the various stages
of the cavitation phenomena. The Rayleigh-Plesset equation, which is the simplest bubble
dynamics equation, has some inherent limitations in that it assumes an incompressible
nature for the cavitating medium and hence underpredicts the cavitational intensity for a
given set of operating parameters. Other bubble dynamics equations consider the
compressibility of the liquid and the time-dependent pressure fields inside the cavitating
bubble [12]. These equations are equally applicable to both sonochemical and hydro-
dynamic cavitation reactors, and the equations for the fluctuating pressure field will be
different in these cases. In the case of sonochemical reactors, the time-varying pressure
field (N/m
2
) is expressed as:
p
1
¼
ð
Þ
p
o
-p
a
sin
2
p
ft
(7.2)
p
2I
where p
o
is ambient pressure, p
a
is the amplitude of driving pressure
¼
r
l
C
, C is the
1500m/s), I is the intensity of ultrasound (W/m
2
)
and
r
l
is the density of the liquid (kg/m
3
).
In the case of hydrodynamic cavitation reactors, the time-varying pressure field is
given as:
velocity of sound in the liquid (in water
V
o
V
tn
ÞD
P
P
1
¼
P
v
þ
1
2
r
l
ð
(7.3)
=
where
D
P is the permanent pressure loss across the constriction, V
tn
is the instantaneous
turbulent velocity, V
o
is the velocity at the constriction and P
v
is the vapour pressure of the
liquid. The instantaneous turbulent velocity (V
tn
) can be calculated by assuming a
sinusoidal velocity variation in the instantaneous local velocity with the frequency of
the velocity perturbations and is given by the following equation:
v
=
sin
V
tn
¼
V
t
þ
ð
2
p
f
T
t
Þ
(7.4)
where V
t
is the local mean velocity and f
T
is the frequency of turbulence. The details of
bubble dynamics approaches to hydrodynamic cavitation reactors can be obtained from
the published literature [19,20].
The solutions of these bubble dynamics equations can aid in quantifying the cavitational
intensity in terms of the magnitudes of collapse pressures and temperatures, as well as
the number of free radicals generated at the end of cavitational events, and also in
determining the type of cavitation - stable or transient - occurring in the reactor. Typically,
stable cavitation gives multiple shock effects with lower intensities and is recommended
for physical processing or reactions involving lower activation energy, whereas transient
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