Microbubble size
The average value of the microbubble diameter Dsa(in the range of 20-170μm) under different experimental conditions is
summarized in Figure 10. The average diameter increases first and then
decreases lightly with increasing Weber number, and it increases with
increasing Bond number. Typical probability distribution of the
normalized microbubble diameter Ds by its average
value is presented in Figure 11. It is clear that the scattered level
reduces as Weber number increases. At lower Weber number, mother bubbles
with different sizes form in one father-bubble cycle, leading to the
pinch-off of satellites with different diameter.
To determine the influence of main factors, we consider the gas flow
rate, nozzle radius, density, viscosity, surface tension, and gravity
for scaling the microbubble size. In fact, the father-bubble departure
diameter, mother-bubble growth rate are also influenced by these
parameters, therefore they can determine the microbubble size. The
density of gas and viscosity of liquid have important effect on the
pinch-off, but they are constant in present study, allowing us to write
the correlation of microbubble diameter,
(5)
Following dimensional analysis, the microbubble size selection is
described in terms of three dimensionless parameters, We, Bo andDs /R 0,
(6)
It should be noted that the nozzle radius R 0 in
the left hand of Equation 6 is a dominant parameter in Bond number.
Therefore, the following equation is substituted into Equation 6,
(7)
where Lb is the capillary length,
(8)
One can obtained,
(9)
By fitting the experimental data, the scaling law for formation of
microbubbles can be established as follows,
(10)
where α is about 0.357. The error bar is also estimated by
fitting the lower and upper bounds and we findH (We)∝We1/4±3/40. The experimental data,
prediction curve given in Equation 10 and both two bounds are shown in
Figure 12(a). Equation 10 is useful for estimating bubble size from the
engineering point of view due to its simplicity. We also estimate the
ratio of the satellite surface
energy to the kinetic energy of the gas flow in one father-bubble cycle
as being,
(11)
where T is the father-bubble cycle period and n is the
number of microbubbles formed in one cycle. This ratio decreases rapidly
and tends to zero with increasing Weber number, as shown in Figure
12(b), indicating that more energy of the gas flow is used to for
microbubbles at lower Weber number. As a result, the lower Weber and
Bond numbers are recommended for generating smaller microbubbles with
higher efficiency. In fact, the size of the generated microbubbles with
present method may be also affected by the liquid viscosity, liquid/gas
density ratio and micro structure of nozzle surface, which will be
further explored in the following works.