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.