Bubbling regimes for downward capillary nozzle
With bubbling regimes identified previously, dimensionless numbers capable of characterizing the regimes are investigated. An illustration of formation mechanism of microbubbles in Regime III is depicted in Figure 6. The associated key processes are (1) the rapid formation of small mother bubbles after departure of the father bubble and (2) the occurrence of partial coalescence:
For the first process, the father-bubble rising speed, waiting time between any two consecutive bubbles and mother-bubble growth rate determine whether the parent bubbles coalesce in the vicinity of the nozzle orifice. The rising velocity of the father bubble increases with the increase in its detachment diameter. The major forces determining detachment of a bubble attached on a downward-pointing nozzle are shown in Figure 6(a) where the dynamic forces on the bubble surface are neglected because of their weak effect23. Before its detachment, the bubble is squeezed onto the orifice wall, resulting in a reacting force acting on it, which is balanced by buoyancy force, gas momentum force and surface tension force in vertical direction. When the gas momentum force and the force induced by the orifice wall exceed the surface tension force in horizontal direction, the neck of the bubble is torn with a sudden contraction. After departure of the father bubble, the gas-liquid interface retracts into the nozzle. During waiting time before formation of the next bubble, the gas in the nozzle is accumulated and its pressure is increased. When the pressure difference between gas and liquid exceeds the capillary pressure, a new mother bubble forms and grows rapidly. Therefore, the waiting time is related to both the gas flow rate and the nozzle size which influence the increasing speed of the gas pressure during the liquid penetration inside the nozzle and the capillary pressure26, respectively. The two parameters also control the mother-bubble growth rate. Previous experiments indicated that the growth rate of the bubble volume increased rapidly at the initial growth stage and trended to fluctuate around a constant value when the nozzle radius is lower than 0.1 mm23, while it was almost a constant when the nozzle radius is over 0.5 mm27-28.
For the second associated process, the partial coalescence, Zhang and Thoroddsen13 verified that it was governed by capillary inertial dynamics. Their experimental data revealed that the partial coalescence time was about 0.68 times the capillary inertial time and was fairly independent of the bubble size. The partial coalescence time tp is the time interval from the first contact of the parent bubbles to the pinch-off, and the capillary inertial time τσ is,
(1)
where Rm is mother-bubble radius, σ andρl are surface tension and liquid density. Figure 7 shows the normalized partial coalescence timetp /τσ versus Ohm obtained in our experiments. The average value is about 0.73 and is larger than that reported by Zhang and Thoroddsen13, which may result from the delay of convergence of the capillary waves by the nozzle. Ohnesorge number of the mother bubble Ohm and relative size of the parent bubbles Df /Dm are suggested to be the dominant dimensionless numbers determining the pinch-off of a satellite12-13. Here, Ohnesorge number is a dimensionless number that compares viscous and surface-tension forces and is defined as,
(2)
where μ is the viscosity of liquid. A phase diagram for pinch-off and no pinch-off in terms of the Ohm andDf /Dm is given in Figure 8. A critical Ohnesorge number about 0.0065 was found for the coalescence of two bubbles with similar size by Zhang and Thoroddsen13, above which the capillary waves would be damped by viscosity obviously and the bubbles coalesced without the pinch-off, as well as this value increased with the increase ofDf /Dm in the range of 0.85-3.5. A similar trend is also observed in our experimental data forDf /Dm between 2 and 6, but the critical Ohm is smaller than that reported by Zhang and Thoroddsen13 for eachDf /Dm . As discussed in section 3.2, the nozzle blocks propagation of capillary waves and delays its convergence obviously, which may suppress the pinch-off for small mother bubbles with large Ohnesorge number. The data also show that the partial coalescence is absent forDf /Dm larger than 6.
Although previous studies have demonstrated that gas inertia and gravity have no considerable influence on the partial coalescence, the sizes of the father bubble and mother bubble in the Ohm andDf /Dm depend on the inertia of the gas flow, surface tension, gravity, gas flow rate and nozzle radius in present air-water experiments. Therefore, we select the Weber and the Bond numbers as the primary dimensionless numbers for microbubble generation in water-air experiments,
(3)
(4)
where u is the average gas injection speed and g is the acceleration due to gravity. Here, Weber and Bond numbers serve as comparisons of inertial and gravitational forces, respectively, to the surface tension. Figure 9 shows a phase diagram for bubble formation and departure from a downward-pointing nozzle depending on Weber and Bond numbers. With increasing the Bond number, the transitional values of Weber number for the upper and lower boundaries of bubble coalescence (regimes II and III) decrease for Bo<0.001 but increase for Bo>0.001. For Bo>0.008, no bubble coalescence occurs near the nozzle and therefore no microbubble is generated. To generate a bubble from a submerged nozzle, the injected gas must overcome the ambient pressure, hydrostatic pressure, and capillary pressure. At low Bond number, the capillary pressure is significant and the main resistance, which gives rise to a high gas overpressure over the liquid pressure and a high growth rate of mother bubble once this resistance is overcome. With increasing Bond number, the capillary pressure decreases, as does the required gas flow rate, leading to a lower transition Weber number. However, for large Bond number, the capillary effects become less important and therefore higher Weber number are necessary to ensure the bubble growth rate.