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.