Daughter bubble size

The daughter bubble size is thus limited by two constraints. The capillary pressure constraint states that if the dynamic pressure of the turbulent eddy Pc v x exceeds the capillary pressure aijd", the fluid particle is deformed and finally breaks up resulting in a minimum breakage fraction /vm, (or bubble size dj min) [69]. d denotes the diameter of the smaller daughter size (or two times the minimum radius of curvature). When breakage occurs, the d3mamic pressure induced by the eddy turbulence kinetic energy satisfy the criterion ... 

The experimental data of Risso and Fabre [99] also indicate that an equal size daughter distribution is more common for bubble breakage than an unequal one. Contrary, Hesketh et al [33, 34] investigated bubble breakage in turbulent flows in horizontal pipes and concluded that an unequal size daughter distribution is more probable than an equal size one. The daughter bubble size distribution model of Luo and Svendsen [74] rely on the assumption that unequal sized is more probable than equal size breakage in accordance with... 

Note that the units of the discrete daughter bubble size distribution variable are different from the units obtained deriving the continuous daughter size distribution function from the microscopic formulations (l/[m]). It is thus not trivial how to adopt the model of Luo [73] within a more fundamental modeling approach. 

Fig. 9.3. A sketch of the bubble breakage surface energy balance. The mean kinetic energy of an eddy of size A breaking a bubble of size di, e di,X), is assumed to be larger than the increase of the bubble surface energy required breaking the parent bubble di into a daughter bubble dj and a second corresponding daughter bubble, es di,dj). Reprinted with permission from [38]. Copyright 2005 American Chemical Society.

In the average microscopic formulations this parameter determines the average number of daughter bubbles produced by breakage of a parent bubble of size d. The breakage of parent bubbles into two daughter bubbles is assumed in most investigations reported (i/ = 2). 

Martinez-Bazan C, Montanes JL, Lasheras JC (1999) On the breakup of an air bubble injected into a fully developed turbulent flow. Part 2. Size PDF of the resulting daughter bubbles. J Fluid Mech 401 183-207 Melzak ZA (1957) A scalar transport equation. Transactions of the American Mathematical Society 85 547-560... 

The upper size limit is made up by the transient cavitation threshold. Smaller bubbles may exist as stable oscillating bubbles that eventually grow in size by rectified diffusion or their path into regions with higher sound pressures. Bigger bubbles than this will collapse and create smaller daughter bubbles. The bubble size Rq is rather small and is of the order of several micrometers. 

Daughter bubbles can be generated by introducing a mother bubble into a highly turbulent liquid flow. Martinez-Bazan et al. [9] found that the bubble size probability density function of the daughter bubbles depends not only on the size of the mother bubble, but also on the value of the dissipation rate of turbulent kinetic energy. 

Minimum energy required to deform a bubble of size d (J) Increase of bubble surface energy required breaking the parent bubble d, into a daughter bubble dj and a second corresponding daughter bubble (J)... 

Daughter particle size probability density function (1/m [x]) Breakage probability function, determining the efficiency of the eddy-bubble collisions (—)... 

The individual rate breaking a parent bubble of size di into the daughter size classes dj is expressed as ... 

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