P

B+g= 2 J2 KBknk<Pgk V $gk = J2 V^kp k=g i=Lg 2_j=l PBj,

where KBk is the total breakage rate of bubbles of group 'k', PBki is the probability of formation of a bubble with mass 'immin' from a bubble of group 'k'. The term Xj-1 PBj,h-1-j represents sum of probabilities of bubble of group 'k' breaking with breakage fraction varying from mmin/mk to (Ik -1)mmin/mk . The term $gk signifies the number of group 'g' bubbles formed per number of group 'k' bubbles broken. Note that this formulation conserves mass but does not conserve bubble numbers. Suitable modifications may however be made to ensure conservation of bubble numbers and/or interfacial area while retaining the same overall structure.

Several authors have proposed different models to describe the bubble coalescence and breakage processes (Prince and Blanch, 1990; Hesketh et al, 1991; Luo and Svendsen, 1996; Carrica et al., 1999; Lehr and Mewes, 2001). Bubble coalescence and breakage may occur by different mechanisms. Local turbulence in the liquid phase, velocity gradients and shear flows are known to cause bubble coalescence. Similarly, collision of a bubble with energetic eddies in the liquid phase or other phenomena such as tip streaming are known to cause bubble breakage. In most gas-liquid reactors, turbulence-induced breakage and coalescence were found to be dominant in comparison with other mechanisms (Prince and Blanch, 1990). Buwa and Ranade (2000) compared magnitudes of different mechanisms and have recommended the turbulence induced collision model of Prince and Blanch (1990) to describe bubble coalescence, and the turbulent eddy interaction model of Luo and Svendsen (1996) to describe the bubble breakage process. These models were used to close the set of mass conservation equations of bubble groups described above:

Pcki = exp

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