## Multigroup Model To Simulate Bubble Size Distribution

In a multigroup model, bubble population is divided into a number of groups and each group is treated as a separate phase. In many cases, it is adequate to associate a single velocity field calculated based on an effective bubble diameter instead of associating separate velocity fields to each bubble group. This approximation significantly reduces the burden on computational resources. In order to simulate bubble size distribution, it is, however, necessary to solve mass balance equations for each bubble group. The multigroup model developed by Buwa and Ranade (2000) is described here.

Their formulation is based on the concept that the entire bubble population can be described in terms of mass (or diameter) of the smallest bubble. It is assumed that the mass of any bubble in the considered population will be an integer multiple of the mass of the smallest bubble, mmin. The simplest way to describe the bubble size distribution in such circumstances will be to define (mmax/mmin) number of bubble groups, where mmax is the mass of the largest bubble that can be envisaged in the considered bubble population. However, it can be seen that such a description will require an inordinately large number of bubble groups (1000 groups to describe a population with smallest bubble of 1 mm and largest bubble of 10 mm). Fortunately such a high resolution

Smallest bubble, m^ Index=1

Smallest bubble, m^ Index=1

Lower limit of Group g, Lg

Upper limit of Group g, Ug

Group 1

Group g,

FIGURE IIA.I Multigroup model to describe bubble population.

Notes: (1) Lower and upper limits are inclusive in a particular group and are integer (multiples of mass of minimum bubble). This means, Lg = Ug_ i + 1. (2) Central index associated with the group, Ig is defined as algebraic mean of lower and upper limits.

of the bubble size distribution is seldom required. It is adequate to lump bubbles of different sizes into a single group and to describe the entire bubble population using 5 to 20 non-uniform sub-groups. The choice of boundaries between these sub-groups depends on bubble size distribution itself and may have to be finalized by examining the predictions (following a procedure similar to that for selection of grid distribution for flow simulations). The approach and notation used by Buwa and Ranade (2000) to describe classification of bubble population in terms of different groups is shown in Fig. 11A.1. The mass conservation equation for any individual bubble group, g, can then be written as:

d d , (Pg£Gg) + —(PGSGgUcgj) = mg{C+ _ C_+ B+_ B_} (11A.1)

where Cg and Bg represent loss (superscript _) or gain (superscript +) in the number of bubbles of group 'g' due to coalescence and breakup respectively. Following the principles of population balance, the loss and gain terms can be expressed as

C+ = 2 !>klPcklnkniXgkl V Lg < Ik + U< Ug, Xgkl =

where ng is bubble number density of group g, NG is total number of bubble groups, KCgk is rate of coalescence of bubbles of group g and k and PCgk is probability of g coalescence of bubbles of group g and k. Xgkl accounts for the intergroup transfer due to coalescence.

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