Ten ranges, or bins, of bubble sizes are considered. Within each range, the bubble diameters are described by a distribution function. Using Boltzmann's equation, the bubbles are allowed to move among the bins as breakup and coalescence processes occur. To do this, source terms are constructed that follow the birth and death of bubbles in each size group so that an overall mass balance in the gas phase is assured. Each of these has its own set of composite source terms that account for the birth and death rates.
Larger eddies are primarily responsible for the transport of bubbles.
To a lesser degree, the variation in rise velocity of differently sized bubbles can also lead to collisions, as can macroscopic recirculation patterns that give rise to radial velocity gradients that transport nearby bubbles at different rates. Large eddies in the center of the column do little to break up rising bubbles. As these small bubbles rise, some get caught in recirculation patterns near the wall while others continue to rise near the center with the larger, more buoyant bubbles. Here, the maximum holdup is shown, where the small bubbles, generated by the breakup, have risen in the column but have not yet branched into either the group that undergoes recirculation or the group that continues to ascend. Without breakup and coalescence, the bubbles are all one size and the gas volume fraction is more uniform in the column.