text: Depending on the regime under investigation, different approaches can be used to model the dispersed gas phase. The simplest is to use a fixed bubble size distribution. This approximation is suitable to simulate the homogeneous flow regime, where the interactions between the bubbles are negligible. In addition, this approach calls for the knowledge of the bubbles diameter since it is an input parameter for the simulations. 
  However, in industrial practice, large-scale bubble columns are typically employed, equipped with gas distributors characterized by large openings, so a heterogeneous flow regime is commonly observed. Bubble coalescence and breakup phenomena are relevant and can not be neglected. In this case, the CFD model can be coupled with a Population Balance Model (PBM) to account for the changes in bubbles size.
  A Population Balance Model consists of a transport equation derived from the Boltzmann statistical transport equation, and it describes the particles entering or leaving a control volume via several mechanisms.  The bubble number density transport equation is also known as Population Balance Equation (PBE):
  Where  is the bubble number density function and represents the probable number density of bubbles at a given time , about a position , with bubble volume between  and , and  is the bubble velocity. The right and side term of the Population Balance Equation is the source/sink term due to bubbles coalescence, breakup, phase change, pressure change, mass transfer, and chemical reactions.