Stability analysis of discrete population balance model for bubble growth and shrinkage

The stability condition for solving the population balance equation (PBE) involving bubble growth and shrinkage within the Eulerian framework is proposed. The particle flux weighted average Courant number, CCFL, which reflects the propagation of particle state of the entire size classes on internal coordinates is first derived. Stability conditions of internal convection for PISO and PIMPLE algorithms are obtained by evaluating a series of tests concerning the constant bubble growth. The proposed stability conditions are then applied to simulate two kinds of laboratory experiments, i.e. the bubble growth in stagnant liquid and condensing steam-water pipe flows. The results show that the stable PBE solutions hold for the cases using either PISO or PIMPLE algorithms combined with the corresponding stability condition. Meanwhile, the conservation of the zeroth and first moments of the number density function can be guaranteed when the stability condition is satisfied. In addition, the effect of heat transfer coefficient correlations is also discussed.