## 1 Introduction
[5] investigated the effect of coalescence and breakup of oil droplet on the size distribution by using the population balance model, and further studied the flow behavior of liquid-liquid system from mechanism. [PERSON] and [PERSON] et al. [6] assumed that the oscillating droplet will break up when the turbulent kinetic energy transmitted by the turbulent vortex to the droplet exceed the surface energy of the droplet, but the two droplets will coalesce when the contact time exceed the film discharge time of the two droplets. [PERSON] and [PERSON] et al. [7] studied the droplet coalescence behavior in the turbulent using high-speed imaging technology and population balance model, proposed a novel collision frequency model, and compared the predicted droplet diameter with the experimental data. The results show that the model is applicable under different discrete phase fraction. [PERSON] and [PERSON] et al. [8] established a bubble breakup model, in which the bubble breakup rate is equal the product of the collision frequency with the turbulent vortex and the bubble breakup efficiency. Regarding the solution of the population balance equation, [PERSON] et al. [9] used the quadrature method of moments (QMOM) to study the droplet breakup and coalescence behavior of the liquid-liquid dispersion system in the stirred reactor. The results show that the droplet coalescence behavior occurs with some difficultly in the low discrete phase fraction, due to the small probability of droplet collision. [PERSON] et al. [10] reviewed the CFD-PBM simulation in extraction column, and the principles and advantages and disadvantages of different solving methods of population balance model were compared. [PERSON] et al. [11] used the experimental method and CFD-PBM coupled model to analyze the influence of viscosity in the bubble bed on the hydrodynamic behavior in the bubble area, such as the total gas holdup, the big and small bubble holdup, and the bubble size distribution. The results show that the CFD-PBM coupled model has a good prediction ability for the hydrodynamic behavior of the bubble bed in a wide range of gas velocities and viscosities. [PERSON] et al. [12,13] used the population balance model to describe the bubble size distribution in the gas-liquid system, considered the different mechanism of bubble coalescence and breakup, established a relatively mature bubble coalescence and breakup model, and calculated the bubble size distribution in the gas-liquid system by numerical solution of PBM. [PERSON] et al. [14] used a multi-Monte Carlo algorithm to simulate the collision and coalescence process of particles in nanoparticle flow, and the simulation results were in good agreement with the direct numerical simulation results. [PERSON] et al. [15] analyzed the parameter relationship between the structural condition of bearing chamber and the oil droplet size distribution, clarified the oil droplet mass distribution based on the continuous oil drop size distribution, and introduced the concept of continuous oil droplet diameter into the analysis of oil droplet movement. [PERSON] et al. [16] measure the size distribution of oil droplet using the phase Doppler particle analyzer (PDPA) technology under various operating condition in the rotating disk chamber. In conclusion, the population balance theory has been widely applied in the analysis of coalescence and breakup between bubbles and droplets. However, no relevant research work has been done to consider the influence of coalescence and breakup on the oil droplet size distribution in the bearing chamber. Therefore, based on the population balance model, the coalescence and breakup behaviors of oil droplets are described through the coalescence and breakup model of the oil droplet. Then, the air-oil two-phase flow model and the population balance model are solved by Fluent software. The numerical simulation of the coalescence and breakup processes of the oil droplet in a bearing chamber is performed.