## 2 Numerical Model
### Mathematical Model
#### 2.3.2 Population Balance Model  
The application of the population balance model [21] is used to calculate the oil droplet diameter distribution in the bearing chamber. The transport equation of the oil droplet number density function can be expressed as  
\[ \begin{array}{l}\frac{\partial}{\partial t}[(n(V,t))]+\
abla\cdot\left[ \stackrel{{\rightarrow}}{{u}}(n(V,t))\right]=\underbrace{\frac{1}{ 2}\int\limits_{0}^{V}a(V-V^{\prime},V^{\prime})n(V-V^{\prime},t)n(V^{\prime},t )dV^{\prime}}_{\text{I}}\\ -\underbrace{\int\limits_{0}^{\infty}a(V,V^{\prime})n(V,t)n(V^{\prime},t)dV^ {\prime}}_{\text{II}}+\underbrace{\int\limits_{0}^{\infty}b(V^{\prime})\beta(V \big{|}V^{\prime})n(V^{\prime},t)dV^{\prime}}_{\text{III}}-\underbrace{b(V)n(V, t)}_{\text{IV}}\end{array}  \tag{8}\]In the right-hand side of the equation in turn are the source term (I) generated by coalescence, the sink term (II) generated by coalescence, the source term (III) generated by breakup, and the sink term (IV) generated by breakup. Where \(n(V,t)\) is the number distribution function (1/m\({}^{6}\)), which means the number of oil droplets in the range of \(V\) and \(V+dV\) in unit volume is \(n(V,t)dV;a(V,V^{\prime})\) is the coalescence rate function (m\({}^{3}\)/s) of oil droplets in volume \(V\) and \(V^{\prime}\) due to collision. \(b(V^{\prime})\) is the breakup rate function (1/s) of oil droplets in volume \(V^{\prime}\). \(\beta(V|V^{\prime})\) is the probability distribution density function (1/m\({}^{3}\)) of oil droplets in volume \(V\), which breaks up from oil droplets in volume \(V^{\prime}\). The population balance equation (PBE) is a hyperbolic integral-partial differential equation; only a few simple cases have an analytical solution, so it is necessary to use a numerical method to solve the PBE. At present, the usually used methods include the discrete method, moment method, and Monte Carlo method. Based on the calculation cost and accuracy, the discrete method is used to solve the population balance equation in this paper. The basic idea of the discrete method is dividing the continuous diameter distribution of oil droplets into \(N\) discrete subintervals, and the size of all the droplets in the subinterval is equal to the node value \(g_{i}\). Meanwhile, the distribution function of the oil droplet number can be approximately as follows,  
\[n(V,t)=\sum_{i=1}^{N-1}N_{i}(t)\delta(V-g_{k}) \tag{9}\]  
where \(N_{i}\) is the number of droplets in unit volume. \[N_{i}(t)=\int_{V_{i}}^{V_{i+1}}n(V,t)dV \tag{10}\]  
In most cases, the oil droplet size calculated by the coalescence and breakup model is not consistent with the node value of the subinterval, so it is necessary to allocate such droplets to the node value of the interval in a certain proportion, and the same time guarantee the conservation of the total oil droplet mass and the total number of oil droplet. Based on the above principles, [PERSON] proposes that the ratio of the newly generated droplets in the interval \((g_{i},g_{i+1})\) to the node \(g_{i}\) and \(g_{i+1}\) is \(\psi(V,g_{i})\) and \(\zeta(V,g_{i})\), and satisfies that  
\[\psi(V,g_{i})g_{i}+\zeta(V,g_{i+1})g_{i+1}=V \tag{11}\]  
\[\psi(V,g_{i})+\zeta(V,g_{i+1})=1 \tag{12}\]  
By substituting Equations (9), (11), and (12) into Equation (8) and through a series of algebraic transformations, the discrete droplet transport equation is given by Equation (13). \[ \begin{split}\frac{dN_{i}(t)}{dt}+\
abla\cdot\tensor{\left[ \right.}_{u}^{\rightarrow}N_{i}(t)]=&\sum_{j,k}^{j\geq k}(1- \frac{1}{2}\delta_{j,k})\alpha_{i,j,k}a(g_{i},g_{k})N_{j}(t)N_{k}(t)\\ &\underbrace{g_{i-1}\leq(g_{i}+g_{k})\leq g_{i+1}}\\ &\underbrace{-N_{i}(t)\sum_{k=1}^{M}a(g_{i},g_{j})N_{k}(t)+ \underbrace{\sum_{k=1}^{M}\gamma_{i,k}b(g_{k})N_{k}(t)}_{\text{III}}}_{\text{III} }\\ &\underbrace{-b(g_{i})N_{i}(t)}_{\text{IV}}\end{split}  \tag{13}\]where  
\[x_{i,j,k}=\left\{ \begin{array}{ll}\frac{g_{i+1}-V}{g_{i+1}-g_{i}}&V_{i}\leq V \leq V_{i=1}\\ \frac{V-g_{i-1}}{g_{i}-g_{i-1}}&V_{i}\leq V\leq V_{i=1}\\ 0&\text{otherwise}\end{array} \right.