# Research on Performance Prediction Model of Impeller-Type Breather
## 4 Data Analysis
### Performance Prediction Model  
In this paper, the dimensional analysis method was used to establish the empirical formula of the ventilation performance calculation for an engineering application. The factors related to the breather performance were sorted into dimensionless numbers using the Buckingham \(\pi\) theorem, forming a dimensionless group equation in general form. By using the least squares method, the engineering formulas for the comprehensive separation efficiency, the ventilation efficiency, and the ventilation resistance were obtained. As shown in Table 1, there are 11 main factors affecting the performance of the breather. Their units and dimensions are listed in Table 4. In order to show the state of the two-phase mixture, \(v\), \(\rho\), and \(\mu\) have been added to Table 4. Considering the relationship between mixture velocity (\(v\)) and ventilation rate, \(v=\frac{q}{\pi(d_{1}/2)^{2}}\), where \(d_{1}\) is the inlet diameter, mixture velocity (\(v\)) can represent the effect of ventilation rate on the performance of breather. Mixture density (\(\rho\)) and dynamic viscosity (\(\mu\)) is related to temperature, which can characterize the influence of temperature. Thus, a total of 12 influencing factors are shown in Table 4.  
The comprehensive separation efficiency \(\eta\), breather separation \(\eta_{0}\), and ventilation resistance \(\Delta P\) are expressed in the functions of various influencing factors, as shown in Equations (6)-(8). The difference between the comprehensive separation efficiency and the breather separation efficiency are given in later section of this paper. \[\eta=f(\overline{d},n,\varepsilon,v,\rho,\mu,\alpha,r_{1},r_{2},\beta,h,d), \tag{6}\]  
\[\eta_{0}=f(\overline{d},n,\varepsilon,v,\rho,\mu,\alpha,r_{1},r_{2},\beta,h,d), \tag{7}\]  
\[\Delta P=f(\overline{d},n,\varepsilon,v,\rho,\mu,\alpha,r_{1},r_{2},\beta,h,d). \tag{8}\]  
In Table 4, the 12 factors contained in Equations (6)-(8) can be expressed by 3 individual physical dimensions, such as M (mass), L (length), and T (time). \(\rho,v\), and \(r_{1}\) are representing the basic dimensions. Dimensionless transformation is carried out using [PERSON]'s \(\pi\) theorem. Dimensionless Equations (9)-(11) can be obtained. \[\eta=f(\frac{\overline{d}}{r_{1}},\frac{n}{vr_{1}{}^{-1}},\varepsilon,\frac{ \mu}{\rho vr_{1}},\alpha,\frac{r_{2}}{r_{1}},\beta,\frac{h}{r_{1}},\frac{d}{r _{1}}), \tag{9}\]  
\[\eta_{0}=f(\frac{\overline{d}}{r_{1}},\frac{n}{vr_{1}{}^{-1}},\varepsilon, \frac{\mu}{\rho vr_{1}},\alpha,\frac{r_{2}}{r_{1}},\beta,\frac{h}{r_{1}},\frac {d}{r_{1}}), \tag{10}\]  
\[\frac{\Delta P}{\rho v^{2}}=f(\frac{\overline{d}}{r_{1}},\frac{n}{vr_{1}{}^{- 1}},\varepsilon,\frac{\mu}{\rho vr_{1}},\alpha,\frac{r_{2}}{r_{1}},\beta,\frac {h}{r_{1}},\frac{d}{r_{1}}). \tag{11}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number** & **Name** & **Symbol** & **Unit** & **Dimension** \\ \hline
1 & Average Inlet Particle Diameter & \(\overline{d}\) & m & L \\
2 & Rotating Speed & \(n\) & r/min & T\({}^{-1}\) \\
3 & Gas-Oil Ratio & \(\varepsilon\) & - & - \\
4 & Mixture Velocity & \(v\) & m/s & LT\({}^{-1}\) \\
5 & Mixture Density & \(\rho\) & kg/m\({}^{3}\) & ML\({}^{-3}\) \\
6 & Mixture Dynamic Viscosity & \(\mu\) & Pa\(\cdot\)s & ML\({}^{-1}\)T\({}^{-1}\) \\
7 & Number of Blades & \(\alpha\) & - & - \\
8 & External Diameter of Blades & \(r_{1}\) & m & L \\
9 & Internal Diameter of Blades & \(r_{2}\) & m & L \\
10 & Blade Angle & \(\beta\) & - & - \\
11 & Thickness of Breather & \(h\) & m & L \\
12 & Diameter of Vent Holes & \(d\) & m & L \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Influencing factors and dimensions.