## 2 Overall Design of Centralized Lubrication and Waste Oil Recovery System for Wind Turbines
### Design of Waste Grease Recycling System
#### 2.2.1 Establish the Mathematical Model of Vacuum Pumping in the Bearing Cavity  
Vacuum differential equation of negative pressure tank:  
\[\sum Q-SP=V\frac{dp}{dt} \tag{1}\]  
In the formula, \(\sum Q=Q_{1}+Q_{2}+Q_{3}+Q_{4}+Q_{5}\) represents the amount of gas entering the vacuum system of the bearing cavity per unit time, \(Q_{1}\) is the amount of gas transported to the bearing cavity by the external atmosphere per unit time due to leakage, \(Q_{2}\) is the amount of gas analysis absorbed by the surface of the inner cavity of the unit time bearing, \(Q_{3}\) is the amount of gas that the cavity wall of the unit time bearing penetrates into the system, \(Q_{4}\) is the amount of gas delivered to the vacuum system of the inner cavity of the bearing in a unit time and \(Q_{5}\) is the amount of gas returned by the unit time to the vacuum system. \(SP\) represents the amount of gas discharged from the vacuum system in the inner cavity of the unit time bearing and \(\sum Q-SP\) represents the increment of the amount of gas in the vacuum system of the inner cavity.  
It can be obtained by conversion:  
\[t=-V\frac{\ln p}{s}+C \tag{2}\]  
Theoretical calculations of vacuums assume that the gas is an ideal gas with Newtonian fluid flow and neglects the heat exchange due to gas motion. When the air in the bearing cavity is pumped, the air flow period interval is set to be very short and the gas motion in the bearing cavity can be approximately treated as a one-dimensional steady flow. Suppose that the air in a cavity of length \(x\) and width \(y\) is flowing smoothly in the--direction, the contact area is \(A\) and the momentum equation of the gas motion is:  
\[\int\rho VdA=\overline{F} \tag{3}\]  
In the formula, \(V\)--gas volume, \(F\)--gas external force.  
Newton friction formula is:  
\[\tau=\mu\frac{du}{dy} \tag{4}\]  
where \(x\)--viscous shear stress on the inner side, \(dp\)--viscous coefficient.  
Ignoring the influence of weight, the resultant force on the control body is zero when the gas flows steadily. Assuming that the pressure distribution on both ends is uniform, the pressure difference between the two ends in the \(x\) direction is \(V\) and the equilibrium equation of the gas control body is:  
\[pdy-(dp+p)dy=\tau dx-(d\tau+\tau)dx \tag{5}\]  
It can be understood as:  
\[dpdy=d\tau dx \tag{6}\]  
Arranged gas volume flow:  
\[Q=\int_{0}^{h}ubdy=\frac{bh^{3}\Delta p}{12\mu} \tag{7}\]  
When the bearing cavity is vacuum, the gas quality of the bearing cavity is changed by the negative pressure tank as follows:  
\[\frac{dM}{dt}=\left(-\frac{\rho bh^{3}\Delta p}{12\mu}+\frac{\rho b_{1}h_{1} ^{3}\Delta p}{12\mu l_{1}}\right) \tag{8}\]\(\Delta p\)--differential pressure between bearing inner cavity and negative pressure tank, \(M\) --air quality, \(\rho\) --air density, \(\mu\) --air viscosity, \(l_{1}\), \(b_{1}\), \(h_{1}\) --length, width and height of leakage area, \(\Delta p_{1}\) --pressure difference between inner cavity and external atmospheric pressure of bearing.  
Bearing cavity volume change rate:  
\[\frac{dV}{dt}=-AV_{A} \tag{9}\]  
\(V\)--bearing cavity volume, \(A\) --cross section area from bearing drain to tank passage, \(V_{A}\)--speed of movement of waste grease.  
Variation of pressure in bearing cavity:  
\[\frac{dp}{dt}=-\frac{kp}{\rho V^{2}}(MAV_{A}-\frac{\rho bh^{3}V\Delta p}{12\mu l }+\frac{\rho b_{1}h_{1}^{3}V\Delta p_{1}}{12\mu l_{1}} \tag{10}\]  
During the solution, time is discretized into minute time units \(\Delta t\). During this time, the initial pressure is used to determine the internal cavity pressure \(p\), air quality, density and other parameters to determine whether the residual oil in the bearing cavity can be reliably discharged.