# Study on Temperature Characteristics of Lubrication Film of Valve Plate Pair in Axial Piston Pumps
## 2 Modeling
### Modeling of Film Temperature  
To calculate the leakage of the wedge film in Figure 6, the wedge film between the cylinder block and valve plate can be divided into the parallel part and the wedge part; therefore, the leakage of the wedge film includes the leakage \(Q_{\rm P}\) of the parallel part and the leakage \(Q_{\rm w}\) of the wedge part [28]. The leakage \(Q_{\rm P}\) of the parallel part is  
\[Q_{\rm P}=\frac{\alpha_{\rm 1}h_{\rm 1}^{3}\Delta P}{12\mu C_{\rm e}\ln(r_{ \rm i}/r_{\rm j})} \tag{15}\]  
where \(\alpha_{\rm 1}\) is the central angle of the parallel part, \(h_{\rm 1}\) is the height of the parallel gap, \(\Delta P\) is the difference between the working pressure \(P_{\rm P}\) and the oil pressure \(P_{\rm out}\) in the shell, \(\mu\) is the dynamic viscosity of oil, \(C_{\rm e}\) is the correction factor and the value is 2.2, and \(r_{\rm i}\) and \(r_{\rm j}\) are the outer and inner diameters of the calculated parallel part, respectively. For the inner sealing belt, \(r_{\rm i}=r_{\rm 2}\) and \(r_{\rm j}=r_{\rm 1}\); for the outer sealing belt, \(r_{\rm i}=r_{\rm 4}\) and \(r_{\rm j}=r_{\rm 3}\); for the auxiliary support belt, \(r_{\rm i}=r_{\rm 6}\) and \(r_{\rm j}=r_{\rm 5}\). Figure 4: Schematic diagram of the oil film shape according to the three-point film thickness. The leakage \(Q_{\rm w}\) of the wedge part is as follows:  
\[\left\{ \begin{array}{l}Q_{\rm w}=\frac{\Delta P/6\mu C_{\rm e}}{\lambda_{1}-B_{ 1}-C_{1}-D_{1}}\\ A_{1}=(b^{2}/a^{3})\ln(r_{1}-r_{\rm j})\\ B_{1}=(b^{2}/a^{3})\ln\bigl{[}(r_{\rm i}-b/a)/(r_{\rm j}-b/a)\bigr{]}\\ C_{1}=(b/a^{2})(r_{\rm i}^{3}-r_{\rm j}^{3})\\ D_{1}=(1/2a)(r_{\rm i}^{2}-r_{\rm j}^{2})\end{array} \right. \tag{16}\]  
where \(a=1/2\bigl{[}tan^{2}\alpha h_{2}(\alpha_{1}+sin\alpha_{1})\bigr{]}\), \(b=tan^{3}\alpha\sin(\alpha_{1}/2)\), and \(h_{2}\) is the center height of the wedge film. The power loss \(E_{\rm L}\) caused by the film leakage between the cylinder block and the valve plate is as follows:  
\[E_{\rm L}=(Q_{\rm P}+Q_{\rm W})P_{\rm P} \tag{17}\]  
The tangential friction resistance results in viscous friction power loss in the lubrication film because of the high-speed rotation of the cylinder block relative to the valve plate. In  
Figure 5: The calculation process schematic diagram for the wedge angle and azimuth angle. Figure 6: Schematic diagram of the wedge gap of the valve plate pair. order to simplify the calculation, the oil in the valve plate film is regarded as the flow in the parallel gap, as shown in Figure 7. And the friction power loss \(E_{\text{V}}\) is  
\[E_{\text{V}}=\int_{\theta_{1}}^{\theta_{2}}\int_{\tau_{1}}^{\tau_{1}}\frac{\mu \omega^{2}r^{3}}{h}drd\theta \tag{18}\]  
where \(\omega\) is the angular velocity of the cylinder block. Thermal interaction occurs between the lubrication film and the contact parts including the valve plate, cylinder block, and the inlet oil after cooling. In Figure 8, the heat interaction occurs between the oil film and the valve plate, and its heat interaction rate \(Q_{\text{vlv}}\) is as follows:  
\[Q_{\text{vlv}}=\frac{T_{1}-T_{0}}{\frac{1}{\lambda_{0}A_{\text{vlv}}}+\frac{ L}{k_{1}A_{\text{vlv}}}+\frac{1}{\lambda_{1}A_{\text{vlv}}}} \tag{19}\]  
where \(T_{1}\) is the oil film temperature, \(T_{0}\) is the starting temperature of the valve plate, \(\lambda_{0}\) is the heat transfer coefficient between the oil film and the valve plate, \(\lambda_{1}\) is the heat transfer coefficient between the oil film and the cylinder block, \(A_{\text{vlv}}\) is the heat transfer area between the valve plate and the oil film, \(k_{1}\) is the heat transfer coefficient of the valve plate material, and \(L\) is the thickness of the valve plate. The heat interaction rate \(Q_{\text{cli}}\) between the oil film and the cylinder block is as follows:  
\[Q_{\text{cli}}=\frac{T_{1}-T_{0}}{\frac{1}{\lambda_{0}A_{\text{cli}}}+\frac{ L_{1}}{k_{2}A_{\text{cli}}}+\frac{1}{\lambda_{1}A_{\text{cli}}}} \tag{20}\]  
Figure 8: Heat transfer model in the oil film.