# Numerical Analysis via Mixed Inverse Hydrodynamic
## 3 Results and Discussion
### Effect of Thermal Analysis  
#### 3.2.1 Effect of Rod Velocity and Sealed Pressure  
Figure 5 shows the distribution of the oil film at each speed when the seal pressure is 35 MPa and the ambient temperature is 25 \({}^{\circ}\)C. As the rod speed increases, a thick film formed owing to improved lubrication performance, and it can be seen that the distribution of the oil film in the outstroke is thicker than that in the instroke; this is because the fluid moves in the direction of seal pressure to atmospheric pressure for the outstroke but moves in the opposite direction for the instroke. Figure 6 shows the amount of leakage according to speed. Here, the reverse pumping rate is the flow rate that occurs from the oil side to the air side and is calculated as the flow rate difference between the outstroke and instroke.  
The outstroke flow direction is defined from the oil side to the air side and the Couette flow--caused by the rod surface--and the Poiseuille flow--increased by the pressure gradient at the seal inlet--are in the same direction. The instroke flow direction is defined as the oil side on the air side, opposite to the outstroke flow direction. The Couette flow has a positive value, and the Poiseuille flow has a negative value. In general, because the Couette flow is larger than the Poiseuille flow, the instroke flow has a positive value. This calculation was performed on the assumption that a thin oil film also exists on the air side [18]. It can be observed that the reverse pumping rate increases as the load speed increases. As the load speed increases, the flow rate increases for both instroke and outstroke because a thick film is formed; however, the increase rate is larger because the instroke is larger and thus the reverse pumping rate increases as the load speed increases. An increase in sealed pressure also increases the reverse pumping rate because a greater sealed pressure creates a greater pressure gradient within the contact area. Figure 7 shows  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter and Symbol** & **Dimensions and Data** \\ \hline \hline Seal material & HNBR \\ \hline Elastic modulus, \(E\) & 43 MPa \\ \hline Poisson’s ratio, \(\
u\) & 0.499 \\ \hline Rod diameter, \(D\) & 25 mm \\ \hline Empirical friction coefficient, \(f\) & 0.25 \\ \hline Seal RMS roughness, \(\sigma\) & 0.3 \(\mu\)m \\ \hline Sealed fluid & API GL-4 \\ \hline Viscosity, \(\mu\) & 0.0771 Pa-s at 25 \({}^{\circ}\)C \\ \hline Viscosity at \(T_{0}\), \(\mu_{0}\) & 0.4690 Pa-s at \(-35\)\({}^{\circ}\)C \\ \hline Viscosity–temperature constant, \(a\) & 0.0301 \\ \hline \(\beta\) & 1 \\ \hline \(\gamma\) & 1.75 \\ \hline Stroke length, \(L\) & 140 mm \\ \hline Sealed pressure, \(P_{s}\) & 20, 25, 30, 35 MPa \\ \hline Rod velocity, \(u\) & 0.1, 0.4, 0.7, 1.0 m/s \\ \hline Ambient temperature, \(T_{a}\) & 25, 55, 85, 115 \({}^{\circ}\)C \\ \hline Reference temperature, \(T_{0}\) & \(-35\)\({}^{\circ}\)C \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the numerical model.  
the reverse pumping rate according to sealing pressure and rod velocity; it can be seen that the reverse pumping rate increases as the sealing pressure and rod velocity increase.