# Thermal-Hydraulic Modeling of Oil-Inmersed Motor Pump
## 2 Mathematical Analysis
### Analytical Results  
Due to the unfavorable flow characteristics of fluids at low speeds and high pressures, this paper aims to investigate the temperature variations of the leakage port and inlet at 1450 rpm under different outlet pressure conditions, as well as the temperature changes in the leakage port and inlet at different speeds when the outlet pressure is 6 MPa.  
As shown in Figure 6(a), an increase in the outlet pressure is observed to cause a gradual rise in the temperature of the leakage port. An increase in the temperature of the leakage port from a minimum of 25.1 to 41.2 \({}^{\circ}\)C is attributed to two factors. Firstly, an elevated viscous friction loss of oil is caused by an increase in pressure, resulting in a rise in the temperature of the oil that leaks into the motor pump. Simultaneously, an increase in motor power and current results in an elevated motor loss, further contributing to the rise in oil temperature at the leakage port. Nevertheless, the temperature of the inlet node remains constant., as the oil absorption rate at the inlet of the motor pump is significantly higher than the leakage rate. While some of the high-temperature oil may leak into the oil inlet, it does not significantly affect the overall temperature of the oil tank, and consequently, the temperature at the inlet remains relatively stable.  
As shown in Figure 6(b), the temperature of the leakage port exhibits a gradual increment with escalating rotational speed. The temperature of the leakage port increases  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Unit** & **Value** \\ \hline \(B\) & T & 0.68 \\ \(d\) & mm & 8.3 \\ \(l\) & mm & 11.5 \\ \(h\) & mm & 79 \\ \(\sigma_{h}\) & W/(kg\(\cdot\)Hz\(\cdot\)T\({}^{2}\)) & 0.136 \\ \(k_{ex}\) & W/(kg\(\cdot\)Hz\({}^{2}\cdot\)T\({}^{2}\)) & 0.005 \\ \(N\) & & 34 \\ \(\rho_{c}\) & \(\Omega\)\(\cdot\)m & 0.0172 \\ \(E\) & V & 270 \\ \(D_{cy}\) & mm & 39 \\ \(t\) & mm & 1 \\ \(l_{cy}\) & mm & 79 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Main parameters of the motor.  
Figure 7: Temperature comparison under different loading conditions: (**a**) leakage and inlet temperature and under different load pressures; (**b**) leakage and inlet temperature under different rotation speeds.  
gradually from an initial value of 25.7 \({}^{\circ}\)C to 85.4 \({}^{\circ}\)C, which can be attributed to two primary factors. As the rotational speed increases, the mixing loss of cylinder block, piston, and rotor intensifies, resulting in elevated levels of viscous friction loss within the pump chamber and motor chamber oil, thereby contributing to the increase in temperature of the leakage port. The rise in motor power and current translates to an amplified motor loss, consequently leading to a further escalation in oil temperature at the leakage port. Notably, the inlet temperature remains relatively unchanged owing to the increased heat exchange rate occurring between the inlet oil and exterior oil per unit time, which in turn preserves thermal equilibrium between the inlet temperature and the tank temperature.