# Thermal-Hydraulic Modeling of Oil-Inmersed Motor Pump
## 2 Mathematical Analysis
### Heat Transfer Analysis  
This paper employs the oil-immersed motor pump as a controlled object to regulate the energy conservation process, as shown in Figure 5.  
According to the first law of thermodynamics, the calculation formula is shown in Equation (10) [10]. \[ \begin{array}{l}\frac{dT}{dt}=\frac{1}{c_{p}m}[\sum\dot{m}_{in}(h_{in}-h)+ \dot{Q}+mT\pi_{p}v\frac{dp}{dt}]\\ h_{in}-h=\overline{c}_{p}(T_{in}-T)+(1-\overline{\alpha}_{p}\overline{T}) \overline{v}(p_{in}-p)\end{array}  \tag{10}\]  
The figures depicting the heat transfer model of oil are presented in Figures 1 and 6. The motor pump model consists of five fluid nodes and one mass node; \(T_{p}\) is the volume of  
Figure 4: Diagram of connector. Figure 5: Control volume. the outlet fluid, \(T_{t}\) is the volume of the intake fluid. \(T_{b}\), \(T_{j}\) and \(T_{l}\) are the volumes of the leakage fluid, and \(T_{w}\) is the shell of the motor pump. The comprehensive thermodynamic calculation model of the pump is elaborately explained in [PERSON]'s [10] article. Nevertheless, the focus of this paper is limited to the presentation of thermodynamic equations pertaining to the connector and leakage port nodes. Oil with a flow rate of \(Q_{el}\) is introduced into the motor via the connector located between the motor and the pump. The temperature of the oil is affected by the power loss occurring at the connector and subsequently rises to \(T_{j}\), as shown in Equation (11). \[ \begin{array}{ll}\frac{dT_{j}}{dt}=&\frac{1}{c_{p}m_{l}}[\rho Q_{el}(c_{p} \big{(}T_{b}-T_{j}\big{)}+\big{(}1-\alpha_{p}\big{(}T_{b}+T_{j}\big{)}/2\big{)} v\big{(}p_{b}-p_{j}\big{)}\big{)}\\ &-k_{fw}A_{fw}\big{(}T_{j}-T_{w}\big{)}+p_{zm}+T_{j}\alpha_{p}V_{j}\frac{dp_{ j}}{dt}]\end{array}  \tag{11}\]  
The motor end receives an inflow of oil which is subsequently discharged via the leakage oil port. A considerable amount of heat is generated due to a succession of power losses occurring at the motor end, resulting in an increase in temperature at the leakage oil port, denoted as \(T_{l}\) in Equation (12). \[ \begin{array}{ll}\frac{dT_{l}}{dt}=&\frac{1}{c_{p}m_{l}}[\rho Q_{el}(c_{p} \big{(}T_{j}-T_{l}\big{)}+\big{(}1-\alpha_{p}\big{(}T_{j}+T_{l}\big{)}/2\big{)} v\big{(}p_{j}-p_{l}\big{)}\big{)}\\ &-k_{fw}A_{fw}(T_{l}-T_{w})+P+T_{l}\alpha_{p}V_{l}\frac{dp_{l}}{dt}]\end{array}  \tag{12}\]  
The thermodynamic modeling of the oil-immersed motor pump is shown in Equation (13). \[\begin{array}{ll}\left\{ \begin{array}{ll}\frac{dT_{l}}{dt}=&\frac{1}{c_{p} m_{l}}\big{[}\rho Q_{il}\big{(}c_{p}\big{(}T_{p}-T_{l}\big{)}+\big{(}1- \alpha_{p}\big{(}T_{p}+T_{l}\big{)}/2\big{)}v\big{(}p_{p}-p_{l}\big{)}\big{)} \\ &+\frac{30}{30}c_{p}\rho\omega D(T_{in}-T_{l})+T_{l}\alpha_{p}V_{l}\frac{dp_{ l}}{dt}]\\ \frac{dT_{p}}{dt}=&\frac{1}{c_{p}m_{l}}\big{[}\frac{30}{\pi}\rho\omega D\big{(} c_{p}\big{(}T_{t}-T_{p}\big{)}+\big{(}1-\alpha_{p}\big{(}T_{p}+T_{l}\big{)}/2 \big{)}v\big{(}p_{t}-p_{p}\big{)}\big{)}\\ \frac{dT_{b}}{dt}=&\frac{30}{\pi}D\omega\big{(}P_{p}-P_{t}\big{)}+L_{p}+T_{p} \alpha_{p}V_{p}\frac{dp_{p}}{dt}]\\ \frac{dT_{b}}{dt}=&\frac{1}{c_{p}m_{l}}\big{[}\rho Q_{el}\big{(}c_{p}\big{(}T _{p}-T_{b}\big{)}+\big{(}1-\alpha_{p}\big{(}T_{p}+T_{b}\big{)}/2\big{)}v\big{(} p_{p}-p_{b}\big{)}\big{)}\\ &-k_{fw}A_{fw}(T_{b}-T_{w})+P_{c}+T_{b}\alpha_{p}V_{b}\frac{dp_{b}}{dt}]\\ \frac{dT_{j}}{dt}=&\frac{1}{c_{p}m_{j}}\big{[}\rho Q_{el}\big{(}c_{p}\big{(}T _{p}-T_{j}\big{)}+\big{(}1-\alpha_{p}\big{(}T_{b}+T_{j}\big{)}/2\big{)}v\big{(} p_{b}-p_{j}\big{)}\big{)}\\ \frac{dT_{l}}{dt}=&\frac{-k_{fw}A_{fw}(T_{j}-T_{w})+p_{zm}+T_{j}\alpha_{p}V_{j} \frac{dp_{j}}{dt}]}{c_{p}m_{l}}\big{[}\rho Q_{el}\big{(}c_{p}\big{(}T_{j}-T_{ l}\big{)}+\big{(}1-\alpha_{p}\big{(}T_{j}+T_{l}\big{)}/2\big{)}v\big{(} p_{j}-p_{l}\big{)}\big{)}\\ &-k_{fw}A_{fw}(T_{l}-T_{w})+P+T_{l}\alpha_{p}V_{l}\frac{dp_{l}}{dt}]\\ \frac{dT_{w}}{dt}=&\frac{1}{c_{w}m_{w}}\Big{[}k_{fw}A_{fw}\big{(}T_{b}+T_{j}+T _{l}-3T_{w}\big{)}\\ &-k_{wa}A_{wu}((T_{w}+273.15)^{4}-(T_{a}+273.15)^{4})\Big{]}\end{array} \right. \tag{13}\]  
Figure 6: Thermodynamic nodal model.