# Thermal-Hydraulic Modeling of Oil-Inmersed Motor Pump
## 2 Mathematical Analysis  
### Heat Generation Mechanism  
The loss of stator and rotor resulting from the fluctuation of magnetic flux is commonly referred to as core loss. [PERSON] conducted a comprehensive investigation on the origin and constituents of core loss and introduced the separation theory of core loss, which posits that core loss is composed of hysteresis loss, eddy current loss, and additional loss.  
As shown in Figure 2, the variation in magnetic field within the iron core will trigger the induction of current.  
Based on Ohm's law and the law of electromagnetic induction, the power loss due to eddy currents, P, can be expressed as follows:  
\[dp_{b}=\frac{E^{2}}{R} \tag{1}\]  
Figure 2: Schematic diagram of a steel sheet eddy current.  
where \(R=\rho_{1}\frac{2h}{ldx}\), \(E=\sqrt{2}\pi f\phi\), \(\phi=2 Bxh\).  
By substituting the values of \(R\), \(E\) and \(\phi\) into Equation (1), the expression for eddy current power loss \(p_{b}\) can be derived.  
\[dp_{b}=\int_{0}^{\frac{d}{2}}dp=\frac{4\pi^{2}f^{2}B^{2}lh}{\rho_{1}}dx \tag{2}\]  
Eddy current loss of the iron core \(P_{b}\) as shown in Equation (3).  
\[P_{b}=\frac{\pi^{2}}{6\rho_{1}}(fB)^{2}d^{3}lh \tag{3}\]  
where \(\rho_{1}\) is the resistivity of the core, \(f\) is the frequency of the magnetic field, \(B\) is the strength of the magnetic field.  
The hysteresis loss is due to the hysteresis effect, and the hysteresis loss \(P_{h}\) is shown in Equation (4).  
\[P_{h}=\sigma_{h}fB^{2} \tag{4}\]  
The extra loss \(P_{ex}\) is shown in Equation (5) [22].  
\[P_{ex}=K_{ex}(Bf)^{1.5} \tag{5}\]  
where \(\sigma_{h}\) is the hysteresis loss coefficient and \(K_{ex}\) is the additional loss coefficient.  
At the same time, the wires that flow through the current also generate a lot of heat, the direct current loss \(P_{Al}\) of the wire is shown in Equation (6).  
\[P_{Al}=\sum{(l^{2}R)} \tag{6}\]  
The motor consists of a stator and a rotor, which are separated by an oil-filled gap. As shown in Figure 3, the rotor and the permanent magnet undergo rotational motion, leading to the generation of viscous friction loss.  
The mathematical model of viscous friction loss \(P_{mj}\) of rotor under low-speed laminar flow and high-speed turbulence is shown in Equation (7) [10].  
\[P_{mj}=\frac{\pi\mu\omega^{2}l_{cy}D_{cy}^{3}}{4t} \tag{7}\]  
where \(l_{cy}\) and \(D_{cy}\) are the length and diameter of the rotor, respectively, and \(t\) is the spacing of the air gap.  
Figure 3: Schematic diagram of fluid domain between stator and rotor.  
The total motor power loss \(P\) is shown in Equation (8).  
\[P=P_{b}+P_{h}+P_{ex}+P_{AI}+P_{mj} \tag{8}\]  
The connector between the high-pressure end of the pump and the low-pressure end of the motor results in throttling power loss when oil flows through it, as shown in Figure 4. The flow of oil can be simplified as the flow between annular gaps.  
The power loss \(P_{2n}\) at the connector is shown in Equation (9).  
\[p_{2n}=q_{V}\Delta p=\frac{12\mu L/q_{V}^{2}}{\pi d/\mu_{2}{}^{3}} \tag{9}\]  
where \(q_{v}\) is the flow rate through the connector, \(d_{f}\) is the diameter of the inner circle, \(L_{f}\) is the gap length, \(h\) is the gap width.