## 2 Model of Friction by Heat Produced by Internal Gear Pump
### Source Analysis of Heat Generation in the Radial Friction Pair Between Gear Ring and Housing  
The clearance between the gear/ring and the crescent plate plays the role of high-pressure oil seal and oil suction and drainage port transport, and its leakage direction is opposite to the rotation direction. However, in actual operation, only a few sealing lines are used for sealing, which leads to a sudden increase in leakage flow at low speed. Conversely, the thin-walled oil film in the gap during high-speed rotation suppresses the leakage but increases the friction loss [29]. The motion of the friction pair between the gear/ring and the crescent plate is shown in Figure 4.  
Figure 4: Oil film sketch: (**a**) between gear ring and crescent plate moving friction pair; (**b**) between gear and crescent plate moving friction pair. The velocity of oil movement in the clearance between gear and crescent plate under different operating conditions can be governed by Equation (5):  
\[u_{1}=\frac{\Delta p_{2}/z_{1}}{2\mu s_{eg1}}\Big{[}\big{(}r_{g1}-r_{g2}\big{)}y_ {1}-y_{1}^{2}\Big{]}-v_{ag}(1-\frac{y_{1}}{r_{g1}-r_{g2}}) \tag{5}\]  
where \(z_{1}\) is the number of gear teeth involved in sealing; \(s_{eg1}\) is the tooth tip thickness of the gear; \(\mu\) is the dynamic viscosity of the hydraulic oil; \(y_{1}\) is the arbitrary distance between the gear tooth tip and the crescent plate; \(v_{ag}\) is the linear velocity at the gear tooth tip; \(r_{g1}\) is the inner radius of the crescent plate; \(\Delta p_{2}\) is the pressure difference between the suction and discharge ports; \(r_{g2}\) is the radius of the gear tooth tip. Equations (6) and (7) describe the leakage flow and volumetric loss between the gear and the friction pair of the crescent plate. \[Q_{2}=B\Bigg{[}\frac{\Delta p_{2}}{12\mu s_{eg1}z_{1}}\big{(}r_{g1}-r_{g2} \big{)}^{3}-\frac{v_{ag}\big{(}r_{g1}-r_{g2}\big{)}}{2}\Bigg{]} \tag{6}\]  
\[P_{Q_{2}}=B\Bigg{[}\frac{\Delta p_{2}{}^{2}}{12\mu s_{eg1}z_{1}}\big{(}r_{g1}- r_{g2}\big{)}^{3}-\frac{v_{ag}\big{(}r_{g1}-r_{g2}\big{)}}{2}\Bigg{]} \tag{7}\]  
The frictional loss between gear and friction pair of crescent plate can be expressed by Equation (8):  
\[P_{f2}=\int_{0}^{r_{g1}-r_{g2}}\mu\frac{\partial u_{1}}{\partial y}dy \tag{8}\]  
The velocity of oil movement in the gap between the tooth ring and the crescent plate under different operating conditions can be expressed by Equation (9):  
\[u_{2}=\frac{\Delta p_{2}/z_{2}}{2\mu s_{eg2}}\Big{[}(r_{e1}-r_{e2})y_{2}-y_{2} ^{2}\Big{]}-v_{eg}(1-\frac{y_{2}}{r_{e1}-r_{e2}}) \tag{9}\]  
where \(z_{2}\) is the number of teeth of ring teeth involved in sealing; \(s_{eg2}\) is the thickness of gear tip; \(y_{2}\) is arbitrary distance between ring tooth tip and crescent plate; \(v_{eg}\) is the linear velocity at the gear ring tooth tip; \(r_{e1}\) is the outer radius of tooth plate; \(r_{e2}\) is the radius of the gear ring tooth tip. The volumetric loss between the gear ring and the friction pair of the crescent plate is expressed by Equation (10), as follows:  
\[P_{Q_{3}}=B\Bigg{[}\frac{\Delta p_{2}{}^{2}}{12\mu s_{eg2}z_{2}}(r_{e2}-r_{e1} )^{3}-\frac{v_{eg}\big{(}r_{e2}-r_{e1}\big{)}}{2}\Bigg{]} \tag{10}\]  
The frictional loss between the gear ring and the friction pair of the crescent plate be governed by Equation (11):  
\[P_{f3}=\int_{0}^{r_{e2}-r_{e1}}\mu\frac{\partial u_{2}}{\partial y}dy \tag{11}\]  
Source Analysis of Heat Generation in the Friction Pair Between the Rotating Shift System and Crescent Plate  
The gear shaft of linear conjugate internal meshing gear pump is supported by deep groove ball bearings, and both ends of the sliding friction pair formed by the journal and the wear-resistant bearing sleeve. As shown in Figure 5. The friction heat generation source consists of two parts: the rolling friction of the bearing and the contact friction of the sliding pair.