## 3 Models for Gerotor Pumps
### Modeling the Viscous Friction Heating  
According to [38], the frictional power generated in the journal bearing is a power loss and is almost completely converted into heat. As already shown by [11], the radial friction surface on the outer gear is the dominant friction power source in regards to the power loss of the pump. Therefore, its influence and the prevailing conditions are focused. In order to investigate both statements and the influence of bearing friction on the temperature of the pump and the resulting friction power, a gerotor pump was modified so that a temperature sensor could be installed in the gap between the housing and the outer gear. The specifications of the gerotor pump are shown in Appendix A. Afterwards, the pump was tested in a climate test cell at various operating points between \(-40\)\({}^{\circ}\)C and \(+20\)\({}^{\circ}\)C. The gear set and the modified pump housing with a mounted temperature sensor can be seen in Figure 9.  
Examples of the measurements are shown in Figure 10, showing how the temperature in the gaps changes significantly in a short time (\(<\)60 s) and how strongly they influence the torque on the pump shaft. Figure 10: Cold start behavior of the gerotor pump with temperature in the gap and corresponding shaft torque. Figure 9: Gerotor pump with additional temperature sensor in gap between outer gear and pump housing. Measurements are shown for a starting temperature of \(-36\)\({}^{\circ}\)C (blue curves) and \(-25\)\({}^{\circ}\)C (orange curves). The almost steady end temperature, approximately \(-25\)\({}^{\circ}\)C with an increase of \(+11\) K, or approximately \(-19\)\({}^{\circ}\)C with an increase of \(+6\) K, also has a significant impact on the required pump torque. At \(-40\)\({}^{\circ}\)C for example, the torque reduces from initially \(6\) Nm (after hydrodynamic lubrication is established) to \(2\) Nm once the steady-state temperature is reached. For the further modeling, it is important to focus on the prevailing flow conditions. If the rotating annular gap is regarded as a \"sliced\" channel with a stationary and a moving wall, the simplified result is Figure 11. The heat generated from the frictional power is partially transferred to the heat sinks (the fixed wall of the housing and the moving wall of the outer rotor). The prevailing flow with a fixed outer cylinder and a rotating inner cylinder is also referred to as Taylor-Couette-Poiseuille flow, where a Taylor-Couette flow exists with an axial Poiseuille flow [41]. The temperature and thereby the kinematic viscosity of the oil varies significantly \(\
u\approx 5000\) cSt (at \(-30\)\({}^{\circ}\)C) and \(\
u\approx 10\) cSt (at \(100\)\({}^{\circ}\)C), which is why the flow indicators also vary. As already shown in Section 3.1, the Reynolds number varies in a range of [0.04; 20] and the Taylor number according to the geometric boundary conditions of the pump from Appendix A due to the gap ratio (inner diameter/outer diameter > 0.99) indicates that no more Taylor vortices occur. Therefore, a laminar Couette-Poiseuille flow can be assumed. If the flow conditions are studied by dimensional analysis, four dimensionless products can be determined using a dimensional matrix (see Appendix B Table A2) shown in Equations (7)-(10). The dimensional matrix is constructed from the dimension exponents of the following dimensional variables: density \(\rho\), dynamic viscosity \(\eta\), thermal conductivity \(\lambda\), thermal diffusivity \(\alpha\), velocity of the fluid \(u\) and the geometric variables of the gap height \(h\) and the gap length \(L\). The resulting four dimensionless products are  
\[\pi_{1}=Re=\frac{\rho uh}{\eta} \tag{7}\]  
\[\pi_{2}=Pr=\frac{c_{p}\eta}{\lambda} \tag{8}\]  
\[\pi_{3}=\frac{L}{h} \tag{9}\]  
\[\pi_{4}=Nu=\frac{\alpha h}{\lambda} \tag{10}\]  
The dimensionless Nusselt number can be used to describe the heat exchange between a flowing fluid and a fixed solid [42]. The Nusselt number can be determined as a function of the Reynolds and Prandtl numbers.