## 3 Models for Gerotor Pumps
### State of the Art Gerotor Pump Models
* \(\delta\) is the clearance between the inner and outer teeth. * \(\mu\) is the oil absolute viscosity. * \(l\) is the gap length. Note that here the gap between gears' teeth is assumed to be known, and constant with the gears' position. In addition, Equation (2) for the Poiseuille flow, as well as the equation for the Couette flow assume a laminar flow: it is possible to estimate some values of the Reynolds number \(Re=UD_{h}/\
u\) (\(l\) being the fluid relative velocity in the gap, \(D_{h}\) the hydraulic diameter of the rectangular gap and \(\
u\) the kinematic viscosity) to verify that this hypothesis is valid for the current study. Let us consider the following values for calculating the Reynolds number in the gap between the gears:  
* \(U\approx 1\,\mathrm{m/s}\) (roughly estimated from the outer diameter of the outer gear given in Table 1 and a shaft speed of 3000 rpm). * \(D_{h}=0.2\,\mathrm{mm}\) (from gear width in Table 1 and a gap of 0.1 mm). * \(\
u\approx 5000\,\mathrm{cSt}\) (at \(-30\,\mathrm{\SIUnitSymbolCelsius}\)) and \(\
u\approx 10\,\mathrm{cSt}\) (at \(100\,\mathrm{\SIUnitSymbolCelsius}\)), for standard lubricating oil. One can then obtain a Reynolds number in the range [0.04; 20], which satisfies the laminar hypothesis. Knowing the flow rates entering or exiting the different chambers will allow us to compute the overall pump flow rate and, as a consequence, the volumetric efficiency. To obtain the total efficiency of the gerotor unit, it is also necessary to evaluate the friction inside the pump, since it will impact the hydro-mechanical efficiency. The friction characteristic of a gerotor pump can be compared with a conventional lubricated radial journal bearing, as shown in Sections 3.3 and 4. Depending on the Gumbel-Hersey number (GHZ), which is calculated by the dynamic viscosity \(\eta\), the angular velocity \(\omega\) and the specific load \(\overline{\rho}\), the friction number is calculated and the three ranges of boundary lubrication, mixed lubrication and hydrodynamic lubrication can be defined [37]. The GHZ is represented in Equation (3) [38]. In the context of the method developed in this paper, only hydrodynamic lubrication is considered, although other friction states may occur at very low pump speeds, high pump pressures or unfavorable viscosity-to-load ratios. \[GHZ=\frac{\eta\omega}{\overline{\rho}} \tag{3}\]  
The chosen approach for modelling the viscous friction in the gerotor pump has similarities with the one described in [22]: the hydro-mechanical friction torque on the shaft \(M_{\mathit{vrfic}}\) is decomposed into four terms:  
\[M_{\mathrm{vrfic}}=M_{\mathrm{igs}}+M_{\mathrm{egs}}+M_{\mathrm{egh}}+M_{ \mathrm{brg}} \tag{4}\]  
with  
* \(M_{\mathrm{igs}}\) the viscous friction torque between the inner gear sides and the housing. * \(M_{\mathrm{egs}}\) the viscous friction torque between the outer gear sides and the housing. * \(M_{\mathrm{egh}}\) the viscous friction torque between the outer gear cylindrical surface and the housing. \(M_{\text{brg}}\) the viscous friction torque at the shaft bearing. The last two torques are modelled as in [22], using [PERSON]'s equation [39]. The use of this equation is based on the fact that the flow distribution in the gap between the cylinders is linear in the radial direction, because of the small gap compared to the cylinders' diameters and because the leakages at both cylinders' ends are neglected. This assumption will be further used when discussing thermal aspects in the oil film in Section 3.3. One limitation that can be noted here is that possible eccentricities of the cylinders are not taken into consideration, even though they can affect the viscous friction. Regarding the friction between gear sides and housing, a different method than the one presented in [22] is developed. The modelling approach described in [22] is an analytical one.