## 3 Models for Gerotor Pumps
### State of the Art Gerotor Pump Models  
The use of 1D or lumped parameter models for the development of gerotor pumps can be justified by two arguments. First, due to the simplicity and low computing resources required by 1D simulations, a large number of model parameters can be modified in a short time and with manageable resources to analyze their influence. Second, the ability to integrate comparatively detailed pump models into complex systems that can only be simulated with 1D simulations. An example of such a complex multi-physical model is shown in the following. In Figure 3, an electric gerotor pump model is represented with the entire electric circuit, the electric motor and the pump gear set [28]. Based on the Figure 4, which illustrates the simplified user interface to change the simulation inputs, the working principle of the gerotor pump can be shortly explained. The gerotor pump is a specific type of internal gear pump and is characterized by the tooth contour consisting of circular arcs on the outer rotor, which is why it is also known as a generated rotor (gerotor) [33]. In the chambers formed between the teeth of the inner and outer gears, the fluid is transported from the suction side (colored in blue in Figure 4) to the pressure side (colored in red), while being pressurized by the volume reduction of the chambers. According to [34], the profile of a gerotor can be described by a small set of parameters, as well as the analytical description of the inlet and outlet geometries of the pump by the use of functions of the rotational angles. Figure 2: Requirements-based development of electric gerotor pump with decomposed requirements. In Simcenter Amesim, a lumped parameter approach to model multi-physical systems is used [28]. In the case of the gerotor pump component seen in Figure 3, it is considered as an assembly of variable control volumes representing the internal fluid volumes between the teeth of the inner and the outer gears, as schematically illustrated in Figure 4 and Figure 5. Mass conservation equations are numerically solved in each of these volumes (called chambers) to obtain the pressure dynamics inside them. Figure 4: Interface App for a parameter-based design of a gerotor pump in Simcenter Amesim. Figure 3: Simcenter Amesim model of an electric oil pump [28]. The chambers are connected to the pump inlet or outlet volumes through flow restrictions (orifices) of varying flow area (see Figure 5). The flow area depends on the rotational angle of the pump, the geometry of the gears and of the inlet and outlet ports. Figure 6 shows the evolution of one chamber volume over a complete rotation of the outer gear, as well as the flow areas corresponding to the inlet and outlet ports. The flow rate \(q_{\mathit{orif}}\) through the orifices connecting internal volumes to inlet and outlet ports is modelled using a well-known law as described in [35]:  
\[q_{\mathit{orif}}=C_{q}A\sqrt{\frac{2|\Delta P|}{\rho}} \tag{1}\]  
where  
* \(C_{q}\) is the flow coefficient (function of the flow regime). This value can be estimated from CFD calculations or measurements. The value is here set to 0.7, based on the authors' experience. * \(A\) is the flow area (function of chamber position and geometries of gears and ports). * \(\Delta P\) is the pressure drop across the orifice. * \(\rho\) is the oil density. Figure 5: Vectorial components structure in Simcenter Amesim with adjustable number of chambers. Figure 6: Hydraulic chamber volume with corresponding intersection areas to inlet and outlet port. Adjacent chambers are also connected together through laminar leakages occurring in the gap between the inner and outer gears' teeth (see Figure 5). Those leakages are modelled as a combination of Poiseuille and Couette laws in a rectangular channel, as described in [36]. The Poiseuille contribution is recalled in Equation (2):  
\[q_{leakP}=\frac{\Delta P\,b\,\delta^{3}}{12\,\mu\,l} \tag{2}\]  
where  
* \(\Delta P\) is the pressure drop across the leakage. * \(b\) is the gears' thickness.