# A Virtual Prototype for Fast Design and Visualization of Gerotor Pumps
## 3 Methodology
### Fluid Dynamics Module  
We discretize the flow domain in several control volumes to obtain a lumped parameter model of the pump, as shown in Figure 6. We assume the fluid properties within each control volume (CV) to be homogeneous, but not constant in time, effectively treating each control volume as the basic domain of simulation. Notice that, as the pump rotates, the geometry of the control volumes changes; therefore, the model requires a constant update of the geometric calculations for each control volume (area, perimeter) as the position of the pump changes (see Figure 2). By the principle of the conservation of mass and energy, along with [PERSON]'s transport theorem, it is possible to derive an expression for the change of pressure within a control volume:  
\[\frac{dp}{dt}=\frac{\beta_{eff}(p)}{V(\theta)}\bigg{(}\sum Q_{i}-\omega\frac {dV}{d\theta}\bigg{)} \tag{4}\]  
We omit the derivation of such an expression, since it is beyond the scope of our paper, the interested reader can find a thorough explanation in Ref. [11]. The net flowrate that flows through the boundary of a control volume needs to be calculated in order to integrate Equation (4), and since the volume and volume derivative at angle \(\theta\) are provided by the  
Figure 5: Locus of the internal profile curve \(C\). Figure 6: Control Volume discretization of the gerotor pump. geometric module. We consider two types of flows through the boundary of a control volume (Figure 7):  
1. _Input/Output flow:_ fluid flowing from the input port to the inside of the pump (charge) or from the inside of the pump to the output port (discharge). 2. _Fluid leak flow:_ fluid flowing from one chamber to another due to imperfect sealing that results from manufacturing defects and design constraints. The flow between the control volume and the input/output port is modeled as the flow through a variable geometry orifice subject to a difference in pressure. The pressure at the input and output ports is fed to the numerical model with data that were collected from the experimental testing bench, while the control volume pressure varies according to Equation (4). The flowrate in this situation can be obtained as:  
\[Q_{in}=C_{d}A_{i,int}\sqrt{\frac{2(P_{i}-P_{in})}{\rho_{eff}}} \tag{5}\]  
\[Q_{out}=C_{d}A_{i,out}\sqrt{\frac{2(P_{i}-P_{out})}{\rho_{eff}}} \tag{6}\]  
Notice that, since a control volume only interacts with one of the ports at any given time, the net port flowrate \(Q\) for a control volume is equal to \(Q_{in}\) or \(Q_{out}\), depending on the position of the control volume at the time of analysis. The calculation of the discharge coefficient \(C_{d}\) depends on the value of the Reynold's number \(Re\) and the hydraulic diameter \(D_{h}\) at such a time. The hydraulic diameter \(D_{h}\) and the Reynold's number \(Re\) are calculated, as follows:  
\[D_{h}=\frac{4A_{i}(\theta)}{P_{i}(\theta)} \tag{7}\]  
\[R_{e}=\frac{D_{h}}{\upsilon}\sqrt{\frac{2\Delta P}{\rho_{eff}}} \tag{8}\]  
Finally, the discharge coefficient \(C_{d}\) is estimated while using an experimental expression (Ref. [11]):  
\[C_{d}=C_{d\max}\tanh\left(\frac{2R_{e}}{R_{crit}}\right) \tag{9}\]  
Figure 7: Types of flows through the boundary of a control volume. where \(C_{d\max}\) is the maximum discharge coefficient and \(R_{cerit}\) is the critical Reynolds number, which indicates the transition between laminar and turbulent regime. Values for constants \(C_{d\max}\) and \(R_{cerit}\) can be found in the literature as a function of conditions of the pump [8]. This flow between adjacent control volumes that should be nominally tight is enabled by the small gap between the rotors at their maximal approximation position. These gaps are necessary for ensuring rotation and limit friction and wear. The resulting fluid migration between adjacent chambers is caused by:  
1. Difference of pressure between adjacent control volumes (Poiseuille flow). 2. Difference in angular speed between inner and outer rotor (Couette flow).