## 2 Simulation Model
### The Discharge Chamber  
Figure 4 depicts a simple hydraulic system for the estimation of the crescent pump's flow-pressure characteristics, which is comprised of a throttle valve utilized as the pump's load, and a delivery line with a constant diameter utilized as the connection between the pump's outlet and the throttle valve. As observed, the pressure in the discharge chamber corresponds to the pump's outlet pressure. Regarding Equation (1), the term \(V\) denotes the volume of the discharge chamber and the delivery line, the term \(q_{\text{in}}\) denotes the flow into the discharge chamber from the trapped chamber (\(q_{\text{trap}}\) in Figure 2) via the relief groove on the floating plate that connects the discharge chamber and the trapped chamber (Figure 3). In this scenario, the term \(q_{\text{in}}\) yields  
\[q_{\text{in}}=q_{\text{trap}}=C_{\text{d}}A_{\text{rel}}\sqrt{2\big{|}p_{\text{ trap}}-p\big{|}/\rho}\cdot\text{sign}(p_{\text{trap}}-p) \tag{2}\]  
where \(q_{\text{trap}}\) is the trapped flow; \(A_{\text{rel}}\) is the flow area of the relief groove; \(p_{\text{trap}}\) is the trapped pressure; \(\rho\) is the oil density dependent on the pressure (the oil temperature was set as a constant and not taken into account in this work), as addressed by [PERSON] et al. (2016); \(C_{\text{d}}\) is the discharge coefficient dependent on the flow velocity and is difficult to be determined. In this work, the value of \(C_{\text{d}}\) was set as 0.7 according to the study by [PERSON] et al. (2017). Regarding the term \(q_{\text{out}}\), it denotes the flow out of the discharge chamber, which consists of three terms: the valve flow through the throttle valve in the hydraulic system (\(q_{\text{out,1}}\) in Figure 4), the triangular flow into the transitional chamber through the triangular grooves on the floating plate (Figure 3), and the internal leakage of the pump. The first part can be given as  
\[q_{\text{out,1}}=q_{\text{val}}=C_{\text{d}}A_{\text{val}}\sqrt{2p/\rho} \tag{3}\]  
where \(q_{\text{val}}\) is the flow through the throttle valve; and \(A_{\text{val}}\) is the flow area of the throttle valve. Figure 4: A hydraulic system for estimating the crescent pump’s flow-pressure characteristics. Figure 3: The triangular grooves and the relief grooves machined on the floating plate. As Figure 3 shows, two triangular grooves are machined on the floating plate, therefore, the triangular flow is further split into two terms accordingly: the flow (\(q_{\text{tri},s}\)) into the gear shaft's tooth space (TS) and the flow (\(q_{\text{tri},r}\)) into the ring gear's TS. The term \(q_{\text{tri},r}\) is depicted in Figure 2, and can be given as  
\[q_{\text{tri},r}=C_{\text{d}}A_{\text{tri}}\sqrt{2(p_{i+1}-p_{i})/\rho} \tag{4}\]  
where \(A_{\text{tri}}\) is the flow area of the triangular groove; and \(p_{i+1}\) and \(p_{i}\) are the fluid pressures in the adjacent tooth spaces. Noting that the term \(q_{\text{tri},s}\) can be evaluated in the same way as Equation (4), the triangular flow yields  
\[q_{\text{out},2}=q_{\text{tri}}=q_{\text{tri},s}+q_{\text{tri},r} \tag{5}\]  
With respect to the internal leakage of the pump, it can be split into three terms according to different types of clearances inside the pump: the lateral leakage through the lateral clearances between the gears' lateral sides and the floating plates, the radial leakage through the radial clearances between the gears' tooth tips and the fillers, and the ring-gear/case leakage through the clearance between the ring gear and the case. Figure 5 depicts the lateral leakage that goes through the gear shaft's lateral side (\(q_{\text{l},s}\)) and the ring gear's lateral side (\(q_{\text{l},r}\)), thus the lateral leakage is further split into two terms accordingly. As observed, due to the complexity of the gears' profiles, the lateral leakage is evaluated utilizing the annular areas bounded by the gears' pitch circles and the floating plate's borders.