## 2 Simulation Model
### Simulation Procedure  
Figure 7 depicts the solution algorithm for the analysis of the tandem crescent pump's flow characteristics implemented in Matlab. It starts with the input parameters of the pump's geometric parameters and the initial values including the initial pressures in the sealing chambers and the initial oil properties, as shown in box 1. Concerning the tandem pump design, it was comprised of two sets of gear pairs characterized by two design parameters, namely the index angle (\(\gamma\)) and the displacement ratio (\(\zeta\)), as shown in boxes 2 and 3 (in the blue dashed box). The key part of the algorithm is represented by the evaluation of the outlet flow of one gear pair depicted in the red dashed box. With respect to a certain time step (the \(i\)th time step), the geometric  
Figure 7: Solution algorithm for analyzing the tandem crescent pump’s flow characteristics. features and the oil properties that vary over time need to be updated first, as shown in box 4: the key points (points A, B, C in Figure 2) for dividing the sealing chambers, the trapped chamber's volume and its time derivative for the evaluation of the trapped pressure, the flow areas of the relief grooves and the triangular grooves on the floating plate for the evaluations of the trapped flow and the triangular flow, respectively, and the oil properties (\(\rho\), \(\beta\), \(\mu\)), which are dependent on the pressure (boxes 4.1 to 4.4). Under the circumstance of two meshing points (C\({}_{1}\) and C\({}_{2}\) in Figure 2, the trapped chamber exists), noting that the lateral leakage of the trapped chamber can be evaluated by Equations (6) and (7) (with different central angles), the trapped pressure in the next time step (the \((i+1)\)th time step) can be evaluated by applying the mass conservation equation written as Equation (15), as shown in box 5. \[p_{\text{trap}}^{(i+1)}=\frac{\beta_{\text{trap}}^{(i)}}{V_{\text{trap}}^{(i)}} \left(-\sum q_{\text{trap}}^{(i)}-\sum q_{\text{leak,trap}}^{(i)}-\left(\frac{ \text{d}V_{\text{trap}}}{\text{d}t}\right)^{(i)}\right)\cdot\Delta t+p_{\text{ trap}}^{(i)} \tag{15}\]  
where \(\Delta t\) is the time interval between time steps. The pressure in the transitional chamber in the \((i+1)\)th time step can be evaluated by Equation (16), the same way as that in the trapped chamber, noting that the term \(\text{d}V/\text{d}t\) yields zero and the leakage of the transitional chamber consists of two parts, namely the lateral leakage and the radial leakage, as shown in box 6. \[p_{\text{tran}}^{(i+1)}=\frac{\beta_{\text{tran}}^{(i)}}{V_{\text{TS}}}\left( \sum q_{\text{tri}}^{(i)}-\sum q_{\text{leak,tran}}^{(i)}\right)\cdot\Delta t+ p_{\text{tran}}^{(i)} \tag{16}\]  
where \(V_{\text{TS}}\) is the volume of the tooth space; and \(p_{\text{tran}}\) is the transitional pressure. As stated above, the internal leakage of the pump consists of three parts: the lateral leakage and the radial leakage of the discharge chamber, and the ring-gear/case leakage, which can be evaluated by Equations (6) to (10) by leveraging the updated points B and C and the \(i\)th outlet pressure as shown in box 7. Noting that the time derivative of the discharge chamber's volume is interpreted as the kinematic flow in Equation (13) (as shown in box 8), the outlet flow of one gear pair yields (box 9)  
\[Q_{\text{out},x}^{(i)}=Q_{\text{kin}}^{(i)}+q_{\text{trap}}^{(i)}-q_{\text{tri} }^{(i)}-q_{\text{leak}}^{(i)} \tag{17}\]  
Hence, the outlet flow of the tandem pump yields by summing the flow produced by the two gear pairs (box 10). \[Q_{\text{out}}^{(i)}=Q_{\text{out},1}^{(i)}+Q_{\text{out},2}^{(i)}(\gamma,\zeta) \tag{18}\]  
Noting that the valve flow can be evaluated by Equation (3), the outlet pressure (pressure in the discharge chamber) in the \((i+1)\)th time step can be evaluated by Equation (19) as shown in box 12.