## 2 Numerical Calculation Model
### Physical and Numerical Model  
The hyper-compressor is a plunger compressor (Figure 1). The central valve (3) is positioned within the spool's hollow (2). Its axial sides are, respectively, connected to the working chamber and discharge pipe socket (4). Its circumferential side links through the spool (2) to the suc-pipe. The central valve (3) is a combination of suction and discharge multi-poppet valves for compact construction, which is composed of (3-1) a suction valve seat, (3-2) suction valve sleeves, (3-3) suc-springs, (3-4) suction spring seats, (3-5) a seal ring, (3-6) a guide sleeve, (3-7) a guide ring, (3-8) a middle piece, (3-9) dis-springs, (3-10) discharge valve sleeves, (3-11) discharge spring seats, (3-12) a discharge valve seat, (3-13) a discharge box, (3-14) screws, (3-15) a spring seat, (3-16) a combined disc spring, and (3-17) a gland nut.  
The suction seat (3-1) is a cylinder with a number of radial through-holes (A and B) connecting it to the spool (2), a working chamber, and guide sleeve (3-7). A gap passage formed between the spool (2) and suction seat (3-1) serves as the gap from the suc-pipe passage through holes (A) to the concave side of the guide ring (3-7), and then into the working chamber through-holes (B). Simultaneously, almost no gas can flee through the clearance between the central valve (3) and the spool (2) because of the seal ring (3-5). The through-holes (B) in the suction seat (3-1), the holes (C) in the guide ring (3-7), and the holes (E) in the middle piece (3-8) are aligned along the same axis to form an internal passage between the cylinder and discharge valve seat (3-12), allowing the gas to flow from the working chamber to the concave side of the discharge valve seat (3-12) and then to the dis-pipe via the hole (G).  
The suction and discharge valve (3-2/10) sleeves are the moving components of the central valve (3), which regulate the fluid flow direction. As illustrated in Figure 2, the  
Figure 1: Schematic view of the hyper-compressor and central valve.  
discharge valve sleeves (3-10), spring seats (3-11), and springs (3-9) are installed in the holes (F) of the discharge valve seat (3-12) along the same axis. The discharge valve seat (3-12) directs the axial motion of the discharge valve sleeves (3-10), while the spring seats (3-11) limit their radial motion lift. For the exhaust process, the discharge valve sleeves (3-10) move to the spring seats (3-11) to connect the flow passage between the dis-pipe and the working chamber, or they are attached to the middle piece (3-7) to prevent gas leaking from the cylinder into the dis-pipe.  
In addition, a small 0.05 mm gap is reserved to accommodate the generation of a new mesh during the valve sleeves' opening phase. Simultaneously, the boundary between the gap and the flow domain of the middle piece (3-8) is either set as an interface to permit gas flow during the exhaust process, or as a wall to prevent gas leaks through the gap. The interface phase and wall events are calculated using LPM with RGM.  
The structured mesh of the entire numerical model is shown in Figure 3. To reduce the effects of the numerical boundaries on the thermodynamic characteristics of the primary domain, the inlet and outlet boundaries were positioned before the suction buffer and reactor, respectively. The dynamic mesh update for the valve motion based on the force analysis could be expressed by the Equation (1).  
\[M_{v}\ddot{x}+C\dot{x}+Kx=F_{s}+\int\limits_{A_{v}}pdA_{v} \tag{1}\]  
where \(M_{v}\) is the valve sleeve equivalent mass, \(K\) is the stiffness of the valve sleeve, \(\ddot{x}\), \(\dot{x}\), and \(x\) are the acceleration, velocity, and lift of the valve sleeve, \(C\) is the damping coefficient, and \(F_{s}\) is the spring force.  
Figure 2: The flow domain of the hyper-compressor.