# A Review on Two-Phase Volumetric Expanders and Their Applications
## 2 General Principles of the Two-Phase Volumetric Expanders  
### Pressure-Volume Diagram of Two-Phase Volumetric Expanders  
A volumetric expander, either single or two-phase, consists of static and moving parts in which these latter parts drive a mechanical shaft. The inner surfaces of these parts define one or more working chambers whose volume variation is due to the relative motion between static and moving surfaces. Depending on the kinematics, the motion of the volumetric machines can be alternative as in the case of piston ones, or rotating as in case of systems based on screws, scrolls, and rotating vanes. In operation, each chamber receives, expands, and discharges the working fluid. The pressure-volume diagram represents the trend of fluid pressure in the chamber with the corresponding volume variation during the sequence of these transformations. The pressure-volume diagram is delimited between a maximum and minimum pressure value, and their ratio is named the pressure-expansion ratio. If we refer to a piston machine, the more general pressure-volume diagram may include a maximum of six transformations (Figure 1), as follows:  
The suction (1-2) occurs when the incoming fluid at constant pressure fills the chamber by flowing through a suitable opening in the stator. The cut-off grade parameter, which characterizes the suction process, is the ratio of the volume difference between the beginning and end of the suction (points 1 and 2, respectively) to the chamber displacement volume (identified as V\({}_{\text{displ}}\) in Figure 1). The suction ends when the opening closes, so that subsequent volume increase in the chamber causes the expansion of the trapped fluid. The expansion process (2-3) is ideally isentropic, and it is prolonged until the maximum volume of the chamber is reached. The following two parameters characterize the expansion process: (i) the built-in volume ratio (BVR), that depends only on the expander geometry because it is defined as the chamber volume at the end of the expansion process (point 3) divided by the volume at the start of the expansion process (point 2); (ii) the specific volumetric fluid ratio (SVFR) that is determined by the operating conditions, since it is found as the ratio of the specific volume of the working fluid at the end and the beginning of the expansion (points 3 and 2, respectively). During the successive free exhaust (3-4), the chamber pressure abruptly reduces because the discharge aperture opens to allow a spontaneous outflow of the fluid. Later, the forced exhaust (4-5) takes place. In particular, the volume chamber decreases while the exhaust aperture is opened so that the trapped fluid is pushed until the closure of the exhaust aperture. The successive recompression process (5'-1') can occur if the exhaust aperture closes before reaching the minimum volume of the chamber (point 5'). In this case, the remaining fluid in the chamber is compressed. The recompression process is characterized by the recompression grade parameter, defined as the ratio of the volume difference between the beginning and end of the recompression (points 5' and 1', respectively) to the chamber displacement volume (identified as V\({}_{\text{displ}}\) in Figure 1). The last transformation is the pressure restoration (1'-1), which occurs when the inlet aperture opens to admit new fluid in the chamber, thus, restoring the initial value of the pressure. In the pressure-volume diagram, the recompression process (5'-1') and pressure restoration process (1'-1) can be absent. In particular, if the forced exhaust process ends at the minimum volume of the chamber (point 5), then the recompression process is absent and the pressure restoration process takes place (5-1). On the other hand, if the forced exhaust process ends at the proper volume of the chamber (point 5\"), then the recompression process takes place (5\"-1), and the pressure restoration process is absent. On the other hand, several factors determine the configuration of the pressure-volume diagram.