Expander stage number problem

The principle of an expander is that it is a machine that generates cooling capacity by utilizing the principle that compressed gas releases mechanical work as it expands and its pressure decreases, thereby reducing the gas temperature. Expanders are commonly used in cryogenic equipment. Expansion machines are classified into piston expansion machines and turbine expansion machines based on their mode of operation and structure. Piston expanders are mainly suitable for small and medium-sized high and medium-pressure cryogenic equipment with high pressure ratios and low flow rates. Compared with piston expanders, turbine expanders feature high flow rates, simple structure, small size, high efficiency, and long operating cycles, making them suitable for large and medium-sized cryogenic equipment. 　　A piston expander is an expander that causes gas to expand in a variable volume to produce external work for cooling (usually, an electric motor is used to absorb this external work). This type of expander comes in two types: discrete and horizontal. The vertical structure is more commonly used; components such as the crankshaft, connecting rods, crosshead, piston, intake valve, and exhaust valve are the moving parts, which are mounted in the frame, cylinders, and intermediate bearings respectively. Its function is similar to that of a reciprocating piston compressor, but its intake and exhaust valves are opened and closed at predetermined times by intake and exhaust cams. Due to losses of cooling capacity caused by flow resistance of the intake and exhaust valves, incomplete expansion, frictional heat, and heat exchange between external and internal sources, the adiabatic efficiency of piston expanders is generally 65–85% for high-pressure expanders, and 60–70% for medium-pressure expanders. The valveless and single-valve expanders, which emerged in the 1950s and do not rely on cam mechanisms, reduced the number of moving parts in the expanders and improved their operational reliability; they have been widely used in small cryogenic devices. In the 1960s, polytetrafluoroethylene sealing elements with fillers were used to replace oil-lubricated metal sealing elements, thereby preventing lubricating oil from entering the deep cryogenic distillation or liquefaction areas and ensuring safety. 　　A turbine expander is an expander that transfers energy through the change in velocity of a gas as it expands. These expanders are available in single-stage and double-stage versions, as well as vertical and horizontal types; they can also be of impulse or reaction type. A single-stage centrifugal radial-flow reaction type is generally used, with the external work generated being absorbed by a generator, blower, or oil brake. It is similar to a single-stage centrifugal compressor, but it has vanes (adjustable blades) for regulating the air intake volume. Low-speed bearings are lubricated by forced oil, while high-speed ones use gas bearings. Due to nozzle losses, impeller losses, residual velocity losses, disk friction losses, leakage losses, cross-flow losses, and external heat intrusion losses, the adiabatic efficiency of turbine expanders is generally: 65–75% for medium-pressure expanders, and 75–85% for low-pressure expanders. In the 1960s, liquid-expanding machines were developed, which were mostly used in natural gas separation equipment.