Interstage matching characteristics of two-stage series-connected helium turbine expander in hydrogen liquefaction system

Hydrogen, renowned as a clean, efficient, safe, and sustainable new energy source, constitutes a pivotal element within the anticipated future energy landscape. However, the primary constraint hindering its widespread adoption as an energy carrier lies in its inherently low energy density. In the context of hydrogen transportation economics, liquid hydrogen has emerged as the most viable and practical means of transportation. The hydrogen liquefaction system represents a pivotal technology that facilitates the cooling of hydrogen gas to exceedingly low temperatures, subsequently compressing it into a liquid state. At the heart of this system lies the turbine expander, which is an essential piece of equipment for liquefying hydrogen gas by generating temperatures below the critical liquefaction threshold. This process enables efficient and compact storage of hydrogen for subsequent utilization, thereby addressing the aforementioned limitations and advancing the realization of a hydrogen-based energy system.

The prevalent helium turbine expander-based refrigeration cycles in the hydrogen liquefaction system primarily involve the Brayton cycle and its inverse counterpart, as detailed in Ref. 1. Originating from heat engine applications, the Brayton cycle has been adapted to serve in refrigeration systems through its inverse variant, which is inherently more suited for cooling processes. The inverse Brayton cycle comprises four fundamental stages: isentropic compression, followed by isobaric cooling, isentropic expansion, and, finally, isobaric heat absorption. Figure 1 presents a schematic illustration of a two-stage series-connected hydrogen liquefaction system that employs the inverse Brayton cycle for refrigeration purposes. This sophisticated system comprises two distinct refrigeration subsystems: one for helium and the other for hydrogen. Initially, the compressed helium undergoes pre-cooling by liquid nitrogen, subsequently traversing a series of heat exchangers and turbine expanders where it is further cooled to temperatures suitable for liquid hydrogen production. Concurrently, the compressed hydrogen gas also undergoes an initial pre-cooling step utilizing liquid nitrogen. This preliminary cooling is followed by a crucial cooling phase within the heat exchangers, where cryogenic helium effectively transfers its cold energy to the hydrogen gas, ultimately resulting in the liquefaction of hydrogen.

The utilization of the inverse Brayton cycle for hydrogen liquefaction offers distinct advantages, notably the compact size of the cryogenic expansion unit, coupled with high operational efficiency and rapid temperature reduction capabilities. Quack² successfully designed a 100 ton/day hydrogen liquefaction system employing a helium–neon inverse Brayton cycle. Gruehagen and Wagner³ reported on the cryogenic helium systems employed by Linde and Air Liquide, which respectively featured 10-turbine and 8-turbine cold box configurations. Linde’s system incorporated three turbine expanders arranged in series for effective cooling, whereas Air Liquide adopted a two-stage turbine expander design incorporating interstage heat exchange for optimized performance. Biglia et al.⁴ proposed a rapid-cooling cryogenic freezing system based on the inverse Brayton cycle, and not only demonstrated its potential for swift cooling, but also provided valuable insights into the design of hydrogen liquefaction facilities. Bi and Ju⁵ introduced an integrated hydrogen liquefaction process that incorporated a helium inverse Brayton cycle with steam methane reforming, offering a comprehensive solution for hydrogen production and liquefaction. Yang et al.⁶ presented an inverse Brayton cycle cryogenic refrigeration machine and calculated the aerodynamic performances of both the compressor and turbine expander. Their analysis provided a solid foundation and reference point for subsequent design optimizations of coupled systems.

The turbine expander plays a pivotal role in the hydrogen liquefaction system by providing the requisite cooling capacity. Its operational performance is instrumental in determining the overall liquefaction efficiency of the system. Additionally, it finds extensive application in establishing cooling environments and recovering pressure energy. Teng et al.⁷ designed a hydrogen turbine expander specifically for a hydrogen liquefaction system capable of liquefying 5 tons of hydrogen per day. Through simulation of the internal fluid flow dynamics, they validated the reliability of their design model. Furthermore, they conducted an exhaustive analysis of the expander's performance under varying operational conditions, maintaining constant speed and characteristic ratio. Their findings revealed that a regulation scheme with fixed characteristic ratio yields superior isentropic efficiency compared with fixed-speed regulation.

As the need for large-scale liquefaction facilities and heightened refrigeration capacities grows, the incorporation of multiple turbine expanders, in either series or parallel configurations, becomes increasingly relevant for augmenting overall cooling capacity. For instance, the Large Helical Device developed by the National Institute for Fusion Science incorporates a helium cryogenic system comprising seven radial-axial turbine expanders, achieving a maximum equivalent refrigeration capacity of 4.4 K.⁸ The KSTAR superconducting magnet tokamak device developed by the National Fusion Research Center of Korea incorporates six turbine expanders that undergo a series of expansion stages, enabling them to effectively cool down to a temperature of 4.5 K under conditions of 70 K and 18 bars.⁹ Meanwhile, the world’s most energetic particle accelerator, the Large Hadron Collider (LHC), at the European Organization for Nuclear Research (CERN), is equipped with the world’s largest cryogenic system. This colossal facility has eight distinct sectors, each featuring a helium expander operating at 4.5 K and another at 1.8 K, ensuring the maintenance of stringent temperature requirements essential for LHC operations.³ Liu and Jiao¹⁰ demonstrated that the adoption of a multistage series expander architecture results in substantially higher power generation capabilities compared with traditional single-stage expanders. This advanced design effectively circumvents the limitation of prolonged low-temperature operation, which can negatively impact the lifespan of the expander, thereby enhancing the overall performance and reliability of the cryogenic system. Dhillon and Ghosh¹¹ improved a refrigeration system based on a helium reverse Brayton cycle by varying the number of stages and the positions of series or parallel turbine expanders. They compared the overall performance of both the enhanced and basic devices across various refrigeration temperatures. Their findings indicated that connecting two expanders in series can significantly improve thermal efficiency. Manolakos et al.¹² investigated the application of a two-stage series-connected turbine expander in an organic Rankine cycle system. They reported that the isentropic efficiency of the two-stage series-connected turbine expander was higher than that of the single-stage one. The series-connected expander could influence the overall performance of the system by adjusting the rotational speed. Lei et al.¹³ developed and investigated the performance of a new intermediate-cooling series-connected refrigeration cycle employing multiple turbine expanders and heat exchangers arranged in a four-stage cascade configuration. Through an analysis of temperature–enthalpy diagrams for various low-temperature systems, they explored how different thermal cycles and unit connections influenced cycle efficiency. Their findings revealed that both serial and parallel expansion cycles could enhance performance, with the intermediate-cooling series-connected cycle exhibiting superior heat transfer efficiency.

In the context of practical implementation, it is a considerable challenge to guarantee that every stage within a multistage series-connected turbine expander operates precisely under its designed conditions. Variations in the outlet conditions of the upstream expander necessitate adjustments to the downstream stage’s operating parameters to optimize overall system efficiency. Failure to adequately align the operational conditions between the upstream and downstream stages of the turbine expander can lead to substantial internal flow energy losses. The primary sources of energy loss in cryogenic turbine expanders encompass a multitude of factors, including nozzle boundary layer loss, nozzle trailing edge loss, rotor incidence loss, rotor passage loss, tip clearance loss, and impeller trailing edge loss.¹⁴ Li et al.¹⁵ conducted a thorough analysis of the internal flow dynamics and the mechanisms underlying secondary flow energy losses within the nozzle of a hydrogen turbine expander. Their findings revealed that the energy loss incurred at the leading edge of the nozzle is significantly lower than those at the throat and trailing edge. Furthermore, the flow loss is more pronounced on the suction surface, accompanied by the presence of trailing vortices at the trailing edge. Persky and Sauret¹⁶ conducted a meticulous analysis of the predictive accuracy of six fundamental energy loss models under off-design operating conditions of an expander. Their research indicated that passage losses and tip clearance losses were the main sources of overall losses, comprising 85.76% and 14.24% of the total, respectively. Li et al.¹⁷ employed a three-dimensional transient viscous flow simulation approach to investigate the effect of impeller radial clearance on leakage mass flow rates and associated energy losses. Their findings underscored the significant influence of this parameter on leakage characteristics. Deng et al.¹⁸ explored the flow dynamics within the tip clearance of a radial turbine impeller, highlighting the dependence on pressure differential between the pressure and suction sides of the impeller tip, in conjunction with the relative motion between stator and rotor. Ebrahimi Saryazdi et al.¹⁹ demonstrated that the incorporation of variable inlet guide vanes can increase the efficiency of a radial expander by as much as 60% in comparison with fixed inlet guide vanes. Hu et al.²⁰ adjusted the inlet pressure and guide vane exit angle to enhance net power output, highlighting the crucial role of blade tip clearance in regulating internal flow patterns and reducing energy losses in the impeller. Ghorani et al.²¹ applied entropy production theory to quantify both direct and turbulent dissipation energy losses in a pump-as-turbine configuration. Their findings revealed that more than half of the energy dissipation takes place within the flow passage, especially at the leading and trailing edges of the blade, as well as in areas where flow separation occurs.

Given the exceptional physical attributes of helium and the intricate nature of multistage series-connected expander circulation systems, contemporary research on hydrogen liquefaction turbines predominantly revolves around numerical simulations and optimal design strategies. However, there exists a notable gap in the literature concerning performance evaluations and interstage matching capabilities of series-connected expanders. To address this limitation, the present study endeavors to delve into the interstage matching characteristics of a two-stage series-connected helium turbine expander in a hydrogen liquefaction system employing the inverse Brayton cycle. An initial step in our investigation involves validating the reliability of the numerical simulation outcomes through a rigorous on-site joint test system involving the helium turbine expander. After establishing the credibility of our simulation framework, we investigate the operational performance of each expander stage, specifically evaluating the efficiency variations of the second-stage expander under altered operating conditions. Furthermore, the energy loss distributions within various through-flow components and the primary loss zones within the impeller are comprehensively analyzed on the basis of entropy production theory. This detailed assessment offers valuable insights into the intricacies of energy dissipation mechanisms within the system. The findings presented in this paper contribute significantly to the design optimization of series-connected helium turbine expanders, as well as providing a foundational reference for the development of other multistage series-connected cryogenic gas turbine expanders.

The helium turbine expander in a hydrogen liquefaction system consists of two coaxial components: the expansion end and the brake end. The expansion end assumes a pivotal role as the primary refrigeration section, governing the isentropic efficiency of the entire expander. The compressed helium gas is introduced into the through-flow components of the expansion end, where it undergoes a cooling and depressurization process via isentropic expansion. Subsequently, this cryogenically cooled helium gas efficiently transfers heat from the hydrogen gas in a stepwise fashion, traversing through the heat exchanger. This energy exchange not only cools the hydrogen but also harnesses the work potential to drive the coaxial brake fan.

The through-flow component of the expansion end of helium turbine expander for each stage, as illustrated in Fig. 2, includes a contracting nozzle and a centripetal impeller. The technical requirements of the expansion end are listed in Table I. The designed expansion ratios of the two-stage turbine expander are set to 2.08 and 3.76, respectively. To ensure that the helium gas in the first-stage turbine expander acquires more kinetic energy to complete the expansion process, the impeller rotational speed of the first-stage expander is increased.

The main structural parameters calculated on the basis of the technical requirements of the helium turbine expanders given in Table I are listed in Table II. The nozzle height and the blade inlet height of the first-stage expander are slightly lower than those of the second-stage one. This difference is attributed to the smaller expansion ratio of the first-stage expander compared with the second-stage one.

The calculational domain for the second-stage turbine expander was established on the basis of the geometrical model of the expansion end of the expander shown in Fig. 2. It is shown in Fig. 3, including the nozzle, impeller, and diffuser domains. The length of the diffuser domain was extended to 200 mm to avoid backflow at the outlet boundary. To enhance grid quality and convergence speed, the calculational domain was divided into hexahedral grids. The grid conditions near the wall of the nozzle and blade, which were refined to ensure computational accuracy, are enlarged in Fig. 3.

As shown in Fig. 3, total pressure inlet and pressure outlet boundary conditions were imposed at the nozzle inlet and diffuser outlet, respectively. The mass flow rate was regulated by fixing the inlet parameters and adjusting the pressure at the outlet. The impeller domain was set as a frozen rotor, while the remaining parts were stationary. Two pairs of interfaces were formed between the nozzle outlet and impeller inlet and between the impeller outlet and diffuser inlet. The walls of the through-flow component were all considered to be no-slip.

The convective term in these governing equations was discretized using an upwind scheme. The convergence criterion was set to 1.0 × 10⁻⁵.

The steady-state numerical simulations were performed using various grid numbers to assess its effect on the simulated performance of the turbine expander. Specifically, the overall grid number was increased by roughly 1.25 times. Figures 4(a) and 4(b) show the calculation results for first-stage and second-stage expanders under their respective design conditions. The simulated isentropic efficiency becomes stable as the grid number rises. Consequently, to conserve computational resources, the final grid totals for the first- and second-stage turbine expanders were set at ∼6.47 × 10⁶ and 6.79 × 10⁶, respectively.

To gain a more precise understanding of simulation discretization errors, the grid convergence index offers a quantitative assessment of how grid discretization affects the convergence of numerical results. Table III lists the parameters used in the discretization error analysis. The results reveal that the grid convergence indices for the efficiency of the first- and second-stage expanders are 0.15% and 0.14%, respectively, suggesting that the grid discretization error is adequate to guarantee the accuracy of the numerical computations.

Figure 5 illustrates the on-site joint test system for the helium turbine expanders. As shown in Fig. 5(a), the experimental flow chart consists of the operation control system of the turbine expander, the operation monitoring system, and the gas supply system of the bearing sealing gas. The working medium inside the gas bearing and turbine expander is pure helium gas without pollution. The turbine expander is controlled by the valves, pressure gauges, tachometers, filters, heat exchangers, connecting pipelines, etc., which are uniformly configured by the overall liquefaction equipment.

The sliding seal is utilized to realize the axial expansion and contraction of the expander. The required rotational speed, efficiency and outlet pressure of the turbine expander are ensured by adjusting the gas intakes of the brake fan and turbine expander. Meanwhile, the intake pressure and volume of gas bearing can be adjusted to maintain its stiffness and stability. To monitor the operation of the cryogenic helium turbine expander, the rotational speed, pressure, and sealing gas pressure measurement points are arranged on the expander body. In Fig. 5(a), P₀ and P₃ indicate the inlet and outlet pressure sensors of the expander. The inlet temperature sensor T₀, outlet temperature sensor T₃, and mass flow sensor G provide analog signals, which are processed by the data acquisition host and presented on the display instrument.

The experimental and simulation results for the two-stage helium turbine expander under the design conditions are shown in Table V. The experimental results were recorded at 30-min intervals once the design operating conditions had been achieved. The calculated average values of three such recordings are shown in Table V. The simulation results slightly underestimate the outlet temperature and overestimate the isentropic efficiency for each stage. The simulation errors in the outlet temperature and isentropic efficiency for the first-stage expander are −0.4 K and +6%, while those for the second-stage expander are −0.7 K and +7.4%. These discrepancies are mainly due to the fact that the turbine expander wall is considered as an adiabatic wall in the numerical simulation. During practical operation, despite the expansion end being placed in a vacuum-insulated chamber, a portion of the cold energy still escapes to the external environment. This results in an increase in the temperature at the expander outlet compared with the idealized conditions. Since the performance prediction error is within an acceptable range, it is confirmed that the numerical model described above is reasonable and the simulation results are reliable.

The output cooling capacity of the helium turbine expander in a hydrogen liquefaction system can be adjusted by changing the rotational speed of the brake fan and the flow rate of the expander. In actual operation, it is not easy to always guarantee the design conditions. To achieve a more comprehensive understanding of the turbine expander performance and determine its reasonable operating range, the variations in the isentropic efficiency with flow coefficient at different rotational speeds for each turbine expander are shown in Fig. 6. The results for flow coefficients Q/Q_d from 0.2 to 1.4 are represented by solid lines and symbols of different colors. For reference, the lower limit of the efficiency of the considered high-efficiency operating region is set as 75% and is drawn as a dashed line.

Figure 6(a) provides the performance curve of the first-stage expander for different flow coefficients and rotational speeds. In the rotational speed range of 49 000–57 000 rpm, the isentropic efficiency generally increases from less than 10% with an increase of the flow coefficient to about 92%, and then decreases once Q/Q_d exceeds 1.2. When the rotational speed is in the range 59 000–61 000 rpm, the isentropic efficiency first decreases as the flow coefficient increases, then gradually increases, and finally decreases. The first-stage expander is less sensitive to variations in rotational speed. Once the flow coefficient exceeds 0.9, the isentropic efficiency of the first-stage expander is improved to more than 75% at different rotational speeds. For the second-stage expander, as shown in Fig. 6(b), the isentropic efficiency increases rapidly with increasing flow coefficient and finally stabilizes above 80%. Compared with the first-stage expander, the isentropic efficiency is more sensitive to the rotational speed, especially in the low flow coefficient range of 0.2–0.6. In this range, the isentropic efficiency shows a generally decreasing trend with increasing rotational speed, which cannot guarantee that the high-efficiency operating region (η ≥ 75%) will be reached. However, it is always possible to maintain a stable and high efficiency exceeding 75% when the flow coefficient increases from 0.6 to 1.4. Therefore, the second-stage expander exhibits a wider adjustment range than the first-stage expander.

Interstage matching ensures that when the outlet conditions of the first-stage turbine expander fluctuate, the operating parameters of the second-stage expander can be adjusted in time to avoid large fluctuations in the outlet temperature of the series-connected system. The matching performance of the second-stage turbine expander was analyzed when the outlet conditions of the first-stage expander and series-connected system fluctuated around the design points.

Figure 7 shows the variation of the outlet temperature of the second-stage expander with its rotational speed at different inlet temperatures. Differences in inlet temperature are caused by fluctuations in the output conditions of the first-stage expander. The blue solid line represents the design condition with an inlet temperature of 28.7 K. The results for two lower and two higher inlet temperatures are plotted with different colors. The design rotational speed of 52 000 rpm and outlet temperature of 18.2 K are shown by red dashed lines for reference. The outlet temperature of the expander exhibits a gradual decrease with increasing rotational speed in the range of 46 000–57 000 rpm. It seems to be impossible to achieve further decreases once the rotational speed exceeds 57 000 rpm. When the inlet temperature decreases by 0.5 and 1 K, the outlet temperature decreases by 0.6 and 0.72 K, respectively. Reducing the rotational speed is beneficial to bring the outlet temperature close to the design value. In other words, the rotational speed of the impeller should be reduced from 52 000 to 49 000 and 46 000 rpm when the inlet temperature decreases to 28.2 and 27.7 K, respectively. However, the outlet temperatures increases are greater than the inlet temperature increases of 0.5 and 1 K. The effect of the rotational speed in reducing the outlet temperature is extremely limited. It has been demonstrated that when the outlet temperature of the first-stage expander increases greatly, it is difficult to match the outlet temperature of the second-stage expander to its design value by adjusting its rotational speed alone.

Figure 8 shows the variation of the outlet temperature of the second-stage expander with its rotational speed at different inlet pressures. The blue solid line represents the design condition with an inlet pressure of 4.7 bars. The results for two lower and two higher inlet pressures are plotted with different colors. At a specific rotational speed, the outlet temperature decreases as the inlet pressure increases. This is because the higher inlet pressure leads to a greater pressure drop, thus increasing the enthalpy drop associated with the greater temperature drop. As the rotational speed increases to be greater than 57 000 rpm, the outlet temperature continuously decreases for the higher inlet pressure conditions of P_in = 4.8 and 4.9 bars, whereas it slightly increases for the lower inlet pressure conditions P_in = 4.5 and 4.6 bars. To ensure that the outlet temperature remains at its design value, the rotational speed should be reduced to 50 500 and 51 000 rpm for P_in = 4.8 and 4.9 bars, respectively, and increased to 52 500 and 54 000 rpm for P_in = 4.5 and 4.6 bars, respectively.

Figure 9 depicts the variation of the outlet temperature of the second-stage expander with its rotational speed at different outlet pressures. The blue solid line represents the design condition with an outlet pressure of 1.25 bars. The results with two lower and two higher outlet pressures are plotted with different colors. Similar to an increase in inlet pressure, a decrease in outlet pressure results in a larger pressure drop, which is favorable for a larger temperature drop. The turning point of the variation trend shifts to a lower rotational speed of 55 000 rpm. In other words, the outlet temperature for the higher outlet pressure conditions of 1.4 and 1.55 bars first decreases until the rotational speed reaches 55 000 rpm and subsequently increases as the rotational speed increases further. When the differential pressure is below the design value of 3.45 bars, an increase in rotational speed does not further reduce the outlet temperature of the turbine expander. Therefore, when the backpressure at the outlet of the second-stage expander becomes higher, simply increasing the rotational speed is insufficient to achieve a low-temperature outlet that meets operational requirements.

Thus, when the inlet pressure of the second-stage turbine expander fluctuates within ±0.2 bar, the design outlet temperature can be achieved by just adjusting its rotational speed. However, once the inlet temperature rises by more than 1 K or the outlet pressure increases by more than 0.15 bar, both the rotational speed and flow rate need to be adjusted synchronously to ensure sufficient cooling capacity for hydrogen liquefaction.

The rotational speed, inlet temperature, and inlet and outlet pressures have different degrees of influence on the performance of a turbine expander when interstage matching differences are present. The reasons for this will be explored from the perspective of internal flow energy loss. Quantifying the energy losses under various interstage matching conditions, identifying the primary loss locations, and elucidating the underlying causes will be instrumental in refining the design of the through-flow components of helium turbine expanders in future iterations. Representative operating conditions and the corresponding simulation results are listed in Table VI. The influences of the inlet temperature and the inlet and outlet pressures at different rotational speeds will be analyzed here on the basis of the simulation results obtained for conditions 1–4, 5–8, and 9–12.

Figure 10 shows the total EPR (S_EPR) distribution of the through-flow components at different inlet temperatures for conditions 1–4. The S_EPR for the nozzle and impeller are provided as well as the total value. Energy loss occurs mainly in the impeller domain, where the higher inlet temperature and lower rotational speed lead to a greater loss. However, the energy loss in the nozzle domain is barely affected by the inlet temperature and rotational speed. The total energy loss is significantly decreased as the rotational speed increases, which may also result in the larger temperature drop and higher isentropic efficiency shown in Table VI.

Figure 11 shows the total EPR distribution of through-flow components at different inlet pressures for conditions 5–8. The higher inlet pressure means a greater pressure drop of the turbine expander. The maximum energy loss occurs for condition 7 with a higher pressure drop and lower rotational speed. The greater energy loss causes a smaller temperature drop and lower isentropic efficiency for conditions 5 and 7. In contrast to the effect of the inlet temperature, the energy loss in the nozzle domain increases sightly as the temperature drop increases. This is because more pressure energy needs to be converted into kinetic energy at the nozzle to drive the rotation of the impeller, thus increasing the energy loss during the energy conversion process.

Figure 12 shows the total EPR distribution of through-flow components at different outlet pressures for conditions 9–12. Similar to conditions 5–8, the lower outlet pressure implies a higher pressure drop. The total energy loss clearly decreases as the rotational speed of the impeller increases from 48 000 to 57 000 rpm. This reduction is due mainly to the decreasing energy loss in the impeller domain. The energy loss in the nozzle domain decreases slightly with increasing rotational speed but appears to be unaffected by the outlet pressure. Under the conditions of a lower rotational speed of 48 000 rpm and a lower outlet pressure below the design value of 1.25 bars, an increase in outlet pressure leads to a reduction in energy loss, accompanied by corresponding increases in temperature drop and isentropic efficiency. However, under the conditions of a higher rotational speed of 57 000 rpm and a higher outlet pressure above the design value, as the outlet pressure increases, the variations in energy loss, temperature drop, and isentropic efficiency exhibit opposite trends.

From above analysis, it can be seen that the impeller is responsible for most of the energy loss in the through-flow components. Therefore, we delve into the mechanisms by which fluctuations in the operating condition affect the temperature drop and isentropic efficiency of the second-stage turbine expander, as exemplified by the volume-averaged EPR (⁠$EPR̄$⁠) and the streamline distributions at half-span of the impeller. The results at different inlet temperatures for conditions 1–4 are shown in Fig. 13(a)–13(d), respectively. The peak $EPR̄$ occurs near both sides of the leading edge of the blade. The energy loss on the suction side is mainly caused by the turbulent vortex, while that on the pressure side is due to the impact of high-speed helium gas. The mainstream impacts the blade again at the middle position of the suction side, creating a local high-$EPR̄$ zone. Another peak $EPR̄$ appears at the blade tail, and this arises from the shedding vortex and further extends to the pressure side around the trailing edge. When the rotational speed increases to 57 000 rpm, as shown in Figs. 13(b) and 13(d), the areas dominated by the turbulent vortex and shedding vortex contract. Therefore, the total EPR in the impeller domain decreases significantly, as shown in Fig. 10. On comparing Figs. 13(a) and 13(c), it can be seen that the increase in inlet temperature at the lower rotational speed of 48 000 rpm only increases the area of the locally high-$EPR̄$ zone near the pressure side of the blade trailing edge. This is the main reason why an increase in inlet temperature leads to a slight increase in the total EPR of the impeller domain and a slight decrease in the isentropic efficiency of the expander.

Figures 14(a)–14(d) show the volume-averaged EPR and streamline distributions at half-span of the impeller at different inlet pressures for conditions 5–8, respectively. The locations of the peak $EPR̄$ are almost the same as those in Fig. 13. As the rotational speed increases, the flow of helium gas in the passage becomes more regular, reducing the area of the local high-$EPR̄$ zone and thus increasing the isentropic efficiency of the expander. Ion contrast to the inlet temperature, the inlet pressure has a more significant impact on the $EPR̄$ distribution inside the impeller at the higher rotational speed of 57 000 rpm. The increase in inlet pressure expands the local high-$EPR̄$ zone on the suction side of the blade leading edge, resulting in a decrease in the efficiency of the expander. In other words, an increase in rotational speed is more favorable to the internal flow inside the impeller passage at lower inlet pressure conditions.

Figures 15(a)–15(d) show the volume-averaged EPR and streamline distributions at half-span of the impeller at different outlet pressures for conditions 9–12, respectively. At the lower rotational speed of 48 000 rpm, the vortex on the suction side of the blade occupies almost half the impeller flow passage. Compared with Fig. 14, the influence of outlet pressure is larger than that of the inlet pressure on the internal flow and the energy loss in the impeller domain. On comparing Figs. 15(a) and 15(b), it can be seen that the higher outlet pressure reduces the peak value of the local high-$EPR̄$ zone in the middle position of the flow passage near the blade leading edge, improving the isentropic efficiency of the expander. At the higher rotational speed of 57 000 rpm, the $EPR̄$ distribution inside the impeller domain appears to be insensitive to changes in outlet pressure.

The interstage characteristics of a two-stage series-connected helium turbine expander in a hydrogen liquefaction system employing the inverse Brayton cycle have been analyzed in depth. Following validation of the reliability of the numerical simulation results using a rigorous on-site joint test system incorporating the helium turbine expander, the operational performance of each expander stage and that of the second-stage expander under various operating conditions have been obtained. In addition, the energy loss distributions within various through-flow components and the primary loss zones within the impeller have been comprehensively analyzed on the basis of entropy production theory. The conclusions of this study can be summarized as follows:

1. The operational performance of the first- and second-stage helium turbine expanders aligns with the technical specifications of the hydrogen liquefaction system. Notably, the second-stage expander demonstrates a broader operational adjustment capability in comparison with the first-stage expander.

2. When the inlet pressure of the second-stage turbine expander fluctuates within ±0.2 bar, adjusting its rotational speed suffices to achieve the designed outlet temperature. However, if the inlet temperature rises by more than 1 K or the outlet pressure increases by more than 0.15 bar, both the rotational speed and flow rate must be adjusted synchronously to ensure a sufficient cooling capacity for hydrogen liquefaction.

3. Among the through-flow components, the impeller domain accounts for most of the energy loss. Increasing the rotational speed effectively weakens the vortex on the suction side of the leading edge and the pressure side of the trailing edge of the blade, thereby enhancing the temperature drop and isentropic efficiency of the turbine expander by reducing the energy losses.

This paper was supported by the Zhejiang Provincial Key Research and Development Project No. 2023C03158.

The authors have no conflicts to disclose.

Ahmed Gomaa: Conceptualization (equal); Data curation (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal). Yang Liu: Funding acquisition (equal); Project administration (equal); Resources (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.