In recent years, the concept of reusable hypersonic vehicles in near space has attracted considerable research attention (Crawford, 2016; Yixiang, 2016). The key technology for such vehicles is a combined engine operable across a wide range of Mach numbers (Wen-hui, 2018; Xuhui, 2020). Inlet flow pre-cooling is an important way to broaden the operating Mach number range of an engine. Several countries have conducted a considerable amount of preliminary research on pre-cooled cycle engines. Among them, the “Synergetic Air-Breathing Rocket Engine” (SABRE) series and Scimitar engine concepts proposed by the British company Reaction Engines have received worldwide attention owing to their superior performance (Varvill and Bond, 2003; Jivraj, 2004). As shown in Figure 1, the SABRE4 and Scimitar schemes evolved from the early SABRE3 scheme by adding bypass and reheating components to improve the cruise-specific impulse of the engine and solve a frost problem in the pre-cooler (Jivraj, 2004; Hempsell, 2010).

With continuous breakthroughs in key technologies and positive evaluation results from the United States Air Force Research Laboratory (Zhicheng, 2015), the SABRE engine has attracted significant research and development funding and attention (Zy, 2015; E.S. Agency, 2011). Numerous research studies have been conducted on pre-cooling combined engines inspired by the progress of the SABRE engine (Zhengping, 2015; Zhang et al., 2017; Yuan, 2020; Zheng-ping, 2021). Fernández-Villacé and Paniagua (2013) indicated that a pre-cooled engine based on a closed cycle improves efficiency by recovering thermal energy from the freestream and aeroshell. However, the classic efficiency figures based on first-principle analyses were inaccurate performance indicators for the resulting combined cycle. Accordingly, the author defined an availability effectiveness parameter and then analyzed the Scimitar cycle, ultimately providing the availability loss distribution of the cycle at typical flight speeds. Zhang et al. (2017) conducted a classical availability analysis on the SABRE cycle and found that unburned hydrogen (H₂) was the largest source of availability loss in the system. The equivalence ratio (ER) is the main influence on the cycle; it can be regulated by adjusting the parameters of the helium (He) cycle. Xuanfei (2020) argued that all pre-cooled engines could be simplified to a fundamentally similar cycle. The author proposed a unified cycle model and conducted a theoretical analysis. Then, the influences of the design parameters on the theoretical pressure ratio and engine performance were studied using numerical methods.

In general, existing research has mainly focused on the evolution and optimization of a pre-cooled engine’s thermodynamic cycle, such as the integration of a parallel heat return system. However, relatively few studies exist concerning the relationship between the performance of a pre-cooled engine and the parameters of its key components for a given thermodynamic cycle. Pre-cooled engines with closed cycles entail numerous components and intricate interrelationships among the coupled parameters. Such research would greatly aid in implementing an effective engine design. Accordingly, based on a simplified pre-cooled core engine model, this study analyzes the relationship between the key thermodynamic parameters and the specific thrust and specific impulse performance of the engine. Then, the relationships between the thermal efficiency of the closed He cycle and the parameters of the closed-cycle components are discussed.

The SABRE3 scheme (as shown in Figure 2) includes air-breathing and rocket modes. The air-breathing mode includes a bypass combustor and the core engine (Varvill and Bond, 2008). This study focuses on analyzing the performance of the core engine in the air-breathing mode of SABRE3. The pre-cooled core engine model can be simplified, as discussed in detail in the following paragraphs.

The main parameters for evaluating the performance of the core engine are the specific thrust and specific impulse.

The previous analysis shows that the engine’s specific thrust and specific impulse are only related to the pressurized coefficient and ER. However, unlike the pressurized coefficient, the ER of the engine is not an independent design parameter. Under certain component performance and thermodynamic cycle conditions, a minimum ER exists. When the ER is lower than this minimum value, the engine cannot be matched. The most significant manifestation of this phenomenon is that the effectiveness of the low-temperature heat exchanger (HX4 in Figure 2) exceeds 1.

Webber et al. (2006) stated that at a thermal capacity ratio of 1 between the pre-cooler and the low-temperature heat exchanger, the entropy increase in the system is minimal and the cycle thermal efficiency is the highest. At this time, the required liquid H₂ flow rate for cooling is approximately 2.5 times that required for stoichiometric combustion.

In actual engineering design, the maximum temperature of He can reach 1,200 K if the heat capacity ratio is kept at 1, which is higher than the current material tolerance limit of 950 K. Therefore, the general design equivalent ratio is higher than 2.5. Taking the thermodynamic cycle shown in Figure 2 as an example, the required liquid H₂ flow for cooling is approximately 2.5–3.5 times the equivalent combustion liquid H₂ flow. Thus, the overall ER is approximately 2.5–3.5.

In the thermodynamic cycle of Figure 2, the closed He cycle system plays a role in the heat-to-power conversion. Q_H denotes the heat absorbed from the airflow path through the pre-cooler and high-temperature heat exchanger, and W_T denotes the power generated to drive the air compressor. Q_C is the heat transferred to the liquid H₂ through the low-temperature heat exchanger. A simplified model of the closed He cycle is shown in Figure 6.

The thermal efficiency is the ratio of the net output power of the cycle to the heat absorbed by the cycle and is calculated as follows: η c y c l e = W n e t Q H = W T − W C Q H = 1 − Q C Q H . (23)

Evidently, improvement in the thermal efficiency of the closed cycle would reduce the amount of heat that needs to be cooled by the liquid H₂, resulting in a lower overall ER. In practical matching, an increase in the closed-cycle thermal efficiency from 20% to 30% reduces the overall ER from 3.8 to 2.6, resulting in a 42% increase in the engine’s total specific impulse.

Assuming that He is an ideal gas with a constant adiabatic index and specific heat at constant pressure, the ratio between the maximum temperature and minimum temperature of the cycle can be defined as the cycle temperature ratio: ε * = T 3 * T 1 * . (24)

Thus, the aforementioned calculations can be transformed as η c y c l e = 1 − ε * 1 − 1 − π H T * 1 − k k ∙ η H T * − 1 ε * − π H C * k − 1 k − 1 η H C * + 1 . (25)

The calculations according to the balance of pressure in the closed cycle are π H T * = δ c y c l e ∙ π H C . * (26)

Here, δ c y c l e is the product of the total pressure recovery coefficient of the other components in the closed cycle except for the compressor and the turbine; it is defined as the cycle pressure recovery coefficient. The closed-cycle efficiency can be expressed as follows: η c y c l e = f ( ε * , δ c y c l e , π H C * , η H C * , η H T * ) . (27)

Figures 7, 8, 9, 10 show the relationships between the cycle efficiency and component parameters.

Figures 7, 8 indicate that under certain component efficiency and cycle temperature ratios, the cycle thermal efficiency increases with an increase in the compressor pressure ratio. The cycle pressure loss coefficient and cycle thermal efficiency demonstrate an approximately linear relationship. For low-pressure ratio cycles (e.g., π H C * = 2 ), for every 10% increase in the pressure loss coefficient, the cycle thermal efficiency increases by 3%–6%. For high-pressure ratio cycles (e.g., π H C * = 10 ), for every 10% increase in the pressure recovery coefficient, the cycle thermal efficiency increases by approximately 2%–3%.

In Figure 9, the cycle efficiency shows an approximately linear relationship with the compressor and turbine efficiency. When the efficiency of turbomachinery increases by 10%, the cycle efficiency increases by approximately 0.5%–2%. The relationship between the cycle temperature ratio and cycle thermal efficiency in Figure 10 shows that the temperature ratio has a greater impact on high-pressure ratio cycles. When the temperature ratio increases from 10 to 20, the efficiency of the low-pressure ratio (e.g., π H C * = 2 ) cycle increases by 1.3%, whereas the efficiency of the high-pressure ratio (e.g., π H C * = 10 ) cycle increases by approximately 4.8%.

We performed engine performance calculations with the system parameters of Fernández Villacé (2013) at Ma5. The comparison of thrust and specific impulse is shown in Table 1. The thrust coefficients of nozzles are considered in the literature, but no specific design parameters are given. Therefore, a thrust coefficient of 0.9–1.0 was selected for calculation in this section. The results show an error of 3.27–12.43% for specific thrust and 4.08–13.63% for specific impulse.

Next, we performed a validation calculation for the ground point of the engine shown in the literature (Cao-Bin et al., 2021) with a thrust factor of 0.9. The results of the calculation are shown in Table 2. The results show that the errors of the thrust and specific impulse are 3.06% and 3.93, respectively.

This study established a performance analysis model for a pre-cooled engine core with a simple closed He cycle. Based on the perfect gas assumption, research was conducted on the relationships between the specific thrust, specific impulse, and engine parameters. The conclusions are as follows:

(1) A new performance analysis method based on analytical methods was proposed for providing overall performance analyses of pre-cooled core engines with closed He cycles, providing a new means for quickly and intuitively establishing the relationships between the overall performances of pre-cooled core engines and their component parameters.