A Novel Curve-and-Surface Fitting-Based Extrapolation Method for Sub-Idle Component Characteristics of Aeroengines

Abstract

The component characteristics of an aeroengine below idle speed are fundamental for start-up process simulations. However, due to experimental limitations, these characteristics must be extrapolated from data above idle speed. Existing extrapolation methods often suffer from insufficient utilization of available data, reliance on specific prior conditions, and an inability to capture unique operating modes (e.g., the stirring mode and turbine mode of compressor). To address these limitations, this study proposes a novel curve-and-surface fitting-based extrapolation method. The key innovations include: (1) extrapolating sub-idle characteristics through constrained curve/surface fitting of limited above-idle data, preserving their continuous and smooth nature; (2) transforming discontinuous isentropic efficiency into a continuous specific enthalpy change coefficient (SECC), ensuring physically meaningful extrapolation across all operating modes; (3) applying constraints during fitting to guarantee reasonable and smooth extrapolation results. Validation on a micro-turbojet engine demonstrates that the proposed method requires only conventional performance parameters (corrected flow, pressure/expansion ratio, and isentropic efficiency) above idle speed, yet successfully supports ground-starting simulations under varying inlet conditions. The results confirm that the proposed method not only overcomes the limitations of existing approaches but also demonstrates broader applicability in practical aeroengine simulations.

1. Introduction

Accurate component characteristics are essential for start-up process simulation of aeroengines. While above-idle data can be obtained through bench tests or generic characteristics, sub-idle characteristics remain experimentally inaccessible, necessitating data-driven extrapolation [1]. Researchers often face the limitation of insufficient data to support the extrapolation [2]. This requires the extrapolation method to be applicable with limited available data and to make full use of it. Furthermore, during aeroengine startup—particularly in windmill starting conditions—components may enter unique operating modes: compressors can exhibit stirring or turbine modes, while turbines may operate in stirring or compressor modes [3]. Ensuring reasonable and complete reflection of these unique modes is crucial in component characteristic extrapolation.

Based on findings from the literature, the extrapolation of aeroengine component characteristics is commonly performed using the following methods:

(1) 

Proportional Coefficient Method

Proposed by Agrawal and Yunis, this approach extrapolates sub-idle component characteristics using similarity principles, establishing proportional relationships between corrected flow, pressure/expansion ratio, specific work, and corrected speed based on known above-idle performance [4]. Sexton later extended its application to turbine characteristics, making it analogous to compressors [5]. Gaudet and Gauthier [6], along with Zhou Wenxiang [7], refined the method’s equations to suit various engine components. While computationally straightforward, it relies solely on one or two known speed lines and fails to reflect unique operating modes.

Researchers have introduced modifications: Rao Gao et al. incorporated Support Vector Machines (SVM) to enhance accuracy, though the method’s inability to capture unique operating modes persisted [8]. Hao Wang et al. replaced isentropic efficiency with corrected torque, achieving better representation of component-specific operating modes but leaving the issue of underutilized data unresolved [9,10].

(2) 

Stage Stacking Method

Developed by NASA [11], this technique derives full compressor characteristics by stacking known single-stage axial compressor data. It has been applied by Yan [12], Ju Xinxing [13], and NASA Lewis Research Center [14] to extrapolate sub-idle characteristics for axial, combined, and fan compressors. Despite its adaptability to various compressors, the method requires detailed design parameters and is not suitable for turbines.

(3) 

Zero-Speed Line Interpolation Method

This approach interpolates sub-idle characteristics using above-idle data and zero-speed characteristics, which are obtained through experiments or CFD. Applications by Zachos [15,16] and Jialin Zheng [17] have demonstrated its effectiveness. However, its reliance on precise zero-speed characteristics makes implementation challenging [18].

(4) 

Backbone Feature Method

Developed collaboratively by General Electric and NASA Lewis Research Center [19,20,21], this method extrapolates component characteristics by parameterizing backbone and non-backbone point data and analyzing their trends across the speed range. With the help of it, Shi Yang established full-state models for a civilian high-bypass turbofan engine [3] and a single-rotor turbojet engine [22]; Wang Jiamei [23] achieved windmilling state simulations for a military turbofan engine, and Fang Yun [24] performed ground start and high-altitude windmilling start simulations for a turbofan engine. Although capable of capturing unique operating modes, it cannot extrapolate to zero speed.

(5) 

β Extrapolation Method

This method was proposed by Dr. Kurzke, the author of the engine performance simulation software GasTurb 14, and is incorporated into the compressor and turbine characteristic processing software Smooth C 9 and Smooth T 9 [25,26]. This method establishes its foundation on the aerodynamic correlations among corrected flow, specific work, torque, and guide vane Mach number at low speeds, as determined from varied compressor maps and CFD data [27]. Using a derived intermediate variable (β), it estimates zero-speed characteristics [28] and has been extended to turbines [29]. Although theoretically robust and validated in ground/windmilling start simulations [30,31], its implementation depends on proprietary software.

In summary, although existing extrapolation methods have been successfully applied to various aeroengines, they suffer from the following issues:

(1)

Insufficient utilization of available data. For example: the Proportional Coefficient Method only uses one or two component characteristic lines above idle.

(2)

Dependence on specific prior conditions, without which implementation is impossible. For instance: the Stage-Stacking Method relies on detailed compressor design parameters, which are often inaccessible to general researchers; the Zero-Speed Line Interpolation Method requires accurate zero-speed component characteristics, obtainable only through locked-rotor tests or CFD simulations; the Backbone Feature Method needs enthalpy change characteristics above idle, which are usually not included in general component maps [32].

(3)

Difficulty in reflecting special operating conditions of components. For example, unmodified Proportional Coefficient Method fails to capture compressor stirring/turbine modes or turbine stirring/compressor modes in its extrapolated results; the Backbone Feature Method cannot directly extrapolate component characteristics to zero speed, the Stage-Stacking Method is applicable only to compressors, not turbines.

The β Extrapolation Method largely resolves these three issues. However, its implementation details have not been fully disclosed, making it nearly impossible to apply without access to Smooth C and Smooth T.

To overcome the limitations of existing methods, this paper proposes a novel curve-surface fitting-based extrapolation method for sub-idle component characteristics of aeroengines. The key innovations are as follows:

(1)

Considering the continuous and smooth nature of aero-engine component characteristics, we creatively perform curve and surface fitting on above-idle performance data. The sub-idle characteristics are then extrapolated from these fitting results. This approach not only ensures full utilization of limited above-idle data but also makes the extrapolation process straightforward to implement.

(2)

When above-idle corrected specific enthalpy change data are unavailable, we define a Specific Enthalpy Change Coefficient (SECC) that is approximately proportional to corrected specific enthalpy change and can be calculated from isentropic efficiency and pressure/expansion ratios. By transforming the extrapolation of isentropic efficiency into SECC extrapolation, we overcome the discontinuity limitation while maintaining method applicability, ultimately achieving comprehensive representation of unique operating modes in the extrapolation results.

(3)

We apply constraints during the fitting process, including strict physical constraints to ensure result validity, as well as component-specific constraints designed according to characteristic behavior. These constraints guarantee both the smoothness of extrapolated results and the flexibility of the extrapolation method.

The remainder of this paper is organized to reflect this methodological progression: Section 2 details the proposed methodology, beginning with component characteristic representation and analysis of full-range operation features, then presenting the complete extrapolation procedure. Section 3 validates the method using a micro-turbojet engine, presenting both extrapolation results and ground-starting simulations across varying inlet conditions. Section 4 concludes from the validation results that the proposed method not only overcomes the limitations of existing approaches but also demonstrates broader applicability in practical aeroengine simulations.

2. Methodology

2.1. General Expression of Aircraft Engine Component Characteristics

In aircraft engine component-level model, component characteristics such as corrected flow, compressor pressure ratio, turbine expansion ratio, and isentropic efficiency are often represented as a 2D array, with the relative corrected speed as the row parameter and intermediate variables as the column parameter. In component characteristic maps, the component characteristics are presented in the form of relative corrected speed lines. Compressor characteristics are plotted on two graphs, with the horizontal axis representing corrected flow and the vertical axis representing pressure ratio and isentropic efficiency, respectively. Similarly, for two turbine characteristic maps, the horizontal axis represents the product of corrected flow and relative corrected speed [33], and the vertical axis represents expansion ratio and isentropic efficiency, respectively.

The primary role of the intermediate variables is to determine the engine operating point’s location on the characteristic map. Two commonly used intermediate variables are the pressure/expansion ratio coefficient and $\beta$. The former includes the compressor pressure ratio coefficient and the turbine expansion ratio coefficient, defined by Equations (1) and (2), respectively.

$$Z_C={\displaystyle \frac{{\pi }_C-{\pi }_{C,\min }}{{\pi }_{C,\max }-{\pi }_{C,\min }}}$$

(1)

$$Z_T={\displaystyle \frac{{\pi }_T-{\pi }_{T,\min }}{{\pi }_{T,\max }-{\pi }_{T,\min }}}$$

(2)

In the equations, $Z$ represents the pressure/expansion ratio coefficient, $\pi$ is the pressure/expansion ratio, and the subscripts $C$ and $T$ represent the compressor and turbine, respectively. The subscripts max and min represent the maximum and minimum values at the same relative corrected speed. From Equations (1) and (2), it can be concluded that the range of values for the pressure/expansion ratio coefficient is $\left(0,1\right)$.

If the compressor pressure ratio and turbine expansion ratio monotonically vary with corrected flow across all relative corrected speed lines, the pressure/expansion ratio coefficient can be used as the intermediate variable. Otherwise, $\beta$ should be used as the intermediate variable [33]. The $\beta$ curves are a set of equidistant parabolas on characteristic maps where the vertical and horizontal axis represents pressure/expansion ratio and corrected flow, respectively. These parabolas are parameterized by the intermediate variable $\beta$, and their intersections with the relative corrected speed lines define the component characteristic points. The values of $\beta$ for the top and bottom parabolas are 1 and 0, respectively, so the range of $\beta$ is also $\left(0,1\right)$.

The validation platform for the method described in this paper is a micro-turbojet engine. Since no published component characteristic data are available for this engine, this study employs generic above-idle characteristics modified based on bench test data [33]. The general procedure for obtaining the above-idle component characteristics is as follows:

(1)

Bench testing was conducted at ambient conditions (approximately 30 °C, 101,325 Pa) to acquire test data at 10 steady-state operating points from idle to maximum thrust.

(2)

Using optimization algorithms, we calculated the working parameters of both the compressor and turbine at each steady-state point, including corrected mass flow, pressure ratio/expansion ratio, isentropic efficiency and inlet total temperature (only for turbine).

(3)

Based on the engine configuration (featuring a single-stage centrifugal compressor and single-stage axial-flow turbine), we selected corresponding generic characteristics from the GasTurb software database. These generic characteristics were then adjusted according to the calculated results to ensure the component-level mathematical model outputs matched the test data with acceptable agreement.

Table 1. Performance parameters of the micro-turbojet engine at maximum thrust condition on the ground.

Mass Flow Rate (kg/s)	Compressor Pressure Ratio	Turbine Inlet Total Temperature (K)	Turbine Outlet Total Temperature (K)	Shaft Speed (rpm)	Thrust (N)
0.692	3.39	1333.04	1165.49	97,070	351.71

presents the performance parameters at maximum thrust condition on the ground, which reflect the engine’s performance. All parameters except turbine inlet temperature were obtained directly from the test data.

The corrected above-idle characteristics of the compressor and turbine are presented in Figure 1 and Figure 2, respectively. Both figures indicate the design point and locations of the 10 steady-state test points. For clarity in displaying the relative corrected speed values corresponding to each speed line, the number of speed lines shown is fewer than the actual count. However, the lowest and highest speed lines from the original component characteristics, along with the design speed line (where the relative corrected speed equals 1.0), are all retained in the plots.

As shown in Figure 1 and Figure 2, the compressor pressure ratio and turbine expansion ratio vary monotonically with corrected flow across all relative corrected speed lines, thus the pressure/expansion ratio coefficient is used as the intermediate variable. The results demonstrate that the corrected generic above-idle characteristics contain only three parameters: corrected mass flow, pressure/expansion ratio, and isentropic efficiency. A key challenge addressed by the proposed method is how to achieve comprehensive sub-idle characteristic extrapolation while making optimal use of these limited known parameters.

2.2. Analysis of the Operating Modes of Aeroengine Compressors and Turbines

Under normal operating conditions, the compressor performs work on the airflow, increasing the total temperature and pressure of the gas, while the turbine extracts work from the gas flow, resulting in decreased total temperature and pressure. However, multiple research institutions and scholars, including NASA Lewis Research Center [19,20,21] and Dr. Kurzke (author of GasTurb) [27,28,30,33], have consistently observed that during aeroengine windmilling conditions or at very low rotational speeds, both compressors and turbines may significantly deviate from their design states. Remarkably, the total temperature and pressure may even decrease after passing through the compressor, while increasing after the turbine in these abnormal conditions. This evidence clearly demonstrates that comprehensive analysis of compressor and turbine operational characteristics across the full operating envelope must precede any component characteristic extrapolation [3].

Throughout the entire operational envelope of an aeroengine, the compressor exhibits three distinct operating modes:

(1)

Compressor Mode

This is the compressor’s normal operating state at higher engine speeds, where it consumes work to compress the airflow. In this regime:

$${\pi }_C={\displaystyle \frac{p_{t,C,out}}{p_{t,C,in}}}>1$$

(3)

$$T_{t,C,out}>T_{t,C,in}$$

(4)

$$\Delta H_{t,C}=H_{t,C,out}-H_{t,C,in}>0$$

(5)

$$\Delta H_{t,is}=H_{t,C,out,is}-H_{t,C,in}>0$$

(6)

$${\eta }_{C,is}={\displaystyle \frac{\Delta H_{t,C,is}}{\Delta H_{t,C}}}\in \left(0,1\right)$$

(7)

In the equations:

$p$ represents pressure, measured in Pa;

$T$ represents temperature, measured in K;

$H$ represents enthalpy, measured in J;

$\Delta H$ represents specific enthalpy change, it has the same unit with $H$;

${\eta }_{C,is}$ represents the isentropic efficiency of the compressor.

The subscripts:

$t$ denote total parameters,

$is$ denote the isentropic process,

$in$ and $out$ denote the component inlet and outlet, respectively.

(2)

Turbine Mode

When the aeroengine operates in windmilling state, the compressor operates in turbine mode, where the gas flow drives the compressor rotation while performing work on it, causing the compressor to function similarly to a turbine [34]. Under this condition:

$${\pi }_C={\displaystyle \frac{p_{t,C,out}}{p_{t,C,in}}}<1$$

(8)

$$T_{t,C,out}<T_{t,C,in}$$

(9)

$$\Delta H_{t,C}=H_{t,C,out}-H_{t,C,in}<0$$

(10)

$$\Delta H_{t,is}=H_{t,C,out,is}-h_{t,C,in}<0$$

(11)

$${\eta }_{C,is}={\displaystyle \frac{\Delta H_{t,C,is}}{\Delta H_{t,C}}}\in \left(1,+\infty \right)$$

(12)

(3)

Stirring Mode

As an intermediate transitional mode between compressor mode and turbine mode, stirring mode represents the compressor’s operating condition at low engine speeds. The kinetic energy loss of the airflow is converted to thermal energy, while the work input by the compressor becomes insufficient to compensate for these losses [35]. At this operating mode:

$${\pi }_C={\displaystyle \frac{p_{t,C,out}}{p_{t,C,in}}}<1$$

(13)

$$T_{t,C,out}>T_{t,C,in}$$

(14)

$$\Delta H_{t,C}=H_{t,C,out}-H_{t,C,in}>0$$

(15)

$$\Delta H_{t,is}=H_{t,C,out,is}-H_{t,C,in}<0$$

(16)

$${\eta }_{C,is}={\displaystyle \frac{\Delta H_{t,C,is}}{\Delta H_{t,C}}}\in \left(-\infty ,0\right]$$

(17)

Similarly, the turbine also exhibits three distinct operating modes across the engine’s full operational envelope:

(1)

Turbine Mode

The turbine operates normally, with the gas flow expanding through it and performing work on the turbine. The conditions are as follows:

$${\pi }_T={\displaystyle \frac{p_{t,T,in}}{p_{t,T,out}}}>1$$

(18)

$$T_{t,T,out}<T_{t,T,in}$$

(19)

$$\Delta H_{t,T}=H_{t,T,in}-H_{t,T,out}>0$$

(20)

$$\Delta H_{t,T,is}=H_{t,T,in}-H_{t,T,out,is}>0$$

(21)

$${\eta }_{T,is}={\displaystyle \frac{\Delta H_{t,T}}{\Delta H_{t,T,is}}}\in \left(0,1\right)$$

(22)

(2)

Compressor Mode

The turbine performs work on the gas flow, functioning similarly to a compressor. The following conditions are applicable:

$${\pi }_T={\displaystyle \frac{p_{t,T,in}}{p_{t,T,out}}}<1$$

(23)

$$T_{t,T,out}>T_{t,T,in}$$

(24)

$$\Delta H_{t,T}=H_{t,T,in}-H_{t,T,out}<0$$

(25)

$$\Delta H_{t,T,is}=H_{t,T,in}-H_{t,T,out,is}<0$$

(26)

$${\eta }_{T,is}={\displaystyle \frac{\Delta H_{t,T}}{\Delta H_{t,T,is}}}\in \left(1,+\infty \right)$$

(27)

(3)

Stirring Mode

Similar to the compressor stirring mode, the turbine exhibits:

$${\pi }_T={\displaystyle \frac{p_{t,T,in}}{p_{t,T,out}}}>1$$

(28)

$$T_{t,T,out}>T_{t,T,in}$$

(29)

$$\Delta H_{t,T}=H_{t,T,in}-H_{t,T,out}<0$$

(30)

$$\Delta H_{t,T,is}=H_{t,T,in}-H_{t,T,out,is}>0$$

(31)

$${\eta }_{T,is}={\displaystyle \frac{\Delta H_{t,T}}{\Delta H_{t,T,is}}}\in \left(-\infty ,0\right]$$

(32)

The micro-turbojet engine discussed in this study has no compressor bleed air and no turbine cooling air, all leakages are neglected, therefore, the following conditions apply:

$${\dot{m}}_{C,in}={\dot{m}}_{C,out}$$

(33)

$${\dot{m}}_{T,in}={\dot{m}}_{T,out}$$

(34)

In the equation, $\dot{m}$ is the gas mass flow rate, measured in kg/s.

Thus, the isentropic efficiencies of the compressor and turbine can be calculated using Equations (35) and (36), respectively:

$${\eta }_{C,is}={\displaystyle \frac{\Delta H_{t,C,is}}{\Delta H_{t,C}}}={\displaystyle \frac{{\dot{m}}_{C,out}{h}_{t,C,out,is}-{\dot{m}}_{C,in}{h}_{t,C,in}}{{\dot{m}}_{C,out}{h}_{t,C,out}-{\dot{m}}_{C,in}{h}_{t,C,in}}}={\displaystyle \frac{\Delta h_{t,C,is}}{\Delta h_{t,C}}}$$

(35)

$${\eta }_{T,is}={\displaystyle \frac{\Delta H_{t,T}}{\Delta H_{t,T,is}}}={\displaystyle \frac{{\dot{m}}_{T,in}{h}_{t,T,in}-{\dot{m}}_{T,out}{h}_{t,T,out}}{{\dot{m}}_{T,in}{h}_{t,T,in}-{\dot{m}}_{T,out}{h}_{t,T,out,is}}}={\displaystyle \frac{\Delta h_{t,T}}{\Delta h_{t,T,is}}}$$

(36)

where, $h$ represents the specific enthalpy of the gas, which is the enthalpy per unit mass of the gas, and $\Delta h$ is the change in specific enthalpy, both of them are measured with J/kg.

When the relative corrected speed is low, considering the relative corrected speed is kept constant, the isentropic efficiencies of the compressor and turbine vary with the corrected mass flow rate, as shown in Figure 3. It can be seen that the isentropic efficiency is discontinuous. Therefore, to make the extrapolation results reasonable and adequately reflect the unique operating modes of the components, it is necessary to overcome the impact of the discontinuity in isentropic efficiency.

2.3. Extrapolation Method for Component Characteristics Below Idle Based on Curve and Surface Fitting

Based on the continuous and smooth nature of aircraft engine component characteristics [33], a method is proposed for extrapolating component characteristics below idle using curve and surface fitting. This method fully utilizes the limited available data to extrapolate component characteristics and ensures that the extrapolation results are reasonable and adequately reflect the unique operating modes of the components. The method includes extrapolation techniques for both the compressor and turbine characteristics. The micro-turbojet engine in this study is used as the application object for the method, and the extrapolation steps are carried out using the Curve Fitting tool in MATLAB R2023b.

2.3.1. Compressor Component Characteristics Extrapolation Method

To extrapolate the compressor component characteristics, the following steps should be followed:

Step 1: Fit the curves of the maximum and minimum pressure ratios above idle as a function of the relative corrected speed. The fitting relationships are given by Equations (37) and (38), and the maximum and minimum pressure ratios below idle are extrapolated from the fitting results.

$${\pi }_{C,\max }=C_{{\pi }_{C,\max }}\left(N_{C,rel,cor}\right)$$

(37)

$${\pi }_{C,\min }=C_{{\pi }_{C,\min }}\left(N_{C,rel,cor}\right)$$

(38)

In the formula, $C$ represents the curve, and $N_{C,rel,cor}$ is the compressor’s relative corrected speed, defined as follows [36]:

$$N_{C,rel,cor}={\displaystyle \frac{N_{C,cor}}{N_{C,cor,des}}}=\left({\displaystyle \frac{N_{mech}}{\sqrt{\raisebox{1ex}{$T_{t,C,in}$}\!\left/ \!\raisebox{-1ex}{$T_{std}$}\right.}}}\right)/\left({\displaystyle \frac{N_{mech,des}}{\sqrt{\raisebox{1ex}{$T_{t,C,in,des}$}\!\left/ \!\raisebox{-1ex}{$T_{std}$}\right.}}}\right)$$

(39)

In the formula, $N_{mech}$ represents the mechanical speed, and $T_{std}$ is the sea-level static temperature under standard atmospheric conditions. The subscript $des$ represents the design point parameters.

The analysis of locked rotor characteristics [15,16] reveals that when the relative corrected speed drops to zero, the compressor ceases to perform work on the airflow. This necessitates imposing two physical constraints:

• The maximum pressure ratio must be constrained to unity (PR ≤ 1), as no compression occurs under static conditions.
• For airflow to pass through the rotor, the minimum pressure ratio should be bounded within (0, 1), ensuring the outlet total pressure remains lower than the inlet value.

For the micro-turbojet engine in this study, we employed the Piecewise Cubic Hermite Polynomial Interpolation (PCHPI) method to fit and extrapolate both maximum and minimum pressure ratios above idle speed, with the results shown in Figure 4. Given the absence of experimental data at zero speed, the minimum pressure ratio at standstill was assigned a value of 0.75 to maintain extrapolation smoothness while respecting thermodynamic plausibility.

Step 2: Based on the extrapolated results of the maximum and minimum pressure ratios, the pressure ratios at other points below idle are calculated using Equation (40).

$${\pi }_C=Z_C·\left({\pi }_{C,\max }-{\pi }_{C,\min }\right)+{\pi }_{C,\min }$$

(40)

Step 3: Fit the curves of the maximum and minimum corrected mass flow rates above idle as a function of the relative corrected speed. The fitting relationships are given by Equations (41) and (42), and the maximum and minimum corrected mass flow rates below idle are extrapolated from the fitting results.

$${\dot{m}}_{C,cor,\max }=C_{{\dot{m}}_{C,cor,\max }}\left(N_{C,rel,cor}\right)$$

(41)

$${\dot{m}}_{C,cor,\min }=C_{{\dot{m}}_{C,cor,\min }}\left(N_{C,rel,cor}\right)$$

(42)

In the equation, ${\dot{m}}_{C,cor}$ represents the compressor corrected mass flow rate, which is defined as follows [36]:

$${\dot{m}}_{C,cor}={\displaystyle \frac{{\dot{m}}_C\sqrt{\raisebox{1ex}{$T_{t,C,in}$}\!\left/ \!\raisebox{-1ex}{$T_{std}$}\right.}}{\raisebox{1ex}{$p_{t,C,in}$}\!\left/ \!\raisebox{-1ex}{$p_{std}$}\right.}}$$

(43)

The locked-rotor characteristics [15,16] demonstrate that when rotational speed reaches zero:

(1)

With a pressure ratio of 1, airflow cannot pass through the rotor, requiring the minimum corrected mass flow rate to be constrained to zero.

(2)

With pressure ratios below 1, airflow can traverse the rotor, necessitating the maximum corrected mass flow rate to exceed zero.

Regarding the micro-turbojet engine in this study, the PCHPI method was applied to perform curve fitting and extrapolation for both maximum and minimum corrected mass flow rates above idle speed, the results are plotted in Figure 5a. Due to the lack of experimental data, the maximum corrected mass flow rate at zero speed remains undetermined; to ensure smoothness of the extrapolated results, it was conservatively assigned a value of 0.1 kg/s.

Step 4: Based on the corrected mass flow rates above idle and the maximum and minimum corrected mass flow rates below idle, perform a surface fitting of the corrected mass flow rate as a function of the relative corrected speed and pressure ratio coefficient. The fitting relationship is given by Equation (44), and the corrected mass flow rates at other points below idle are extrapolated from the fitting results.

$${\dot{m}}_{C,cor}=S_{{\dot{m}}_{C,cor}}\left(N_{C,rel,cor},Z_C\right)$$

(44)

In the equation, $S$ represents the surface.

Concerning the micro-turbojet engine in this study, Thin-Plate Spline (TPS) interpolation was employed to perform surface fitting and extrapolation of the corrected mass flow rate characteristics, with the results presented in Figure 5b.

Step 5: To address the issue of discontinuity in isentropic efficiency, the extrapolation of isentropic efficiency should be converted into the extrapolation of corrected specific enthalpy change. However, the corrected specific enthalpy change above idle is unknown. Therefore, the compressor specific enthalpy change coefficient (SECC) is defined as follows:

$${\psi }_C=\left\{\begin{array}{cc}\left({{\pi }_C}^{{\displaystyle \frac{{\tilde{\overline{\gamma }}}_C-1}{{\tilde{\overline{\gamma }}}_C}}}-1\right)/{\eta }_{C,is}& \left({\eta }_{C,is}\ne 0\right)\\ \underset{\left({\pi }_C,{\eta }_{C,is}\right)\to \left(1,0\right)}{\lim }\left[\left({{\pi }_C}^{{\displaystyle \frac{{\tilde{\overline{\gamma }}}_C-1}{{\tilde{\overline{\gamma }}}_C}}}-1\right)/{\eta }_{C,is}\right]& \left({\eta }_{C,is}=0\right)\end{array}\right.$$

(45)

In the equation, ${\psi }_C$ represents the compressor SECC, ${\tilde{\overline{\gamma }}}_C$ is the approximate value of the average specific heat ratio of the gas in the compressor, and in this study, ${\tilde{\overline{\gamma }}}_C=1.4$.

For the compressor, at any relative corrected speed, if ${\pi }_C=1$, then ${\eta }_{C,is}=0$, and ${\eta }_{C,is}=f\left({\pi }_C\right)$ is continuously differentiable at this point. According to L’Hôpital’s rule, $\underset{\left({\pi }_C,{\eta }_{C,is}\right)\to \left(1,0\right)}{\lim }\left[\left({{\pi }_C}^{{\displaystyle \frac{{\tilde{\overline{\gamma }}}_C-1}{{\tilde{\overline{\gamma }}}_C}}}-1\right)/{\eta }_{C,is}\right]$ exists and is a constant. Thus, it can be concluded that ${\psi }_C$ is continuous.

The compressor corrected specific enthalpy change can be calculated using the following equation [27]:

$$\Delta h_{t,C,cor}={\displaystyle \frac{\Delta h_{t,C}}{T_{t,C,in}}}=\left\{\begin{array}{cc}{\overline{C}}_{p,C}\left({\pi }^{{\displaystyle \frac{{\overline{\gamma }}_C-1}{{\overline{\gamma }}_C}}}-1\right)/{\eta }_{C,is}& \left({\eta }_{C,is}\ne 0\right)\\ {\overline{C}}_{p,C}\underset{\left({\pi }_C,{\eta }_{C,is}\right)\to \left(1,0\right)}{\lim }\left[\left({{\pi }_C}^{{\displaystyle \frac{{\overline{\gamma }}_C-1}{{\overline{\gamma }}_C}}}-1\right)/{\eta }_{C,is}\right]& \left({\eta }_{C,is}=0\right)\end{array}\right.$$

(46)

In the equation, $\Delta h_{t,C,cor}$ represents the compressor corrected specific enthalpy change, while ${\overline{C}}_{p,C}$ and ${\overline{\gamma }}_C$ are the average specific heat capacity at constant pressure and the average specific heat ratio of the gas in the compressor, respectively. From Equations (45) and (46), it can be concluded that ${\psi }_C$ is positively correlated with $\Delta h_{t,C,cor}$.

In summary, converting the extrapolation of isentropic efficiency into the extrapolation of the SECC resolves the issues of isentropic efficiency discontinuity and the unknown corrected specific enthalpy change above idle. From Equation (45), Equation (47) can be derived. Using Equation (47), the corresponding isentropic efficiency can be calculated from the extrapolated results of the SECC, thereby indirectly achieving the extrapolation of isentropic efficiency.

$${\eta }_{C,is}=\left({{\pi }_C}^{{\displaystyle \frac{{\tilde{\overline{\gamma }}}_C-1}{{\tilde{\overline{\gamma }}}_C}}}-1\right)/{\psi }_C$$

(47)

With the help of Equation (45), we can calculate the SECCs for each point above idle. Perform curve fitting of the SECCs corresponding to the maximum and minimum pressure ratios above idle as functions of the relative corrected speed. The fitting relationships are given by Equations (48) and (49), and the SECCs corresponding to the maximum and minimum pressure ratios below idle are extrapolated from the fitting results.

$${\psi }_{C,\pi \max }=C_{{\overline{\psi }}_{C,\pi \max }}\left(N_{C,rel,cor}\right)$$

(48)

$${\psi }_{C,\pi \min }=C_{{\overline{\psi }}_{C,\pi \min }}\left(N_{C,rel,cor}\right)$$

(49)

When the speed is 0, if the pressure ratio equals 1, the flow rate is 0, and there is no energy exchange between the compressor and the airflow. Consequently, the SECC should theoretically be 0. However, for the SECC corresponding to the maximum pressure ratio, a value greater than 0 must still be specified as a constraint when the relative corrected speed is 0. This is because the compressor at zero speed can only operate in the stirring mode or turbine mode. As the pressure ratio gradually decreases from 1, the stirring mode should appear first, followed by the turbine mode. Correspondingly, the isentropic efficiency should first decrease from 0 to $-\infty$, and then continue to decrease from $+\infty$. From Equation (47), it can be deduced that only when the SECC at zero speed starts from a value greater than 0 and decreases with the pressure ratio until it becomes less than 0, the isentropic efficiency at zero speed can exhibit such a variation.

When the speed is 0 and the pressure ratio is less than 1, the flow rate is greater than 0. The airflow loses energy as it passes through the compressor, and the SECC should be less than 0. Therefore, during fitting, for the SECC corresponding to the minimum pressure ratio, a value less than 0 should be specified as a constraint when the relative corrected speed is 0. After extrapolation, if the isentropic efficiency is found to be less than 1 when the pressure ratio is less than 1, the constraint conditions should be adjusted, and the extrapolation process should be repeated.

In the case of the micro-turbojet engine in this study, quartic Gaussian fitting and PCHPI were implemented to perform curve fitting and extrapolation of the SECC values corresponding to maximum and minimum pressure ratios above idle speed, respectively. The resultant profiles are graphically presented in Figure 6a.

Given the absence of experimental data, while ensuring extrapolation smoothness, the SECC at minimum pressure ratio was conservatively assigned a value of −0.04 when relative corrected speed reaches zero. Notably, the unconstrained extrapolation for maximum pressure ratio conditions naturally satisfied the request of positive SECC (SECC > 0) at zero relative corrected speed.

Step 6: Based on the SECCs above idle and those corresponding to the maximum and minimum pressure ratios below idle, perform surface fitting of the SECC as a function of the relative corrected speed and pressure ratio coefficient. The fitting relationship is given by Equation (50), and the SECCs at other points below idle are extrapolated from the fitting results.

$${\psi }_C=S_{{\psi }_C}\left(N_{C,rel,cor},Z_C\right)$$

(50)

As pertains to the micro-turbojet engine in this study, TPS interpolation was utilized to conduct surface fitting and extrapolation of the SECC characteristics, with the resulting three-dimensional profile illustrated in Figure 6b.

2.3.2. Turbine Component Characteristics Extrapolation Method

For the turbine component characteristics extrapolation, the following procedure applies:

Step 1: Perform curve fitting for the maximum and minimum turbine expansion ratios above idle as functions of the relative corrected speed. The fitting relationships are given by Equations (51) and (52). The maximum and minimum turbine expansion ratios below idle are extrapolated based on the fitting results.

$${\pi }_{T,\max }=C_{{\pi }_{T,\max }}\left(N_{T,rel,cor}\right)$$

(51)

$${\pi }_{T,\min }=C_{{\pi }_{T,\min }}\left(N_{T,rel,cor}\right)$$

(52)

In the equation, $N_{T,rel,cor}$ represents the turbine relative corrected speed, which is defined as follows [36]:

$$N_{T,rel,cor}={\displaystyle \frac{N_{T,cor}}{N_{T,cor,des}}}=\left({\displaystyle \frac{N_{mech}}{\sqrt{\raisebox{1ex}{$T_{t,T,in}$}\!\left/ \!\raisebox{-1ex}{$T_{std}$}\right.}}}\right)/\left({\displaystyle \frac{N_{mech,des}}{\sqrt{\raisebox{1ex}{$T_{t,T,in,des}$}\!\left/ \!\raisebox{-1ex}{$T_{std}$}\right.}}}\right)$$

(53)

Analogous to the compressor case, when the relative corrected speed approaches zero, the following constraints must be imposed:

(1)

The minimum expansion ratio should be constrained to unity.

(2)

The maximum expansion ratio must exceed unity.

For the micro-turbojet validation platform, the PCHPI method was applied to perform curve fitting and extrapolation of both maximum and minimum expansion ratios above idle speed, as demonstrated in Figure 7. Without imposing explicit constraints on the maximum expansion ratio, the extrapolated results inherently satisfy the fundamental requirement of ${\pi }_{T,\max }>1$ at zero relative corrected speed.

Step 2: Based on the extrapolated results of the maximum and minimum expansion ratios, calculate the expansion ratios at other points below idle using Equation (54).

$${\pi }_T=Z_T·\left({\pi }_{T,\max }-{\pi }_{T,\min }\right)+{\pi }_{T,\min }$$

(54)

Step 3: Perform curve fitting for the maximum and minimum corrected mass flow rates above idle as functions of the relative corrected speed. The fitting relationships are given by Equations (55) and (56). The maximum and minimum corrected mass flow rates below idle are extrapolated based on the fitting results.

$${\dot{m}}_{T,cor,\max }=C_{{\dot{m}}_{T,cor,\max }}\left(N_{T,rel,cor}\right)$$

(55)

$${\dot{m}}_{T,cor,\min }=C_{{\dot{m}}_{T,cor,\min }}\left(N_{T,rel,cor}\right)$$

(56)

In the equation, ${\dot{m}}_{T,cor}$ represents the turbine corrected mass flow rate, which is defined as follows [36]:

$${\dot{m}}_{T,cor}={\displaystyle \frac{{\dot{m}}_T\sqrt{\raisebox{1ex}{$T_{t,T,in}$}\!\left/ \!\raisebox{-1ex}{$T_{std}$}\right.}}{\raisebox{1ex}{$p_{t,T,in}$}\!\left/ \!\raisebox{-1ex}{$p_{std}$}\right.}}$$

(57)

Similar to the compressor case, physical constraints must be applied when the relative corrected speed reaches zero:

(1)

The minimum corrected mass flow rate should be constrained to zero.

(2)

The maximum corrected mass flow rate must remain positive.

Regarding the micro-turbojet validation platform, the PCHPI method was implemented to perform curve fitting and extrapolation of both maximum and minimum corrected mass flow rates above idle speed, with the results shown in Figure 8a. Due to the absence of experimental data at standstill conditions, the maximum corrected mass flow rate at zero speed was assigned a conservative value of 0.4 kg/s to ensure extrapolation smoothness, while maintaining physical consistency with throughflow requirements.

Step 4: Based on the corrected mass flow rates above idle and the maximum and minimum corrected mass flow rates below idle, perform surface fitting of the corrected mass flow rate as a function of the relative corrected speed and expansion ratio coefficient. The fitting relationship is given by Equation (58), and the corrected mass flow rates at other points below idle are extrapolated from the fitting results.

$${\dot{m}}_{T,cor}=S_{{\dot{m}}_{T,cor}}\left(N_{T,rel,cor},Z_T\right)$$

(58)

As for the micro-turbojet validation platform, a Biharmonic Interpolation Method is used for surface fitting and extrapolation of the corrected mass flow rate, with the results plotted in Figure 8b.

Step 5: Similar to the compressor, the turbine SECC is defined as follows:

$${\psi }_T={\eta }_{T,is}\left(1-{\displaystyle \frac{1}{{{\pi }_T}^{\frac{{\tilde{\overline{\gamma }}}_T-1}{{\tilde{\overline{\gamma }}}_T}}}}\right)$$

(59)

In the equation, ${\psi }_T$ is the turbine SECC, and ${\tilde{\overline{\gamma }}}_T$ is the approximate value of the average heat capacity ratio of the gas in the turbine, which is taken as ${\tilde{\overline{\gamma }}}_T=1.33$ in this study. According to the definition, it is known that ${\psi }_T$ is continuous.

The turbine corrected enthalpy change can be calculated as follows [27]:

$$\Delta h_{t,T,cor}={\displaystyle \frac{\Delta h_{t,T}}{T_{t,T,in}}}={\overline{C}}_{p,T}{\eta }_{T,is}\left(1-{\displaystyle \frac{1}{{{\pi }_T}^{\frac{{\overline{\gamma }}_T-1}{{\overline{\gamma }}_T}}}}\right)$$

(60)

In the equation, $\Delta h_{t,T,cor}$ is the turbine corrected enthalpy change, ${\overline{C}}_{p,T}$ and ${\overline{\gamma }}_T$ are the average specific heat at constant pressure and the average heat capacity ratio of the gas in the turbine, respectively. From Equations (59) and (60), it can be concluded that ${\psi }_T$ and $\Delta h_{t,T,cor}$ are positively correlated.

From Equation (59), Equation (61) can be derived. By using Equation (61), the corresponding isentropic efficiency can be calculated from the extrapolated results of the SECC, thus indirectly achieving the extrapolation of the isentropic efficiency.

$${\eta }_{T,is}={\psi }_T/\left(1-{\displaystyle \frac{1}{{{\pi }_T}^{\frac{{\tilde{\overline{\gamma }}}_T-1}{{\tilde{\overline{\gamma }}}_T}}}}\right)$$

(61)

By the use of Equation (59), we are able to calculate the SECCs for each point above idle. Perform curve fitting of the SECCs corresponding to the maximum and minimum expansion ratios above idle as a function of the relative corrected speed, with the fitting equations being (62) and (63). Extrapolate the SECCs corresponding to the maximum and minimum expansion ratios below idle based on the fitting results.

$${\psi }_{T,\pi \max }=C_{{\psi }_{T,\pi \max }}\left(N_{T,rel,cor}\right)$$

(62)

$${\psi }_{T,\pi \min }=C_{{\psi }_{T,\pi \min }}\left(N_{T,rel,cor}\right)$$

(63)

When the speed is 0, if the expansion ratio is 1, the flow rate is 0, and there is no energy exchange between the turbine and the airflow. The SECC should be 0. However, for the SECC corresponding to the minimum expansion ratio, a value smaller than 0 should still be specified as the constraint at this point. This is because the turbine at zero speed can only be in either the stirring state or turbine state. As the expansion ratio gradually increases from 1, the stirring state should appear first, followed by the turbine state. Accordingly, the isentropic efficiency should increase from $-\infty$ until it becomes greater than 0. From Equation (62), it can be seen that only by starting with an SECC less than 0 at zero speed, and increasing it as the expansion ratio increases until it exceeds 0, can the extrapolated isentropic efficiency show this pattern of change.

When the speed is 0, if the expansion ratio is greater than 1, the flow rate is greater than 0, and there is energy loss as the airflow passes through the turbine. The SECC should be greater than 0. Therefore, when fitting, a value greater than 0 should be specified for the SECC corresponding to the maximum expansion ratio at this point.

Concerning the micro-turbojet validation platform, Quadratic Polynomial Fitting and PCHPI were respectively applied to perform curve fitting and the extrapolation of the SECC values corresponding to both maximum and minimum expansion ratios above idle speed. The results are presented in Figure 9a. Given the lack of experimental data at standstill conditions, conservative SECC values of 0.12 and −0.012 were, respectively, assigned to the maximum and minimum expansion ratio conditions at zero relative corrected speed to ensure smooth extrapolation behavior while preserving thermodynamic plausibility.

Step 6: Based on the SECCs above idle and the SECCs corresponding to the maximum and minimum expansion ratios below idle, surface fitting of the SECC with respect to the relative corrected speed and expansion ratio coefficient is performed. The fitting relationship is given by Equation (64), and the extrapolation results for other points below idle can be calculated from the fitting results.

$${\psi }_T=S_{{\psi }_T}\left(N_{T,rel,cor},Z_T\right)$$

(64)

In the case of the micro-turbojet engine validation platform, TPS interpolation was used for surface fitting and extrapolation of the SECC, with the resulting three-dimensional profile shown in Figure 9b.

Step7: If the minimum expansion ratio above idle is not less than 1, the turbine extrapolation results will not include the compressor state after the above extrapolation process and will not fully cover the stirring state region. Therefore, further extrapolation of the turbine component characteristics is required.

First, for each relative corrected speed line, a value less than 1 is specified as the new minimum expansion ratio. Then, based on the extrapolated results obtained from Steps 1–6, for each relative corrected speed line, curve fitting of the corrected flow and SECC with respect to the expansion ratio is performed. The extrapolated results for the corrected flow and SECC between the original minimum and new minimum expansion ratios are then used. The SECC extrapolated results and Equation (63) are used to calculate the isentropic efficiency. Finally, all corrected flow values less than 0 in the extrapolated results are set to 0. If the isentropic efficiency falls below unity under conditions where the expansion ratio is less than 1, the extrapolated SECC results must be adapted according to Equation (65) to ensure physically meaningful extrapolation of the isentropic efficiency characteristics.

$${\psi }_{T,old}=AF·\left({\psi }_{T,new}-{\psi }_{T,\pi \min ,old}\right)+{\psi }_{T,\pi \min ,old}$$

(65)

In the equation, $AF$ is the adaption factor, ${\psi }_{T,old}$ and ${\psi }_{T,new}$ represent the SECCs before and after correction, respectively, and ${\psi }_{T,\pi \min ,old}$ is the SECC corresponding to the original minimum expansion ratio.

As pertains to the micro-turbojet engine validation platform, implementation of the PCHPI method for the curve fitting described in Step 7 revealed that extrapolation without adaption would yield non-physical results—specifically, isentropic efficiencies below unity at expansion ratios less than 1. Consequently, the SECC extrapolation results were systematically adapted according to Equation (65), with $AF=2$.

3. Results and Simulation Verification

3.1. Component Characteristic Extrapolation Results and Discussion

The extrapolated component characteristics of the micro-turbojet engine are shown in Figure 10 and Figure 11. It should be noted that, to better present the extrapolation results and the regions corresponding to different operating modes in the component characteristic plots, the relative corrected speed values for each corresponding relative corrected speed line are marked in Figure 10 and Figure 11, and the number of relative corrected speed lines above idle speed in the plots has been reduced. As can be seen, the extrapolation results cover all operating modes of the compressor and turbine. Moreover, on each relative corrected speed line below idle speed for both the compressor and turbine, the trend of isentropic efficiency changes is consistent with that described in Figure 3.

It is worth mentioning that the extrapolation method proposed in this study is not only applicable to micro-turbojet engines but also to other types of gas turbine engines. It is also applicable to component characteristics with pressure/expansion ratio coefficients as intermediate variables, as well as to component characteristics with $\beta$ as intermediate variables.

3.2. Preparation Before Simulation Validation

We programmed in C++ using Visual Studio 2022. First, we imported the extrapolated component characteristics into the micro-turbojet engine component-level model. We then developed the mathematical models for the actuators and sensors. The actuators include the starter motor, electric fuel pump, ignition fuel valve, main fuel valve, and igniter, while the sensors include the speed sensor, compressor outlet total pressure sensor, and turbine outlet total temperature sensor.

After completing the model program, we encapsulated it into an S-function and called it in Simulink. Then, we wrote the control algorithm program using the MATLAB Function block, which includes the open-loop control algorithm for the start-up process and the closed-loop control algorithm for above-idle conditions. The logic of the open-loop control algorithm for the start-up process is as follows:

(1) 

Starter Motor-Driven Acceleration Phase

After pressing the start switch, the starter motor’s PWM duty cycle is set to 100%, causing the rotor to accelerate from a standstill. When the speed reaches 6000 rpm, the start fuel valve is opened, but the main fuel valve and igniter remain closed, allowing a small amount of fuel to flow into the ignition fuel path without ignition. Once the speed reaches 14,000 rpm, the igniter is activated, while the main fuel valve stays closed. The fuel in the ignition path is ignited, heating the internal gases and causing the turbine outlet total temperature to rise. When the turbine outlet total temperature sensor reads above the main fuel valve opening temperature, the main fuel valve opens, the start fuel valve closes, and a larger amount of fuel flows into the combustion chamber to participate in combustion, allowing the turbine to start doing work. The main fuel valve opening temperature is determined by the engine’s inlet conditions.

(2) 

Starter Motor and Turbine-Driven Acceleration Phase

After the main fuel valve opens, the fuel pump’s PWM duty cycle increases linearly, raising the fuel flow and allowing the rotor to continue accelerating. The starter motor ceases to operate once the speed reaches 25,000 rpm.

(3) 

Turbine-Driven Acceleration Phase

The fuel pump’s PWM duty cycle continues to increase linearly, accelerating the rotor further. When the speed reaches 35,000 rpm, the above-idle closed-loop control algorithm is activated, maintaining the rotor speed at 35,200 rpm (idle speed), marking the end of the start process.

Considering the poor convergence of the component-level model below 5000 rpm (about 5.15% of the design point speed), we introduced a behavioral simulation program into the model: Using engine corrected speed, inlet altitude, and inlet temperature deviation, the compressor outlet total pressure, turbine outlet total temperature, thrust, and rotor damping torque are determined through three-dimensional linear interpolation. The remaining torque is calculated from the rotor damping torque and starter motor torque, and the rotor acceleration is determined using the rotor dynamics equation. Additionally, using inlet altitude and temperature deviation, two-dimensional linear interpolation is used to determine the initial guess values for the component-level model at 5000 rpm, enabling the transition from the behavioral simulation program to the component-level model during the start-up process.

3.3. Simulation Results and Discussion

To further validate the usability of the extrapolated results, we conducted ground start process simulations under different inlet conditions in Simulink, as follows:

Inlet Condition 1: Altitude 0 m, temperature deviation 30 K;

Inlet Condition 2: Altitude 2000 m, temperature deviation 0 K;

Inlet Condition 3: Altitude 4000 m, temperature deviation −30 K;

The simulation results are shown in Figure 12 and Figure 13.

It should be noted that during all the simulations mentioned above, the start switch was pressed at 5 s.

In all the simulation results shown in Figure 12, the following phenomena can be observed:

• At the start of the process, the starter motor torque rapidly rises to its maximum value, and the rotor begins to accelerate.
• The ignition fuel valve opens between 6 and 7 s, causing a slight increase in fuel flow at that time.
• Both the ignition and main fuel valve openings occur between 8 and 11 s. During ignition, the turbine outlet total temperature rises quickly, enhancing turbine power output. The compressor, driven by the turbine, increases its capacity to compress the air, causing the compressor outlet total pressure to rise quickly. The temperature and pressure rise rate then significantly slows down until the main fuel valve opens, and fuel flow increases linearly, leading to rapid increases in temperature and pressure, with the rotor continuing to accelerate.
• The starter motor disengages between 11 and 13 s, with rotor speed and thrust continuing to increase after slight fluctuations.
• The start process ends between 14 and 16 s when the control algorithm switches from open-loop control to closed-loop control for above-idle operations. Rotor speed fluctuates under different inlet conditions, which is due to the fact that the closed-loop control for above-idle operations is based on a PI controller, and the control parameters were tuned under standard atmospheric conditions with Altitude 0m. The further the inlet conditions deviate from these reference conditions, the worse the control performance.

This indicates that the actuator and engine model responses during the start-up process align with the intended effects of the open-loop control algorithm, confirming the validity of the simulation result and the success of the start-up process simulation.

It should be noted that Figure 12 additionally displays the variation of relative corrected speed for both the compressor and turbine. Since the engine inlet total temperature experiences minor variations during acceleration and the model neglects inlet airflow losses (making compressor inlet total temperature equal to engine inlet total temperature), Equation (39) dictates that the compressor’s relative corrected speed should follow a similar trend as the rotor mechanical speed—a relationship confirmed by the simulation results across all inlet conditions.

The turbine’s relative corrected speed curve exhibits non-smooth characteristics, which Equation (53) attributes to fluctuations in turbine inlet total temperature. Significant temperature variations occur during engine ignition, starter disengagement, and transition from acceleration to stable idle operation. The absence of thermal inertia modeling for high-temperature components further amplifies these temperature fluctuations in the simulation results.

From Figure 13, it is evident that under all inlet conditions, the compressor’s operating point remains within the compressor mode region, while the turbine’s operating point initially falls within the stirring mode region during the early stages of the start process, and then enters the turbine mode region, which corresponds to the expected operating modes of the compressor and turbine during a ground start. Additionally, with increasing inlet altitude and decreasing temperature deviation, the total inlet temperature of all components decreases, causing the relative corrected speed to rise, thereby shifting the start process operating line to the right.

In conclusion, the curve-and-surface fitting-based extrapolation method proposed in this study for sub-idle component characteristics can fully utilize known data when only above-idle corrected flow, pressure/expansion ratio, and isentropic efficiency characteristics are available. The extrapolated results cover all operating modes of the components. The simulation results demonstrate that the extrapolation method satisfies the requirements for conducting ground start simulations of aeroengines under different inlet conditions, further proving the method’s validity and usability.

4. Conclusions

To address the limitations of existing extrapolation methods and fully utilize limited known data for sub-idle characteristic extrapolation while ensuring the results properly reflect component special operating modes, this paper proposes an innovative curve-and-surface fitting-based extrapolation method, which demonstrates three key advancements over conventional approaches:

First, it leverages the continuous and smooth nature of aeroengine component characteristics through constrained curve/surface fitting, enabling direct extrapolation from limited above-idle data while maintaining physical plausibility.

Second, the introduction of the Specific Enthalpy Change Coefficient (SECC) circumvents isentropic efficiency discontinuity, permitting consistent characterization across all operating modes using standard performance parameters.

Third, physically informed fitting constraints ensure result validity while preserving the flexibility required for diverse component behaviors.

Validation was conducted on a micro-turbojet engine, demonstrating that:

(1)

The proposed method successfully achieves extrapolation using only limited above-idle data (corrected mass flow, pressure/expansion ratios, and isentropic efficiency), whereas conventional methods fail due to either insufficient data (Stage-Stacking Method, Zero-Speed Interpolation Method, Backbone Method) or reliance on proprietary software (β-Extrapolation). Even applicable methods (Proportional Coefficient Method) underutilize available data, highlighting this method’s superior applicability.

(2)

As shown in Figure 10 and Figure 11, the extrapolated results fully and rationally capture all special operating states, including the compressor’s stirring and turbine modes, as well as the turbine’s stirring and compressor modes.

(3)

When applied to the engine’s component-level model (Section 3.3), the extrapolated data enabled successful ground-start simulations under varying inlet conditions, confirming their validity and practicality.

In summary, this method overcomes the key limitations of existing approaches—incomplete data utilization, dependency on prior conditions, and inadequate representation of unique operating modes of aeroengine components. The scientifically sound results fully meet the requirements for aeroengine ground-start simulations, providing a reliable foundation for control law design and hardware-in-the-loop testing.

Note: While the extrapolated results partially enable windmilling-start simulations by characterizing special operating modes, full windmilling simulation requires further matching of compressor and turbine characteristics under these conditions—a direction for future refinement.

5. Patents

The method employed in this study has been filed for a patent in China under the title: “A Method for Extrapolating Characteristics of Aeroengine Components Below Idle Speed Based on Curve Fitting and Surface Fitting” (Patent Application No. CN202410578419.9). The first and second inventors listed on the patent application is the same as the first author and second (corresponding) author of this paper, respectively. Currently, the patent application is under review and has not yet been granted.