LPT rotor speed after low-pressure shaft failure

When designing engine safety and compliance with airworthiness regulations, it is important to determine the variation of rotor rotational speed and other parameters of aero-engine under shaft failure scenarios. In this paper, a transient process simulation model is established using the component method for a high bypass ratio two-shaft direct-drive turbofan engine. It helps analyze the response of engine rotor speed and key temperature parameters after the failure of the low-pressure shaft, the influence of the fuel supply, low-pressure turbine efficiency, and rotor resistance torque on the response of the parameters. The results demonstrate that the sudden increase of the low-pressure shaft speed at the turbine end after the failure of the low-pressure shaft is the most important factor affecting the engine safety design. The fuel supply, LPT efficiency, and resistance torque have significant effects on this sudden increase in speed.


Introduction
Aero-engine is a sophisticated product in the field of equipment manufacturing, which is known as the jewel in the crown of industry.The reliability and safety of aero engines are extremely important for air safety.Exceptionally harsh working environments have a great risk of causing engine failure.
The compressor and turbine of the aero engines are responsible for the compression and expansion of the working fluid in the engine.The turbine and the compressor are connected by a shaft element.In normal operation, the turbine rotor extracts power from the high-temperature, high-energy gas in the main flow path and transfers it to the compressor through the shaft element.Shaft failure in civil high bypass ratio turbofan engines will cause a series of aero-engine failures during development and operation [1].The hazard is releasing high-energy debris from rotor overshoot.
A series of studies are carried out to analyze the mode of shaft failure events, the changes in engine system response caused by shaft failure, and the design of the protection system in the case of such failure.Psarra [2] introduced a modeling method of contact friction wear in a gas turbine shaft failure scenario.The proposed modeling approach converts the complex problem of shaft failure fracture into a discrete model, which investigates the effects of friction between the turbine and surrounding stators and explores the relationship between frictional structure and energy dissipation.Gallar [3] modeled the aero-thermal performance of turbine components under extreme conditions during a shaft failure overshoot event by simulating turbine blade performance at high negative incidence angles.Soria [4] developed an analytical model of a gas turbine engine in a shaft over-running or failure condition and established an aerodynamic and thermodynamic model to study the system response of the whole engine during a turbofan engine shaft failure event.
Liu et al. modeled the performance of a turbofan engine after a high-pressure turbine shaft fracture and the strong transient process of the air system [5,6].Gong et al. [7] conducted a predictive analysis IOP Publishing doi:10.1088/1742-6596/2764/1/012025 2 of the rotational speed of the turboshaft engine power turbine rotor under loss of load to conform to CCAR33.27 [8], which investigated the relationship between the over-rotation transient aerodynamic efficiency and the rotational speed of the turboshaft engine.Li and Wang demonstrated the superiority of the limit strain method applied to airworthiness certification, which is based on the requests from FAR 33.27 [9].Zhang et al. [10] studied the effects of material property and temperature gradient on over-rotation speed characteristics under high temperatures.
Zhang et al. simulated the engine performance after the shaft break under the most conservative conditions, which assumed the total enthalpy of high-pressure compressor outlet airflow before the shaft break as the airflow steady state parameter for a short time period after the shaft break [11].Pawsey et al. investigated the characterization of the turbine over-speed behavior to be integrated into an engine over-speed model capable of predicting the terminal speed of the HPT in the event of a high-pressure shaft failure [12].Tang and Tong established a transient model based on the volumetric method and investigated the real-time validity of the calculation of the engine model, which lacks the study of the strong transient process under the shaft failure scenario [13].Researchers have studied the engine response after shaft failure in three-rotor engines, which focused on the component characteristics under extreme conditions.However, this model is less involved for specific types of complete engine models (e.g., two-axis direct-drive turbofan) and specific shaft failures (e.g., low-pressure shaft failure).Liu et al.'s study mainly focuses on low bypass ratio turbofan engines and turboshaft engines but not on the shaft failure response of high bypass ratio engines [5].
This paper focused on the dynamic response of the whole engine and the rotor speed variation regularity after low-pressure shaft failure from a performance simulation model for a two-shaft directdrive high bypass ratio civil aircraft engine.It conducted a sensitivity analysis of the important factors affecting the rotor speed design and airworthiness-related compliance design.

Materials and methods
The overall engine performance simulation model includes engine rotor speed, section temperature, and section pressure under the failure scenario, which is improved on the basis of the normal engine overall performance simulation model.

Normal engine overall performance simulation model
The modeling process is based on the characteristic data of the engine components.The engine performance model is based on the common working relationship between components.The engine components are described as a compression process, expansion process, combustion process, etc. Volumetric effects of components are considered.The residual test equations are established based on the common operating conditions, such as continuous flow, pressure balance, speed balance, and power balance, and solved numerically iteratively to achieve the whole engine performance simulation by constraining the specific common operating conditions [14][15][16][17][18].The overall engine performance calculation model mainly includes four parts, which are the air gas property calculation module, component performance calculation module, engine aerodynamic and thermal calculation module, and engine co-working iteration calculation module.

Gas property calculation module
The main function of this module is to calculate gas thermal property parameters, such as specific enthalpy, specific entropy, constant pressure specific heat, constant volume specific heat and specific heat ratio.

2.1.2
Component performance calculation module This module is to calculate the performance parameters of components based on the operating parameters of engine components, outlet temperature, pressure, and flow rate.The module includes all the components of a high bypass ratio two-axis direct-drive turbofan engine, such as a fan, compressor, combustion chamber, and turbine.An example of the calculation principle and process of the fan IOP Publishing doi:10.1088/1742-6596/2764/1/0120253 component is shown in Figure 1, which is similar to the calculation principle and process of the remaining components.The fan operates with known values of its inlet temperature and pressure, which are noted as T 1 and P 1, respectively.If the fan physical speed and the operating point interpolation auxiliary parameters are given, the fan operating point performance parameters can be obtained by interpolating the compressor characteristic diagram to convert the flow rate, W_cor, boost ratio π and efficiency η.Thus, the fan outlet temperature, pressure, and air-flow rate can be calculated by the fuel gas property calculation module.

2.1.3
Engine aerodynamic thermal calculation module The main function of this module is to calculate performance parameters such as engine rotor speed, thrust, and fuel consumption rate, as well as section parameters, which include temperature, pressure, and gas flow rate of each component section.The engine aerodynamic thermal calculation is according to the upstream and downstream relationship of the engine components in the airflow, as shown in Figure 2, until the airflow velocity at the nozzle exit is calculated.Therefore, the engine thrust, fuel consumption rate, rotor speed, and other performance parameters are obtained.

Test equation
Equation Parameter symbols β1

A18_cal=A18_Set
The calculated area of the bypass nozzle is equal to the set area.

A8_cal=A8_Set
The calculated area of the core flow nozzle is equal to the set area.
Booster outlet air flow is equal to the highpressure compressor inlet airflow.
β4 WgHPT_Cal=WgHPT_Fig The calculated high-pressure turbine converted flow rate is equal to the converted flow rate from the characteristic diagram.
β5 WgLPT_Cal=WgLPT_Fig The calculated low-pressure turbine converted flow rate is equal to the converted flow rate from the characteristic diagram.

N2
PWHPT=PWHPC+J ω Power from a high-pressure turbine is equal to the power consumed by a high-pressure compressor.

Pw J ω dω dt Pw
The power emitted by the low-pressure turbine is equal to the power consumed by the low-pressure shaft (turbine end).

Engine common working iterative calculation module
As shown in Figure 3, eight parameters, low-pressure rotor speed N1, high-pressure rotor speed N2, fan operating point interpolation auxiliary parameter β1, boost stage operating point interpolation auxiliary parameter β2, high-pressure compressor operating point interpolation auxiliary parameter β3, combustion chamber fuel flow rate Wf, high-pressure compressor operating point interpolation auxiliary parameter β4, high-pressure compressor operating point interpolation auxiliary parameter β5 are the testgiven parameter.The engine components need to work together to test whether the test-given parameters are consistent with the parameters of the real engine operation.If all the test parameters meet the test equation, the test-given value of the test parameters is the real value of the engine.Generally, a nonlinear equation calculation model is used for numerical solutions, such as the Newton-Raphson method, particle swarm algorithm, genetic algorithm, etc. Taking the engine with N1=const regulation law as an example, seven test-given parameter is required for performance calculation of the high bypass ratio two-axis direct-drive turbofan engine.At the same time, seven check equations are also required to make the set of equations closed, as shown in Table 1 and Figure 3.

Overall engine performance simulation model in case of low-pressure shaft failure
In the event of low-pressure shaft failure, the low-pressure turbine rotor loses connection with the fan/boost stage rotor, and the speeds of the two are no longer equal.At this point, the engine test gives one more parameter than the normal engine: the LPT rotor speed.At the same time, the LPT no longer drives the fan booster stage, and the test equation condition that the power of the LPT is equal to the power of the fan booster stage also fails, requiring two additional test equations.Thus, the overall engine performance simulation model when the low-pressure shaft fails has a total of eight test-given parameters and correspondingly eight test equations, which are shown in Table 2 and Figure 4.

Test equation
Equation Parameter symbols β1

A18_cal=A18_Set
The calculated area of the bypass nozzle is equal to the set area.

A8_cal=A8_Set
The calculated area of the core flow nozzle is equal to the set area.
Booster outlet air flow is equal to the highpressure compressor inlet airflow.
β4 WgHPT_Cal=WgHPT_Fig The calculated high-pressure turbine converted flow rate is equal to the converted flow rate from the characteristic diagram.
β5 WgLPT_Cal=WgLPT_Fig The calculated low-pressure turbine converted flow rate is equal to the converted flow rate from the characteristic diagram.

N2
PWHPT=PWHPC+J ω Power from a high-pressure turbine is equal to the power consumed by a high-pressure compressor.

Pw J ω dω dt Pw
The power emitted by the low-pressure turbine is equal to the power consumed by the low-pressure shaft (turbine end).

Pw J ω dω dt
The power consumed by the fan and booster stage is equal to the power consumed by the lowpressure shaft (fan end).

Low-pressure turbine power test equation
The power generated by the low-pressure turbine is mainly used for rotor acceleration and overcoming the power loss generated by the structure touching and grinding,

Fan boost stage power test equation
The low-pressure shaft at the fan end loses the low-pressure turbine shaft drive, and the power consumed by the fan booster stage mainly comes from the virtual power generated by the low-pressure shaft deceleration at the fan end.Compared with the normal engine performance calculation model, the calculation model of the overall engine performance under the shaft failure scenario added an extra testgiven parameter.Accordingly, two power check equations are changed.The fuel gas property calculation module, the component performance calculation module, and the engine aerodynamic and thermal calculation module are the same.Therefore, the accuracy of the overall engine performance calculation under the shaft failure scenario is the same as the accuracy of the normal engine performance calculation model.

Results and discussion
This paper is based on the constructed transient performance simulation model of low-pressure shaft failure.The analysis of the dynamic response of the whole machine and the rotor speed variation regularity after low-pressure shaft failure will be carried out with the high-temperature take-off operating point in a typical flight envelope as the analysis benchmark, for example, 0ft/ISA+15K/0Ma.

Analysis of rotational speed variation pattern
After the low-pressure shaft failure, the LPT rotor instantly enters the rotational process of load dumping.The power generated by the high-temperature gas expansion through the LPT is entirely used for the LPT rotor acceleration.In addition, after the shaft break, the LPT moves backward under the axial force, which leads to the collision and touching of the rotor stators and generates a shock-type resistive moment; meanwhile, considering the detection and diagnosis of the shaft failure event in FADEC, the fuel will be cut off quickly.Under the effect of the above two, on the one hand, the resistive torque will consume the energy generated by the LPT.On the other hand, the high-temperature gas entering the LPT will be reduced.This eventually leads to a sharp increase in LPT rotor speed followed by a rapid peak and then a gradual decrease.The maximum speed that occurs throughout the process is the one that applies to the assessment of the LPT rupture margin and is used to demonstrate compliance with the provisions of CCAR/FAR 33.27.The variation regularity of resistance torque is shown in Figure 5.The time required from the start of the fuel valve to the completion of the fuel cut is set to 800 ms, and the low vortex efficiency is set as the reference value.Based on the engine component test, the rotor torque variation regularity is shown in Figure 5.The time required from the start of the fuel valve to the completion of the fuel cut is set to 800 ms.The value of each parameter (pre-turbine temperature, fuel flow, temperature, etc.) at the moment before the shaft break is selected as the reference (i.e., 100%).Based on the above settings, the transient simulation model constructed to obtain N_LPT and WFE, N_HPShaft, and T4 as a function of time is shown in Figure 7   sequence in time after low-pressure turbine shaft failure.Figure 7(a) shows that after the failure of the LPT shaft, the physical speed of the LPT shaft will rise rapidly to reach the maximum speed within 1s, which is about 172% of the starting speed.From Figure 7(b), the physical speed of the high-pressure rotor rises very little, not more than 101% of the starting speed.Figure 7(c) shows that the temperature T4 before the turbine rises rapidly before the fuel starts to cut off.When the fuel flow rate starts to decrease, T4 also starts to decrease, and its maximum value is about 110% of the starting moment.This appears earlier than the highest over-rotation speed, and both do not reach the maximum at the same moment.

Sensitivity analysis of fuel variation pattern
By analyzing the remaining power of LPT after the failure of the low-pressure shaft, it was found that the flow rate of engine-supplied fuel would affect the power generated by LPT.Under the premise of keeping the resistance torque and LPT efficiency unchanged, the fuel cut-off times of 800 ms, 500 ms, and 300 ms were selected to analyze the effects of different fuel cut-off speeds on the change of LPT speed, as shown in Figure 6.The fuel flow cut-off speed will directly affect the fuel supply into the combustion chamber after the shaft failure.The faster the cut-off is, the smaller the core work is, and the smaller the power generated by the LPT is.The images show that the effect of fuel flow cut-off speed on maximum speed is significant, with a reduction in cut-off time of 300 ms, effectively reducing the speed by about 22% (a percentage relative to the starting speed before shaft failure).At the same time, the moment of maximum over-revolution can be brought forward by about 32%, reducing the secondary damage caused by prolonged over-revolution.

Conclusion
In this paper, a transient model applicable to a high bypass ratio turbofan engine under a low-pressure shaft failure scenario is established by analyzing the physical process after low-pressure shaft failure.A simulation analysis of low-pressure shaft failure at a high-temperature take-off operating point in a typical flight envelope is carried out.Overall, the engine performance model is improved.The test parameters and test equations applicable to the low-pressure shaft failure scenario have been constructed, and the transient performance simulation model has been built, which realizes the dynamic performance simulation under the whole engine failure scenario.Engine performance simulation was carried out in a typical flight envelope at the high-temperature take-off operating point.The LPT speed variation law was obtained.Reduction of cut-off time by 300 ms can effectively reduce the speed by approximately 22%.The moment of appearance of the maximum over-rotation speed is advanced by 32%, reducing the secondary damage caused by prolonged over-rotation.
In summary, the research in this paper can realize the speed prediction under the low-pressure shaft failure scenario and guide the design and optimization of low turbine rotor margin.It can be used to guide the design of the low-pressure turbine rotor rupture margin, the turbine overspeed protection system, as well as the airworthiness-related compliance design.

Figure 2 .
Figure 2. Flow chart of aerodynamic thermal calculation for a high btpass ratio two-axis direct-drive turbofan engine.Figure 3. Schematic of calculation flow chart.

Figure 3 .
Figure 2. Flow chart of aerodynamic thermal calculation for a high btpass ratio two-axis direct-drive turbofan engine.Figure 3. Schematic of calculation flow chart.

Figure 5 .
Figure 5. Resistance moment and axial force variation regularity.

Table 1 .
Engine nonlinear equation set iterative test-giving parameters and check equations (normal calculation, N1=const).

Table 2 .
Engine nonlinear equation set iterative test parameters and test equations (low-pressure shaft failure, N1=const).