Interaction of Combined Module Variances and Influence on the Overall System Behaviour

Abstract

Within the Collaborative Research Centre 871, geometrical variances caused by repair procedures and deterioration are evaluated for the turbomachinery of a high-bypass aircraft engine. Part of this evaluation is the investigation of the influence of isolated and combined geometric variances on the overall aircraft engine performance. For this purpose, a virtual twin of a research aircraft engine is developed in sub-project D6. This virtual aircraft engine is based on the Pseudo Bond Graph approach, which allows for transient manoeuvres and the effects of interactions to be simulated with a higher degree of accuracy compared to conventional methods. After validation of the model, a design of experiments is performed to analyse the sensitivities between the variances of modules and engine performance. Within the sensitivity analysis, it is shown that the evaluated steady-state and transient performances are mainly influenced by the high-pressure modules, especially by the mass flow and efficiency variances. Furthermore, it is shown that the sensitivities strongly depend on the operating points. However, significant interactions are found which can be attributed to both the high-pressure and low-pressure modules.

1 Introduction

Commercial aircraft engines have long service times of several decades (Weiner 2015). To maintain safe, reliable and efficient functionality, aircraft engines undergo extensive maintenance, repair and overhaul (MRO) activities during their lifetime.

More than 40% of the direct maintenance costs of an active aircraft are accounted for by the aircraft engines, which thus represent the most expensive and significant part (Markou and Cros 2021). Thus, a fundamental understanding of the influence of aircraft engine deterioration on performance has a significant economic viability for manufacturers, operators and maintainers. An effective and accurate regeneration or recovery of components can significantly reduce operating costs as well as the risk of failure.

The spectrum of maintenance work on aircraft engines is caused by wear and damage mechanisms. The mechanisms include, in particular, damage and contamination of the engine blading (erosion, corrosion and fouling) as well as deterioration of blade tip gaps (Müller 2013; Saravanamuttoo et al. 2001). In general, these local variances of the different modules have a direct impact on the overall performance, propulsion efficiency and safety margins of the aircraft engine. The condition monitoring of an aircraft engine requires performance-specific data as well as knowledge about the relationship between module deterioration and performance output (Fentaye et al. 2019; Spieler et al. 2008; Volponi 2014). To gain this knowledge, the non-linear behaviour of the aircraft engine has to be reproduced in the model and its ability to capture the influence and the interaction of combined module variances must not be neglected.

Therefore, the Collaborative Research Centre 871 “Regeneration of Complex Capital Goods” (CRC 871) develops a scientific basis for novel technologies and approaches to analyse, evaluate and determine the causes and effects of wear and to transfer the experience based approaches into knowledge-based approaches (Aschenbruck et al. 2014; Kellenbrink et al. 2022). Hence, these methods are exemplified using the V2500-A1, a mature high-bypass turbofan jet engine owned and operated by the Institute of Jet Propulsion and Turbomachinery (IFAS). In order to evaluate the degradation, a virtual process is developed in the CRC 871, which uses blades of the high-pressure turbine to determine the influence of geometrical variances on the performance output. This automated virtual process is represented in the flow chart in Fig. 1 (Goeing et al. 2022b).

Here, the geometry of the HPT blades is digitised (1), parameterised (2), and analysed (3) automatically. Furthermore, the influence on the local aerodynamics of the reconstructed blade is investigated using a computational fluid dynamics (CFD) simulation (4–6). The results of aerodynamic analyses are transferred into a simulation of the performance of the entire aircraft engine (7–9).

The main focus of this sub-project “Interaction of combined module variances and influence on the overall system behaviour” is on the investigation of sensitivities between module degradation and overall aircraft engine performance. Therefore, this sub-project is focused in particular on the blue-framed parts (7–9) in the virtual assessment process of Fig. 1. In order to investigate these sensitivities between performance and module deterioration, a design of experiments (DoE) with a virtual twin of the research aircraft engine is carried out. The virtual twin is developed with the in-house performance analysis tool ASTOR (AircraftEngine Simulation for Transient Operation Research) to simulate the performance of the aircraft engine and validated experimentally and numerically. The module variances are integrated via performance maps of compressors and turbines in ASTOR. The impact of module variances on overall performance is evaluated by a variance-based sensitivity analysis that is able to evaluate the sensitivities and interactions of combined module variances.

Fig. 1

Virtual performance evaluation process of the system demonstrator

Since the conventional 0D performance simulation approaches have fundamental limitations in dealing with the non-linear behaviour of the aircraft engine (Kurzke and Halliwell 2018; Fentaye et al. 2019), ASTOR is based on the quasi 1D consider- ation of the gas path, whereby volume effects as well as interactions in the gas path are taken into account.

Below, the developed and validated virtual twin and results of the sub-project D6 are briefly described and presented (for further details (Goeing et al. 2018, 2019a, b, 2020a, b, c, 2022a, b; Salomon et al. 2021; Lück et al. 2022).

2 Setup

In this section, the methods used to develop the virtual twin of the aircraft engine are presented. These are carried out on the basis of the IFAS research aircraft engine, the V2500-A1. The V2500 turbofan from IAE (International Aero Engines) usually powers the Airbus A320 medium-range aircraft. It is equipped with a low-pressure compressor (LPC), an intermediate-pressure compressor (IPC), a high-pressure compressor (HPC), a high-pressure turbine (HPT), a low-pressure turbine (LPT), and a common thrust nozzle.

2.1 ASTOR

The ASTOR model is able to simulate on- and off-design steady-state operations, as well as transient and highly dynamic engine responses, such as compressor surge. Furthermore, ASTOR is developed using Pseudo Bond Graph theory, which is a powerful unified approach to model interdisciplinary and dynamic systems. ASTOR is implemented in a Matlab/Simulink environment. The system of ordinary differ- ential equations (ODE) is solved with an explicit multistep solver (Shampine et al. 1999) and a variable time step.

2.1.1 Dynamic Model

For the simulation of transient performance and dynamic phenomena, ASTOR considers main effects (dynamics of rotating machines) as well as secondary effects (heat transport, interactions and volume dynamics). For this system dynamic approach, the mass (1), momentum (1.2) and energy (1.3) conservation of fluids and solid bodies are considered on fixed control volumes:

$${\iiint }_{V}\frac{\partial \rho }{\partial t}dV=-{\iint }_{S}\rho \cdot {\nu }_{j}\cdot {n}_{j}dS$$

(1)

$${\iiint }_{V}\frac{\partial }{\partial t}\left(\rho \cdot {v}_{i}\right)\cdot dV+{\iint }_{S}\rho \cdot {{v}_{i}\cdot v}_{j}\cdot {n}_{j}\cdot dS=-{\iint }_{S}p\cdot {n}_{i}dS+{\iiint }_{V}\rho \cdot {f}_{i}dV$$

(2)

$$\begin{aligned} & {\iiint }_{V}\frac{\partial }{\partial t}\left[\rho \left(e+\frac{{v}^{2}}{2}\right)\right]dV+{\iiint }_{V}\frac{\partial }{\partial {x}_{j}}\left[\rho \left(e+\frac{{v}^{2}}{2}\right)\right]dV={\iiint }_{V}\frac{\partial }{\partial {x}_{j}}\left(\rho \cdot {v}_{j}\right)dV \hfill \\ & +{\iiint }_{V}\frac{\partial }{\partial {x}_{i}}\cdot \dot{q}dV \end{aligned}$$

(3)

Based on the above integro-differential equation system, an ODE-system is spatially discretised (see Eqs. 4–8), which represents the quasi 1D gas path by finite control volumes. The turbomachinery component characteristics are included via the surface forces F_P and F_T and are equal to zero in a non-turbomachinery control volume. The pressure ratio π and efficiency η are imposed on the compressor and turbine volumes, while the burner efficiency and pressure losses are imposed on the burner volume. The energy and momentum contributions of gravitational forces have been neglected and the frictional shear stress tensor is replaced by the Fanning and Darcy friction coefficient λ. Thus, interaction effects within the gas path as well as with the boundaries of the control volume and within the control volume (inertia and capacity of the gas) can be taken into account

$$\frac{d}{dt}\left(\rho \right)=\frac{1}{V}({\dot{m}}_{in}-{\dot{m}}_{out})$$

(4)

$$\begin{aligned}\frac{d}{dt}\left(\dot{m}\right) & =\frac{1}{L}\cdot \left({A}_{in}\cdot {p}_{in}-{A}_{out}\cdot {p}_{out}+p\cdot \left({A}_{in}-{A}_{out}\right)+{\dot{m}}_{in}\cdot {v}_{in}-{\dot{m}}_{out}\cdot {v}_{out}+{F}_{pt}\right) \hfill \\ & \quad -\rho \frac{\lambda }{D}\cdot \frac{v}{2}\cdot A \end{aligned}$$

(5)

$${F}_{pt}=\frac{{A}_{in}+{A}_{out}}{2}\cdot \left({p}_{out}-{p}_{in}\right)+{\dot{m}}_{in}\cdot ({v}_{out}-{v}_{in})$$

(6)

$$\frac{d}{dt}\left(\rho \cdot {h}_{t}-{p}_{t}\right)=\frac{1}{V}({\dot{m}}_{in}\cdot {h}_{t,in }-{\dot{m}}_{out}\cdot {h}_{t,out }+{F}_{Tt}+{\dot{m}}_{f}\cdot {h}_{f}+\dot{Q})$$

(7)

$${F}_{Tt}={\dot{m}}_{in}\cdot {(h}_{t,out}-{h}_{t,in})$$

(8)

$$p=\rho \cdot R\cdot T$$

(9)

Equation 4–8 are first-order nonlinear state equations relating density, velocity and internal energy derivatives for the control volume. Equation 9 is the ideal gas equation.

Based on the energy conservation for solid bodies, the wall temperatures T_W of the blades, casing and disks are simulated using Eq. 10. The convective heat transfer is integrated into the ODE system with Eq. 11. For the heat transfer coefficient α, substitute models are created for the casing, blades and disks (see Stephan et al. (2019))

$$\frac{d{T}_{w}}{dt}=\frac{1}{V\cdot \rho \cdot {c}_{v}}\cdot \left({\dot{Q}}_{i}-{\dot{Q}}_{o}\right)$$

(10)

$$\dot{Q}=\alpha \cdot A\cdot \left({T}_{tx}-{T}_{W}\right)$$

(11)

Furthermore, the dynamic of rotating machines is included by the momentum equation for discretised solid bodies

$$\frac{dN}{dt}=\frac{1}{J\cdot 2\pi }\cdot \left({\tau }_{T}-{\tau }_{C}\right)$$

(12)

Based on the thermal and mechanical stress, the radial expansion during a transient manoeuvre is also considered (Fiola 1993; Kypuros 2003).

2.1.2 Boundary Conditions

Together with the sub-projects “Exhaust Jet Analysis”, “Loss Behavior of Complex Surface Structures” and “Coupled geometric variances”, the machine-specific performance maps are generated, e.g. through CFD studies. Apart from the performance maps of the miscellaneous modules, the geometry, material information as well as the initial steady-state operating point are required for the simulation of dynamic manoeuvres. Therefore, a full CAD-model based on the IFAS research V2500-A1 engine is designed (see background of Fig. 2). The mass of fan, HPC, HPT and LPT blades are weighed. The weight of the booster blades, the two spools and discs are approximated based on technical sketches. Mostly, titanium and nickel alloys are used as material for these components. The resulting moment of inertia J of the LP-system is 57 kgm² and that of the HP-system is 11 kgm².

Fig. 2

CAD model and Pseudo Bond Graph of the V2500-A1

In order to model the initial steady-state operating point and therefore the starting point for ASTOR, a global iterative 0D engine matching framework is developed. Matching in this context means iterating within the performance maps, turbine entry temperature, bypass ratio (BPR), the rotational speed N₁, until: (1) the turbine power output matches the compressor power, (2) the mass flow into all modules is the same (bypass, secondary air and fuel) and (3) the nozzle pressure is equal to the pressure downstream of the LPT (including friction). The convergence criterion E for all three equations is E < 10⁻⁶.

2.1.3 Pseudo Bond Graph

The Pseudo Bond Graph theory is used to superimpose complex systems into a unified system-dynamic notation and categorises all quantities into efforts e and flows f. The advantage is that the complex compressible system of the gas turbine can be connected to other physical domains, which may be advantageous in view of future hybrid electric propulsion systems (Wahler et al. 2022). The Pseudo Bond Graph schematic is shown in Fig. 2. Here, efforts e (e.g. pressure p, temperature T or torque τ) and flows f (e.g. mass flow m, energy flow, heat flow or rotational speed) of the physical domains represented within a jet engine are layed out. The 0—and 1—junctions are applied to connect these efforts e and flows f of the miscellaneous domains. In the gas path, the conservation of momentum is solved at the 1—junction and the conservation of energy and mass, which take into account heat transfer and secondary air flow, at the 0—junction. Therefore, gas effects, which are based on inertia and capacity (mass storage and dynamic volumes), are directly included in the system of differential equations.

3 Results and Discussion

In this section, the results of sub-project D6 are discussed. First, ASTOR will be validated with the real IFAS research aircraft engine. Second, ASTOR is compared to conventional performance simulation methods to illustrate the improvement in performance prediction capabilities that can be achieved by using higher order approaches. Finally, the results of the DoE study and the sensitivity analysis are shown.

3.1 Validation

Numerous experiments were carried out to validate ASTOR. In Goeing et al. (2020b), ASTOR was shown to accurately predict the steady-state and transient performance of the IFAS experimental turbojet engine. However, the main focus of the validation study in this section is the comparison to the IFAS two-spool, high-bypass V2500- A1 engine. For this purpose, a complex pass-off test is conducted at the MTU-H-71 test bed. During the test run, the rotational speeds (N₁, N₂), thrust and temperatures behind the compressors (T_t21, T_t25, T₃) are recorded at a sampling rate of 10 Hz. In addition to the standard instrumentation, additional total pressure and total temper- ature probes are used in the HPC, behind the HPT (T_t45, p_t45) and behind the LPT (T_t5, p_t5) for a higher data acquisition. The test setup is depicted in Fig. 3a.

Fig. 3

a IFAS research aircraft engine in the MTU-H-71 test bed. b Test procedure and individual thrust curves

Furthermore, customised thrust curves are defined in order to obtain information at the steady-state operating points and to perform various transient manoeuvres. In Fig. 3b these thrust curves are shown. In general, four different thrust levels are approached (Band D/41 kN, C/72 kN, B/92 kN and A/101 kN) in this test campaign. Starting from the lowest level D, the different thrust levels are approached by accelerations. To reach steady-state operating conditions, the thrust levels are held for 2–3 min. After that, decelerations are carried out until the Band D is reached again. Finally, a slam acceleration and deceleration between Band D and Band A is performed.

The steady-state performance data is then used to calibrate the virtual aircraft engine in ASTOR. Therefore, the performance maps of different turbomachinery components are modified iteratively until the deviation between measured and simulated steady-state data is below 5%. In the illustration 1.3b, the real engine data (green) is compared to the virtual engine data (blue). In general, the graphs are consistent with each other. The mean deviation during the whole manoeuvre is 1.1% with a 95% confidence interval of ±0.27%. During the fast acceleration between Band D and Band A, ASTOR overestimates the thrust by 2.5% compared to the steady-state operating point. This effect does not occur during the measurement.

In Fig. 4 the variations of temperatures T_t45, T_t5 and the rotational speeds N₁, N₂ during the fast manoeuvre between Band D and Band A of the real and virtual aircraft engine are shown.

Fig. 4

Comparison of the transient performance of the real and virtual V2500-A1 turbofan during slam acceleration and deceleration

In general, the transient performance of the virtual twin matches the real aircraft engine. In the comparison of T_t45 (see Fig. 4a), the mean deviation is 1.5 ± 0.2% during the acceleration and 1.7 ± 0.3% during the deceleration. In addition, the transient peak during the acceleration as well as the behaviour during the deceleration are reproduced by ASTOR (see magnification).

The mean deviation of T_t5 (see Fig. 4b) is 1.2 ± 0.2% during the acceleration and 1.3 ± 0.3% during the deceleration. Similar to T_t45, the location and amplitude of the temperature peaks correspond to each other. Furthermore, the influence of the heat soakage, which dampens the temperature rise, is detected in both curves.

The reaction of the N₁ shaft is very sluggish due to the high moment of inertia, which means that there are no significant dynamic effects to the N₁ speed (see Fig. 4c). The largest deviation is 3% and is located at the steady-state operating point at Band A. The deviation can be traced back to the calibration of the performance maps.

In contrast to the N₂ speed where the steady-state deviation is between 0.3 and 0.7%. Furthermore, the mean deviation during acceleration is 0.6 ± 0.2% and during the deceleration 0.5 ± 0.2%. However, the steady-state N2 at Band A is overestimated during the acceleration, which is not detected in the measurement. Finally, the deceleration is consistent except for an offset of 100 rpm.

3.2 Comparison of Performance Models

In this section, ASTOR is compared to a conventional engine matching (EM) approach, which is used in commercial performance tools (Kurzke and Halliwell 2018). This global engine matching (EM) method is implemented in Matlab to use same parameter and modelling errors. The EM approach iteratively determines the transient operating point by integrating the changes in rotational speeds dN/dt while matching mass flow rate inside the modules and the nozzle pressure and the pressure downstream of the LPT. On one hand, the most significant influence on the transient performance is exerted by the inertia of rotating machines as well as the convective heat transport. These effects can be taken into account by EM. On the other hand, through the ODE system, ASTOR allows a more accurate simulation of the interaction between the different effects (e.g. feedback between heat flow, gas path, material and system) as well as the influence of the volume dynamics.

To quantify the impact of the higher order approach, a slam acceleration and deceleration between idle and takeoff operating points is performed using ASTOR and the EM method. In Fig. 5 the mean deviation and 95% confidence interval between ASTOR and the EM are shown. Furthermore, the impact of interdependencies (green) due to the use of ODE and the impact of volume dynamics (yellow) are shown separately. In Fig. 5a, the temperature deviation of the full manoeuvre in each module is illustrated. The maximum deviation is observed downstream of the combustion chamber (2.8 ± 0.57%), followed by the HPT (2.6 ± 0.52%) and the LPT (2.2 ± 0.44%). Furthermore, it can be seen that the mean deviation increases particularly in high temperature ranges. In Fig. 5b, the mean deviation of the com- pressor surge margins is presented. These deviations have a magnitude of 5 to 7.5% with the maximum at the HPC. The impact of the volume dynamics is represented by the yellow area of the plot in both diagrams and is significantly lower compared to the total deviation (see Eq. 4–8). For the temperature, this deviation is up to 0.35% in the compressors, 0.7% in the turbines and 1.2% for the surge mar- gin. The remaining and thus evidently significant part of the deviation is based on the feedback between gas temperature T_t, wall temperature T_t,w, heat flow Q^· and their interaction with the entire system. These dynamics are inherently included in ASTOR, whereas they are neglected in the EM method.

Fig. 5

Comparison between ASTOR and EM methods during a slam acceleration and deceleration. a Deviation in temperature. b Deviation in SM

3.3 Numerical Experiment

After validating the model and quantifying the improvement, a DoE is carried out to determine the interaction of combined module variances and its influence on the overall performance.

3.3.1 Latin Hypercube Sampling

For this purpose, the parameter space of the possible wear and tear and their combinations were first explored. Hence, a gas path analysis (GPA) is used to explain the performance-specific differences between a baseline (no specific V2500-A1 engine) and the deteriorated IFAS V2500-A1 engine. In the GPA, the individual characteristic curves of machine-specific performance maps of compressors and turbines are modified until the performance matches. The obtained scaling factors SF (see Eq. 13) thus represent the deterioration of the turbomachines

$$SF\pi =\frac{\pi }{{\pi }_{Ref,{N}_{corr}}},SF\dot{m}=\frac{\dot{m}}{{\dot{m}}_{Ref,{N}_{corr}}},SF\eta =\frac{\eta }{{\eta }_{Ref,{N}_{corr}}}$$

(13)

The performance of both aircraft engines is listed in Table 1. The table shows that the IFAS research aircraft engine requires 3.1% more fuel flow for the maximum thrust condition and the exhaust gas temperature T_t5 has increased by 44 K. Moreover, the temperature T_t5 of 850 K is 30 K above the EGT limit of 820 K, therefore the IFAS research aircraft engine is very well suited to define the limits of the parameter space for a deteriorated engine.

Table 1 Corrected aircraft engine conditions at maximum thrust

In addition to the thrust requirement F_N, the rotational speeds N₁ and N₂, the engine mass flow rate m, the fuel flow m_f, the temperatures T_t25, T_t3, T_t45, T_t5, as well as the pressure ratios of the compressors π_C are varied during the matching process. Since wear has a variety of effects on different areas of the performance maps, the GPA is performed at four thrust levels, based on the results of a pass-off test (see Fig. 3a). The scaling factors for the intermediate characteristic curves are interpolated. The resulting scaling factors for the higher and lower operating points of the turbomachines are shown in Table 2.

Table 2 Parameter range and scaling factors SF for the characteristic lines at thrust level Band D and A

In general, the results show that the influence of the scaling factors increases with decreasing rotational speed. The mass flow rate, pressure ratio and efficiency of all compressors are reduced. Compared to the HPC, the performance of the LPC and IPC decreases less significantly. In general, the efficiencies of all compressors decrease less than the mass flow and pressure ratio. The deterioration of the turbines leads to a decrease of efficiency and pressure ratio while the mass flow increases. For the HPT, the same scaling factors are used as a consequence of the minor reduction of corrected HPT rotational speed between the considered operating points. In order to further analyses the performance and sensitivities, several simulations are carried out based on the parameter space from Table 2. The combinations were selected using Latin hypercube sampling (McKay et al. 2000) to efficiently sample the parameter space.

3.3.2 Performance Simulation

In order to analyse the interaction of combined module variances, the steady-state and transient performances are analysed in this section. For this purpose, the temperature load in the hot gas path and the surge margin of the HPC is investigated in more detail. Hence, different aircraft engine configurations Θ are first compared to each other:

As a representative example of a degraded component, the engine state from Table 2 is used.

In order to determine the influence of interactions, these results are compared to a simplified model Γ. As shown in a previous investigation (Goeing et al. 2020c), this model is able to represent the qualitative performance of combined deterioration, but not the interaction through system response. The combination of a deteriorated HP and LP system called Γ is calculated as below:

$$\Gamma =\frac{{\Theta }_{4}{\cdot\Theta }_{5}}{{\Theta }_{1}}$$

(14)

In Fig. 6, the transient performance during a slam acceleration from the Band D to Band A, for the defined engine configurations, is shown. The first configuration Θ₁ is represented in blue, Θ₂ in orange, Θ₃ in green and Θ₆ in purple. The simplified combined engine Γ is shown in red.

Fig. 6

Transient performance during slam acceleration from Band D to Band A of different aircraft engine configurations. a Exhaust Gas Temperature T_t5; b Surge margin SM. Annotations for definitions of ∆N, ∆SM and ∆T_t. Steady-state operating points highlighted by markers

In Fig. 6a, the EGT T_t5 is plotted against rotational speed N₂. It can be clearly seen that the temperature T_t5 increases with the level of deterioration at all operating points. Moreover, the distance ∆T between the transient temperature peak and the final EGT at Band A is influenced by deterioration. The ∆T_t for configuration Θ₂ is increased by 40% compared to the new engine. Configuration Θ₃ shows a nearly unchanged ∆T_t. Furthermore, the simplified combined engine overestimates the overload ∆T_t with 30% compared to Θ₆. This deviation is caused by interaction between the different module deterioration which is neglected in the simplified approach.

While the EGT increases with the degree of deterioration, the difference in rotational speeds between the two operating points has shown a different performance. On one hand the deterioration of HPT Θ₃ shifts the operating points to lower rotational speeds and decreases the operating range ∆N₂ compared to the new engine. On the other hand, configuration Θ₂ and Θ₆ have shown an increase in the rotational speed at the operating points and an increase in the operating range ∆N₂.

In the Fig. 6b the HPC surge margin is shown over the rotational speed N₂. Here, the surge margin SM is defined as below:

$$SM=\frac{{\pi }_{SL}}{{\pi }_{OP}}\cdot \frac{{\dot{m}}_{OP}}{{\dot{m}}_{SL}}-1 \; \text{with}\; \dot{m}=const.$$

(15)

In contrast to the EGT, the deterioration has positive and negative effects on the surge margin. The deterioration of the HPC in Θ₂ results in a shift of steady-state operation point and minimum surge margin to the lowest margin (0.12) among all configurations. In configuration Θ₃ the surge margin is shifted to higher values for all operating points. Configuration Θ₆ with the highest degree of deterioration produces a surge margin behaviour in between the configuration of isolated deteriorated modules. In addition, the configuration Θ₂ and Θ₃ shows an increase in ∆SM com- pared to the new engine. The configuration Θ₆ causes a decreased ∆SM but less than the configuration with single deterioration. ∆SM is overestimated by the simplified combined engine Γ of 7% compared to Θ₆.

3.3.3 Sensitivity Analysis

In order to evaluate the influence and interactions of combined module variances on the overall performance, a global variance-based sensitivity study is carried out (Saltelli et al. 2010). Therefore, the results of ASTOR are used to determine the first and total indices. These indices can be used to identify not only the direct effects of particular inputs on the variances of output variables (first order S_y), but also any potential effects due to interaction between the variances of the inputs and the outputs (total order S^T).

Here, the considered steady-state output variables are T_t3, T_t5, N₁, N₂, p_t3 and specific fuel consumption (SFC). For the sake of simplicity, only the 5 highest values are considered. In Fig. 7 the first (dark) and total (bright) order indices are presented for Band D in green and Band A in red.

Fig. 7

First S_y and total S^T effect order Kucherenko indices for thrust level band D and band A

The main impact on the temperatures T_t3 and T_t5 is from the HPC efficiency at both operating points, which is quantified by a first index value of 0.42 and total index of 0.67. At the operating point Band D the sensitivity is increased to a first order index of 0.74 and total index of 0.8. The scaling factor with the second highest sensitivity for the temperature T_t3 is the mass flow rate of the HPT and for T_t5 it is the HPT efficiency. The temperature T_t3 is influenced by the HPT mass flow as a consequence of a lower HPC pressure ratio. In addition, a higher temperature T_t3 is caused by a lower efficiency of the compression. The decrease in HPT efficiency causes a higher temperature T_t5, which in terms results from a higher fuel flow to provide the required power to the compressor. Similarly, the decrease of HPC efficiency leads to an increase of the downstream temperature T_t5. Both types of deterioration cause an increase of the steady-state temperature T_t5, as described in Fig. 6. The SFC has almost the same sensitivities as the temperature T_t5. As a result of the decrease in HPT efficiency, a higher fuel flow is required to provide the power to the HPC. A deteriorated HPC requires more power input which causes an additional increase in fuel flow. This results in an further rise of the temperature T_t5 at steady-state, as determined before. The pressure p_t3 is most sensitive to the HPT mass flow rate. The first order index is determined to be 0.6 for Band A and 0.3 for Band D. The sensitivity of the HPC efficiency is the second highest. The HPC is shifted to lower pressure ratios due to the increased mass flow rate of the HPT. A reduced HPT efficiency provides less power to the HPC which causes a decrease in corrected mass flow and pressure ratio. The main effect on the low pressure spool speed N₁ is caused by the influence of the LPC mass flow and pressure ratio. All remaining scaling factors have no significant impact on N₁. The high pressure spool speed N₂ is mainly influenced by the mass flow of the HPC, which is shown in Fig.6. With the sensitivity analysis it can be determined that the increase of rotational speed is caused by the HPC mass flow. The HPC responds to a decrease in HPC mass flow with a shift of the steady-state operating points to higher rotational speeds. One reason for the decreasing sensitivity of the HPT and the increasing sensitivity of the HPC for Band D in comparison to Band A is due to the higher degree of deterioration of the HPC compared to the HPT at this operating point.

In Fig. 8 a global sensitivity analysis of the transient performance is presented. Therefore, the sensitivity of the temperature loads ∆T are determined. Furthermore, the impact of HPC surge margin and acceleration dN/dt are investigated. In general, all investigated temperature loads are mainly influenced by the HPC efficiency. The first order effect is between 0.7 and 0.8 for all ∆T. The remaining scaling factors show a minor impact on the temperature loads. However, the total order index of 0.2 indicates strong interaction effects. In the investigation of isolated deterioration in Fig. 6, it is shown that the HPC has a significant influence on the temperature load while the HPT has got a minor impact. The increased temperature load is caused by the effect of the deteriorated HPC on the acceleration of the high pressure spool speed. Due to the loss in HPC efficiency, a higher power input of the HPT is necessary. This results in a lower acceleration at the beginning of the transient manoeuvre. As a consequence of the lower acceleration of N₂, the temperature load increases. Because of the higher temperature load, the HPT provides more power, which allows a higher acceleration rate and thus causes the increase in ∆N₂.

Fig. 8

First S_y and total S^T effect order Kucherenko indices for transient performance. ∆T_t according to definition in Fig. 6a

The minimum surge margin of the HPC is influenced significantly by different scaling factors. The main impact is caused by the mass flow of the HPT, with a first order index of 0.3 and the HPC efficiency, whose first order index amounts to 0.2. The existence of interactions for the five scaling factors are indicated by the total order indices, which are significantly larger than the first order indices. Similar to the first order index, the total order indices indicate that the surge margin is mainly influenced by interactions that result from changes in HPT mass flow rate and HPC efficiency. With the results of the sensitivity analysis, it is observed that the increased HPT mass flow has a positive influence on the surge margin. The HPC operating point is shifted to lower pressure ratios due to an increase of the HPT mass flow, which results in a higher surge margin (see Fig. 6). In contrast to the HPT mass flow, the surge margin is decreased the most when the HPC efficiency is deteriorated, which is mentioned in the configuration Θ₂. This decreased surge margin is caused by the lower acceleration during the beginning of the transient maneuver. Mostly, the acceleration of the LP-spool is influenced by the pressure ratio of the LPC. The first order and the total order indices have reached a value of 0.95.

The rotational speed N2 is significantly influenced by the scaling factors compared to the low pressure spool speed. The acceleration of the spool is significantly influenced by the mass flow of the HPC. The first order index is calculated to be 0.65 and the total index to 0.7. Similar to the surge margin, the acceleration is also influenced by the efficiency of the HPC. However, the indices have shown that the influence of the HPC efficiency is less significant with a first order index of 0.25 and a total order index of 0.3. The sensitivity of the acceleration of the HP spool is shown in Fig. 6. The HPC rises the rotational speed between the two operating points. In conclusion it is observed that the investigated steady-state and transient performance are mostly influenced by HP-system.

4 Conclusions

A virtual twin of the IFAS V2500-A1 research aircraft engine was developed to investigate the interaction of combined module variances and their influence on the overall system performance in the sub-project D6 of the Collaborative Research Centre 871. For this task, the aircraft engine was transferred into a quasi 1D environment of the Pseudo Bond Graph notation in order to simulate the non-linear system dynamics in higher order. The virtual twin was validated with the IFAS research aircraft engine and the performance could be predicted with high accuracy (see Fig. 3a). In addition, ASTOR is compared with conventional performance analysis methods and it was found that the dynamic model leads to an improvement of the transient performance simulation capabilities by up to 8% in the surge margin during a slam acceleration (see Fig. 1.4), due to the volume dynamics and the interaction effects. Based on the validated virtual twin, a design of experiments was performed to investigate the sensitivities for steady-state and transient performance of combined module variances in a turbofan aircraft engine. The sub-project D6 showed that:

• 1. The qualitative and quantitative relationship between module variances and per- formance output strongly depends on the operating point (see Table 2 and Fig. 7).

• 2. The impact of a module with geometrical variances on the aircraft engine per- formance depends on the condition of the remaining modules (see Figs. 6 and 7).

• 3. The high-pressure system dominates the sensitivities in the steady-state and tran- sient performance (see Figs. 7 and 8).

• 4. Strong interaction effects have been detected, especially due to the low-pressure system (see Figs. 6, 7 and 8).

These results can be used to improve detection and understanding of the sensitivities of the overall engine performance to the module variances, which is an essential field in production, maintenance, repair, and overhaul (MRO) as well as in the operation of an aircraft engine. The virtual twin and the results obtained from the sensitivity analysis are used in the virtual process from Fig. 1 to evaluate the scanned high-pressure turbine blades. In general, it is possible to integrate such a performance assessment analysis using the virtual twin for a condition-based decision making on component regeneration. This knowledge and the method of sensitivity analysis will become more important for modern aircraft engines due to more complex system architectures. In addition, recognising the interaction between low-pressure and high-pressure systems will become increasingly important with the progress towards ultra-high-bypass ratio aircraft engines.