2.1 Gas turbine performance, stress and thermal models

The gas turbine performance modelling is based on the fundamental thermodynamics and empirical component maps. As the performance modelling is not the focus of this study and is only used to set up a gas turbine performance model to obtain the boundary conditions for the lifing calculations of HP turbine rotor blades for different flight missions, no details are provided and the interested readers may refer to references [Reference MacMillan20, Reference Li and Singh21] for further details.

A stress model has been set up to calculate the stresses of gas turbine HP turbine rotor blades mainly considering the effects of centrifugal force due to turbine rotation and gas bending moment caused by both the change in gas momentum and the presence of static pressure difference. Temperature variations on the blades cause extra thermal stresses particularly at transient operation processes but they are ignored as only steady-state flight conditions are considered in this study, and the thermal stresses are much smaller than the stresses caused by centrifugal force and gas bending moment. To calculate the blade stresses, a blade is divided into several sections along blade radial direction. The centrifugal stress at each blade section is calculated by Equation (3).

(3) \begin{align}{\sigma _{CFSec}} = \frac{{\sum \rho \times {A_{AvCs}} \times {h_{sec}} \times {\omega ^2} \times {d_{CG}}}}{{{A_{CS}}}}\end{align}

where $\rho $ is the blade density, ${A_{AvCs}}$ is the average cross section area between the top and the bottom sections, ${h_{sec}}$ is the blade section height, $\omega $ is the angular speed, and ${d_{CG}}$ is the distance between the rotating axis and the section centre of gravity, and ${A_{CS}}$ is the cross section area of the corresponding section.

The pressure force on each blade section, $P{F_{Sec}}$ is calculated by Equation (4) and the pressure bending moment at each section, $BM{P_{sec}}$ is calculated by Equation (5),

(4) \begin{align} P{F_{Sec}} = {{{A_{AnSec}} \times \Delta {p_{AvSec}}} \over {{N_b}}}\end{align}

(5) \begin{align} BM{P_{sec}} = \sum \left( {P{F_{Sec}} \times {d_{CGsec}}} \right)\end{align}

where ${A_{AnSec}}$ is the blade section annulus area, $\Delta {p_{AvSec}}$ is the average section static pressure difference, ${N_b}$ is the number of blades and ${d_{CGsec}}$ is the distance between the section centre of gravity to the respective section.

The thermal model is used to calculate the blade metal temperature distribution along turbine blades considering the radial gas temperature distribution profile (RTDF) (assuming to be 0.1) and blade cooling. The metal temperature is calculated by Equation (6):

(6) \begin{align}{T_M} = {T_G} - \lambda\ {\rm{*}}\ \left( {{T_G} - {T_C}} \right)\end{align}

where ${T_G}$ is the gas temperature, ${T_C}$ is the coolant temperature and $\lambda $ is the cooling effectiveness.

For the calculation of blade metal temperature, it is assumed that part of the air flow from compressor exit is taken as the coolant for turbine cooling that flows along a single cooling channel within each of the HP turbine rotor blades and is discharged continuously along the blade span for film cooling.

As the blades are divided into several sections in span-wise direction, the average metal temperature and the maximum stress within each blade section are considered for the life calculations of the blade sections.

As the stress model and the thermal model are adopted from the work reported in [Reference Abdul Ghafir, Li, Singh, Huang and Feng14] so the interested readers may refer to the reference for further details.

2.2 Creep life estimation

The Larson-Miller parameter (LMP) method [Reference Larson and Miller1] has been used in this study for creep life estimation due to its simplicity and satisfying predictions since it was first proposed in 1952. Abdul et al. [Reference Hanumanthan, Stitt, Laskaridis and Singh15] demonstrated the use of the LMP method in estimating the creep life of HP turbine rotor blades and the same method has been adopted in this paper for creep life estimation.

Based on the LMP method [Reference Larson and Miller1], the time to failure due to creep can be calculated by Equation (7)

(7) \begin{align}CL = {10^{\left( {\frac{{1000P}}{{{T_M}}} - C} \right)}}\end{align}

where $CL$ is the time to failure or the creep life, $P$ is the Larson-Miller parameter, ${T_M}$ is the blade metal temperature (K) and $C$ is a material constant which is taken as 14.07 [Reference Vöse, Becker, Fischersworring-Bunk and Hackenberg11]. The LMP master curve for CMSX-4 can be found in Ref. (Reference Bullough, Toulios, Oehl and Lukas22).

Since the blades have been divided into several span-wise sections and the maximum stresses are calculated respectively for all blade sections, we can expect different creep lives along the blade span. The minimum time to failure among all blade sections will be selected as the creep life of the blades.

2.3 Low-cycle fatigue life estimation

There are mainly two types of approaches for LCF life estimation [Reference Suresh23], the stress-life approach and the strain-life approach. Considering the strain-life approach can better capture the LCF properties, the Smith-Watson-Topper (SWT) parameter strain-life approach [Reference Smith, Watson and Topper6] regarded as one of the most promising LCF life prediction models in terms of prediction accuracy and computational efficiency, is used in this study for the estimation of the LCF life of HP turbine rotor blades. The SWT approach considers the mean stress effect by including the maximum stress, ${\sigma _{max}}$ in the estimation as shown in Equation (8)

(8) \begin{align}{\sigma _{max}}\varepsilon = {\frac{{\sigma _f'}}{E}^2}{\left( {2{N_f}} \right)^{2b}} + \sigma _f'{\varepsilon _f}'{\left( {2{N_f}} \right)^{b + c}}\end{align}

where $\varepsilon $ is the true strain of the turbine blades, ${\sigma _{max}}$ is the maximum true stress among the stresses of various locations on the blades, $\;{N_f}$ is the low cycle fatigue life of the blades, $\sigma _f'$ , $\;E$ , $\;{\varepsilon _f}'$ , $b$ and $c$ are the material properties. According to Ref. (Reference Sun, Yang and Yuan24), the coefficients for CMSX-4 take values of $\sigma _f' = 1700.7\;MPa$ , ${\varepsilon _f}^{\rm{'}} = 191688.9\;MPa$ , $b = - 0.1892$ , and $c = - 3.356$ . Young’s modulus for CMSX-4 takes a value of $E = 80\;GPa$ at 1000 °C as a representative value based on Ref. (Reference Epishin, Fedelich, Finn, Kunecke, Rehner, Nolze, Leistner, Petrushin and Svetlov25). Again, the minimum number of cycles to failure ${N_f}$ calculated by Equation (8) corresponding to the maximum stress ${\sigma _{max}}$ on the blades is selected as the LCF life of the blades.

2.4 Cycle counting

The gas turbine low cycle counting method used in this study is the rainflow counting (RFC) method first proposed by Endo and Matsuishi [Reference Endo and Matsuichi26] in 1968. The idea is to mirror how rain flows down a pagoda roof, hence it has the name as rainflow counting. It offers the capability of calculating the fatigue life of an object by converting its complicated cyclic loading history into a set of equivalent simple stress cycles.

Figure 1(a) shows a typical simplified flight mission profile where typical flight sections are included, including take-off and climbing, cruise, approach, loitering and final approach and landing. The rainflow counting method is applied to discretise the flight mission profile into equivalent cycles and constant flight sections in the following procedure as shown in Figs. 1(b) and (c):

1. 1. Turn the original flight mission profile shown in Fig. 1(a) clockwise at 90 $^\circ $ as shown in Fig. 1(b) and let the earliest time be on the top of the graph.

2. 2. Imaging the load history is a series of pagoda roofs, and a water flow is coming down from the top of the profile, Fig. 1(b).

3. 3. Count the number of half-cycles if the water flow either:

   1. a. Reaches the end of time history; or

   2. b. Merges with a flow that started at an earlier tensile peak; or

   3. c. Flows down when an opposite tensile peak has a greater magnitude.

   The magnitude of the half-cycle is the stress difference between the starting point and termination point.

4. 4. Repeat Step 3 for compressive valleys.

5. 5. Pair up half cycles with the same magnitude until all paired cycles are collected, as shown in Fig. 1(c). There could be some half-cycles left and should be delt with separately.

6. 6. All constant flight sections are collected and listed separately as shown in Fig. 1(c).

After such a conversion, a complicated flight mission profile can be dissembled into several equivalent cycles and several constant flight sections.

Figure 1. Schematic discretisation of mission profile using rainflow counting method (a) typical simplified flight mission; (b) water flows according to rainflow counting method; (c) equivalent discrete cycles and constant flight sections.

2.5 Creep-LCF interaction

For a certain loading pattern of a flight mission of a gas turbine engine as shown schematically in Fig. 1(a), the flight sections where the engine is kept at steady-state conditions are dominated by time-dependent creep damage and the flight sections with varying loads are dominated by low cycle fatigue damage. After dissembling the mission into creep damage sections and fatigue damage cycles as schematically shown in Fig. 1(c), the creep and the LCF lives at corresponding conditions can be calculated separately. In these calculations, the creep life is calculated by LMP method shown in Section 2.2 and the low cycle fatigue life is calculated by the SWT model described in Section 2.3.

For each creep section and fatigue cycle, the concept of life fraction [Reference Stephens, Fatemi, Stephens and Fuchs27], or the creep damage and the LCF damage are introduced and can be calculated by Equations (9) and (10), respectively:

(9) \begin{align}{D_{Creep}} = \mathop \sum \limits_j^m \frac{{{t_j}}}{{C{L_j}}}\end{align}

(10) \begin{align}{D_{LCF}} = \mathop \sum \limits_i^k \frac{{{n_i}}}{{{N_{f,i}}}}\end{align}

where ${D_{Creep}}$ is the creep damage, ${D_{LCF}}$ the LCF damage, $m$ the total number of steady-state operation sections, ${t_j}$ the operating time at the $j$ th section, $C{L_j}$ the creep life at the $j$ th section, $k$ the total number of cycles, ${n_i}$ the number of the $i$ th cycle type, and ${N_{f,i}}$ the LCF life of the $i$ th cycle type.

Since research [Reference Sun and Yuan12, Reference Troha and Stabrylla13, Reference Pohja, Holmström and Lee28] have shown that the linear damage summation rule may not be adequate for some materials experiencing cyclic loading under high temperature conditions and could lead to inaccurate estimation of the combined damage, the creep-LCF interaction diagram [Reference Wright, Carroll and Wright29] as shown in Fig. 2 is adopted to estimate gas turbine component life consumption. In this study, it is proposed that the combined creep-fatigue damage represented by Equation (10) determines the failure criteria.

(10) \begin{align} x = \sqrt {{D_{Creep}}^2 + \;{D_{LCF}}^2} \; \lt y\end{align}

where y is the maximum allowed combined creep-fatigue damage represented by the distance between the no-load Point 0 and the Point B on the failure line. In Fig. 2, the failure line E-G-F determined by the intersection point G is material dependent and its position is set to be (c, d). Due to lack of the creep-fatigue interaction diagram for CMSX-4, it is assumed in this research that c = d = 0.5 according to Ref. (Reference Wang, Zhang, Gong, Zhu, Tu and Zhang31) for Nikel-based super alloy. For more accurate failure line for CMSX-4, the point G should be determined via creep-LCF experiments for the specified material.

Figure 2. Creep-LCF interaction diagram [Reference Wright, Carroll and Wright29].

After calculating the creep and LCF damages of the turbine rotor blades separately for a flight mission profile in concern, the combined damage caused by the repeated loading may be represented by point A in Fig. 2. If the loading is repeated until failure, the failure point would be the intersection between the red failure line and the extended line of 0-A, denoted as point B. The total damage represented at point A caused by the repeated loading is then defined by Equation (11) as

(11) \begin{align}D = \frac{x}{y}\end{align}

where $D$ is the total damage caused by the creep and the LCF interaction, x the distance between point A and the original no-load point 0, and $y$ the distance between failure point B and the original no-load point 0.

The total damage is ranging between 0 and 1. All points located on the red failure line with a total damage of 1 indicate a complete failure, and the original no-load point with a total damage of 0 indicates no damage at all.

Figure 3. Reference creep operating condition and LCF cycle.

2.6 Equivalent operating hours (EOH)

2.6.1 Scaling of creep and LCF lives

Due to many assumptions made in the predictions of creep life and low cycle fatigue life for the HP turbine rotor blades, the predicted lives may have significant errors. To make the life predictions useful and also consider the lives given by OEMs for the gas turbine engines in concern, the following scaling factors are introduced to convert the predicted life consumptions to those of the real engines in concern:

• • Creep life scaling factor $\boldsymbol{SF}_{\boldsymbol{creep}}$ defined as the ratio between the predicted creep life ( $C{L_{pre}}$ ) at a reference operating condition (Fig. 3) and the creep life ( $C{L_{giv}}$ ) given by OEM represented by Equation (12).

  (12) \begin{align}S{F_{creep}} = \frac{{C{L_{pre}}}}{{C{L_{giv}}}}\end{align}

• • LCF life scaling factor $\boldsymbol{SF}_{\boldsymbol{LCF}}$ defined as the ratio between the predicted number of LCF life cycles ( ${N_{f,pre}}$ ) for a reference flight cycle (Fig. 3) and the number of LCF life cycles ( ${N_{f,giv}}$ ) of the real engine in concern given by OEM represented by Equation (13)

  (13) \begin{align}S{F_{LCF}} = \frac{{{N_{f,pre}}}}{{{N_{f,giv}}}}\end{align}

Consequently, the creep and LCF lives of the engine in concern can then be calculated by Equations (14) and (15).

(14) \begin{align} CL = C{L_{pre}}/S{F_{creep}}\end{align}

(15) \begin{align}{N_f} = {N_{f,pre}}/S{F_{LCF}}\end{align}

It is worth mentioning that the pre-defined reference creep operating condition may not be the same as that of the peak operating condition of the pre-defined reference LCF cycle. By using the introduced life scaling factors defined by Equations (12) and (13), the engine life (CL and ${N_f}$ ) at any flight conditions and flight cycles can be obtained by scaling the engine life predicted by the lifing models, which is schematically shown in Fig. 4 where the solid line represents the predicted life directly from the lifing model while the dotted line represents the scaled engine life obtained from Equations (14) and (15). Such an approach ensures that the scaled predictions of both the creep and the LCF lives represent the actual lives of the engine in concern at different operating conditions and operating cycles.

Figure 4. Schematic of predicted and scaled engine life.

2.6.2 Equivalent operating hours (EOH)

EOH of a flight mission for a gas turbine engine is defined as the operating hours of the engine at a pre-defined reference operating condition denoted with a subscript “ref” to reach the same level of damage done by the engine operating at an actual flight mission. The mathematical formula for EOH definition may be represented by Equation (16).

(16) \begin{align}D = \frac{{\sqrt {{D_{Creep}}^2 + \;{D_{LCF}}^2} }}{y} = \frac{{EOH}}{{{L_{ref}}}} = {D_{ref}}\end{align}

where D is the total damage caused by the actual flight mission, ${L_{ref}}$ is the total engine life at the pre-defined reference operating condition given by OEM, and D _ref is the total damage caused at the pre-defined reference operating condition. Therefore, the EOH can be expressed by Equation (17).

(17) \begin{align}EOH = {L_{ref}} \cdot D = {L_{ref}} \cdot \;{{\sqrt {{D_{Creep}}^2 + \;{D_{LCF}}^2} } \over y}\end{align}

The total engine life ${L_{ref}}$ at the reference condition may be set being equal to the engine creep life $C{L_{giv}}$ when no low cycle operations are considered. However, the low cycle operations will result in LCF damage alongside creep damage and contribute to the total life consumption as shown by Equations (10) and (11).

The following are discussions at two special scenarios:

• • In a scenario where a gas turbine engine operates continuously at one or more stabled operating conditions without any stops, the start-stop cycles can be ignored and therefore the life consumption is mainly determined by creep damage ( $y = 1$ ) and EOH can be calculated by Equation (18).

  (18) \begin{align}EOH\; \approx \;C{L_{giv}} \cdot {D_{Creep}}\end{align}

• • In a scenario where a gas turbine experiences frequent low cycle operations and stabled operating conditions are very short and can be ignored, the engine life consumption is mainly determined by LCF damage ( $y = 1$ ) and equivelant operating cycles (EOC) can be calculated by Equation (19).

  (19) \begin{align}EOC\; \approx \;{N_{f,giv}}\; \cdot {D_{LCF}}\end{align}

2.7 Integrated EOH prediction system

The individual models described above are integrated into a physics-based EOH calculation system as shown in Fig. 5.

Figure 5. Gas turbine engine EOH prediction system.

The reference operating condition, the reference cycle and an actual engine flight mission profile representing their loading patterns are used as input for the EOH prediction system. An engine performance model is set up using Cranfield’s performance and diagnostics software, Pythia [Reference Li and Singh21] and used to predict the thermodynamic parameters at the inlet and outlet of the HP turbine rotor blades. Consequently, the stress and temperature distributions on the HP turbine rotor blades are predicted using the stress model and the thermal model, respectively. Then, a cycle counting module based on the rainflow counting method [Reference Endo and Matsuichi26] is used to discretise the loading patterns of the flight mission into a number of sub-cycles and sub-sections of the continuous flight. Afterwards, a creep model based on the LMP method and an LCF model based on the SWT parameter approach are used to estimate the creep and LCF lives, respectively. Then, the predicted lifing results are sent to the creep-LCF interaction model to calculate creep, LCF and total damages. Finally, an EOH prediction model based on the total damage together with the creep and LCF scaling factors are applied to deliver an estimated EOH. It is suggested that the pre-set reference flight condition for the creep scaling factor calculation and the peak operating condition of the pre-set reference cycle for the LCF scaling factor calculation may be close to the take-off condition for the gas turbine engine in concern.