New model-based method for aero-engine turbine blade tip clearance measurement

Hanlin SHENG, Tong LIU, Yan ZHAO, Qian CHEN, Bingxiong YIN,Rui HUANG

Jiangsu Province Key Laboratory of Aerospace Power System, College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

KEYWORDS Aero-engine real-time model;Gas turbines;Model buildings;Tip clearance measurement;Turbine components

Abstract Active control of aero-engine turbine tip clearance is one of the best chances for engine performance uplift currently.To do that,the first requirement is real-time measurement of tip clearance in aero-engine working environment.However, turbine complexity makes it unlikely for tip clearance sensors to be loaded.In recognition of that, this paper proposed a model-based method for tip clearance measurement.Firstly,by considering previously wrongly neglected factors such as load deformation, a mathematical model to monitor dynamic tip clearance changes is built to improve calculation accuracy.Then,after clarifying the coupling relationship between engine models and tip clearance models, this paper builds a component-level mathematical model integrating dynamic characteristics of turbine tip clearance, which helps realize accurate measurement of tip clearance in working environment.How tip clearance affects turbine efficiency is studied afterwards and reported to aero-engine model, so as to mitigate performance difference between aero-engine model and real engines caused by turbine tip clearance.Lastly,by hardware-in-the-loop simulation,tip clearance model demonstrates 15.9% better accuracy than previously built models in terms of turbine centrifugal deformation calculation.As tip clearance measurement model takes averagely 0.34 ms in calculation, meeting the operation requirement, it proves to be an effective new way.

1.Introduction

Turbine tip clearance is a small gap between turbine blades and a case of an aero-engine.The leakage of gas through the tip clearance will reduce the power capacity of turbine components,severely affecting the performance and fuel consumption rate and service life of the aero-engine.1Active closed-loop control of turbine tip clearance throughout the operating conditions of the aero-engine was performed to ensure turbine rotors and stators do not rub together while maintaining tight tip clearance, which is a decisive technology required for the future aero-engines with outstanding performance.2,3Currently, the turbine tip clearance can be measured by the fiber optic method,4an eddy current sensor,5a non-contact probe,6and a capacitive regulator circuit7.However, since turbine components of the aero-engine are in an extremely harsh environment of high temperature, high pressure,and high speed (vibration) for a long time, it is arduous for the current measurement methods of tip clearance to be applied on board.Therefore,it is urgent to develop a new perception method for tip clearance.8The aero-engine model-based predictive method 9 is a critical tool for precepting unmeasurable parameters.Hence,a model-based perception method for turbine tip clearance was put forth herein.

The model-based perception method for the tip clearance is based on the engine’s operating condition, and the tip clearance was calculated in real time according to the on-board model.The main technical challenge lies in establishing a mathematical model with high confidence of the tip clearance that is adapted to the on-board computing environment and deeply coupled with data of the engine’s on-board model.Although many researchers10–19have modeled the turbine tip clearance,they have not considered the computational requirements for airborne real-time applications.For instance, Bindon10conducted a detailed experimental study on the internal flow of turbine clearance for the first time and proposed a model of clearance gap flow, which revealed the effect mechanism of the tip clearance on engine operation and clarified the necessity of tip clearance control, but did not carry out a study on the change process of tip clearance.Agarwal et al.11developed a simplified model for predicting variations of the tip clearance in gas turbines.The radial distance between a blade end and a case shroud could be estimated by the theoretical variations of the engine’s state parameters, but the modeling process is relatively simple.It does not reflect the deformation of components under real operating conditions of the engine.Kypuros et al.12,13studied the mechanism of changes in the tip clearance;extracted the main factors affecting changes in the turbine tip clearance; calculated the deformation of cases, rotors, and blades respectively by using simplified thermodynamic and kinetic equations; established a mathematical model adopted to calculate the dynamic changes in the turbine tip clearance, and reflecting the‘‘pinch point”of clearance changes, but they did not deliberate over the accuracy of the discretized solution to the model under airborne conditions.Peng et al.14also developed a tip clearance model based on simplified assumptions, but they focused on a study on active control algorithms for the tip clearance and did not improve the modeling process based on others’previous efforts.Jaw15provided a new way to model the tip clearance by modifying the mathematical model of tip clearance through a neural network.However, the neural network-based approach could not apply calculation with high confidence in the tip clearance over the entire flight segment due to the unavailability of engine data in the full envelope.Andreoli et al.16calculated and solved the effect of tip clearance on component efficiency by using finite elements based on the previous models for turbine tip clearance.Although the calculation is highly accurate, the model is complex and compute-intensive, so their approach can only be adopted for the offline study of the tip clearance and cannot be further developed for the airborne application.Chapman et al.17analyzed the heat transfer factors such as cooling airflow within the key components in detail, based on the previous work.They finally obtained turbine components’thermal and mechanical deformation by solving the temperature field distribution through discretized nodes.However,their study did not consider the material properties that change with temperature.Kratz and Chapman18established a clearance model through thermodynamic and kinetic equations and introduced the properties of material parameters that change with temperature, but their modeling process did not consider the influence of blades on the deformation produced by rotors, which still cannot satisfy the accuracy requirements of airborne computers.Liu et al.19,based on the artificial intelligence,corrected the creep deformation of blades during the degradation of engine performance and obtained a more accurate way to calculate the blade deformation.However, their method cannot be leveraged to calculate the deformation of other components and still cannot meet the requirements of calculating the tip clearance.

Therefore, the following innovations and improvements were put forward in this paper to address the shortcomings of current methods: (A) Component deformation of turbine tip clearance was improved in terms of the calculation accuracy: the influence of the centrifugal force of blades on rotor deformation, which should not have been neglected in the previous modeling process,10–19and the material properties changing with temperature were considered based on previous research; (B) A model-based perception method for turbine tip clearance in the airborne environment was proposed:an all-in-one model integrating the engine and the tip clearance was constructed to realize a new way to sense the tip clearance in the airborne environment, based on the coupling mechanism of the engine and the tip clearance.The model enables airborne sensing and active closed-loop control of the tip clearance of aero engines; (C) An on-board model of aeroengines that can reflect the effect of turbine tip clearance on overall performance was proposed to correct errors between the engine model and the actual engine due to turbine tip clearance: changes in the turbine efficiency caused by the tip clearance was calculated and fed back to the on-board engine model, so that the engine model can match changes in the turbine performance during the dynamic changes in the tip clearance, thus further improving the accuracy of the engine’s on-board model.

The primary research of this paper is presented as follows:firstly, the physical mechanism of changes in turbine tip clearance was studied based on the first principle according to the structural characteristics,working principle,and installation environment of turbine components.In addition, the main and secondary factors affecting changes in the tip clearance were sifted out according to the importance of the influence on changes in the tip clearance by combining with the previous research; secondly, the dynamic model and the steady state model characterizing changes in the tip clearance were established.The calculation of the secondary factors was reasonably simplified under the premise that the requirements of calculation accuracy were satisfied, thus improving the computation speed of the model and supplementing the major influencing factors.Besides, the influence of the centrifugal force of blades on the rotor deformation was introduced, and the influence weight was calculated through simulation experiments;Thirdly, a model-based on-bornd perception method was set up as shown in Fig.1 according to the coupling mechanism between the aero-engine and turbine tip clearance to analyze the effect of changes of the tip clearance on the turbine efficiency, and feed the efficiency back to the engine’s component model, so that the engine can sense the efficiency degradation caused by the tip clearance in real time; finally, simulation experiments were conducted based on the model-based perception method, aiming to verify the feasibility of the model in performing calculation under the airborne environment in the form of hardware-in-the-loop.The experiments provide the future active closed-loop control with a new accurate perception method for the airborne tip clearance.Moreover, the variation law of the tip clearance under the entire flight segment was studied based on the simulation data, which provides more accurate data support for the current open-loop heating regulation.20,21

2.Structure and change mechanism of turbine tip clearance

2.1.Component structure of an aeroengine’s high-pressure turbine

The component structure of a high-pressure turbine is presented in Fig.2.22There are three core components, namely blades, rotor,and case.Blades are rotating parts that obtain kinetic energy from the high-temperature and high-pressure gas, whose roots are connected to a rotor.The blade and the rotor revolve around the highpressure axis together.A case, a shell structure outside the main gas passage of the high-pressure turbine, consists of an external supportive casing and an internal shroud serving as a thermal barrier as well as an abradable seal coating.Tip clearance exists between a blade of the high-pressure turbine and the shroud (an abradable seal coating) inside the case.Although the clearance is small, gas leakage caused by it will decrease the turbine’s power capacity when the aero-engine works.When tip clearance is too large, it will negatively influence the engine’s Specific Fuel Consumption (SFC), pollutant emissions, and service life; when it is too small, it will lead to friction and scuffing between the blade and the case, affecting the flight safety of the aircraft.Hence,dynamic changes of tip clearance should be monitored in a realtime manner when the engine operates.

Changes in the size of turbine tip clearance occur mainly due to the displacement or deformation of an engine turbine’s components.The tip clearance used for control mainly includes axisymmetric clearance variations generated by thermal stress and centrifugal force loading.In contrast non-axisymmetric clearance variations are not dominant and hard to estimate, so they are beyond the scope of this paper.

In Fig.3,the change mechanism of turbine tip clearance is displayed.It is observed that the clearance is determined mainly by the radial distance between a blade and a shroud: a turbine blade is installed on the outer edge of a rotor as a rotating part, whose radial displacement can be calculated by superposing the deformation of the rotor and that of the blade;a turbine shroud is installed inside a case as a stationary part, and its radial displacement can be simplified to the radial deformation of the case due to its minor deformation, for its thickness is thin, and it is unaffected by centrifugal force.

Therefore, turbine tip clearance can be determined by the geometry size of the three core components, namely: the inner radius of case, the blade length and the outer radius of rotor, as in the expression:

Fig.1 Proposed model-based on-board perception method for tip clearance and its application scenarios.

Fig.2 Structure of high-pressure turbine components of aeroengine.

where TC0represents the initial tip clearance when components are invariant, TC（t) the actual dynamic value of tip clearance,rsand δsthe initial inner radius and deformation of the case (in the abradable seal coating), rr,outand δr,outthe initial outer radius and deformation of the rotor, Lband δbthe initial length and deformation of the blade.

Thermal stress and centrifugal force dominate among the many factors affecting changes in the tip clearance:(A)deformation of a case is not affected by centrifugal force,so only the thermal strain generated by the temperature field should be considered;(B)as for a rotor and a blade who are rotor parts,both centrifugal and thermal strains should be considered simultaneously.

2.2.Deformation mechanism of turbine case

The turbine case is the outermost cylindrical shell structure of an engine’s turbine components, which is crucial to encapsulate the main gas passage with turbine components,subjected to the internal and external pressure difference and thermal stress.As shown in Fig.4,since the case is a stationary part,it is not subject to centrifugal force, and its deformation is less affected by the internal and external pressure difference but mainly affected by the thermal stress.When the thermal strain was analyzed, only the case was considered due to the extremely thin thickness and small deformation of the shroud and the abradable seal coating, that is.It is assumed that the radial displacement at the shroud δsis equal to the inner radius of the case δc,in,δs=δc,in.

Fig.3 Change mechanism of turbine tip clearance.

As in Fig.4,the shroud and the abradable seal coating insulate the high-temperature gas exiting from the combustion chamber,so the thermal strain on case components is affected by both the cooling airflow on the outer surface and the internal cooling airflow.The outer surface of the case is cooled by the fan bleed from the outer duct The temperature field distribution of the case is calculated based on the convective heat transfer model between the airflow and the solid wall,23and the thermal strain of the case is calculated according to the material properties.In the end, the material strain is integrated into the radial direction to characterize the deformation process of the turbine case accurately.

During a flight, the temperature and flow of the compressor bleed and the fan bleed on the outer surface of the case will change drastically, greatly influencing the mechanical properties of materials and the convective heat transfer coefficient.Hence,the above factors should be considered in the calculation, and parameters should be corrected in a function or a linear interpolation table.For instance, the convective heat transfer coefficient h is mainly affected by the airflow temperature and the airflow rate when the shape of the solid surface is determined, and is less influenced by the gas pressure.In addition,the airflow rate is associated with the airflow in the bleed passage, so24

where hdes,Tdesand Wdesare the convective heat transfer coefficient, the gas temperature and the gas mass flow at the designed point, respectively.

2.3.Deformation mechanism of turbine blades

Blades of a high-pressure turbine are high-speed rotating components that extract kinetic energy from high-temperature gas and are mounted on the outer surface of a turbine rotor by means of a structure such as a mortise.The deformation of a blade is mainly affected by thermal and centrifugal stresses.In studying the thermal deformation mechanism of a blade,the heat transfer model of the blade’s gas film-cooling process should be accurately characterized.As presented in Fig.5, bleed for gas film-cooling derives from the high-pressure compressor, so the main factors affecting the blade temperature distribution include the gas temperature T4of the turbine inlet where the blade is in and the cooling temperature of the gas film T3.A geometric configuration with small blade volume and surface was considered, and its Biot number Bi?1, that is, the ratio of thermal-conduction resistance on the unit thermal conductivity area of the blade to the heat transfer resistance on the unit area is minimal,and its internal temperature gradient can be ignored in any transient state.Therefore, the lumped parameter method applies to analyzing the zerodimensional thermal conductivity, thus obtaining the blade thermal deformation.

Fig.4 Schematic diagram of heat transfer in turbine case.

In the working process of the engine, the centrifugal deformation of the blade is more significant than the thermal deformation because the blade rotates around the high-pressure shaft at high speed, and the center of mass is far from the rotating shaft.The installation position of the blade is located at the outer edge of the rotor.Therefore,when calculating the centrifugal deformation of blades, the rotation radius should take into account the deformation of the rotor and the change in the length of the blade itself.Then, according to the current state of the engine high-pressure shaft speed N3, the accurate value of the centrifugal force on the blade is calculated, and the deformation of the blade under the action of the centrifugal force is calculated in consideration of the characteristics that the Young’s modulus E and other parameters change with temperature.Finally, the total deformation of the blade can be obtained by superposition of thermal deformation and centrifugal deformation.

2.4.Deformation mechanism of turbine rotor

A turbine rotor is a rotating part installed on a turbine shaft to fix turbine blades.When the engine operates, the rotor transmits the power generated by blades driven by gas to the rotating shaft, so there are significant service loads between the rotor and the blades.As displayed in Fig.6, it is assumed that the outer surface of the rotor with blades installed is adiabatic in studying the thermal strain,and the temperature changes inside the rotor are completely influenced by the convective heat transfer of the cooling airflow on both sides.Given the geometric characteristics of the rotor, the cooling airflow on both sides comes from the bleed of highpressure compressor, and the flow and the temperature are the same, so the temperature fields of the rotor will be symmetrically distributed along the center plane.In the following modeling process,a certain degree of simplification was made based on this law to improve the real-time ability of the model’s calculation.

In the calculation of the centrifugal strain,the centrifugal force on the rotor itself under high-speed rotation is the dominant factor of its centrifugal deformation.Previous modeling studies of the tip clearance17–19simplified the centrifugal force of the turbine rotor itself as a single-value function of the rotor’s centrifugal deformation.However, the centrifugal force of rotating blades will also bring a great stress on the rotor when the engine is operating at a higher speed due to the interconnection between the blades and the rotor.Therefore,the influencing factors mentioned earlier were introduced in this paper to improve the calculation method of the centrifugal deformation of a turbine rotor.In addition,the external load calculation was introduced when the turbine rotor is rotating, and the Young’s modulus E and the Poisson’s ratio v changing with the temperature were considered, so as to calculate the rotor’s centrifugal strain according to the properties of elastic materials.Finally, the obtained centrifugal strain of the rotor is superimposed with the thermal stress to characterize the deformation mechanism of the rotor more accurately.

Fig.5 Turbine blade and film cooling.

Fig.6 Heat transfer diagram of turbine disc.

3.Modeling method of turbine tip clearance

3.1.Dynamic model of turbine tip clearance

As presented in Fig.7, the dynamic model of the tip clearance established in this paper contains three essential parts: the case model,the rotor model and the blade model.Deformation of each component was calculated separately by the thermodynamic calculation module and the rotor dynamics calculation module, and then deformation of the tip clearance was calculated by Eq.(2).The deformation modeling methods of each component were introduced as follows.

3.1.1.Dynamic modeling of a turbine case

Thermal expansion deformation is mainly considered for a case when turbine tip clearance changes.It is assumed that the case deforms only in the radial direction.According to the principle of heat transfer, the differential equation of one-dimensional heat transfer is17

where T,cp,k,x represent the temperature,heat capacity,thermal conductivity and position.The value of the parameter a is determined by the model’s structural form:a=0 for the planar heat transfer model and a = 1 for the cylindrical structural form.Since the turbine case is a thin-walled annular structure,the thickness of its wall is much smaller than the case’s radius,so the heat transfer model of the case was simplified as the planar heat transfer model.Eq.(4)was discretized using the onedimensional finite difference method,and the case components were divided into temperature state nodes with uniform space.Each node is thought a one-dimensional planar thermal conductivity model with different temperature boundary conditions along the wall thickness direction and on both sides.The number of computational nodes M for heat transfer is chosen to be 20 (see Fig.8).

In the calculation, the average temperature of the case was determined by Eq.(5):

The minor thermal deformation of the case at each time step was obtained from the following:

where rc,inrepresents the inner radius of case and αcrepresents the coefficient of thermal expansion of case that varies with temperature.

Finally,the real-time dynamic output results related to thermal deformation of the turbine case’s wall thickness at different temperature conditions were obtained:25

Based on the methods described above, the thermal expansion model of a case was successfully established, based on which the relevant studies on the thermal deformation of the turbine case under different operating conditions of an aero-engine can be carried out.

3.1.2.Dynamic modeling of turbine blades

Due to its structural characteristics, a turbine blade is modeled based on the simplification as a lumped system with even temperature and consistent temperature changes everywhere inside in the thermal deformation modeling in this section.Then the centrifugal deformation modeling is made based on the rotor centrifugal equation, and the final blade deformation is the superposition of the two.

As the blade adopts gas film-cooling, flow fields on its surface are complex.When using the lumped parameter method to simplify the model, it is necessary to determine the effective temperature of gas on the blade surface, which was calculated by the empirical Eq.(8):

where Tref,Tcore,Tcool,ηcoolrepresent the effective airflow temperature on the blade surface, gas temperature of the turbine inlet,gas temperature in the gas film-cooling and the efficiency of gas film-cooling respectively.

Fig.7 Dynamic model architecture of turbine tip clearance.

Fig.8 Schematic diagram of one-dimensional heat conduction node.

After the effective temperature of gas on the blade surface was obtained,the heat balance of the blade in the high-temperature gas should be further considered, to acquire the differential equation of lumped parameter method for zero-dimensional thermal conductivity:

where Tb,hb,Ab,cp,b,mbrepresent the temperature,heat transfer coefficient, surface area, heat capacity and the mass of blade.

After the above equation was discretized,the real-time temperature of the blade Tbat each time node was solved, and the method to solve the subsequent thermal deformation is the same as that to solve the thermal expansion deformation of the case,so the method was not repeated here.Thermal deformation of the blade ub1finally obtained is presented in Eq.(10):

where αbrepresents the coefficient of thermal expansion of blade that varies with temperature.

The centrifugal force of the blade is mainly and positively correlated to the rotational angular velocity ωH（t),the mass mbof the blade and the distance of the center of mass of the blade from the center of rotation.26Among them, the mass is constant, and the angular velocity and the position of the center of mass is changing with time, so the centrifugal force on a single blade Fcentris as follows:

where Eb,ρb,σbis Young’s modulus, density and stress of blade.Asecrepresents the blade’s effective cross-sectional area.Given the unit of rotational angular velocity,centrifugal deformation of the blade ub2was obtained as in Eq.(13) after organization:

where N3represents speed of high-pressure-shafts, r/min.

The specific meaning of each parameter in the equation is displayed in the annotated table.Besides, parameters of material properties involved in this section have considered the effect of temperature variation, and were obtained during the modeling process by using the same method of interpolation in the previous section.

Thermal deformation and centrifugal deformation of the blade were obtained separately by the above methods, and the actual deformation of the turbine blade was generated by superimposing the two.

3.1.3.Dynamic modeling of a turbine rotor

Like blades, turbine rotor is rotating, whose deformation is also affected by thermal and centrifugal deformation.However, the temperature difference between internal parts of rotor is significant, and the heat transfer process should be assumed as a onedimensional heat transfer model along the thickness direction,thus solving the equivalent temperature and deformation of the rotor;the centrifugal deformation of the rotor is positively related not only to its own centrifugal force, but also to the external load brought by the centrifugal force of blades installed on its outer edge, and the influence brought by the centrifugal force of the blades is more critical and cannot be ignored.

At first, thermal strain of the rotor was calculated.Based on the analysis of the heat transfer mechanism of the rotor,the outermost circumferential surface of the rotor was assumed to be adiabatic.The dominant factor of temperature changes in the rotor refers to convective heat transfer between the front and rear surfaces of the rotor.In the modeling process,the equivalent temperature of the rotor was obtained by dividing the nodes along the thickness direction of the rotor, and calculating the node temperature and the average temperature.The thermal expansion deformation of rotor ur1:

where A and B are both integration constants.No external force acts on the inner edge of the rotor during rotation, and stress boundary conditions on the inner surface of the rotor are

where C is the process parameter and Fcentrcomes from Eq.(11).The above boundary conditions were substituted into Eq.(20)to solve the constants A and B,which were again substituted back into Eq.(20)to obtain the equation for the stress of rotor.Constants A and B were substituted into Eq.(19),and r=rr,outwere set to obtain the radial displacement of the outer edge of the rotor under the centrifugal force of blades ur3:

Rotor deformation at different speeds was calculated,as shown in Fig.9.When Power Lever Angle(PLA) reached 80°,The maximum radial expansion of the rotor was 0.3 mm after the centrifugal force of blades was considered, while the radial expansion of the rotor without considering the centrifugal force of blades was only 0.25 mm.That means the centrifugal force of blades makes the rotor produce 0.05 mm more radial expansion, accounting for 15.9% of its total expansion.After calculation, the radial expansion of the rotor under the centrifugal force of blades at each speed accounts for about 15.9%of its total expansion,so the centrifugal force of blades critically influences the radial deformation of the rotor, which occupies a large proportion of the total deformation of the rotor and cannot be ignored.The modeling method proposed in this paper can calculate the deformation of the rotor in a more accurate way.

Finally, the thermal expansion deformation of the rotor, the deformation under the centrifugal force of the rotor itself and the radial deformation of blades on the rotor were superimposed to obtain a more accurate total deformation output of the rotor than that in the previous studies10–19:

3.2.Steady-state modeling of turbine tip clearance

The previous subsection described approaches for the dynamic modeling of turbine tip clearance were described in detail.The model can characterize the dynamic variation of turbine tip clearance with engine operating conditions changing in real-time.However, it is necessary to evaluate further the impact of changes in turbine tip clearance on engine performance, efficiency and other indices in practical engineering applications.A common practice in current research17,18is to compare the theoretical steady-state values of turbine tip clearance under a given condition with the actual dynamic values of the tip clearance calculated by the model and analyze them.However, the thermal effect is relatively slow among the main factors affecting the variation of tip clearance.A rotor was taken as an example in Fig.10.A large amount of time for computation is needed to determine the ultimate values of the rotor’s temperature, which is unacceptable due to the real-time need of the model.Therefore, it is crucial to study the steady-state modeling of turbine tip clearance and directly solve the steady-state value of the tip clearance from the current engine state and the operating environment of turbine components.

The steady-state model also consists of the case model, the rotor model, and the blade model, but the steady-state model’s modeling mechanism is more straightforward than the dynamic model of tip clearance.Components of turbine tip clearance affected by the centrifugal force include turbine blades and rotor,both of which are rotating parts, while the component deformation caused by the centrifugal force is a transient process, and there is no long-term deformation accumulation.Therefore, the ideas and approaches of the steady-state modeling in this part are identical to those of the dynamic modeling of tip clearance,which were not repeated here.

Fig.9 Centrifugal deformation of rotor when its rotational speed quickly steps up.

In calculating steady-state heat deformation, the solution process for the above three components varies due to differences in the structural characteristics and heat transfer of blades, case,and rotor.The turbine blade can be regarded as a lumped system whose final temperature is the effective temperature of gas expressed in Eq.(8).The steady-state heat deformation of blade can be solved in the combination of Eq.(10)based on the effective temperature; as for the rotor, its front and rear surfaces have the same cooling airflow, and its steady-state temperature can be regarded as equivalent to the temperature of cooling airflow.The final steady-state heat deformation of the rotor can be solved by combining the steady-state temperature with Eq.(15);as for the case, since the temperature of cooling airflow differs on the inner and outer surfaces the ultimate temperature of the case should be solved according to the steady-state thermal conductivity by the plate method the model’s computational performance was verified by simulation experiments.

The engine parameters during the step change of throttle lever parameters were input into the model of turbine tip clearance established in this paper to study the dynamic and the steadystate responses of the turbine case,blade and rotor models respectively,and to analyze the dynamic changes of turbine tip clearance in the case of continuous step signals.The simulation results are demonstrated in Fig.11.

It is observed that the dynamic model,in the form of real-time tracking,truly reflects the dynamic deformation under the thermal and the centrifugal stresses, while the steady-state model can quickly calculate the corresponding stable deformation of each component under different engine operating conditions.Although the modeling principles of the two models are different, as shown in Fig.11,the component deformation values of the dynamic and the steady-state models eventually converge with the calculated values of the tip clearance.The two validate each other’s confidence during the modeling process.

4.Real-time dynamic model of an aero-engine with the capability to estimate tip clearance

A mathematical model capable of accurately assessing the dynamic changes and the final steady-state value of tip clearance was developed above.However, to study the actual condition of tip clearance when the aero-engine operates not only relies on the model of tip clearance alone, but also requires a set of engine condition parameters with high confidence and real-time performance not slower than the model of tip clearance, which may derive from the engine component-level model, the engine onboard model or even the measurable parameter sensors installed on the engine.In this paper, the most comprehensive engine component-level model was selected as the vehicle, and the model of tip clearance and the engine model were integrated based on the coupling relationship between the real-time state parameters of the engine and changes of tip clearance,so as to evaluate the variation pattern of tip clearance and its impact on various engine performance parameters in a more intuitive and convenient way.In practical applications, the model of turbine tip clearance can be integrated with the engine on-board model to achieve modelbased real-time accurate sensing of high-pressure turbine tip clearance based on the coupling mechanism between the engine and turbine tip clearance studied in this chapter,thus solving the problem that tip clearance sensors are difficult to be applied on-board.

Fig.10 Dynamic heat transfer of rotor at each node.

4.1.Aero-engine component-level dynamic model

This paper adopted the engine of a geared fan with a large bypass ratio as the research object.Its structure and cross-section were defined as presented in Fig.12.The main components include its inlet duct, fan, gearbox, low-pressure compressor, high-pressure compressor, combustion chamber, high-pressure turbine, lowpressure turbine, inner duct nozzle, and bypass nozzle.The part where the high-pressure turbine tip clearance is located will directly contact the gas exiting from the combustion chamber and meet the cooling airflow from the high-pressure compressor.Parameters, including the temperature, the pressure, and the flow of each part, can be obtained by calculating each part’s aerodynamics and thermal force.

In addition, each component should satisfy the equilibrium equation of mass flow or power.Hence, to establish the mathematical model of engine by the component method requires calculating the aerodynamic heat of each component successfully during the engine’s intake process, thus establishing and solving the equilibrium equation reflecting the collaborative relationships among the components during the steady-state operation of the engine:

(1) Continuity equation of the fan inner duct’s outlet W21and low-pressure compressor’s inlet flow W22:

where nLPT,nLPC,nFrepresent the power of low-pressure turbine, low pressure compressor and accessory.ηLrepresents the efficiency of low-pressure-shaft.

The dynamic model of an aero-engine characterizes the dynamic process of the engine transitioning from one steadystate to another, which should consider the engine’s inertia,including the inertia of each rotor part on the rotary shaft and the thermal inertia of high-temperature components such as the combustion chamber and turbine.Therefore, the collaborative equation proposed above should be modified during the dynamic simulation, and an equilibrium equation reflecting the collaborative relationship among the components in the transient operation of the engine should be established and solved.The flow continuity equation and the pressure balance equation still hold when the engine is in a dynamic process, and the power balance equation should be further modified by introducing acceleration and rotor inertia into Eqs.(32) and (33) to obtain the equation sets Eqs.(34) and (35), which describes rotor dynamics:

where N,J represent the speed and the moment of inertia of shafts.

In the simulation of the engine’s dynamic model,the nonlinear algebraic Eqs.(26)–(33) and differential Eqs.(34)–(35) should be solved to calculate the acceleration of the rotary shaft, thus obtaining the rotor speed by integration, and the operating and state parameters of all engine components were further acquired.The specific modeling process is unveiled in Fig.13, in which Eqs.(26)–(35) is the equilibrium equation mentioned above.

Fig.11 Simulation experiment results of dynamic and steady-state models of turbine tip clearance.

Fig.12 Schematic diagram of engine structure.

When the engine is running each component is at its specific Capacity Operating Point(COP),and the operating states of each component will affect and constrain each other.27Hence,to determine parameters of the engine model requires the COP of each component to match each other.Since the specific COP of each component are not known at the beginning of the calculation,the component parameters cannot be determined.Therefore,some unknown parameters should be predicted and an equal number of errors must be defined,as shown in Fig.13.As an example,Error1 represents the error between the fan flow after conversion and calculation by aerodynamics and thermodynamics and the flow obtained by interpolation of fan components’characteristics:

where WC2is the converted flow, T2and P2the temperature and the pressure of the current gas,TSTDand PSTDthe gas temperature and pressure at standard atmosphere.WCalc2is obtained by the interpolation from the component characteristic diagram,Cw2is the correction factor,and WCMapis the component characteristic diagram flow.

During each iteration, the current error is reduced by continuously updating the predictive value.When each error reaches the restricted condition, the dynamic and thermodynamic parameters of the engine at the current time can be output and proceed to the next time step, which is calculated by the Newton-Raphson (NR)method.In this paper,the model-based perception method for tip clearance was studied by sending the output parameters of the engine’s dynamic model at each time step to the tip clearance model to drive the model for calculating a real-time solution.

4.2.Efficiency correction model

Fig.13 Flowchart on calculation of aero-engine component-level dynamic model.

When the engine is operated,the efficiency of turbine components will impact on the overall engine performance parameters.The nominal turbine efficiency ηnomwas defined in the engine’s dynamic model to reflect the turbine efficiency at the designed point.Since the gas leakage caused by the tip clearance will reduce the turbine’s power capacity, the actual efficiency of turbine components will deviate from the nominal efficiency ηnomwhen the actual tip clearance is different from the nominal clearance at the designed point.As presented in Fig.14, the turbine efficiency will gradually increase when the tip clearance changes from the maximum point to the minimum ‘‘pinch point”, and the turbine can rotate faster with lower flow.In Fig.14, 1 bbm/s = 0.454 kg/s.

To measure the variation of turbine efficiency, the correction factor K of turbine efficiency was defined, and its empirical relationship is presented as follows:17

where TC is the tip clearance, η the turbine efficiency, and the subscript nom the nominal value.In the simulation operation of the engine component model, the high-pressure turbine characteristics can be corrected by K to reflect the deviation of the actual turbine efficiency from the nominal efficiency.K was determined by the following equation:

where ψZtipis the Zweifel loading coefficient, defined as the ratio of the actual tangential load to the ideal load in the blade cascade, whose value was retrieved from reference.17

4.3.Coupling mechanism of engine and turbine tip clearance

Engine performance and state parameters directly affect the thermodynamic and rotor dynamic deformation of the case, blade,and rotor, thus changing tip clearance.Therefore, the real-time dynamic model of turbine tip clearance should be integrated with a system functioned with the output of engine state parameters.In this paper, the engine component-level dynamic model was adopted to study the coupling mechanism between the engine and turbine tip clearance, and the data from the engine dynamic model drives the real-time calculation of the turbine tip clearance model.In addition, the correction coefficient of turbine efficiency calculated by the tip clearance model was fed back to the engine dynamic model in real-time to realize the closed-loop of data,whose structure is displayed in Fig.15.

Firstly, the initial state parameters and the control parameters of the engine were input, and the engine’s dynamic model solved each component’s thermodynamic and kinetic parameters.Secondly, parameters that influence the variation of the tip clearance were selected (see Section 2 of this paper for more information)and sent to the tip clearance model in real-time.According to the modeling principles stated in Section 3.1, the heat transfer characteristics and material properties of the turbine components were updated from the thermodynamic parameters,and the deformations of the case, rotor and blade were solved.Thirdly, the value of tip clearance and the correction factor K of turbine efficiency were calculated from the component deformations.Unlike previous studies17,18, the correction factor K was fed to the highpressure turbine module in the engine’s dynamic model in realtime so that the engine model could sense the degradation or improvement of turbine performance during the dynamic changes in tip clearance, thus obtaining more accurate parameters of engine performance.Finally, these performance parameters were output with the parameters of tip clearance to commence the simulation calculation in the next step.

4.4.Simulation of integrated tip clearance model

Fig.14 Characteristics of high-pressure turbine components under influence of tip clearance.

The tip clearance model established in this paper is universal for engines with turbine components of the same structure, and the simulation experiments should be conducted by configuring an engine with the corresponding parameters of part sizes, material properties of components, and flow field environments according to the engine type.The parameters of the AGTF30 engine18were taken as the baseline model in this paper to investigate whether the turbine tip clearance of different engines has the same change pattern during operation.In addition, the convective heat transfer coefficient h, as well as the thermal conductivity k and the heat capacity cpof the case, blade, and rotor were adjusted by ± 50% to analyze the sensitivity of the tip clearance model in terms of the differences in convection and heat transfer of different engines, as presented in Fig.16.As heat transfer of blades was quite speedy and posed little influence on tip clearance in a long term, it was not taken into consideration here.

Fuel oil of the engine model stepped up at 100 s to adjust the engine from an idle state to a peak state.Variation of the tip clearance under different parameter configurations demonstrates that changes in the tip clearance at the beginning when the engine accelerates implied clear pinch points, and the minimum values were all around 0.7 mm.The maximum deviation from the baseline did not exceed 10%.After that, the tip clearance started to increase slowly and gradually reached the steady-state.Different parameters might impact blade tip clearance quite differently.cpand h changes would significantly affect blade tip clearance variation range, which could come up to ±0.2 mm with a maximum deviation of about ±15% compared to baseline model.As such,it was quite close to the magnitude of required for active control of tip clearance.The above study shows that the various patterns of tip clearance are similar for engines with different parameters such as material properties of components and flow field environments,so the results of the simulation experiments adopting baseline parameters in this paper are universal and representative.

4.5.Hardware-in-the-loop test

To validate real-time calculation and data exchange performance of blade tip clearance measurement method on onboard computing platform,a simulation experiment on HIL test bench as shown in Fig.17 was performed.Here, component assemblies were (A)upper computer for monitoring developed by LINKS-RT platform, (B) engine component-level dynamic model developed in MATLAB/Simulink environment,

(C)real-time simulator-a,28(D)real-time simulator-b28(E)Switch.When simulation started, (A) input flight and control parameters and sent to other components by (E).Engine model in (B),after being compiled, would operate in (C) and simulate a real engine, with parameters coming from (A).Blade tip clearance model operated in (D), after integrating with engine onboard model and receiving data from (A), would make out variations of blade tip clearance directly, which would be output to (A) by(E)and be reflected real-timely there.Alternatively,(D)also could receive data from simulation engine in(C),which could be seen as measurable engine data drove computation of blade tip clearance model.HIL simulation showed that as blade tip clearance model operated at time step of 20 ms in (D),within 100 s,its calculation time per step was 1.273 ms maximally, 112.503 μs minimally and 339.293 μs averagely, which met the calculation requirements in working environment.

5.Analysis of variation law and impact on aero-engine of turbine tip clearance

Hereinabove a blade tip clearance model,with good accuracy and real-time reflection was already built.This paper would base on that and study how blade tip clearance variated over the entire flight segment and its effect on engine performance.Similar studies were available, however were inaccurate due to poor model precision and restricted only to the effect on engine performance by closed-loop control.Under the condition that fuel consumption was unchanged, blade tip clearance would cause engine performance degradation, which even more straightforwardly and precisely, was unfortunately not evaluated.This paper would make a supplement to previous studies.

Fig.16 Variation of tip clearance of different engines.

5.1.Variation of aero-engine performance parameters in full flight segment

In this paper,the engine of a turbofan with large bypass ratio was adopted as the research object, and the flight parameters for the engine were set according to the flight conditions of the transport aircraft as shown in Fig.18.

Data related to aircraft flight altitude and Mach number were set based on reference24accordingly.During the flameout of the engine, it was assumed that the flight control system commands the other engines to maintain the flight altitude and speed, so no Mach number or altitude changes were considered18.

After the simulation over the entire flight segment commenced,engine high pressure turbine speed N3, turbine inlet temperature T4, and engine thrust Fgrossand Fnetdisplayed similar variation trends as shown in Fig.19.

5.2.Variation law of turbine tip clearance during full flight segment

Changes in the temperature of each component of the highpressure turbine over the full flight segment mentioned above are demonstrated in Fig.20.

The corresponding dynamic deformation process over the full flight segment is unveiled in Fig.21.Since the model simulation in this paper commenced from the engine’s idle state, the dimensions of turbine components were somewhat different compared with the initial dimensions installed when the engine’s highpressure turbine had started to rotate and output power under the propulsion of high-temperature and high-pressure gas, the case,blade and rotor started to change with non-zero deformation at the moment of 0 s.

As demonstrated in Fig.21, the deformation of the blade is smaller than that of other components due to its small size in the full flight segment, so variation of turbine tip clearance is mainly dominated by the deformation of the case and rotor.As the temperature of turbine components is positively correlated with the engine’s rotational speed,so the temperature of each component will rise when the engine accelerates, and the centrifugal and thermal stresses will cooperate to make the blade and the rotor extended radially.At the same time the case will also be subject to thermal expansion.The typical process is unveiled in Fig.21, in which each component at the beginning of the simulation before 500 s will expand rapidly,and the expansion of the case and rotor is close to 1 mm;conversely,the temperature of turbine components will drop when the engine decelerates, and the radial deformation of the blade, rotor, and case will lessen.The typical situation is presented in Fig.21, where the components shrink to a larger extent at 2500 s.Therefore, the deformation direction of each component is the same when the engine’s operating condition changes, and the variation of the tip clearance is caused by the uncoordinated deformation rate and deformation of the components, as presented in Fig.22.

The maximum turbine tip clearance 1.0 mm in the full flight segment appears at about 250 s when the aircraft takes off and climbs, for the deformation of the case by thermal expansion is larger than that of the rotor and blade.At this time,the gas leaks most through the tip clearance, and the turbine efficiency drops significantly.As shown in Fig.23, turbine efficiency deteriorated and reached the lowest point in the entire flight envelope, about 1.5%worse compared to that of baseline model.At such moment,clearance reduction by tip clearance active control would improve working performance of turbine components and thus uplift engine performance.

Fig.17 Hardware-in-the-loop simulation experiment equipment.

As for the minimum clearance, the pinch point appears in Fig.22.Smaller clearance will easily cause scuffing and friction of blades and cases at this point, thus affecting the engine life and flight safety.The minimum clearance point under the full flight segment is only 0.3 mm in the transient phase of engine Reacceleration at high altitude about 3000 s.At this time, the high-pressure rotational speed rises.The expansion rate of the blade and rotor is larger than that of the case, and the tip clearance gradually decreases because the thermal response lags behind the response of rotational speed; the centrifugal deformation no longer continues after the engine’s rotational speed becomes stable, while thermal expansion gradually plays a leading role,and the thermal expansion of case is faster than that of rotor and blade,increasing the increase in the clearance again;therefore,there will be a minimal value of clearance called ‘‘pinch point”.The‘‘pinch point”often appears when the rotary shaft accelerates from the current state, while the turbine efficiency will be improved if the smaller tip clearance appears with a‘‘pinch point”,as presented in Fig.23.However, margins of the tip clearance are insufficient in this state given the asymmetric clearance and other factors, so abrasion may easily occur.Hence, this value will be adopted as a reference indicator for extreme cases in the design of the active control method.

Fig.18 Flight envelope data during full flight segment.

5.3.Effect of changes in turbine efficiency due to tip clearance on engine performance

After the variation of turbine tip clearance over the entire flight segment was obtained, the effect of changes in the turbine efficiency due to the tip clearance on the engine performance was studied in this paper, which was compared with the flight simulation data without efficiency correction,as displayed in Fig.24 and Fig.25.

In Fig.24, as simulation ramped up within 100–400 s and tip clearance was close to its maximum value as shown in Fig.23,the tip clearance posed a severely negative impact on engine performance with actual rotational speed of the high-pressure turbine N3reduced by about 200 r/min after correction.Also it could be seen from Fig.25 (a), the corrected engine thrust Fgrosswas reduced nearly 2% and failed to reach the real-time thrust of the engine dynamic model.As simulation came to 3000 s shown in Fig.24,due to appearance of‘‘pinch point”,tip clearance reached its minimum value and comparatively better turbine efficiency showed up, with corrected engine thrust about 10% better than the calculated real-time thrust of engine dynamic model.Therefore, tip clearance active control was believed to be a useful technology, by keeping tip clearance around ‘‘pinch point”, for improving rotational speed of engine high-pressure turbine and total engine thrust, and in the end uplifting the overall aeroengine performance.

At the end of this paper, by applying tip clearance efficiency correction model, sensitivity analysis of several groups of engine performance parameters at engine idle and cruise state were made,and whether engine closed-loop control system was working or not was tested, with results presented in Fig.26.Taking TC at cruise state as an example,as TC reduced 0.5 mm,turbine efficiency went up 3.28%, engine fan speed Nf, total thrust Fgross, engine static thrust Fnet, thanks to engine control system, remained nearly unchanged, while engine Specific Fuel Consumption (SFC) and Exhaust Gas Temperature (EGT) saw a reduction respectively of 1.02%and 1.56%,and turbine inlet temperature was slightly lowered as well.Those positive results contributed positively to enable engine to serve longer, be stealthier, more environment-friendly and fuel economical.Meanwhile, as SFC decreased, speed N3of engine high-pressure shaft and outlet pressure P3of highpressure compressor increased by about 1%, which told that engine working was strengthened along with tip clearance decrease, thus getting better engine performance.In turn, as tip clearance increased, above processes would turn to the opposite and bring even more negative impact on engine performance.Tip clearance change impacted engine performance even more significantly when engine was at idle state with low power.

Fig.19 N3, T4, Fgross and Fnet in whole flight segment.

Fig.20 Temperature variation of turbine components in full flight segment.

Regardless of effect from engine control system,at cruise state and fuel consumption pre-given, as TC reduced by every 0.5 mm,turbine efficiency uplifted 3.28%,EGT reduced about 1.28%,turbine inlet temperature T4slightly dropped, while engine net trust improved about 0.84%,speed of high-pressure shaft N3,fan speed Nf, total engine thrust Fgrossand outlet pressure P3of highpressure compressor saw improvements of different degrees.In turn, as tip clearance increased, above processes would turn to the opposite and bring even more negative impact.Likewise,above effects were even more obvious when engine was at idle state with low power.

Fig.23 Correction coefficient of turbine tip clearance efficiency at full flight stage.

Fig.24 Effect of changes in turbine tip clearance on rotational speed of high-pressure turbine.

Fig.25 Influence of turbine tip clearance variation on engine thrust.

Fig.26 Sensitivity analysis of engine performance parameters.

Therefore, it explained why it’s essential to perform tip clearance control over the entire flight segment of engine operation and keep tip clearance within a small safety margin, as it made positive effects on engine performance and enabled engine to serve longer, be stealthier, more environment friendly and more fuel economical.

6.Conclusions

This paper proposed a new idea to resolve the long-existing problem that tip clearance was challenging to measure under engine working conditions by offering a model-based intelligent tip clearance measurement method of high confidence and good real-time reflection.

At first, the paper clarified the primary factors and secondary factors which affected tip clearance changes by first principles.With the aim of building a simpler and more accurate model in mind, this paper also included factors that were wrongly ignored by the previous modeling, such as blade centrifugal force impact on rotor deformation, material characteristics changes, temperature change, etc.By doing that, the rotor deformation amount was calculated about 15.9% more accurately than before.

Later,tip clearance impact on turbine efficiency was quantified and reflected in the engine on-board model, which mitigated the performance difference between the engine on-board model and real engines caused by tip clearance and thus improved the accuracy of the engine on-board real-time model.

Next, by considering the coupling mechanism between tip clearance changes and engine state parameters, a hardware-inthe-loop simulation platform was set up, to simulate interactions between real engines at work and the tip clearance model.The simulation showed that as the tip clearance model operated at a time step of 20 ms, it took an average of 0.34 ms per time step in realtime simulators, meeting the real-time reflection and calculation requirements at the engine working state.

Lastly, a simulation of tip clearance dynamic changes over the entire flight segment by tip clearance measurement model was performed, which made corrections and supplements to previous studies and analyzed tip clearance impact on engine performance quantitatively.Simulation results showed that as tip clearance at cruise state reduced by every 0.5 mm, turbine efficiency uplifted 3.28%, Specific Fuel Consumption (SFC) decreased by 1.02%given the same engine thrust while engine net thrust Fnetimproved by 0.84% given the same specific fuel consumption.Thus, active control of tip clearance over the entire flight segment to maintain a close enough tip clearance contributed positively to uplifting engine performance,enabling it to serve longer,be stealthier,more environment-friendly, and have a better fuel economy.Therefore,starting from this model-based tip clearance measurement method proposed in the paper,further study on designing an active closedloop control system for tip clearance would be of great scientific significance.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was supported by the National Natural Science Foundation of China (Nos.51906103, 52176009).