Model Adaptation of a Mixed Flow Turbofan Engineltu.diva-portal.org/smash/get/diva2:1463578/FULLTEXT01.pdf · Model Adaptation of a Mixed Flow Turbofan Engine Oskar Lindkvist Space

Model Adaptation of a Mixed Flow

Turbofan Engine

Oskar Lindkvist

Space Engineering, master's level

2020

Luleå University of Technology

Department of Computer Science, Electrical and Space Engineering

INDUSTRIAL SUPERVISORSven Anzen

EXAMINERProf. Lars-Goran Westerberg

i

Abstract

Gas turbine performance models are usually created in an object oriented manner, where dif-ferent standard components are connected to form the complete model. The characteristicsof these components are often represented by component maps and empirical correlations.However, engine specific component characteristics are seldom available to anyone outside ofthe manufacturers. It is therefore very common for researchers to use publicly accessible orgeneric component maps instead. But in order to reduce prediction errors the maps have tobe modified to fit any specific engine. This thesis work investigates the process of adaptinga parametric turbofan engine model to a limited amount of test-data using the propulsionprogram EnVironmental Assessment (EVA). Steady state test-data was generated using aninitial reference model with Sea Level Static (SLS) operating conditions. Another enginemodel with different fan, compressor and turbine maps was then used in the adaptation.

An initial on-design model was adapted to the highest power test-data point. Thismodel is based on aerothermodynamic equations and is used as a reference to scale thegeneric component maps to. A sensitivity analysis was done at this point in order to finddependencies between unknown component parameters and test data. These were thenincluded in the cycle solver which employs a version of the Newton-Raphson method. Afterthe fan and compressor maps had been scaled to the design point they were adapted totest-data by adjusting the mass flow parameters in a direct search optimizer. Finally, speedlines in the fan and compressor maps were relabeled to reduce rotor speed errors. Theadapted performance model was then validated against the reference model at a few flyingconditions.

The performance model results demonstrate that it is possible to greatly reduce predic-tion errors by only adjusting the corrected mass flow in fan and compressor maps. Addi-tionally, rotor speed errors could successfully be corrected as a final step in the adaptationby relabeling speed lines in the component maps. When validated, the adapted model hada maximum parameter error of 1.5%.

ii

Acknowledgements

I would like to express my sincere gratitude to my supervisor Sven Anzen, who has beena constant source of help throughout this thesis. Your valuable inputs from an industryperspective really sparked my interest in aircraft engines. I would also like to acknowledgeMikael Stenfelt for his much appreciated advice and guidance during the initial phases ofthis project. In addition, I wish to express my deepest gratitude to Saab and everyone atthe department of Propulsion Systems, for making me feel very welcome and for providingme with valuable feedback along the way.

Finally, a big thank you goes to my family and friends for all the support and encour-agement.

Oskar Lindkvist, Linkoping, August 2020

iii

Contents

1 Introduction 11.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Engine Model 32.1 Component Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Design Point Adaptation 53.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2.1 Gas Path Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2.2 Broyden’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2.3 Compressor and Turbine Matching . . . . . . . . . . . . . . . . . . . . 8

3.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3.1 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3.2 Selection of Targets and States . . . . . . . . . . . . . . . . . . . . . . 10

3.4 Estimations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4.1 Shafts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4.2 Intake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4.3 Combustor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4.4 Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4.5 Mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Off-Design Adaptation 154.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2.1 Component Map Basics . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2.2 Modifying Component Maps . . . . . . . . . . . . . . . . . . . . . . . 184.2.3 Component Map Matching . . . . . . . . . . . . . . . . . . . . . . . . 18

4.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3.1 Estimating Design Point Map Location . . . . . . . . . . . . . . . . . 194.3.2 Selection of Targets and States . . . . . . . . . . . . . . . . . . . . . . 194.3.3 Component Map Corrections . . . . . . . . . . . . . . . . . . . . . . . 204.3.4 Relabeling Speed Lines . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Validation and Discussion 27

6 Conclusions 29

References 30

Appendix A Test Data 32

Appendix B Design Point Parameters 34

iv

List of Figures

2 EVA component setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Flowchart describing the matching procedure . . . . . . . . . . . . . . . . . . 84 Sensitivity in the design point . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Estimated cooling flow requirements . . . . . . . . . . . . . . . . . . . . . . . 136 β-lines in a compressor map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 β-lines in a compressor map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Component map table structure . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Adaptation errors with N1 as target . . . . . . . . . . . . . . . . . . . . . . . 1910 Adaptation errors with P5 as target . . . . . . . . . . . . . . . . . . . . . . . 2011 Flow correction factors for the fan map . . . . . . . . . . . . . . . . . . . . . 2112 Flow correction factors for the compressor map . . . . . . . . . . . . . . . . . 2213 Speed lines before and after the flow correction factors has been applied . . . 2214 Adaptation errors after flow corrections . . . . . . . . . . . . . . . . . . . . . 2215 N1 and N2 before relabeling of speed lines . . . . . . . . . . . . . . . . . . . . 2316 N1 and N2 after relabeling of speed lines . . . . . . . . . . . . . . . . . . . . . 2317 Adaptation errors after mass flow and speed corrections . . . . . . . . . . . . 2418 Prediction errors after mass flow and speed corrections . . . . . . . . . . . . . 2419 P21 and T21 model agreement . . . . . . . . . . . . . . . . . . . . . . . . . . . 2520 P3 and T3 model agreement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2521 P46 and T46 model agreement . . . . . . . . . . . . . . . . . . . . . . . . . . . 2522 P5 and T5 model agreement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2623 Thrust and specific fuel consumption model agreement . . . . . . . . . . . . . 2624 Model validation, test-data parameters . . . . . . . . . . . . . . . . . . . . . . 2725 Model validation, performance parameters . . . . . . . . . . . . . . . . . . . . 28

List of Tables

1 Available engine data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Manually selected design point targets and states . . . . . . . . . . . . . . . . 103 Test data not used by design point matching scheme . . . . . . . . . . . . . . 104 Parameter values for different levels of turbine cooling technology . . . . . . . 135 Flow correction factors for a few off-design points . . . . . . . . . . . . . . . . 216 Flight conditions used for model validation . . . . . . . . . . . . . . . . . . . 27

v

Acronyms

BPR Bypass Ratio

BVR Beyond Visual Range

CAW Flow Correction Factor

CFD Computational Fluid Dynamics

DAE Isentropic Efficiency Delta

EVA EnVironmental Assessment

GPA Gas Path Analysis

ICM Influence Coefficient Matrix

ISO International Organization for Standardization

LPT Low Pressure Turbine

NGV Nozzle Guide Vane

RMSE Root Mean Square Error

SAS Secondary Air System

SFC Specific Fuel Consumption

SLS Sea Level Static

Symbols

β auxiliary coordinate used in component maps

Bi biot number

E combustion efficiency

ε effectiveness

η efficiency

F engine thrust

h specific enthalpy

H influence coefficient matrix

J jacobian matrix

K cooling flow factor

N rotor speed

P total pressure

R pressure ratio

T total temperature

W mass flow rate

x independent variable

z dependent variable

vi

Subscripts

bc blade cooling

c cooling

des design point value

f fuel

fc film cooling

g mainstream gas

hpc high pressure compressor

hpt high pressure turbine

∞ polytropic

is isentropic

lpt low pressure turbine

map component map value

met metal

ref reference value

rel relative value

tbc thermal barrier coating

Superscripts

* corrected quantity

+ pseudoinverse

Engine Station Numbering

a ambient conditions

0 ram conditions in free stream

1 engine intake front flange

2 fan front face

21 fan exit face

27 compressor front face

3 compressor exit face

31 combustor inlet plane

4 combustor exit plane / HPT front face

46 HPT exit face / LPT front face

5 LPT exit face

vii

1 INTRODUCTION

1 Introduction

Gas turbine engines are very common machines in the modern world. They are most gen-erally known to propel commercial aircraft in the form of turbofan engines. However, gasturbines are also used to power industrial generators, watercraft and even tanks. The pur-suit of better and more efficient engines has also led to an increase in their complexity. Gasturbine performance modelling has therefore become both common and challenging problemfor engineers as the ever growing industry is looking for higher quality models to predict andsimulate engine designs. Accurate performance models are also, for example, very soughtafter in the field of gas turbine diagnostics, as engine components degrade over time and giverise to changes in measurable thermodynamic parameters within the engine [1]. Gas turbinemanufacturers typically develop very complex performance models for their own internaluse. These proprietary models are based on a large amount of parameters measured duringdevelopment and are, as such, not accessible to outsiders.

For third parties wanting to build a performance model out of an existing engine, thedifficulty lies in the lack of thermodynamic parameters available for monitoring. Typicalparameters needed to determine an engine cycle are pressure ratios, component efficiencies,mass flow rates, etc. It is possible to obtain some of these parameters, even for the enginecustomer; although, some requires expensive equipment and special facilities to be measured.Consequently, the information available to third parties usually only consists of parametersneeded for engine control. This makes gas turbine engine modelling an inherently non-deterministic problem, where the amount of unknown parameters greatly outweighs theknown. For this reason, it is not unusual that the process of adapting an engine model to asmall amount of data partly consists of making simplifications and estimations of unknownparameters based on previous knowledge or literature.

Engines are typically designed with a specific point of operation in mind, where enginecomponents are optimized to operate at maximum efficiency or to reach certain performancecriteria. This point is called the engine design point and every other point within theoperational envelope of the engine falls into the category off-design points. Gas turbineengine modelling is therefore often broken up into two parts: design point modelling andoff-design modelling. Design point modelling, sometimes referred to as on-design modelling,specifies the state of the working fluid throughout an engine at a specific flight condition inorder to relate performance parameters to engine design choices. In other words, a designpoint model defines the sizing of various engine components at a specific key mission point.

An off-design model on the other hand, determines the engine performance with fixeddesign choices at all operating points. Off-design modeling is therefore sometimes calledperformance modelling.

1.1 Objectives

This thesis will study the model adaptation process of a generic mixed flow turbofan engineusing the propulsion program EnVironmental Assessment (EVA), developed by Kyprianidis[2]. However, the general methodology could still be applied to other types of engines andsimulation programs. The adaptation will be based on steady state Sea Level Static (SLS)test-data with the following main objectives:

• Perform a sensitivity analysis at the design point in order to investigate how designparameters affect other measurable parameters. This sensitivity analysis aims to pro-vide a set of dependent and independent parameters that can be used in EVAs designpoint solver.

• Provide an off-design adaptation technique that, given a set of component maps andtest-data, can find the map-modifications needed for maximum fitness with minimalhuman input. This kind of automation could prove very useful for engineers lookingto modify already existing component maps to fit their engine test-data.

1

1 INTRODUCTION

• Validate the adapted model at a few operational points within the engine envelope.Ultimately, one of the reasons to adapt an engine model is to be able to predictperformance at points other than those used for the adaptation itself.

1.2 Thesis Overview

Chapter 2 presents the engine model used in simulation program EVA. Additionally, thechapter has a few paragraphs covering the set of available test-data parameters and howthey were acquired.

Chapter 3 consists of some background surrounding design point simulations, a theorysection covering EVAs way of simulating engine cycles as well as a section covering the designpoint adaptation method. Design point parameter estimations are also presented. Thesecan be used as rough estimates of some common component parameters when no other datais available.

Chapter 4 presents some background and theory of component maps, which is the mostcommon way to represent engine components in off-design models. Then, a method to mod-ify compressor maps is suggested and implemented. Finally, the error between predictionsand test-data is presented.

Chapter 5 validates the adapted model at a few selected points of operation within theenvelope. This is done in order to test the prediction accuracy at points that is not part ofthe test-data. The results are presented and discussed.

Chapter 6 presents the main conclusions drawn from this study along with some futureresearch recommendations.

2

2 ENGINE MODEL

2 Engine Model

An engine model is constructed using different standard components, connected in the flowdirection as well as though shafts. Firstly, this chapter presents the set of components usedto form the complete model in EVA. Then, the method which was used to retrieve test-datais presented.

2.1 Component Setup

A necessary step in any simulation is to decide upon what level of detail to include in themodel. A simple engine model might disregard all pressure losses in-between componentsto reduce complexity. Compromises can also be made on, for example, the Secondary AirSystem (SAS) in order to further reduce model complexity.

The engine used in this thesis will be a mixed flow, low bypass ratio fighter enginewith two spools. An illustration of the component setup used within EVA can be seenin figure 2. Ducts are placed in-between the main components of the engine, where nonnegligible pressure losses are to be expected. A SAS is also included in the model, consistingof customer bleed as well as cooling air for the high and low pressure turbines. Air for theSAS is taken from the high-pressure compressor.

Military engines of this type usually have an afterburner installed after the mixer. How-ever, this study will only consider non-afterburning operation and pressure losses associatedwith a non-lit afterburner can be accounted for with a duct.

DUCT

DUCT

INTAKE

NOZZLE

MIXER

BYPASS DUCT

LPTFANDUCT

DUCT

SPLIT HPTHPCDUCT

COMB

CUSTOMERBLEED

HPT-cooling

LPT-cooling

HP-shaftLP-shaft

Figure 2: EVA component setup.

Each type of component has it’s own set of exposed design point parameters within EVA.Components are then simply connected to form the complete design point thermodynamicmodel. The off-design model, on the other hand, is based on component maps. Eachcomponent map describes that components behaviour at every operating condition, actingas a look-up table for off design calculations.

The design point model is used as a first step in the simulation to determine the thermo-dynamic cycle at design point operating conditions. This information can later be used toscale component maps using the design point performance as a reference point; producingan off-design model that match the design point. A more detailed description of design pointmodels is presented in chapter 3. EVA has a built in set of generic component maps thatare used for the adaptation. Since the maps are of a generic type, they will be different inscale and shape compared to the maps of a specific real engine. Consequently, they will haveto be modified in order to match the test-data. A more detailed description of componentmaps and map modifications is presented in section 4.2.

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2 ENGINE MODEL

2.2 Data Acquisition

When measurements are done on a real engine some uncertainties are unavoidable. Theseuncertainties can come from the sensor itself in the form of measurement noise, miscalibra-tions or malfunctions. But the positioning of the sensors can also introduce uncertaintiesdue to the non-uniformity of gas properties at any specific engine cross section. All dataused in this thesis comes from a simulated engine rather than a real one. This is done toprovide a good basis for adaptation validation, but it also means that the data is noise-free.

Only a set of common parameters which realistically could be measured by on-boardsensors are included in the data, presented in table 1. In a real scenario, the amount ofparameters available would vary between manufacturers and engine types.

Parameter Description Unit

Alt Altitude m

M0 Free stream Mach number -

Pa Ambient total pressure kPa

Ta Ambient temperature K

P21 Fan discharge total pressure kPa

T21 Fan discharge temperature K

P3 Compressor discharge total pressure kPa

P5 Low pressure turbine discharge total pressure kPa

T5 Low pressure turbine discharge temperature K

N1 Low pressure shaft speed rpm

N2 High pressure shaft speed rpm

Wf Fuel flow rate kg/s

Table 1: Available engine data.

The data consists of 31 different SLS operating points generated though a sweep of differ-ent power settings with the component setup in figure 2. This is meant to be representativeof steady state test-bed runs and will result in a clean operating line with evenly spacedpoints. Only having one operating line also makes the figures in this report a bit moreclearer.

When the test-data was generated, design point parameters were assigned realistic valuesthat fit a set of GasTurb component maps. These maps were purposely selected to bedifferent in shape and scale to EVAs built in maps that later on are going to be used in theadaptation. A complete list of all the test-data can be found in appendix A. Componentparameters used in the design point of the reference model can be found in appendix B.

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3 DESIGN POINT ADAPTATION

3 Design Point Adaptation

Design point modelling largely consists of specifying a lot of component parameters for var-ious stages in the engine (pressure rations, efficiencies, losses, etc), combined with boundaryconditions such as the flight Mach number and altitude. Because of limitations on theamount of data available, some parameters either has to be estimated through guesswork,or through some adaptive algorithm. This chapter will include an introduction to relevantdesign point modelling techniques along with their application in EVA.

3.1 Introduction

As different engines are designed for different applications, it stands to reason that the designpoint very much depend on the operational conditions surrounding that application. Forcommercial aeroengines, the design point typically corresponds to cruise conditions, as thisis where the plane spends most of its time during a flight. Industrial engines, on the otherhand, normally has their design point at International Organization for Standardization(ISO) base load [3]. The real design point isn’t necessarily known by the user. However,any operating point can theoretically be chosen as a basis for the design point model. Thechoice of assumed design point is therefore often dictated by the availability and qualityof engine data. Sea Level Static (SLS) operations can facilitate additional instrumentationon the engine, providing measurements of parameters otherwise not accessible to the user.Thus making SLS-points a good basis for design point models.

Even though some of the engine parameters can be measured by users, some are stillleft to be estimated in order to determine the design point cycle. When design point sim-ulations are done using estimated component parameters, some errors are to be expected.These errors cause other parameters in the engine model to deviate from their correct value.Depending on the amount of engine data available, a subset of these deviations can be cal-culated through comparison between predicted and measured data. In the early days ofengine modelling, engineers had to manually adjust design parameters until the errors be-tween predicted and measured parameters was adequately small [4]. This can become a verycumbersome task, especially for engineers lacking sufficient engine performance knowledge.Newer methods, usually derivatives of one or more older methods, has since been developedin order to automate this procedure. Adaptation techniques utilizing internal cycle solvershas become very common. These solvers use variations of Newton–Raphsons method alongwith gradient information to tune component parameters until the difference between pre-dictions and measurements (residuals) are close to zero. Derivatives of measurable (target)parameters are calculated with respect to component parameters to form the Influence Co-efficient Matrix (ICM). Corrections to the component parameters can then be obtained bymultiplying adaptation errors with the inverted ICM. Although this kind of method auto-mates the iteration process, a user still has to provide the right type of parameters in orderfor the solver to find a solution. The method has successfully been applied to the modeladaptation of a GE LM2500+ industrial gas turbine [5] by Li et al. Genetic algorithms waslater explored by Li [6] and compared to the previous ICM-method [7]. It was found thatgenetic algorithms, which doesn’t require gradient information, provided a more numericallyrobust adaptation at the cost of computational speed and prediction accuracy. Roth et al.introduced an adaptation technique based on a minimum variance estimator algorithm [8].This method addressed the noise and uncertainties in measurements, providing a quicklyconverging solution with small errors that takes advantage of probabilistic information inthe matching process.

The simulation program used in this thesis (EVA) implements gradients and a versionof Newton-Raphsons method to solve on- and off-design problems. The next section will goover some basic theory concerning these methods.

5


3.2 Theory

3.2.1 Gas Path Analysis

Gas Path Analysis (GPA) was first introduced by Urban [9] and the method has since success-fully been applied to both gas turbine performance prediction and performance diagnostics.Linear GPA simplifies the non-linear relationships between measurable parameters and non-measurable component parameters through linear approximation. Dependent, measurableparameters are typically pressures, temperatures, mass flow rates etc. These are assumedto be linearly interrelated to independent component parameters (such as efficiencies, pres-sure ratios and flow capacities). With the use of an Influence Coefficient Matrix (ICM),the relationship between dependent and independent parameters can be expressed as thefollowing:

~z = H~x (1)

Where ~z ∈ Rm is the dependent parameter vector, ~x ∈ Rn the independent parameter vectorand H ∈ Rm,n the Influence Coefficient Matrix, also called Jacobian. This matrix containsall first-order partial derivatives of the vector valued function z(x) at some operating point(for example maximum power or cruise). Rewriting equation (1) in terms of parameterdeltas:

∆~z = H∆~x (2)

In the case of performance diagnostics, the deviation of non-measurable component param-eters ∆~x can be found by taking the inverse of H.

∆~x = H−1∆~z (3)

A small change in measurable parameters can therefore be used to calculate degradation inengine components. This application rely heavily on a accurate influence coefficient matrix,which in order to even be calculated, requires experimental data of engines with knowndegraded components or an engine model. Moreover, the amount of uncorrelated and noise-free measurements has to be the same as the amount of engine component parameters [10].

Equation (2) is under-determined if the amount of component parameters n is largerthan the amount of measurements m. In this case, the inverse of H doesn’t exist. But apseudoinverse can be utilized. Where H+ denotes the pseudoinverse and HT denotes thetranspose of H.

H+ = HT (HHT )−1 (4)

If instead (m > n), then equation (2) is over-determined. In this case a pseudoinverse, whichrepresents the least square solution, is defined as:

H+ = (HHT )−1HT (5)

This type of linearized relationship between changes in dependent and independent vari-ables can also be used for adaptation purposes. ∆~z can be seen as a vector containingprediction errors, also known as residuals. Corrections to the component parameters ∆~x arethen found using equation (3). This assumes linear engine behaviour around the baselineoperating condition, which often isn’t the case. However, non-linearity can be taken intoaccount through the use of Newton–Raphsons method. Where the linear prediction processis executed iteratively with re-evaluations of H at every iteration. The process is stoppedwhen the residuals are deemed small enough.

6


3.2.2 Broyden’s Method

While it is possible to solve the non-linear equations described in the previous section usingNewton-Raphsons method, it has some shortcomings. The method produces better guessesof independent variable x using equation (6). However, this requires re-evaluations of thederivative at every iteration, which can be expensive in terms of computing time.

xn+1 = xn −z(xn)

z′(xn)(6)

In multiple dimensions we have the following equation:

xn+1 = xn − J−1z(xn) (7)

Where, J is the Jacobian matrix, previously referred to as the Influence Coefficient Matrix.Since J contains all the first order derivatives of x with respect to z, the amount of functionevaluations required to compute J can become increasingly large, depending on the numberof variables. Remember, a function evaluation in this case is a complete cycle computation.

Secant methods, or quasi-Newton methods, does not require function evaluations atevery iteration. Instead, they make approximations to the jacobian, which is updated everyiteration. Broyden’s Method is one of the most common secant methods for solving non-linear equation systems. Updates to the Jacobian are found using the following equation:

Jn = Jn−1 +∆zn − Jn−1∆xn

||x||2∆xTn (8)

Here ∆xn = xn−xn−1 and ∆zn = zn−zn−1. ||x|| is the norm of x and superscript T denotesthe transpose. Using approximations to update the jacobian instead of re-evaluating it everyiteration can speed up the convergence when function executions are expensive.

7


3.2.3 Compressor and Turbine Matching

The steady state performance is obtained by matching the turbines to the compressors.EVA uses Broyden’s method to do this in both design- and off-design simulations. How-ever, different sets of dependent and independent variables are utilized for the two types ofmodels. From here on dependent parameters will be referred to as targets and independentparameters will be referred to as states in order to be consistent with EVA. A flowchartshowing the matching procedure can be seen in figure 3.

InitialStates EngineModel

ComputeResiduals

SmallEnough?

Results

Targets

Estimatenewstates

Stop

Yes

No

Figure 3: Flowchart describing the matching procedure. New state estimates arefound using the methods described in previous sections.

Design point simulations will iterate turbine pressure ratio until conservation of energyis satisfied for the shaft. This is because the compressor work has to equal the turbine work,minus any power off-takes and mechanical losses in the shaft connecting the two. Turbinepressure ratios are therefore assigned as states and shaft energy residuals as targets. This isdone automatically by EVA. Thus removing turbine pressure ratios from the list of unknowncomponent parameters in the design point model (an initial guess still has to be specifiedfor the solver to start at). The conservation of shaft energy is something that always has tobe satisfied for the thermodynamic model to be valid. However, additional constraints canbe placed on the model by introducing more target/state-pairs into the matching procedure.These can for example be rotor speeds and fuel flows, i.e change the fuel flow until thetarget rotor speed has been reached. All user-defined targets and states will be solvedsimultaneously along with those automatically added by EVA.

Worth noting is that the values obtained for state variables doesn’t necessarily have tobe correct; an adapted state variable is only as good as the remaining component estimates.This is because the adaptation is based on the assumption that all other parameters notincluded in the matching procedure is correct. While in reality, a large amount of themmost likely are estimates.

8


3.3 Method

This section will cover the method used to adapt the design point model. The first stepis to select an operating point from the data to base the design point model on. Ideallythis point would be the actual engine design point, but any high power operating point canbe selected [11]. In order to simplify matters, the highest power operating point from thetest-data is chosen for the design point model.

3.3.1 Sensitivity Analysis

A sensitivity analysis can be performed in the design point in order to identify which modelparameters influence the test data parameters. The idea is to find coupled target and statevariables. By doing so, the amount of unknown parameters in the design point model canbe further reduced.

Firstly a baseline model has to be established. However, no component parameters areknown at this stage other than the boundary conditions, rotor speeds and fuel flow; so aninitial estimated model has to be used. While the magnitude of some sensitivities mightdiffer depending on how good the estimates are, the general behaviour of targets and statesis expected to stay relevant. For example, if the fan pressure ratio was estimated with anerror of 10%, a decrease in fan pressure ratio would still result in a decrease in compressordischarge pressure if all other component parameters stayed constant. Thus, making thesensitivity analysis applicable even if the simulated design point might differ from the targetengine. To perform the sensitivity analysis, a small change is made to one componentparameter at the time and its effect on potential target parameters is observed.

Figure 4: Sensitivity in the design point, 1% decrease in design parameters.

Figure 4 is showing the sensitivities for some temperatures and pressures when commoncomponent parameters are changed. Most sensitivities are of magnitude 1:1; meaning, 1%change in component parameters give rise to 1% change in target parameters. However, thisisn’t true for every parameter, for example the bypass ratio BPR doesn’t affect the selectedtarget candidates nearly as much; which makes sense as bypass ratio only defines how muchairflow goes through the bypass duct in relation to the core. Also note how a deviation incomponent parameters far downsteam doesn’t affect pressures and temperatures upstreamwhen all other component parameters are kept constant.

9


3.3.2 Selection of Targets and States

When choosing parameters to be used as targets and states, one has to consider whichcomponent parameters are most easily estimated. If the value of a component parametercan be estimated with comparatively small uncertainty, it might be a good idea to selectsome other parameter to be adapted in order to get the most out of the matching scheme.For example, the mechanical efficiency of a shaft is seldom known. However, these efficienciesare usually very high and they typically lie within a small range of possible values. Thus,choosing mechanical efficiency as a state variable would in this case be unnecessary. Asummary of the selected targets and states for the design point adaptation can be seen intable 2.

Target Description State Description

P21 Fan discharge pressure Rfan Fan pressure ratio

T21 Fan discharge temperature η∞,fan Fan polytropic efficiency

P3 Compressor discharge pressure Rhpc Compressor pressure ratio

P5 LPT discharge pressure η∞,hpc Compressor polytropic efficiency

T5 LPT discharge temperature Wintake Intake mass flow

Table 2: Manually selected design point targets and states.

The fan pressure ratio was selected as a state, using P21 as a target. Since P21 islocated so far upstream it’s usability is very limited and Rfan is the only major componentparameter it can be used for. T21, also located far upstream is used to tune the fan polytropicefficiency, completely defining the fan performance. P3 and P5 was assigned as targets forthe compressor pressure ratio and polytropic efficiency respectively. Finally T5 is used totune the intake mass flow.

Even though it helps to see targets and states as pairs, that is a simplification of theactual case. For example, P5 isn’t only used to tune the compressor polytropic efficiency. Infact, P5 is also affected by every other component parameter in figure 4. EVA will iterate onall states at the same time, where multiple states might affect multiple targets as dictatedby the jacobian matrix, described in section 3.2.1.

Parameter Description Unit

Alt Altitude m

M0 Free stream Mach number -

Pa Ambient total pressure kPa

Ta Ambient temperature K

N1 Low pressure shaft speed rpm

N2 High pressure shaft speed rpm

Wf Fuel flow rate kg/s

Table 3: Test data not used by design point matching scheme.

A summary of the remaining test data not used in the matching scheme is presentedin table 3. These parameters are simply employed in the design point model as they are,without any matching.

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3.4 Estimations

Even though a few component parameters can be determined through the adaptive processdescribed above, some remains to be estimated. This section will cover some rough estima-tions gathered from open literature that can be used when no other data is available. A listof design point parameter values used in the reference model can be found in appendix B.

3.4.1 Shafts

Mechanical efficiencyBearings are used to support the rotating shafts in the engine. These can be of severaltypes but the most common ones are ball or roller bearings due to their low power loss andrelatively small oil flow. Larger industrial engines usually utilize hydrodynamic bearings asthese have a longer life-span and is capable of larger thrust loads, but either way the shaftwill be subject to mechanical losses in the form of friction. A mechanical efficiency between99-99.9% is to be expected from a ball or roller bearing. Whereas hydrodynamic bearingscan have a mechanical efficiency as low as 96% [3].

Power extractionAuxiliary components such as fuel and oil pumps has to be powered by the gas turbine. Thisis often done by extracting power from the high pressure shaft in the case of a two-spoolgas turbine. The effect of removing power from a shaft is that the turbine output powerhas to increase for a given compressor speed, which in turn requires the an increase in fuelflow. An increase in fuel flow means a higher turbine inlet temperature, which results in theoperating point moving towards surge and an increase in SFC [12].

The amount of power extraction varies between aircraft and will therefore not be con-sidered in this study.

3.4.2 Intake

Pressure recoveryIntake pressure losses very much depend on the aircraft application and their design phi-losophy. Supersonic intakes has to decelerate the airflow to subsonic speeds before the firstfan or compressor stage. This is done through a series of shock waves with the side effectof additional intake pressure losses. Theoretically a large amount of weak oblique shockswould be preferred over a few strong shocks, with the motivation that an infinite amountof weak shocks represents the ideal isentropic intake [13]. The stronger normal shocks onthe other hand tend to cause large drops in total pressure, which is undesirable. However,several aircraft manufacturers still opted for fixed-geometry pitot intakes featuring a singlenormal chock due to their simplicity, good subsonic performance, low weight and potentiallysmall capture area. As an example, the General Dynamic F-16 featuring a pitot intake hasa pressure recovery around 98% at flight Mach numbers ranging from 0.4 to 1. At speedslower than Mach 0.4 and higher than 1.2 the pressure recovery factor will start to drop [14][15].

Intake pressure recovery maps can be used to model the pressure losses for different flightMach numbers. However, this study will employ a simple constant pressure recovery factor.

3.4.3 Combustor

Pressure lossesIdeally, the combustor in turbofan engines would burn fuel with an isobaric (constant pres-sure) combustion process. However, this is not the actual case as there are two types ofpressure losses present within the combustor:

1. Cold losses in the form of skin friction and turbulence. These are the result of forcingair through the combustor wall. This type of pressure loss is strongly dependent on

11


the combustion zone inlet velocity, where lower velocities yields less losses. In order toreduce the velocity of air coming from the compressor, a diffusion section is introducedbetween the last compressor stage and the combustor. However, the diffusion ductitself will introduce some loss in total pressure. At their design point, good combustordesigns usually have cold pressure losses between 2-4% of the combustor inlet totalpressure. Though, in some cases this loss in can be as high as 7% [3].

2. Hot losses, also called fundamental losses. These are caused by the increase in temper-ature in the combustion zone and are of significantly smaller magnitude than the coldlosses. A typical combustor has a design point hot pressure loss of around 0.05-0.15%,depending on the temperature ratio [3].

Combustion efficiencyCombustion efficiency is the ratio of burnt fuel to total fuel input. Combustion efficienciesat high power settings has always been near 100%, but historically that number dropped offat lower power. However, developments during the 70s and 80s improved the near idle com-bustion efficiencies significantly, leaving small rooms for further improvements. The focuswas instead shifted towards shortening the combustor length and reducing nitrogen oxides(NOx) emissions [16]. Reference [17] contains combustion efficiencies for a few turbofan en-gines with annular combustion chambers such as the General Electric F101, General ElectricJ85 and Pratt & Whitney F100. All with efficiencies above 99% at high power settings.

LoadingCombustor loading is related to the mass flow W31, inlet temperature T31, inlet pressureP31 and combustor volume. In order to achieve a high combustion efficiency the loadingshould be less than 10 kg/s atm1.8m3 at SLS maximum rating conditions. Preferably lessthan 5 kg/s atm1.8m3 [3].

3.4.4 Turbines

CoolingCooling flow requirements for the Nozzle Guide Vanes (NGVs) and turbine blades depend ona number of factors such as lifetime requirements, material and cooling technology, turbineinlet temperature, cooling air temperature, etc [3]. However, one way to estimate coolingflow requirements is through the methodology described in reference [18]. In this methodthe cooling air mass fraction ψ is calculated using the following equation:

ψ =W c

W hpc=

K

1 +B

(εbc − εfc[1− ηc(1− εbc)]

ηc(1− εbc)

)(9)

Where K is the cooling flow factor, εfc is the film cooling effectiveness and ηc is the efficiencyof the internal cooling. B can be computed with the following equation:

B = Bitbc −(εbc − εfc1− εbc

)Bimet (10)

where, Bitbc is the thermal barrier coating Biot number and Bimet is the metal Biot number.However, the exact values of the above mentioned parameters has to be determined throughsimulations or experiments. Instead, as an estimation, reference [19] contains recommendedvalues for these parameters based on the level of technology. These are presented in table 4.Finally, the blade cooling effectiveness εbc can be computed with the following equation:

εbc =T g − Tmet

T g − T c(11)

where, T g and T c are the temperatures of the main stream gas and coolant respectively.Coupled with Tmet, which is the maximum allowed metal surface temperature.

12


Cooling Technology Level K ηc εfc Bimet Bitbc

Current 0.045 0.70 0.40 0.15 0.30

Advanced 0.045 0.75 0.45 0.15 0.40

Super-Advanced 0.045 0.80 0.50 0.15 0.50

Table 4: Parameter values for different levels of turbine cooling technology

Figure 5: Estimated cooling flow requirementsfor different technology levels.

Using equations (9) and (10) along withtable 4, the mass flow fraction can be plot-ted against blade cooling effectiveness fordifferent technology levels. Figure 5 high-lights how more advanced cooling systemsrequire less mass flow to achieve the same ef-fectiveness. Equation (11) can then be usedto determine the necessary blade cooling ef-fectiveness for a given turbine inlet temper-ature, coolant temperature and blade metaltemperature.

Increased cooling capabilities has al-lowed engines to operate at gas tempera-tures well above the melting temperatureof the turbine materials [20]. The turbinematerial capabilities and inlet temperaturespresented in reference [21] can be used to es-timate the maximum allowed metal surfacetemperature Tmet and turbine gas temper-ature T g. As an example, the material tem-perature was limited to around 1325K during the 90s. It can also be assumed that theturbine generally isn’t operating at its maximum allowed surface temperate. Tmet is there-fore arbitrary selected to be 1150K in order to increase the turbine lifetime and accountfor non-uniformity in the combustion gas temperature. In the mid 90s, turbine inlet gasseshad reached a temperature T g of around 1800K. Using equation (11) and a cooling flowtemperature of 850K, the required blade cooling effectiveness is approximately 0.69.

Looking at figure 5, the required mass flow fraction for a blade cooling effectivenessof 0.69 is about 7% with the current technology. However, considering that the currenttechnology in table 4 refers to the year of 2005, it can be assumed that the required massflow fraction is a bit higher for a 1990s turbine. This mass flow fraction is of course onlyfor the NGVs, but the same calculations can be done for the rotors which will require lesscooling flow since the temperature is lower at the rotor entry.

3.4.5 Mixer

Turbofan engines might employ a mixer, where the hot and cold streams are joined togetherbefore exiting through a common nozzle. Mixing is done for several reasons, for militarypurposes mixing prior to an afterburner will increase the thrust boost significantly. Further-more, the bypass flow can be used to cool the afterburner wall, reducing infrared signaturesof the combined exhaust stream while at the same time provide cooling for the jet-pipeand nozzle [3]. Mixers can have very different designs depending on their main purpose.For example, lobed annular mixers can achieve good mixing with relatively short mixingchambers and low pressure losses. These can increase the thrust significantly; in low bypassturbofans at a flight Mach number of 2 the thrust gain can be up to 4% [3]. Whereas someturbofans simply mix the two streams through perforations in the jet pipe, with the mainpurpose of cooling the afterburner walls.

13


Some of the parameters that might be used to model these mixers are impulse scalingfactors, pressure ratio between the two streams, cold and hot stream Mach numbers andmixing pressure losses. These parameters can be hard to estimate since the they very muchdepend on the mixer design. However, for optimal performance the pressure of the coldstream should only slightly exceed the hot stream pressure [22]. Additionally, the mixingMach number should be between 0.35 and 0.5 to achieve adequate mixing with limitedpressure losses [3].

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4 OFF-DESIGN ADAPTATION

4 Off-Design Adaptation

With an established design point model, the thermodynamic cycle and performance at de-sign conditions is determined. The remaining problem is to find the gas turbine performanceover its complete operating range, at conditions other than the design point. The most com-mon method of steady state off-design modelling utilizes thermodynamic matching betweenconnected components, where each components behaviour at all off-design conditions is rep-resented by a map. However, engine customers typically don’t have access to any of theseengine specific maps. This has led researchers to generate their own component character-istics or opt for already existing component maps, modified to fit their engine data. Thischapter will include an introduction to different off-design methods along with componentmap theory and an implementation of a map modification technique in EVA.

4.1 Introduction

Simple map modifications in the form of constant scaling factors to the map parameters hasbeen applied with relatively good results [23]. However, this technique only proved reliable ifthe scaling factors were close to 1, meaning the original performance maps had to be similarto the scaled maps. Another problem with the this method is that the scaling factors arecalculated in the design point. Points further way from the design operating condition willconsequently produce worse results. Stamatis et al. introduced map modifications based ongas path experimental data [24]. The method adapted map parameters to measured dataand thus reduced the required accuracy of the initial component maps. Another map scal-ing method utilizing experimental data was explored by Kong et al. [25]. They constructedpolynomial scaling factors from performance data obtained at different operating conditionsand saw great improvements in adaptation errors compared to the traditional constant scal-ing approach. Miste et al. proposed map modifications not based on any regression models;scaling functions were instead generated using an optimizer along with constraints in theform of a penalty function [26]. They achieved good results by matching generic componentmaps to several operating points of a small turbojet engine. However, the operating rangeof the adaptation is still bounded by the experimental data and even though the methodremoved complexities such as regression models; complexities were added in the form of anextensive penalty function.

Methods to completely circumvent the need for existing performance maps has also beenexplored. Saravanamuttoo and Maclsaac [27] developed generalized component maps for thepurpose of gas turbine diagnostics back in 1983. Kong et al. used polynomial functions todescribe the relationships between component map parameters [28]. Coefficients to the poly-nomials were then found using experimental data and genetic algorithms. When comparedto traditional scaling methods the generated compressor map showed great improvements interms of agreement with test-data. Tsoutsanis et al. introduced a map generation techniquethat fitted elliptical curves to compressor map data though a multi-objective optimizationalgorithm [29]. This further improved the accuracy and suggested that the most mathemat-ically robust way of describing compressor lines is by elliptical curves. Data-driven methodsin the form of neural networks has also been exploded as a means of retrieving componentcharacteristics from experimental data [30][31]. However, the accuracy of these methods relyheavily on the quality and quantity of the engine data provided by manufacturers or users.More recently, work has been done utilizing support vector machine algorithms for nonlinearregression [32]. The method was compared to other artificial neural network techniques andshowed improvements to the Root Mean Square Error (RMSE) when extrapolating to higherand lower speed regions in the map.

This thesis will utilize EVAs built in mass flow correction factor CAW to adjust alreadyexisting compressor maps through a point by point optimization scheme. The suggestedmethod aims to demonstrate the effectiveness of only adjusting the mass flow parameter incompressor maps while at the same time removing the need for regression models.

15


4.2 Theory

4.2.1 Component Map Basics

Off-design performance calculations are most commonly done utilizing empirical componentmaps. A component map acts as a complete numerical representation of that componentsbehaviour in off-design simulations. There are several methods of obtaining such a map; theycan either be generated through Computational Fluid Dynamics (CFD) if the componentgeometry is known, or by measuring the necessary parameters during rig tests. Anotherway of obtaining a component map is by modifying already existing maps. The use of pre-generated maps, modified to fit engine data, can be a fast and convenient way to obtaincomponent maps for unknown turbomachinery.

The parameters needed to create compressor or turbine maps are listed below.

Pressure ratio

R =Pout

Pin(for compressors) R =

Pin

Pout(for turbines) (12)

Isentropic efficiency

ηis =∆his

∆hreal(for compressors) ηis =

∆hreal

∆his(for turbines) (13)

Corrected mass flow rate

W * = W

√T/Tref

P/Pref(14)

Corrected speed relative to design point

N*rel =

N√T/Tref

Ndes√Tdes/Tref

(15)

A component map contains all four of the parameters listed above for every operatingpoint. However, only two parameters are needed to specify a point in the map. This hasseveral implications:

1. The two parameters has to be independent from each other in order to represent aunique point in the map.

2. The other two parameters has to be determined through some relationship with theformer.

Point number one usually does not hold true for any combination of parameters. Espe-cially in the case of standard compressor maps, where pressure ratio is plotted over correctedmass flow for several lines of constant speed and lines of constant efficiency. It is not possibleto use this kind of map in simulation programs without some modifications. Speed lines can,for example, go horizontal due to stall and produce several values of corrected mass flow fora given speed and pressure ratio. Speed lines are also vertical due to choking in some partsof the map, producing multiple values of pressure ratio for a given set of speed and massflow. An illustration of a compressor map can be seen in figure 6.

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Efficiency

Surge LineSpeed Line

Corrected mass flowPressureratio

1.11

0.50.6

0.70.8

0.9

Figure 6: β-lines in a compressor map.

This non-uniqueness in component maps has led to the introduction of β-lines by Kurzke[33]. They have no specific physical meaning, but act as an auxiliary coordinate that can beused along with relative corrected speed to completely define an operating point. Becauseβ-lines doesn’t hold any physical significance, it is possible to select their shape and spacingin the map arbitrarily, as long as they provide unambiguous intersections with the speedlines.

Surge LineSpeed Line

Corrected mass flow

Pressureratio

1.11

0.50.6

0.70.8

0.9

Beta Line β=1

β=0

Figure 7: β-lines in a compressor map.

W1,1

Wm,1

W1,n

η1,1

ηm,1

η1,n

r1,1

r1,n

rm,1

rm,n

Beta

N

Figure 8: Component map table structure

The addition of β-lines allows pressureratio, efficiency and corrected mass flowto be determined using relative correctedspeed and β.

[R, η,W ∗] = map(N∗rel, β) (16)

Component maps are therefore stored astabulated values for corrected mass flow, ef-ficiency and pressure ratio, where each ta-ble entry corresponds to a specific β andcorrected relative speed. This is the stan-dard format used by gas turbine perfor-mance software Gasturb [34].

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4.2.2 Modifying Component Maps

Scaling to the design pointWith the design point adaptation finished, there should now exists one operating point inevery map where all component parameters are defined. But the component maps doesn’tnecessarily reflect the behaviour of the simulated engine since they are pre-generated andindependent of the design point model. A point in the map with the same values of N*

rel

and β as the design point might have completely different values of pressure ratio, mass flowand efficiency compared to those calculated in the design point simulation.

Component maps are therefore scaled to match to design point simulation using thefollow set of equations.

R =Rdes − 1

Rdes,map − 1(Rmap − 1) + 1 (17)

W * =W *

des

W *des,map

W *map (18)

η =ηdes

ηdes,mapηmap (19)

Where the left hand side is the new parameter value, subscript des denotes the design pointvalue and subscript map denotes values taken from the component map. EVA does all thedesign point component map scaling automatically, but the user still has to provide valuesfor the design point relative corrected speed and beta.

Parameter correctionsAfter each map has been scaled to fit the design point, additional modifications can beapplied. The initial scaling only makes sure that the map values are correct at the designpoint. However, the maps will most likely not be in agreement with the test-data at regionsfurther away from the design operating point. EVA has a pair of built in parameters thatcan be employed to make modifications to the maps at off-design points, these are the FlowCorrection Factor (CAW) and the Isentropic Efficiency Delta (DAE). They allow the massflow and efficiency to be modified, effectively altering values read from the map.

For an off-design point with relative corrected speed N*rel, beta β and flow correction

factor CAW, the corrected mass flow W ∗ will be computed as follows:

W ∗ = CAW ·W ∗map(N∗rel, β) (20)

Similarly, for a point with efficiency delta DAE, the efficiency η will be computed as follows:

η = ηmap(N∗rel, β) +DAE (21)

This suggests that CAW should have a value of 1 at the design point in order for the mapscaling to remain correct. DAE should therefore, for the same reason, have a value of 0 atthe design point.

4.2.3 Component Map Matching

Just like the design point, the off-design model utilizes matching between turbines andcompressors. The design point matching was done by making sure the energy applied toeach shaft is being conserved. Then, by adding additional targets and states to the matchingscheme, more constraints could be applied to the system. Off-design matching is done in asimilar way, but instead of only driving shaft energy residuals to zero, EVA also has to findthe correct operating point on each map. Target residuals and states will automatically beassigned to account for this; however, the user still has to manually set the mixer residualas a target for the matching. Off-design points are then defined by providing at least onemore target/state-pair.

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4.3 Method

This section will cover the off-design method used to adapt component maps to test-data.In order to validate the off-design method itself, all design point parameters are set to theircorrect values. This is only possible because of the engine model used to generate test-data;if a real engine was used, the correct design point parameters wouldn’t be available.

4.3.1 Estimating Design Point Map Location

Because component maps are scaled to match the design point it is important that thelocation of the design point in each map is correct. The first step in the off-design adaptationis therefore to decide where in each map the design point should be located, so that the mapscan be scaled accordingly. This can be adjusted using two quantities, relative correctedspeed and beta. Relative corrected speed decides along which speed line the design pointshould be placed. A relative corrected speed of 1 corresponds to that component mapsdesign point speed. In theory, the design point model could be created using data from anoperating point that does not necessarily correspond to the engine design point. In this casea different relative corrected speed can be selected. However, in this study the design pointrelative corrected speed is set to 1.

The position along a certain speed line can be adjusted using beta values. Design pointbeta values are in this thesis estimated through engineering judgement. It can be assumedthat the design point is located in a high efficiency region of the map that is not too closeto surge. Beta values for each map are therefore selected through inspection.

4.3.2 Selection of Targets and States

First off, a parameter that is part of the test data has to be assigned as a target to define off-design points. This target parameter can for example be low pressure shaft speed or enginethrust, depending on what data is available. With the data-set in this thesis, LPT dischargepressure P5 was selected as a target and fuel flow Wf as state. Even though fuel flow also ispart of the test-data, having it assigned as a state makes for a good measurement of how wellthe adaptation fits the target engine. The reason for not choosing N1 as a target is becauseof the thermodynamic irrelevance of spool speeds. Spool speeds are for example not neededin the calculation of cycle efficiency or thrust. Additionally, spool speeds can be adjustedwith minimal side effects by re-labeling the speed lines in the turbine and compressor maps.A more detailed explanation of why spool speeds only are of secondary importance can befound in Kurzkes paper [11]. Figure 9 and 10 demonstrates how, in this case, the initialadaption errors can be reduced when using P5 as target instead of N1.

Figure 9: Adaptation errors with N1 as target. Every engine parameter has 30 error-bars, each representing an off-design point, starting from the point closest to thedesign point.

19


Figure 10: Adaptation errors with P5 as target. Every engine parameter has 30error-bars, each representing an off-design point, starting from the point closest tothe design point.

Even though N1 has become the largest adaptation error in figure 10, it can be fixedwith minimal side effects by relabeling the compressor speed lines. This will be covered insection 4.3.4 as a final step in the adaptation.

4.3.3 Component Map Corrections

While the correction parameters CAW and DAE can be applied to all components, thisthesis will focus on adjusting the fan and compressor maps exclusively. In the case of aturbofan engine, turbine maps usually have a very narrow operating range. Especially if thedownstream nozzle is choking [11]. This means that one generally should put more focuson getting the fan and compressor maps right. Furthermore, the isentropic efficiency deltaDAE will not be utilized in this thesis. When making corrections to compressor maps onemust also be careful not to break any laws of physics, which can be a difficult task withadaptation methods based solely on mathematics [35]. Map efficiencies are therefore left asthey are in order to reduce the risk of producing component maps that are not in line withthe laws of physics.

That leaves the fan and compressor flow correction factors. These are found using MAT-LABs built in minimization function fminsearch. This function takes in a starting point forthe to-be-adapted parameters and an objective function to minimize. The minimization isdone point by point, meaning fminsearch will adjust CAWfan and CAWhpc at one off-designpoint at the time. The objective function will run EVA and return the Root Mean SquareError (RMSE) of test-data parameters. The RMSE is calculated using only a subset of thetest-data; parameters that aren’t affected by the mass flows are simply excluded from thissubset. Excluded parameters are typical boundary conditions, in this case Pa, Ta, M0 andaltitude. The spool speeds are also not included as these will be adjusted later on.

At this stage it is possible to assign different weights to each parameter in the RMSEcalculation. This enables the minimization function to pay extra attention to the errors ofspecific parameters deemed more important. However, for this study the weights are left atunity.

fminsearch will then iterate on the flow correction parameters using Nelder-Meads methoduntil a minimum of the objective function has been found. There is however no guaranteethat this minimum will be the global minimum, so the starting values for each point is setto the results of the previous point, with the assumption that consecutive points will havesimilar solutions. The points are therefore processed in order, starting from the design point;which will have starting values of 1.

Every off-design point in the test-data is included in this process, but in order to savecomputational time only a few points could be selected. The results of these computationsare presented in table 5.

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Point number 5 10 14 18 22 27 31

Fan flow correction factor 1.02 0.96 0.97 0.96 0.95 0.89 0.81

Fan relative corrected speed 0.92 0.85 0.77 0.70 0.61 0.51 0.41

HPC flow correction factor 0.99 0.98 0.97 0.94 0.92 0.84 0.87

HPC relative corrected speed 0.99 0.97 0.95 0.93 0.91 0.89 0.80

Table 5: Flow correction factors for a few off-design points.

So far the CAW parameter has been used to modify the component mass flows. However,the maps themselves have remained unchanged, only values which has been read from themaps are adjusted by the minimization function. Instead, it would be beneficial to modify theactual component maps using the previously retrieved CAW-values. This way it is possibleto run off-design simulations with adapted maps instead of having to run the minimizationfor every operating point.

In order to apply the CAW-values in table 5 to the component maps, some simplificationshas to be made. First off, it is assumed that the flow correction factors doesn’t depend onbeta values. Whole speed lines in the component maps will therefore be multiplied withtheir corresponding correction factor according to the following equation.

W ∗new(N∗rel, β) = W ∗old(N∗rel, β) · CAW (N∗rel) (22)

Additionally, CAW-values corresponding to the speed lines are linearly interpolated fromthe minimization-data and speed lines that are outside the range of the data are set to theclosest available correction factor. The results of these simplifications are visible in figure 11and 12.

Figure 11: Flow correction factors for the fan map.

The horizontal parts in figure 11 and 12 are the regions where no test-data is available.Instead of trying to extrapolate the correction factors they are left constant.

A comparison of the component maps before and after the flow correction factors hasbeen applied can be seen in figure 13. In some parts of the maps the new speed lines arelooking unrealistic, especially in the compressor map. This is the regions where no CAW-values are available. However, those parts of the maps are currently not in use and willtherefore not cause any problems in the adaptation. But if points with lower speeds are tobe simulated this would have to be addressed.

21


Figure 12: Flow correction factors for the compressor map.

(a) Fan (b) HPC

Figure 13: Speed lines before and after the flow correction factors has been applied.

Looking at Figure 14, most adaptation errors have been reduced with the flow correctedmaps. The largest errors are still the spool speeds N1 and N2. These will be adjusted in thenext section.

Figure 14: Adaptation errors after flow corrections. Every engine parameter has 30error-bars, each representing an off-design point, starting from the point closest tothe design point.

22


4.3.4 Relabeling Speed Lines

After the mass flow correction factor has been applied to each speed line, the remainingproblem is now to correct the spool speeds. Figure 15 reveals how N1 and N2 in the modelstarts to deviate from the test-data at lower speed regions.

Figure 15: N1 and N2 before relabeling of speed lines. Only half of the data pointsare included in the figure.

The speed line relabeling is done by calculating the errors between model predictionsand test-data for each off-design point. However, relative corrected speed isn’t part of thetest-data, so it too has to be calculated. Rewriting equation (15).

N∗rel =

√T des

T

N

Ndes(23)

In this equation, T is the component inlet temperature and subscript des denotes designpoint values. Component inlet temperatures are usually not part of the test-data. Instead,Ta and T21 are utilized as estimations of T for the fan and compressor respectively.

The value of each speed line is then adjusted according to the errors through linearinterpolation. This is done for the fan and compressor without significantly affecting otherparameters. Figure 16 displays the shaft speeds after the speed lines have been relabeled.

Figure 16: N1 and N2 after relabeling of speed lines. Only half of the data points areincluded in the figure.

23


4.4 Results

This section will present the off-design adaptation results. Normally, only parameters thatare part of the test-data can be compared to the model. However, this adaptation was madeusing data from another engine model instead of a real engine. One major advantage of thisis that parameters that are not part of the test-data also can be compared between the twomodels.

Firstly, in figure 17 are the adaptation errors of test-data parameters after the componentmaps have been adapted using the previously explained methods. N1 and N2 errors arereduced to less than 1% with the relabeled speed lines. This improvement was made withoutsignificantly affecting any of the other parameters, making the largest adaptation error lessthan 1.5%. Then, in figure figure 18 are the prediction errors for the thrust, SFC and intakemass flow. Even though the model wasn’t adapted to these parameters, the errors are stillwithin the same range as for the test-data parameters.

Figure 17: Adaptation errors after mass flow and speed corrections. Every engineparameter has 30 error-bars, each representing an off-design point, starting from thepoint closest to the design point.

Figure 18: Prediction errors after mass flow and speed corrections. Every engineparameter has 30 error-bars, each representing an off-design point, starting from thepoint closest to the design point.

Following this, figure 19 to 22 presents pressures and temperatures for different stationsalong the engine. The adapted model match the reference model with less than 1.5% errorin any parameter over the whole range of data points.

24


Figure 19: P21 and T21 model agreement. Only half of the data points are includedin the figure.

Figure 20: P3 and T3 model agreement. Only half of the data points are included inthe figure.

Figure 21: P46 and T46 model agreement. Only half of the data points are includedin the figure.

25


Figure 22: P5 and T5 model agreement. Only half of the data points are included inthe figure.

Finally, figure 23 shows the specific fuel consumption and thrust. These parameters arenot part of the test-data and have not been used in the adaptation, but the adapted modelstill match the reference model with errors similar to the test-data parameters.

Figure 23: Thrust and specific fuel consumption model agreement. Only half of thedata points are included in the figure.

26

5 VALIDATION AND DISCUSSION

5 Validation and Discussion

Ultimately, one of the main reasons to adapt an engine model is to be able to predictperformance at operating conditions other than those used in the adaptation itself. So far,the model has only been validated against the SLS test-data. This chapter will test how wellthe model can predict performance at a few common flight conditions, presented in table 6.

Point Cruise1 Cruise2 Take-off Dash1 Dash2 BVR

Alt 9km 11km 0km 0km 0km 11km

Mach 0.8 0.9 0.3 0.6 0.9 1.3

Power 70% 75% 100% 100% 100% 100%

Table 6: Flight conditions used for model validation. Power refers to a percentage ofthe maximum thrust at that point.

At this time there exists two models, the first one is the reference model which was usedto generate test-data and the second one is the adapted model. For validation, the differentoperating points in table 6 are simulated using both models. This allows for comparisonof all parameters deemed interesting, including performance parameters such as thrust andSFC. Firstly, the result of these simulations in terms of test-data parameters is presented infigure 24.

Figure 24: Model validation, test-data parameters.

Here, N1 was assigned as a target to set the thrust percentage, where 11500 rpm ismaximum thrust. As previously, fuel flow Wf is set as a state to achieve the target N1. It isevident from figure 24 that the magnitude of these errors is about the same as the previouslyexamined test-data points in figure 17. This is to be expected as operating points shouldn’tdeviate very much from the SLS operating line. Cruise1, Cruise2 and BVR have the largesterrors, these are also the points located the farthest away from the model design point.

A maximum parameter error of 1.5% is still significant, but considering that the correctedmass flow and corrected speed are the only parameters that have been adjusted, the resultsare looking promising. Furthermore, this result was achieved using non-adapted genericturbine maps. Some small improvements could still be made by tuning the design pointbeta values since these were estimated based on engineering judgement as a first step in theadaptation process.

27

5 VALIDATION AND DISCUSSION

Perhaps more interesting in terms of performance prediction is the thrust and SFC errors.The simulation results for these parameters are presented in figure 25. Take-off, Dash1 andDash2 are still the points with least errors.

Figure 25: Model validation, performance parameters.

While the prediction errors certainly are larger than in some previously cited studies,this method could still be used to quickly get a decently adapted model from a very limitedamount of test-data. Additionally, since only the rotor speeds and mass flows have beenadjusted, the risk of producing physically inaccurate component maps is reduced. Although,one should still verify the adapted maps using the guidelines outlined in Kurzkes paper [35].

The results are also expected to be dependent on the initial choice of component maps.However, only a few combinations of component maps have been tested due to the limitedamount of maps available.

28

6 CONCLUSIONS

6 Conclusions

This thesis has focused on adapting a parametric turbofan engine model to a limited amountof SLS test-data. The main results of this study can be summarised as follows:

1. Performing a sensitivity analysis in the design point can be of great help when lookingfor dependencies between measurements and component parameters. These can thenbe used in design point solvers like the one in EVA to adapt component parameters.

2. Prediction errors will most likely be unacceptably large when non-adapted componentmaps are utilized in the off-design model. With N1 as a target the prediction errorswere in this case as large as 18% at some operating points.

3. It is possible to greatly reduce prediction errors by only adjusting the mass flow infan and compressor maps. Most commonly, component maps are modified throughsome correction function based on a regression model. The method developed in thisthesis calculates optimal mass flow correction factors for a range of operating pointsusing the Nelder-Mead algorithm. Thus, removing the need for regression models andassumptions on the shape of the correction function.

4. Rotor speeds can be adjusted to match test-data by relabeling speed lines in thefan and compressor maps. Though, the method implemented in this thesis requirescomponent inlet temperatures to be estimated.

Recommendations:

1. The off-design method improved model predictions significantly. However, those re-sults were achieved with an exactly matching design point. In a real scenario thiswould be highly unlikely. It would therefore be a good idea to test the method withslightly misplaced design points.

2. Only relative corrected speed and corrected mass flow is adjusted in the fan andcompressor map. One could look into the possibility of extending the method withefficiency and pressure ratio adjustments.

3. Another possible progression would be to adapt the turbine maps. Unmodified turbinemaps are also expected to contribute to the prediction errors.

29

REFERENCES

References

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[9] Louis A. Urban. Gas path analysis applied to turbine engine condition monitoring.Journal of Aircraft, 10(7):400–406, 1973.

[10] Y. G. Li. Performance-analysis-based gas turbine diagnostics: A review. Proceedings ofThe Institution of Mechanical Engineers Part A-journal of Power and Energy, 216:363–377, 08 2002.

[11] Joachim Kurzke. How to create a performance model of a gas turbine from a limitedamount of information. volume 1, 01 2005.

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[17] Arthur Henry Lefebvre. Fuel effects on gas turbine combustion. Technical report, 1983.

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[19] R. C. Wilcock, J. B. Young, and J. H. Horlock. The Effect of Turbine Blade Coolingon the Cycle Efficiency of Gas Turbine Power Cycles . Journal of Engineering for GasTurbines and Power, 127(1):109–120, 02 2005.

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[21] Konstantinos Kyprianidis. Future Aero Engine Designs: An Evolving Vision, pages3–24. 11 2011.

[22] T. H. Frost. Practical bypass mixing systems for fan jet aero engines. AeronauticalQuarterly, 17(2), 1966.

[23] J. F. Sellers and C. J. Daniele. Dyngen: A program for calculating steady-state andtransient performance of turbojet and turbofan engines. 1975.

[24] A. Stamatis, K. Mathioudakis, and K. D. Papailiou. Adaptive Simulation of Gas Tur-bine Performance. Journal of Engineering for Gas Turbines and Power, 112(2):168–175,04 1990.

[25] C. Kong, J. Ki, and M. Kang. A New Scaling Method for Component Maps of GasTurbine Using System Identification . Journal of Engineering for Gas Turbines andPower, 125(4):979–985, 11 2003.

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[27] H. I. H. Saravanamuttoo and B. D. Macisaac. Thermodynamic models for pipeline gasturbine diagnostics. 1983.

[28] Changduk Kong, Seonghee Kho, and Jayoung Ki. Component Map Generation of aGas Turbine Using Genetic Algorithms. Journal of Engineering for Gas Turbines andPower, 128(1):92–96, 03 2004.

[29] Elias Tsoutsanis, Nader Meskin, Mohieddine Benammar, and K. Khorasani. An efficientcomponent map generation method for prediction of gas turbine performance. volume 6,06 2014.

[30] Neural-network based analysis and prediction of a compressor’s characteristic perfor-mance map. Applied Energy, 84(1):48 – 55, 2007.

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[33] How to Get Component Maps for Aircraft Gas Turbine Performance Calculations, vol-ume Volume 5: Manufacturing Materials and Metallurgy; Ceramics; Structures andDynamics; Controls, Diagnostics and Instrumentation; Education; General of TurboExpo: Power for Land, Sea, and Air, 06 1996. V005T16A001.

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31

https://www.gasturb.de/software/gasturb.html

A TEST DATA

Appendix A Test Data

No. Pa [kPa] P21 [kPa] P3 [kPa] P5 [kPa] Ta [K]

1 1.0133E+02 3.7535E+02 2.3783E+03 3.3817E+02 2.8815E+02

2 1.0133E+02 3.6434E+02 2.2910E+03 3.2679E+02 2.8815E+02

3 1.0133E+02 3.5328E+02 2.2039E+03 3.1537E+02 2.8815E+02

4 1.0133E+02 3.4225E+02 2.1176E+03 3.0404E+02 2.8815E+02

5 1.0133E+02 3.3156E+02 2.0306E+03 2.9288E+02 2.8815E+02

6 1.0133E+02 3.2091E+02 1.9449E+03 2.8186E+02 2.8815E+02

7 1.0133E+02 3.1025E+02 1.8619E+03 2.7106E+02 2.8815E+02

8 1.0133E+02 2.9954E+02 1.7782E+03 2.6017E+02 2.8815E+02

9 1.0133E+02 2.8878E+02 1.6939E+03 2.4917E+02 2.8815E+02

10 1.0133E+02 2.7797E+02 1.6097E+03 2.3819E+02 2.8815E+02

11 1.0133E+02 2.6798E+02 1.5331E+03 2.2812E+02 2.8815E+02

12 1.0133E+02 2.5820E+02 1.4582E+03 2.1825E+02 2.8815E+02

13 1.0133E+02 2.4850E+02 1.3834E+03 2.0851E+02 2.8815E+02

14 1.0133E+02 2.3896E+02 1.3100E+03 1.9929E+02 2.8815E+02

15 1.0133E+02 2.2990E+02 1.2406E+03 1.9087E+02 2.8815E+02

16 1.0133E+02 2.2238E+02 1.1832E+03 1.8416E+02 2.8815E+02

17 1.0133E+02 2.1500E+02 1.1262E+03 1.7775E+02 2.8815E+02

18 1.0133E+02 2.0771E+02 1.0700E+03 1.7166E+02 2.8815E+02

19 1.0133E+02 2.0057E+02 1.0143E+03 1.6591E+02 2.8815E+02

20 1.0133E+02 1.9350E+02 9.5907E+02 1.6044E+02 2.8815E+02

21 1.0133E+02 1.8653E+02 9.0371E+02 1.5524E+02 2.8815E+02

22 1.0133E+02 1.7954E+02 8.4866E+02 1.5028E+02 2.8815E+02

23 1.0133E+02 1.7260E+02 7.9314E+02 1.4556E+02 2.8815E+02

24 1.0133E+02 1.6567E+02 7.3712E+02 1.4106E+02 2.8815E+02

25 1.0133E+02 1.6068E+02 6.9677E+02 1.3797E+02 2.8815E+02

26 1.0133E+02 1.5582E+02 6.5502E+02 1.3500E+02 2.8815E+02

27 1.0133E+02 1.5104E+02 6.1295E+02 1.3216E+02 2.8815E+02

28 1.0133E+02 1.4634E+02 5.7025E+02 1.2945E+02 2.8815E+02

29 1.0133E+02 1.4170E+02 5.2804E+02 1.2690E+02 2.8815E+02

30 1.0133E+02 1.3722E+02 4.8283E+02 1.2443E+02 2.8815E+02

31 1.0133E+02 1.3289E+02 4.3577E+02 1.2209E+02 2.8815E+02

32

A TEST DATA

No. T21 [K] T5 [K] Wf [kg/s] N1 [rpm] N2 [rpm]

1 4.4317E+02 1.0234E+03 1.2246E+00 1.1500E+04 1.6500E+04

2 4.3789E+02 1.0026E+03 1.1551E+00 1.1250E+04 1.6353E+04

3 4.3260E+02 9.8102E+02 1.0871E+00 1.1000E+04 1.6203E+04

4 4.2732E+02 9.5940E+02 1.0210E+00 1.0750E+04 1.6052E+04

5 4.2216E+02 9.4047E+02 9.5970E-01 1.0500E+04 1.5912E+04

6 4.1736E+02 9.2331E+02 9.0238E-01 1.0250E+04 1.5782E+04

7 4.1301E+02 9.0779E+02 8.4879E-01 1.0000E+04 1.5664E+04

8 4.0858E+02 8.9155E+02 7.9538E-01 9.7500E+03 1.5540E+04

9 4.0404E+02 8.7466E+02 7.4249E-01 9.5000E+03 1.5411E+04

10 3.9938E+02 8.5757E+02 6.9083E-01 9.2500E+03 1.5277E+04

11 3.9506E+02 8.4168E+02 6.4478E-01 9.0000E+03 1.5152E+04

12 3.9074E+02 8.2545E+02 6.0036E-01 8.7500E+03 1.5025E+04

13 3.8632E+02 8.0875E+02 5.5683E-01 8.5000E+03 1.4891E+04

14 3.8189E+02 7.9224E+02 5.1520E-01 8.2500E+03 1.4754E+04

15 3.7757E+02 7.7638E+02 4.7673E-01 8.0000E+03 1.4617E+04

16 3.7382E+02 7.6276E+02 4.4535E-01 7.7500E+03 1.4498E+04

17 3.7006E+02 7.4907E+02 4.1486E-01 7.5000E+03 1.4373E+04

18 3.6627E+02 7.3546E+02 3.8559E-01 7.2500E+03 1.4244E+04

19 3.6248E+02 7.2284E+02 3.5795E-01 7.0000E+03 1.4107E+04

20 3.5864E+02 7.1047E+02 3.3140E-01 6.7500E+03 1.3964E+04

21 3.5479E+02 6.9844E+02 3.0584E-01 6.5000E+03 1.3812E+04

22 3.5088E+02 6.8659E+02 2.8129E-01 6.2500E+03 1.3652E+04

23 3.4693E+02 6.7512E+02 2.5765E-01 6.0000E+03 1.3481E+04

24 3.4291E+02 6.6405E+02 2.3490E-01 5.7500E+03 1.3295E+04

25 3.3972E+02 6.5527E+02 2.1873E-01 5.5000E+03 1.3151E+04

26 3.3662E+02 6.4925E+02 2.0394E-01 5.2500E+03 1.2991E+04

27 3.3349E+02 6.4350E+02 1.8964E-01 5.0000E+03 1.2821E+04

28 3.3036E+02 6.3883E+02 1.7607E-01 4.7500E+03 1.2637E+04

29 3.2717E+02 6.3404E+02 1.6305E-01 4.5000E+03 1.2446E+04

30 3.2413E+02 6.3487E+02 1.5165E-01 4.2500E+03 1.2234E+04

31 3.2122E+02 6.4486E+02 1.4262E-01 4.0000E+03 1.2026E+04

33

B DESIGN POINT PARAMETERS

Appendix B Design Point Parameters

Parameter Value Unit

Speed LP Shaft 11500 rpm

Speed HP Shaft 16500 rpm

Power Off-take LP Shaft 0 kW

Power Off-take HP Shaft 0 kW

Mechanical Efficiency LP Shaft 0.999 -

Mechanical Efficiency HP Shaft 0.999 -

Altitude 0 m

Flight Mach Number 0 -

Intake Mass Flow 82 kg/s

Fan Pressure Ratio 3.78 -

Fan Polytropic Efficiency 0.87 -

Bypass Ratio 0.35 -

Compressor Pressure Ratio 6.4 -

Compressor Polytropic Efficiency 0.882 -

Combustion Efficiency 0.9995 -

Combustor Fuel Flow 1.2246 kg/s

HP Turbine Pressure Ratio 2.6 -

HP Turbine Polytropic Efficiency 0.885 -

LP Turbine Pressure Ratio 2.6 -

LP Turbine Polytropic Efficiency 0.877 -

Mixer Inlet Static Pressure Ratio 1.05 -

Mixer Bypass Momentum Scaling Factor 0.5 -

Mixer Core Momentum Scaling Factor 1 -

Mixer Mach Number 0.5 -

Customer Bleed Mass Flow Fraction 0.025 -

HP Turbine Cooling Mass Flow Fraction 0.09 -

LP Turbine Cooling Mass Flow Fraction 0.03 -

Convergent Nozzle Coefficient of Discharge 1 -

Intake Fractional Pressure Loss 0.02 -

Splitter Inlet Duct Fractional Pressure Loss 0.005 -

Splitter Fractional Pressure Loss 0 -

Compressor Inlet Duct Fractional Pressure Loss 0.005 -

Combustor Inlet Duct Fractional Pressure Loss 0.002 -

Combustor Fractional Pressure Loss 0.04 -

Mixer Core Inlet Duct Fractional Pressure Loss 0.007 -

Bypass Duct Fractional Pressure Loss 0.07 -

Afterburner Duct Fractional Pressure Loss 0.002 -

34

Documents

Model Adaptation of a Mixed Flow Turbofan Engineltu.diva-portal.org/smash/get/diva2:1463578/FULLTEXT01.pdf · Model Adaptation of a Mixed Flow Turbofan Engine Oskar Lindkvist Space