Propulsion and Power (eBook)

Joachim Kurzke spent his engineering life dealing with gas turbine performance, first at the Technical University of Munich, Institute for Flight Propulsion. 1976 Kurzke joined the performance department of the company which is now MTU Aero Engines and stayed there for 28 years. He worked on a multitude of different engine projects, designed the engine performance program MOPS for MTU, and applied it to the everyday problems in the performance department.MOPS is due to the modular design very flexible and can be adapted easily to new requirements, however, it’s use requires significant training in gas turbine performance. This observation resulted in the development of the performance program GasTurb™. This software concentrates on the user interface, without neglecting any detail which is required for professional gas turbine performance work. After more than 20 years in the public domain, GasTurb™ is well known and acknowledged all over the world. Kurzke has published numerous papers dealing with gas turbine performance, and was a member of several RTO (former AGARD) working groups, as well as the SAE E33 committee “Thrust in Flight”. He is still a member of the ASME/IGTI Aircraft Engine and Education Committees. Ian Halliwell obtained his B.Sc. in Aeronautical Engineering and M.Sc. in Aerodynamics from Imperial College, London, followed by a Ph.D. in Experimental Gas Dynamics from the University of Southampton.  His professional career began in 1975 at Rolls-Royce, Derby in Turbine Aerodynamics Research.  He then crossed the Atlantic to work for Pratt and Whitney Canada in Mississauga and subsequently GE in Cincinnati, where he moved into the preliminary design of complete engine systems and spent a few years on the High Speed Civil Transport program.  During that period, he also began teaching in GE after-hours education.  While continuing to model complete engine systems, his teaching activities continued after moving to the small business world, as a contractor at the NASA Glenn Research Center and expanded through involvement with AIAA and ASME/IGTI.  He chaired the AIAA Air Breathing Propulsion and Gas Turbine Engine Technical Committees and is still an active member of AIAA. He is also a member of the ASME/IGTI Aircraft Engine and Education Committees.  His connection to students and university faculty was enhanced during the 14 years he organized the AIAA International Engine Design Competition for undergraduate teams.  He met Joachim Kurzke while presenting a tutorial on Preliminary Engine Design at ASME TurboExpo in 2001 and the seeds were sown for this book a few years later.  His current special interests are in new engine architectures, involving vaneless counter-rotation and exoskeletal architectures for compressors and turbines – both axial and radial.  The past 10 years or so, back in aerodynamics, involved the development and application of new design tools. 

Preface 5
Contents 7
About the Authors 18
Acknowledgements 20
Introduction 21
Simulation Tasks 23
1 New Engine Design 24
1.1 Nomenclature 24
1.2 Generation of Shaft Power 26
1.2.1 Ideal Thermodynamic Cycles 26
1.2.1.1 Methods to Increase the Power Output 28
1.2.1.2 Methods for Reducing Fuel Consumption 31
1.2.2 The Efficiency of Shaft Power Generation 32
1.2.2.1 Ideal Cycles 32
1.2.2.2 Real Cycles 36
1.2.2.3 Back to the Definition of Efficiency 38
1.2.3 Combined Cycle 40
1.2.3.1 The Heat Recovery Steam Generator (HRSG) 40
1.2.3.2 Steam Turbine 42
1.2.3.3 Combined Cycle Output 43
1.3 Aircraft Propulsion 43
1.3.1 Turbojet 43
1.3.1.1 Ideal Turbojet Cycle 44
1.3.1.2 A Method to Increase Turbojet Thrust 44
1.3.1.3 Effect of Flight Velocity 46
1.3.2 More Definitions of Efficiency 47
1.3.2.1 Real Turbojet Cycle 49
1.3.2.2 Efficiency of the Turbojet with Reheat (Afterburner) 51
1.3.3 Turbofan 55
1.4 Fundamental Design Decisions 63
1.4.1 Turbofan: Mixed Flow or Separate Flow? 64
1.4.1.1 Separate Flow Turbofan 64
1.4.1.2 Mixed Flow Turbofan 65
1.4.1.3 Comparison at Constant Propulsive Efficiency 65
1.4.2 Dry or Reheated Turbofan? 69
1.4.2.1 Reheated Turbofans for Supersonic Flight 70
1.4.2.2 Dry Turbofans for Supersonic Flight 73
1.4.3 Convergent or Convergent-Divergent Nozzle? 74
1.4.4 Single or Two Stage High Pressure Turbine? 77
1.4.4.1 Simple Cycle Study 79
1.4.4.2 Realistic Optimization 80
1.4.4.3 Engines with Single Stage Turbines 81
1.4.4.4 Design with Prescribed Rotor Blade Metal Temperature 83
1.4.4.5 Engines with Two-Stage Turbines 85
1.4.4.6 Conclusion 86
1.5 Conceptual Turbofan Design 87
1.5.1 Flow Annulus 87
1.5.1.1 Local Mach Numbers 87
1.5.1.2 Hub/Tip Radius Ratio 87
1.5.1.3 Relationships Between Components 88
1.5.1.4 Spool Speed 89
1.5.1.5 Core Size 89
1.5.2 Direct Drive or with a Gearbox? 91
1.5.3 Conventional Turbofans with Bypass Ratios Between 6 and 14 93
1.5.3.1 Fan and Booster 94
1.5.3.2 Bypass 95
1.5.3.3 Low Pressure Turbine 96
1.5.3.4 Effect of Spool Speed 99
1.5.4 Turbofan with Gearbox 102
1.5.5 Comparison 103
1.5.5.1 Mechanics 105
1.5.5.2 Aerodynamics 106
1.5.6 The Fundamental Differences 109
1.6 Mission Analysis 111
1.6.1 General Requirements 111
1.6.2 Single Point Design 112
1.6.3 Multi-point Design 114
1.6.3.1 Commercial Aircraft 115
1.6.3.2 Fighter Aircraft 116
1.6.4 High Speed Propulsion 121
1.6.4.1 Point Performance 121
1.6.4.2 Turbojet 121
1.6.4.3 Turbojet with Reheat (Afterburner) 125
1.6.4.4 Turbofan with Reheat 127
1.6.4.5 Ramjet 128
1.6.4.6 Acceleration to High Mach Numbers 128
1.7 References 135
2 Engine Families 136
2.1 Baseline Engine 137
2.2 Derivative Engine 139
2.2.1 Fan and Booster 139
2.2.2 Core Compressor 141
2.2.3 Combustor 142
2.2.4 High Pressure Turbine 142
2.2.5 Low Pressure Turbine 143
2.3 Optimizing the Growth Engine 143
2.3.1 Design Variables 143
2.3.2 Design Constraints 144
2.3.3 Figure of Merit 145
2.3.4 Ranges for the Design Variables 145
2.3.5 Starting Point 146
2.3.6 Graphical User Interface 146
2.4 Exploring the Design Space 149
2.5ƒReferences 151
3 Modeling an Engine 152
3.1 Sources of Data 153
3.1.1 Magazines and Marketing Brochures 153
3.1.2 Official Engine Data 153
3.1.3 Calculated Engine Cycle Data 154
3.1.4 Measurements Made by the Gas Turbine User 154
3.1.5 Measurements in an Engine Maintenance Shop 155
3.1.5.1 Contractual Performance 155
3.1.5.2 Thermodynamic Performance 155
3.2 Data Correction 156
3.2.1 Correction to Standard Day Atmosphere 157
3.2.1.1 Humidity Corrections 157
3.2.1.2 Condensation Corrections 158
3.2.2 Data Enrichment 160
3.2.2.1 Indirect Test Data 161
3.2.2.2 Hybrid Test Data 162
3.3 Cycle Reference Point 162
3.3.1 Trial and Error Method 162
3.3.2 Multi Point Analysis 163
3.3.3 Optimized Data Match 163
3.3.4 Unable to Create a Reasonable Model? 164
3.4 Off-Design 164
3.4.1 Compressor Maps 165
3.4.1.1 Corrected Flow—Efficiency Correlation 165
3.4.1.2 Corrected Flow—Corrected Speed Correlation 166
3.4.2 Turbine Maps 166
3.4.3 More Simulation Details 167
3.5ƒReferences 167
4 Engine Model Examples 168
4.1 J57-19W 168
4.1.1 Cycle Reference Point 169
4.1.2 Off-Design Simulation 174
4.1.2.1 Simulation with GasTurb Standard Maps 174
4.1.2.2 Replacing the LPC Map 178
4.1.2.3 The High Altitude Case 179
4.1.2.4 Spool Speed Model 180
4.1.2.5 Final Remarks 187
4.2 CFM56-3 187
4.2.1 Check of the Data 188
4.2.1.1 Humidity 191
4.2.1.2 Mass Flow 191
4.2.1.3 Thrust 191
4.2.1.4 Fuel Flow 193
4.2.1.5 Temperatures 195
4.2.1.6 Pressures 196
4.2.1.7 Spool Speeds 197
4.2.2 Cycle Reference Point 200
4.2.2.1 Some Remarks 204
4.2.3 Off-Design 204
4.2.3.1 Test Data Enrichment 205
4.2.3.2 Fan Map 205
4.2.3.3 Booster Map 206
4.2.3.4 HPC Map 209
4.2.3.5 HPT Map 209
4.2.3.6 LPT Map 209
4.2.4 Preliminary Model Calibration 210
4.2.4.1 Booster Map 211
4.2.4.2 Second Thoughts About the Booster Map 211
4.2.4.3 HPC Map 213
4.2.4.4 Bypass Ratio 213
4.2.4.5 Fan and LPT Map 214
4.2.4.6 Spool Speeds 215
4.2.4.7 Model Check 216
4.2.5 Refined Model 218
4.2.6 Some Final Remarks 220
4.3 F107-WR-400 222
4.3.1 Cycle Reference Point Max Continuous ISA SLS 222
4.3.2 Off-Design Model 228
4.4 References 233
5 Model-Based Performance Analysis 234
5.1 The Analysis by Synthesis Methodology 235
5.1.1 The Model 236
5.1.1.1 Degree of Detail 236
5.1.1.2 Calibration 236
5.1.2 Data Preprocessing 236
5.1.2.1 Data Correction to Standard Day Conditions 237
5.1.2.2 Comparing the Measurements with the Model 237
5.1.3 Definition of the AnSyn Factors 237
5.1.3.1 Compressor 237
5.1.3.2 Turbine 240
5.1.4 A Simple Analysis Example 241
5.1.5 How to Deal with Missing and Additional Measurements 243
5.1.6 AnSyn and Optimization 244
5.1.7 Application of the AnSyn Factors 245
5.1.7.1 ISA Correction 245
5.1.7.2 EGT Margin 246
5.1.7.3 Rated Power 246
5.2 AnSyn During Engine Development 247
5.2.1 Sensor Checking 248
5.2.2 Test Analysis 250
5.2.3 Model Improvement Potential 250
5.3 AnSyn in Engine Maintenance Shops 253
5.3.1 Baseline Model for Diagnostics 254
5.3.2 Engine Diagnostics 254
5.4 AnSyn for Engine Performance Monitoring 256
5.4.1 Baseline Model for Monitoring 256
5.4.2 Trend Monitoring 257
5.5 Interpretation of the AnSyn Factors 258
5.5.1 Component Degradation 259
5.5.2 Model Faults 261
5.5.3 Sensor Faults 261
5.5.4 Measurement Errors 264
5.6 Concluding Remarks 267
5.7 References 268
6 Inlet Flow Distortion 269
6.1 Types of Inlet Flow Distortion 270
6.1.1 Pressure Distortion 270
6.1.2 Temperature Distortion 273
6.2 Parallel Compressor Theory 273
6.2.1 Theory and Experiment 277
6.2.2 Compressor Coupling 281
6.3 Impact of Distortion on Thermodynamics 283
6.4 Changes Due to Control System Actions 285
6.4.1 Unintended Reactions 286
6.4.2 Intended Actions 286
6.5 Reprise 286
6.6 References 287
7 Transient Performance Simulation 288
7.1 Transient Basics 289
7.1.1 Overcoming Rotor Inertia 289
7.1.2 Transient Control Strategies 290
7.2 Engine Geometry 292
7.2.1 Steady State Geometry 292
7.2.2 A Turbofan Example 293
7.3 An Enhanced Approach 296
7.3.1 Tip Clearance 296
7.3.2 Heat Transfer 297
7.3.3 Burner 300
7.3.4 Other Transient Phenomena 300
7.4 Transient Behavior of a Turbofan 301
7.4.1 Accelerating the Cold Engine 301
7.4.2 Decelerating the Hot Engine and Re-slam 305
7.5 Concluding Remarks 309
7.6 References 310
Preliminary Design 311
1 Engines 312
1.1 The Role of Preliminary Design in Systems Studies 312
1.2 Approach and Implementation 316
1.2.1 Building an Engine Model 316
1.2.2 Component Models 316
1.2.3 Design Constraints 318
1.2.4 Trade Studies Revisited 319
1.2.5 Component Hierarchy 321
1.2.6 The Cycle Design Point 322
1.2.7 Individual Component Design Points 323
1.2.7.1 LP Compressor 324
1.2.7.2 HP Compressor 324
1.2.7.3 Combustion Chamber 325
1.2.7.4 HP Turbine 325
1.2.7.5 LP Turbine 325
1.3 Engine Development—The Role of Performance 326
2 Compressors 330
2.1 Function, Environment, and Basic Efficiency 330
2.1.1 Introduction 330
2.1.2 Limitations of Isentropic Efficiency 334
2.1.3 Polytropic Efficiency 335
2.1.4 Additional Operational Functions 337
2.2 Velocity Diagrams 338
2.2.1 Introduction 338
2.2.2 Sign Convention for Angles and Circumferential Velocities 341
2.2.3 Construction 343
2.2.4 Use of Velocity Diagrams 343
2.2.5 Stage Characteristics 346
2.3 Preliminary Compressor Design 350
2.3.1 Flow in a Blade Passage 351
2.3.2 Mean Line Analysis 352
2.3.3 Three-Dimensional Flow and Radial Equilibrium 352
2.3.4 Diffusion, Turning and Blockage 353
2.3.5 Mean Line Loss Models 359
2.3.6 The Structure of the Mean Line Design Code CSPAN 361
2.3.7 Structure of the Mean Line Analysis in GasTurb 363
2.4 Compressor Design Envelopes 364
2.4.1 Introduction 364
2.4.2 Specification of Design Space 365
2.4.3 Primary Design Variables 367
2.4.4 A Core Compressor with 11 Stages—An Example 368
2.4.5 A Core Driven Fan—A More Complex Example 372
2.5 References 373
3 Turbines 374
3.1 Function, Environment and Basic Efficiency 374
3.1.1 Limitations of Isentropic Efficiency 379
3.1.2 Polytropic Efficiency 381
3.2 Velocity Diagrams 384
3.2.1 Introduction 384
3.2.2 Sign Convention for Angles and Circumferential Velocities 387
3.2.3 Construction 388
3.2.4 Use of Velocity Diagrams 388
3.2.5 Stage Characteristics 391
3.3 Preliminary Turbine Design 397
3.3.1 HP Turbine 397
3.3.2 LP Turbine 398
3.3.3 Mean Line Analysis 399
3.3.4 Development of a Mean Line Code 402
3.3.5 Structure of a Mean Line Code 403
3.3.6 Mean Line Loss Models 405
3.3.7 Loss Components 408
3.3.7.1 Profile Loss 408
3.3.7.2 Secondary Loss 409
3.3.7.3 Trailing Edge Loss 409
3.3.7.4 Over-Tip Leakage Loss 409
3.3.8 Effects of Cooling Air 410
3.4 Turbine Design Envelopes 411
3.4.1 Introduction 411
3.4.2 Specification 412
3.4.3 Primary Design Variables 413
3.4.4 Solution and Interpretation 414
3.5 Vaneless Counter-Rotation 418
3.6ƒReferences 425
4 Mechanical Design 427
4.1 Introduction 427
4.2 Flow Path 429
4.2.1 Compressors 429
4.2.2 Low-Bypass-Ratio Fan or LP Compressor 429
4.2.3 High-Bypass-Ratio Fan 430
4.2.4 Splitters 432
4.2.5 Booster 432
4.2.6 HP Compressor 432
4.2.7 Combustor 434
4.2.8 HP Turbine 434
4.2.9 LP Turbine 436
4.2.10 Afterburners 437
4.2.11 Nozzles 437
4.2.11.1 Subsonic Nozzle 438
4.2.11.2 Supersonic Nozzle 438
4.3 Frames and Ducts 438
4.3.1 Front Frame 438
4.3.2 Main Frame 439
4.3.3 Turbine Center Frame 439
4.3.4 Rear Frame 440
4.4 Shafts 440
4.5 Disks 441
4.5.1 Disk Design Methodology 442
4.5.2 Rim Load 443
4.5.3 Disk Temperature 445
4.5.4 Disk Stress 446
4.5.5 Material Properties 447
4.5.6 Design Margins 448
4.5.7 Stress Distribution 449
4.6 Engine Weight 449
4.7 References 452
Off-Design 453
1 Component Performance 454
1.1 Inlet 454
1.1.1 Aircraft Engines 454
1.1.1.1 Loss Description 455
1.1.1.2 Subsonic Aircraft 456
1.1.1.3 Supersonic Aircraft 459
1.1.1.4 Spillage Drag 462
1.1.2 Power Generation 463
1.2 Off-Design Behavior of Compressors 466
1.2.1 About Compressor Maps 467
1.2.1.1 The Shape of Speed Lines 469
1.2.1.2 The Zero-Speed Line 470
1.2.1.3 “Supersonic” Speed Lines 471
1.2.1.4 Specific Work 473
1.2.1.5 Torque 475
1.2.2 Compressor Map Coordinates 476
1.2.2.1 Efficiency Correlations 479
1.2.2.2 Work and Flow Correlations with Spool Speed 482
1.2.3 Compressors with Variable Guide Vanes 483
1.2.4 Fan Maps 488
1.2.4.1 The Flow Field in a Single Stage Fan 490
1.2.4.1.1 Core Section 490
1.2.4.1.2 Bypass Section 493
1.2.4.2 Extended Fan Map 494
1.2.4.2.1 The Algorithm 494
1.2.4.2.2 How to Get Such a Map 494
1.2.5 Secondary Effects 498
1.2.5.1 Bleed Air from an Intermediate Stage 498
1.2.5.2 Tip Clearance 498
1.2.5.3 Blade Untwist 501
1.2.6 Scaling Compressor Maps 502
1.2.6.1 How to Find the Mach Number Scale in a Fan Map 506
1.2.6.2 Mach Number Scale for Other Compressor Maps 510
1.2.7 The Map Preparation Program Smooth C 511
1.2.8 A Simple Map Scaling Procedure 513
1.2.9 Advanced Map Scaling 514
1.2.10 Map Scaling During Off-Design 516
1.3 Turbine Performance 518
1.3.1 Operational Behavior 518
1.3.1.1 Corrected Mass Flow 518
1.3.1.2 Specific Work 522
1.3.1.3 Efficiency 524
1.3.1.4 Exit Angle 527
1.3.2 The Map Preparation Program Smooth T 527
1.3.3 Turbine Map Format 530
1.3.3.1 Beta Lines 531
1.3.3.2 Turbine Map Scaling 532
1.3.4 Tip Clearance 533
1.3.5 Variable Geometry Turbines 535
1.3.6 Vaneless Counter-Rotating Turbines 535
1.4 Combustor 537
1.4.1 Efficiency 537
1.4.2 Pressure Loss 540
1.4.3 Temperature Distribution at the Combustor Exit 543
1.5 Mixer 544
1.5.1 How Mixing Increases Thrust 545
1.5.2 Mixer Geometry 546
1.5.3 Fully Mixed Thrust 547
1.5.4 Unmixed Thrust 548
1.5.5 Mixer Efficiency and Mixer Velocity Coefficient 548
1.5.6 Thrust Gain Potential 549
1.5.7 Practical Mixers 551
1.5.8 Mixer Design Example 552
1.5.9 Mixer Off-Design 554
1.6 Afterburner 556
1.6.1 The Need for a Precise Afterburner Simulation 557
1.6.2 Geometry and Nomenclature 558
1.6.3 Afterburner Operation 559
1.6.3.1 Pressure Losses 560
1.6.3.2 Flow Field 560
1.6.4 Reheat Efficiency 561
1.6.4.1 Definition 561
1.6.4.2 Methods of Determining Efficiency 563
1.6.4.3 Efficiency at Part Load 564
1.6.5 EJ 200 Example 565
1.6.5.1 Modeling Results from the Altitude Test Facility 566
1.6.5.2 Conclusions 573
1.7 Nozzles 573
1.7.1 Convergent Nozzles 574
1.7.1.1 Discharge Coefficient 574
1.7.1.2 Thrust Coefficient 575
1.7.2 Convergent-Divergent Nozzles 578
1.7.2.1 Theory 578
1.7.2.2 Reality 579
1.7.2.3 Implementation 587
1.8 References 588
2 Understanding Off-Design Behavior 591
2.1 Turbojet 592
2.1.1 Off-Design Behavior of the Components 592
2.1.1.1 Compressor 592
2.1.1.2 Burner 594
2.1.1.3 Turbine 595
2.1.1.4 Exhaust Nozzle 596
2.1.2 Component Synergy 596
2.1.2.1 Flow Conservation Between Compressor and Turbine 598
2.1.2.2 Flow Conservation Between Turbine and Nozzle 599
2.1.2.3 Flow Conservation Between Compressor Exit and Turbine Inlet 600
2.1.2.4 Energy Balance Between Compressor and Turbine 601
2.1.2.5 Compressor and Turbine Operating Lines 602
2.1.2.6 How Accurate Are These Equations? 602
2.1.2.7 Variable Compressor Geometry 604
2.1.3 Booster Operating Line 607
2.2 Turbofan 608
2.2.1 Fan Operating Line 609
2.2.2 Turbofan Booster Operating Line 611
2.2.2.1 Shape of the Operating Line 612
2.2.2.2 Map Selection and Scaling 614
2.2.2.3 Variable Geometry 615
2.2.2.4 Wrap up 619
2.2.3 Low Pressure Turbine 619
2.3 Multi-spool Turboshaft 621
2.4 Single Spool Turboshaft 622
2.5 References 624
Basics 625
1 Gas Properties and Standard Atmosphere 626
1.1 The Half Ideal Gas 626
1.1.1 Enthalpy 627
1.1.2 Entropy Function 628
1.2 Numerical Values 628
1.2.1 Specific Heat, Enthalpy and Entropy Function 628
1.2.2 Temperature Rise Due to Combustion 629
1.2.3 Fuel 629
1.3 Standard Atmosphere 630
1.4 Reference 631
2 Spreadsheet Calculations 632
2.1 Frequently Needed Equations 632
2.1.1 Some Simple Correlations 632
2.1.2 Compressor 633
2.1.3 Turbine 635
2.1.4 Isentropic and Polytropic Efficiency 638
2.1.5 Combustor 638
2.1.6 Nozzle 639
2.2 Cycle Calculation for a Turbojet Engine 640
2.2.1 Requirement 640
2.2.2 Solution 640
2.2.3 Summary 650
3 Non-dimensional Performance 651
3.1 Non-dimensional Compressor Performance 651
3.2 Non-dimensional Engine Performance 656
3.2.1 Practical Correction to Standard Day Conditions 659
3.2.2 How to Determine the Exponents 660
3.2.3 Aircraft Engines with Afterburners 661
3.2.4 Gas Turbines with Heat Exchanger 662
3.3 References 662
4 Reynolds Number Corrections 663
4.1 Reynolds Number Index 663
4.2 Turbomachinery Loss Correlations with Reynolds Number 665
4.2.1 Compressor 666
4.2.2 Turbine 667
4.2.3 Some Additional Remarks 668
4.3 Applying the Pipe Flow Analogy in Performance Programs 669
4.4 Variations of the Pipe Flow Analogy 670
4.5 Mass Flow Correction 671
4.6 References 672
5 Efficiency of a Cooled Turbine 673
5.1 Single Stage Turbine 675
5.1.1 Simulation Principle 675
5.1.2 About NGV Cooling Air 678
5.1.3 Exchange Rates 679
5.2 Two-Stage Turbine 680
5.3 Equivalent Single Stage Turbine 682
5.3.1 The Virtual RIT Method 685
5.3.2 The Virtual T4 Method 686
5.3.3 Sensitivity Analysis 686
5.3.4 Application 689
5.4 Thermodynamic Efficiency 689
5.4.1 Comparison of Turbine Efficiency Estimates 691
5.4.2 Efficiency Definition Affects the Results of Cycle Studies 693
5.5 Efficiency Losses Due to Cooling 694
5.5.1 Some Numbers 694
5.5.2 A Real-World Example 695
5.6 References 697
6 Secondary Air System 698
6.1 SAS in the Performance Model 698
6.2 SAS Calculation 701
6.2.1 Interstage Bleed 702
6.3 Turbine Cooling Air 703
6.3.1 Multi-stage Turbines 705
6.4 References 707
7 Mathematics 708
7.1 The Off-Design Simulation Task 708
7.2 Basic Algorithms 711
7.2.1 Newton 711
7.2.2 Regula Falsi 712
7.2.3 Newton-Raphson 713
7.3 Application to Performance Calculations 715
7.4 More About the Analytical Technique 715
7.4.1 Hierarchy of Iterations 717
7.4.2 Steady State Performance 718
7.4.3 Limiters 719
7.4.4 Dynamic Engine Simulation 720
7.5 About Convergence Problems 722
7.5.1 A Solution Exists, but the Program Does not Find It 722
7.5.1.1 Poor Variable Estimates 722
7.5.1.2 Reasonable Variable Estimates 722
7.5.1.3 Problem Formulation 723
7.5.1.4 Problematic Component Maps 724
7.5.2 No Solution Exists 724
7.5.2.1 Operating Point Is Outside One or More Component Maps 724
7.5.2.2 Cycle Is not Feasible 724
7.5.2.3 Garbage in—Garbage Out 724
7.6 References 725
8 Optimization 726
8.1 Parametric Studies 726
8.2 Numerical Optimization 727
8.2.1 A Gradient Strategy 728
8.2.2 Adaptive Random Search Strategy 729
8.2.3 Constraints 731
8.2.4 Application 731
8.3 References 733
9 Monte Carlo Simulations 734
9.1 Statistical Background 734
9.1.1 Normal Distribution and Standard Deviation 734
9.1.2 Probability Distribution and Confidence Level 737
9.2 Measurement Uncertainty 738
9.2.1 Systematic Errors 738
9.2.2 A Conventional Test Analysis Procedure 738
9.2.3 Core Flow Analysis 739
9.3 Engine Design Uncertainty 741
9.4 Engine Manufacturing Tolerance 746
9.4.1 Random Variations 747
9.4.2 Correlations 747
9.4.3 Control System Tolerance 747
9.4.4 A Turboshaft Example 748
9.5 References 749
Appendix 750
Nomenclature* 750
Index 756