Fundamentals of Jet Propulsion With Applications

viii

Contents 2.2.6 Shaft 2.2.7 Combustor 2.2.8 Afterburner 2.2.9 Primary Nozzle 2.2.10 Fan Nozzle 2.2.11 Bypass Duct 2.2.12 Bypass Mixer 2.2.13 Exhaust for a Power-Generation Gas Turbine 2.3 Cycle Analysis 2.3.1 Ramjet 2.3.2 Turbojet 2.3.3 Turbofan 2.3.4 Turboprop 2.3.5 Power-Generation Gas Turbine 2.4 Summary

3 Non-ideal Cycle Analysis 3.1 Introduction 3.1.1 Variable Specific Heats 3.2 Component Losses 3.2.1 Diffuser 3.2.2 Compressor 3.2.3 Fan 3.2.4 Turbine 3.2.5 Propeller 3.2.6 Shaft 3.2.7 Combustor 3.2.8 Afterburner 3.2.9 Primary Nozzle 3.2.10 Fan Nozzle 3.2.11 Bypass Duct 3.2.12 Bypass Mixer 3.2.13 Power Turbine Exhaust 3.2.14 Summary of Nonideal Effects and Simple Parameter Models in Components 3.3 Cycle Analysis 3.3.1 General Approach 3.3.2 Examples 3.4 Use of Cycle Analysis in Preliminary Design 3.5 Summary

59 59 61 63 65 66 67 68 70 71 78 91 113 119 124 134 134 134 135 135 137 141 141 143 144 145 146 147 150 151 152 153 154 155 155 156 182 182

Part II Component Analysis 4 Diffusers 4.1 Introduction 4.2 Subsonic 4.2.1 External Flow Patterns 4.2.2 Limits on Pressure Rise 4.2.3 Fanno Line Flow

209 209 210 210 211 214

Index

adiabatic processes, 46–50, 592–593 adiabatic flame temperatures, 61, 146, 449, 453, 454, 455, 483 enthalpy and, 454 fuel ratios and, 450, 455 stoichiometric condition and, 450, 453, 455 See also specific components aeroderivative, 31 afterburners, 19, 90, 146, 166, 167–168, 440, 445, 462–463 chemistry and, 450 efficiency and, 146–147 enthalpy and, 451–452 fuel ratios and, 63 ideal, 61 ideal characteristics of, 46–47 ignition of, 449 isobaric processes in, 61 materials in, 447 objective of, 19, 445 pressure ratios and, 147 ramjets and, 446 screech, 447 stoichiometric condition and, 450 thermodynamics of, 451 thrust-TSFC and, 168, 445 turbofans and, 25, 98–102, 106–108, 112, 113 turbojets and, 19, 84 airfoils, 51, 331–332, 378. See also propellers; compressors; turbines altitude, 83, 258–259, 264 area changes friction and, 608, 610 heat addition and, 610 See also specific components atmosphere, standard, 527 blades, 141 angle of, 56, 58 attachment methods, 279 camber, 281 cascade, 277 chord, 281 cooling of, 142, 428 data for, 332 disk and, 277, 279 materials in, 277–279 resonance and, 281 rotor, 51, 277, 407

631

stator, 51, 277, 407 See also compressors; impellers; mixers; specific components boundary layer compressors and, 137 diffusers and, 135–136, 209 See also separation Brayton cycle, 10 intercooling and, 14 regeneration and, 13 Buckingham Pi theorem, 261, 461, 620 burners, 88, 118, 145, 440 adiabatic flame temperature and, 61, 146 CFD method for, 440 chemistry and, 450 component matching and, 483 counter-flow, 374 design criteria for, 440–441 dimensional analysis and, 461 drag and, 456, 459 efficiency and, 145–146, 461, 462, 483 enthalpy and, 451–452 exit pressure from, 157 exit temperatures from, 75–76 Fanno line flow and, 456, 457, 461 flame speed, 448 flame holders. See flame holders flame stability, 447 friction and, 456, 458 fuel mass-flow rate, 60, 146 fuel ratios and, 61, 447, 462, 483 fuel types, 463 gas velocities and, 448 generalized flow method, 461 geometries of, 441 heat addition and, 458 heating value method, 60, 146 ideal, 59 ideal characteristics of, 46–47 ignition, 449 imperfect combustion and, 145 materials for, 445 modeling of, 488 percent air, 450 performance maps for, 461 pressure losses and, 145, 146, 456, 483 pressure ratios, 146, 157, 445, 461, 462 primary, 59, 61–62, 145 Rayleigh line flow, 456, 461

632 burners (cont.) starting and, 448 temperature ratios and, 60 types of, 441 See also adiabatic flame temperature; afterburners; flameholders bypass air, 20–21 bypass duct, 151, 471 frictional losses, 151–152, 471 ideal, 66 ideal characteristics of, 46–47 pressure ratio and, 152 split-ratio, 151 bypass mixer, 67, 152, 471 adiabatic flows and, 68, 153 bypass ratio and, 68 enthalpy and, 68, 153 frictional losses and, 153 ideal characteristics of, 46–47 mixing losses and, 153 pressure ratio, 153 stagnation pressure and, 67 bypass ratio, 22, 26, 53–55, 56, 68 CFD. See computational fluid dynamic methods closed-form equations, 46, 487 combustors. See burners component matching, 134, 481 burners and, 483 closure equations for, 484, 487 compressors and, 482 dynamic-transient responses, 499 modeling and, 491 power generation gas turbines and, 486 solution procedures for, 492 turbines and, 481, 483 component modeling, 487 burners and, 488 component matching and, 491 compressors and, 488 diffusers and, 489 nozzles and, 489 shafts and, 489 turbines and, 489 components cycle analysis and, 47, 155 ideal characteristics of, 47 non-ideal effects in, 154 simple parameter models, 154 See also specific components compressible flow, 591. See also specific components compressors, 53–55, 137 boundary layer and, 137 component matching and, 482 diffusers and, 135–136, 137, 209 efficiency in, 138, 139 efficiency, polytropic, 139 enthalpy and, 138

Index flow direction and, 137 friction and, 138 ideal, 51 ideal characteristics of, 46–47 modeling of, 488 off-design performance, 137–138 peak design condition, 140 pressure ratios and, 12, 51–52, 55, 138–141 specific heats and, 138 small stage efficiency, 139 compressors, axial flow, 276 auxilliary functions of, 276–277 basic premise of, 276 “best efficiency point”, 302–303 blade attachment, 279 blade deflection, 347, 351 blade drag and, 335, 343–344 blade heights and, 283, 285, 296 blade incidence, 51, 307, 342 blade lengths and, 336, 339 blade lift and, 335, 349 blade loading, 288, 341 blade pitch, 281 blade tip, 277 blade tip-hub ratios, 280, 289, 316 blade tip, pressure losses and, 279–280 blades, forces on, 340 boundary layer and, 281 camber-chord ratio, 281 cascades, 288, 346 centrifugal compressors and, 276 CFD methods, 320, 321, 332, 350 constant reaction, 318 deflection coefficient, 334 deviation, 285, 352 differential analysis and, 316 diffusers and, 281–283 dimensional analysis and, 299 efficiency and, 281, 283, 287, 299, 301–303, 307, 332, 342–345, 346, 347, 349 efficiency, “best efficiency point”, 302–303 empirical approaches and, 346, 351 exit guide vanes, 280 experimental data on, 301 fan blades and, 277–279 flow coefficient and, 334, 342, 343–345, 347 free vortex, 317, 323 geometry of, 277 history of, 276 high pressure, 280 incompressible flow, 288 inlet guide vanes, 280 low pressure, 280 Mach numbers and, 293 mapping conventions, 302 mass flow, corrected, 299–302 materials in, 277–279 maximum pressure coefficient, 306

Index mean line control volume approach, 286 multiple spool engines and, 280–281, 312 operating line, 302 percent reaction, 287, 289, 294, 342, 343 performance, by stages, 331 performance maps, 299 pressure changes in, 283 pressure coefficient, 305 pressure ratios, 276, 277, 281, 287, 289, 299, 300–301, 302, 307 pressure ratio limits, 306 pressure ratios, by stages, 299 radial equilibrium and, 316 rotational speed and, 290, 299–301, 302 rotor blades, 277 separation and, 281 single stage energy analysis, 286 single stage performance predictions, 354 solidity parameter, 281, 346–347 speed, corrected, 299–302 stator vanes, 277, 307, 312 streamline analysis method, 320 surge, 283, 299, 301–303, 306 testing, 301 thermodynamic efficiency and, 276 trends, 300 TSFC and, 276 turbines and, 406–407, 409 variable stators, 307 velocity polygons and, 283, 287, 289, 332 compressors, centrifugal, 374 blade geometries, 377 blade number, 386 blade stresses, 392 cross over, 377 diffusers, 375–378, 394, 396 dimensional analysis and, 390 durability of, 374 efficiency, 374, 383, 385, 391, 392–393, 395 geometry of, 374 impellers, 378–380, 391, 392, 396–397 incompressible flow, 381 inducers and, 375 Mach numbers and, 396–397 mapping conventions for, 390 percent reaction and, 381, 396 performance maps, 390 pre-swirl, 375, 379, 383, 385 pressure ratios and, 381, 384, 386, 390–391, 395 resonance and, 395 rotors and, 380 scroll and, 376–377 separation and, 395 single stage energy analysis, 380 slip factor, 382, 385, 386, 392 surge and, 390, 391 swept blades, 377, 392

633 swirl free, 378 vaned diffusers, 394 velocity polygons and, 378, 386 computational fluid dynamics (CFD) method, 209, 227, 231–232, 276, 320, 321, 332, 350, 440 cycle analysis, 10 example problems of, 156 general approach of, 155 ideal, 46 non-ideal, 134 de Laval, Carl, 5 diffusers, 48, 135, 209, 380, 394 additive drag and, 227, 229, 232 adiabatic processes and, 209 area changes and, 215, 216 area ratios and, 210–211, 212–213, 214, 225, 226, 227–228 boundary layers and, 135–136, 209, 210, 214, 230 CFD method and, 216, 227, 231–232 compressors and, 135–136, 137, 209, 281–283 conical ramps and, 220 conical shocks and, 220 critical condition, 227 enthalpy and, 135, 209, 211 external flow patterns, 210 Fanno line flow and, 214, 215, 216, 236 free stream air, 209 friction and, 215 ideal, 48 ideal characteristics of, 46–47 incompressible flow, 213 internal area of, 225 internal diffuser recovery factor, 217 isentropic processes and, 209, 211, 213 lambda shocks, 226 Mach numbers and, 210, 212, 213, 216–217, 218–220, 221–223, 224 mass flow ratio, 227 modeling of, 489 non-ideal, 135 normal shocks and, 216, 217 objectives of, 209 oblique-conical shocks and, 220 oblique-planar shocks and, 218, 227 oblique shocks and, 225, 226, 232 off-design operation of, 227 one-dimensional flow method, 215 operation modes of, 226 performance map of, 235 pressure coefficient, 211, 213 pressure losses and, 209, 210, 214, 215, 217 pressure ratios and, 213, 214, 217, 235 pressure recovery factor, 135–137, 209, 210, 236 pressure rise limits, 211 separation and, 210, 211–212, 213 sonic condition, 232 spikes, 220

634 diffusers (cont.) starting and, 232 subcritical condition, 227 subsonic conditions, 210 supercritical condition, 227 supersonic conditions, 216 vaned, 394 vaneless, 375–376 viscous effects and, 211, 214 See also inlet disks, 277, 411. See also impellers drum, 51, 277, 407 efficiency polytropic, 139–143 See also specific components engines aircraft-frame matching, 508 comparisons of types, 32 complexity of, 46 ideal models of, 69 types of, 16 enthalpy, 560 of formation, 451 Euler, Leonhard, 4 fan nozzle efficiency, 150 ideal, 65 non-ideal, 150 fan pressure ratio, 54, 141 Fanno line flow, 214, 215, 216, 236, 456, 457, 461, 471, 597, 600 fans, 53, 100, 141, 276 blade design and, 141 efficiency, 141 ideal, 53 ideal characteristics of, 46–47 non-ideal, 141 polytropic efficiency, 141 pressure ratio, 55 See also compressors flameholders, 146, 441, 446, 456, 460 friction, 138, 215, 456, 471, 597 area changes and, 608, 610 burners and, 458 generalized method for, 458 heat addition and, 609, 610 impellers and, 393 turbines and, 408 See also Fanno line flow; specific components fuels properties, 463 ratio, 61 types of, 463 gas dynamics, 591 gas generator, 482

Index gases, ideal, 591 equations for, 591 gas constant, 591 generalized 1-D flow, 607 heat addition area changes and, 610 friction and, 609, 610 heating value, 60, 146, 452 hydraulic pumps, 288, 381 hydraulic turbines, 418 impellers, 58, 375–378, 385–390, 396–397 blade design, 394 blade number, 393 blade shapes, 392 boundary layer and, 393 compressors and, 378–380 eye diameters, 392 frictional effects, 393 geometries of, 391 Mach numbers and, 392 shock waves and, 389–390 splitter blades and, 394 See also blades; disks; centrifugal compressors incompressible flow, 288, 381, 418, 618 inlet, 69 ideal characteristics of, 46–47 water spray injection system, 82 See also diffusers inverse engine design, 182, 511 isentropic processes, 595. See also specific components iteration methods, 585 progressive, 585 successive substitution method, 588 JANAF tables, 134, 452 jet-wakes, 137–138, 142 kerosene, 450 mass flow corrected, 260, 299, 390, 425, 462 matching of components, 481. See also specific components materials afterburner, 447 burner, 445 compressor, 279 fan, 279 nozzle, 244 turbine, 429 materials, thermal limits of, 74, 427 method of characteristics, 249 mixers, 67, 152, 471, 605 drag and, 475 materials for, 471 mixing process, 473

Index pressure losses and, 471 See also bypass mixer, by pass duct multiple spool engines, 18, 78, 91, 499 burners and, 448 compressors and, 312 critical speeds, 314 mechanical implementation of, 314 off-design operation and, 312 power and, 314 three spooled, 315 turbines and, 407 turbofans and, 20, 22 Newton, Isaac, 4 nozzles, 63, 65, 147, 150, 244 adiabatic processes and, 63–65, 147 afterburners and, 446 altitude and, 258–259, 264 area ratios, 245 choked, 149–150, 233–234, 245–246, 252–254 converging, 150, 245–246, 259 converging-diverging, 150, 161–163, 247, 259 converging-diverging, variable, 258, 262, 263, 490 corrected mass flow, 260 dimensional analysis and, 260 efficiency and, 148–149, 150, 245 engine performance and, 256 equations for, 244, 245 exit area, fixed, 262, 490 exit pressure, 147 fan, 65, 150, 245 fixed-converging, 261, 490 fixed minimum area, 263, 490 ideal, 63, 65 ideal characteristics of, 46–47 iris type, 258–259 isentropic processes and, 63–65 lambda shocks and, 249 Mach numbers and, 260, 261, 262, 263 mass flow and, 149, 260–262 materials used in, 244 modeling of, 489 overexpanded, 248–249 performance maps for, 260 performance of, 245 performance trends, 261 plug type, 258–260 pressure losses and, 148 pressure ratios and, 245, 256 schedules, 262–264 shock diamonds, 249 sonic conditions, 149 sonic line, 251 specific heats and, 245–246, 262 underexpanded, 247 See also fan nozzles; thrust reversers; vectoring

635 Ohain, Hans von, 6 Papin, Denis, 4 percent reaction, 618. See also specific components performance maps, 482 burners, 461 compressors, axial, 299 compressors, centrifugal, 390 diffusers, 235 nozzles, 260 turbines, 425 performance measures, 41 power generation gas turbines, 30, 119 component matching and, 486 compressor pressure ratio, 124 exhaust, 68, 153, 486 fuel ratios and, 499 inlet, 119, 486, 490 load system, 486 modeling of, 490, 491, 495 power of, 120 shaft loading and, 491 specific fuel consumption and, 121–122 thermal efficiency of, 121–122 power generation measures, 42 power turbine exhaust, ideal, 68 non-ideal, 153 pressure ratios. See specific components propellers, 55, 143 advance ratio, 144 efficiency and, 143–144 fixed blades and, 144 flow direction, 143 ideal, 56 ideal characteristics of, 46–47 Mach numbers and, 58, 144 nonideal, 143 power coefficient, 58, 144 reversible pitch, 265 schedules for, 56 stresses on, 56 thrust and, 144 thrust coefficients and, 58 work coefficient and, 57 variable blades, 144 propulsion performance measures, 41 propulsive efficiency, 41 pumps, 286, 288, 381, 384 slip and, 384 ramjets, 16 afterburners and, 446 ideal, 71 Rankine cycle, 15–16 Rankine-Hugoniot equations, 602 Rayleigh line flow, 456, 461, 598, 600 Regula Falsi method, 585

636 resonance, 281, 395, 412 rotational inertial effect, 499 Schlieren photographs, 249 screech, 446–447 separation, 136, 137, 154, 209, 288, 301, 304, 395. See also boundary layers SFC. See specific fuel consumption shadowgraphs, 249 shafts, 59, 144 bearings, 144 efficiency and, 144–145 friction and, 144 ideal, 59 modeling of, 489 squeeze film dampers, 144 shocks, 216 normal, 600 oblique-planar, 601 oblique conical, 220 See also specific components single stage turbomachine energy analysis, 613 specific fuel consumption (SFC), 42 power generation gas turbines and, 121 turboprops and, 28, 116 specific heat, 591 ratio, 591 values for, 593, 594 variable, 593 stagnation pressure, 591 bypass mixer and, 67 stagnation properties, 591 stagnation temperature, 591 standard conditions, 47, 527 starting an inlet, 232 starting an engine, 449 steam topping cycle, 15 stoichiometry, 74–76, 447, 450, 453, 455. See also fuels, ratios successive substitutions method, 588 superposition analysis, 216 Taylor-Maccoll flow, 227 thermodynamic efficiency. See specific components thrust equations, 34 reversers, 265 See also thrust specific fuel consumption thrust specific fuel consumption (TSFC), 41 afterburners and, 168 by-pass ratio, 174 See also specific engine types thrust-weight ratios, 26, 41 TSFC. See thrust specific fuel consumption turbines, 54, 101 blade cooling and, 142 choking, 142 component matching and, 481, 483

Index efficiency and, 142–143 efficiency, polytropic, 142–143 enthalpy and, 142 ideal, 55 ideal characteristics of, 46–47 modeling of, 489 nonideal, 101 peak values, 142 pressure ratios and, 55–56, 142–143 specific heats, 142 See also power generation gas turbines turbines, axial flow, 406 aerodynamics and, 406, 409 blade cooling and, 406, 409, 410, 427, 428 blade heights and, 411 blade manufacture, 430 blade materials, 429 blade slip, 408 blade thicknesses, 411 blade tip clearance, 409 blades, 427 blades, passage areas between, 409 blades per stage, 412 cascades, 408 choke line, 426 compressors and, 406–407, 409 configuration of, 407 dimensional analysis, 425 efficiency and, 406, 409, 412, 426 frictional losses and, 408 geometry of, 407 high pressure, 407 hydraulic turbines, 418 impulse turbines, 419 incompressible flow in, 418 low pressure, 407 mapping conventions, 425 mass flow, corrected, 425, 426, 427 materials in, 406, 410, 427 multiple spool engines and, 407 operating line, 426 percent reaction, 417, 418, 419, 420 performance, 419 performance maps, 425 pressure ratios, 409, 417, 420, 426 reaction turbines, 419 resonance and, 412 separation, 409, 410 shocks, 409 shrouding, 411 single stage energy analysis, 416 solidity factor, 408 speed, corrected, 425 stages, number of, 410 streamline analysis method, 433 temperatures in, 409, 410 thermal limits, 427

Index thermodynamic efficiency, 427 turning angles, 410 velocity polygons, 413, 419 viscous shear, 409 turbofans, 20, 25, 91, 471 afterburners and, 25, 98–102, 106–108 bypass ratios, 53, 93, 94, 95–98, 105–106 compressor pressure ratio, 93, 94–96, 105 efficiency of, 22 fan exhausted, 20, 25, 38, 91–102 fan mixed, 24, 25, 102–108 fan total pressure ratio, 94–98 fuel economy, 25 high bypass and, 406 ideal, 91 ideal thrust, 93, 94, 98, 102, 103–105, 107, 108, 111, 113 ideal TSFC and, 93–102, 105, 108–111, 113 Mach numbers, 94, 98–99, 103–105 mass flow and, 105 mixed-exhausted comparisons, 108 nonafterburning, 92–98, 102–106 pressure ratios and, 108 turbojets, compared, 108 turbojets, 17 afterburners and, 84 altitude and, 83 compressor pressure ratio, 81–84, 86–87 fuel economy, 25 ideal, 78 ideal thrust, 81, 89–91 ideal TSFC and, 86–87, 89–91

637 inlet temperatures, 82–83 Mach numbers and, 83–84 mass flow and, 81 nonafterburning, 79 ramjets and, 84, 86–87 turbofans, compared, 108 turbomachines fundamentals of, 613 history of, 3 turboprops, 27, 40, 113, 374 compressor pressure ratio, 116 gearboxes, 27, 56, 58, 374 ideal, 58 ideal thrust for, 113–116, 119 ideal TSFC and, 116, 119 reversible pitch propellers, 265 specific fuel consumption and, 116 work coefficient for, 115–116, 119 turboshafts, 29, 374 UDF. See unducted fan unducted fan (UDF), 29, 58, 113 V2 rocket, 8 vectoring, thrust, 267 velocity polygons compressors and, 283, 289, 332, 378, 386 turbines and, 413, 419 See also specific components Whittle, Sir Frank, 6 windmills, 3 wind tunnels, 231–232, 234, 508