A Design Study of Single-Rotor Turbomachinery Cycles
by
Manoharan Thiagarajan
A thesis submitted to the faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of
Master of Science
in
Mechanical Engineering
Committee
Dr. Peter King, Chairman Dr. Walter O’Brien, Committee Member Dr. Clint Dancey, Committee Member
August 12, 2004 Blacksburg, Virginia
Keywords: Auxiliary power unit, single radial rotor, specific power takeoff, compressor, burner, turbine
A Design Study of Single-Rotor Turbomachinery Cycles by Manoharan Thiagarajan
Dr. Peter King, Chairman Dr. Walter O’Brien, Committee Member Dr. Clint Dancey, Committee Member
(ABSTRACT)
Gas turbine engines provide thrust for aircraft engines and supply shaft power for various applications. They consist of three main components. That is, a compressor followed by a combustion chamber (burner) and a turbine. Both turbine and compressor components are either axial or centrifugal (radial) in design. The combustion chamber is stationary on the engine casing. The type of engine that is of interest here is the gas turbine auxiliary power unit (APU). A typical APU has a centrifugal compressor, burner and an axial turbine. APUs generate mechanical shaft power to drive equipments such as small generators and hydraulic pumps. In airplanes, they provide cabin pressurization and ventilation. They can also supply electrical power to certain airplane systems such as navigation. In comparison to thrust engines, APUs are usually much smaller in design. The purpose of this research was to investigate the possibility of combining the three components of an APU into a single centrifugal rotor. To do this, a set of equations were chosen that would describe the new turbomachinery cycle. They either were provided or derived using quasi-one-dimensional compressible flow equations. A MathCAD program developed for the analysis obtained best design points for various cases with the help of an optimizer called Model Center. These results were then compared to current machine specifications (gas turbine engine, gasoline and diesel generators). The result of interest was maximum specific power takeoff. The results showed high specific powers in the event there was no restriction to the material and did not exhaust at atmospheric pressure. This caused the rotor to become very large and have a disk thickness that was unrealistic. With the restrictions fully in place, they severely limited the performance of the rotor. Sample rotor shapes showed all of them to have unusual designs. They had a combination of unreasonable blade height variations and very large disk thicknesses. Indications from this study showed that the single radial rotor turbomachinery design might not be a good idea. Recommendations for continuation of research include secondary flow consideration, blade height constraints and extending the flow geometry to include the axial direction.
Acknowledgement The author wishes to express his sincere gratitude to Dr. Peter King, major professor, for contributing valuable time, advice, and assistance to the research and to the preparation of this manuscript. Sincere thanks are due to the members of the author’s graduate committee composed of Dr. Walter O’Brien, and Dr. Clint Dancey for their advice and constructive criticism. The author also is grateful to Phoenix Integration for allowing him to use Model Center for the purpose of optimization to help in the completion of this research project. Very special thanks are due to the author’s parents for their understanding, patience, and encouragement throughout the course of this study. Heartiest thanks are also due to Rene Villanueva, An Song Nguyen, and Kevin Duffy for all their encouragement. Special appreciation goes out to Ms. Lisa Stables for all her assistance during this research. To all turbolabbers, warp speed ahead. Space is the final frontier.
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Table of contents TABLE OF FIGURES ............................................................................................................................................VII LIST OF TABLES.................................................................................................................................................... XI NOMENCLATURE .............................................................................................................................................. XIII CHAPTER 1
INTRODUCTION.........................................................................................................................1
1.1
ABOUT SMALL GAS TURBINE ENGINES ..........................................................................................................1
1.2
AUXILIARY POWER UNIT (APU) AND PURPOSE OF RESEARCH .......................................................................4
CHAPTER 2 2.1
LITERATURE REVIEW.............................................................................................................6
HISTORY OF THE APU...................................................................................................................................6
2.1.1
Project A .........................................................................................................................................6
2.1.2
The Black Box .................................................................................................................................6
2.1.3
The GTC43/44.................................................................................................................................8
2.2
IDEAL BRAYTON CYCLE AND IDEAL JET PROPULSION CYCLE ........................................................................9
2.3
HOW CURRENT APUS WORK .......................................................................................................................11
CHAPTER 3
FORMULAS USED FOR THE APU ........................................................................................15
3.1
GENERAL INFORMATION .............................................................................................................................15
3.2
AMBIENT AIR AND DIFFUSER .......................................................................................................................17
3.3
COMPRESSOR ..............................................................................................................................................18
3.4
BURNER AND TURBINE ................................................................................................................................22
3.4.1
Burner equations...........................................................................................................................23
3.4.2
Burner input parameters and method of solving equations ..........................................................25
3.4.3
Turbine equations .........................................................................................................................28
3.4.4
Turbine input parameters..............................................................................................................29
3.4.4.1
Subsonic turbine............................................................................................................................................ 30
3.4.4.2
Supersonic turbine ........................................................................................................................................ 31
3.4.5
Method of solving turbine equations.............................................................................................32
3.4.6
Burner and turbine output summary .............................................................................................35
3.5
OVERALL APU PROPERTIES ........................................................................................................................37
CHAPTER 4 4.1
RESULTS OF ANALYSIS.........................................................................................................39
SIMPLE ONE-DIMENSIONAL FLOW ...............................................................................................................39
4.1.1
Burner ...........................................................................................................................................40
4.1.1.1
Constant area flow with drag and heat addition ............................................................................................ 40
4.1.1.2
Constant area flow with only heat addition................................................................................................... 41
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4.1.2 4.2
Variable area flow ........................................................................................................................42 SINGLE ROTOR APU RESULTS .....................................................................................................................43
4.2.1
Model Center and input/output constraints ..................................................................................43
4.2.2
Results from Model Center............................................................................................................46
4.2.2.1
Case 1: Without the stress and |(P0-P5)/P5| constraints. ................................................................................. 46
4.2.2.2
Case 2: With the stress constraint but without the |(P0-P5)/P5| constraint...................................................... 48
4.2.2.3
Case 3: Without the stress constraint but with the |(P0-P5)/P5| constraint...................................................... 49
4.2.2.4
Case 4: With the stress and |(P0-P5)/P5| constraints ....................................................................................... 50
4.2.3
Rotor material and size .................................................................................................................51
CHAPTER 5
CONCLUSION............................................................................................................................52
5.1
SUMMARY ...................................................................................................................................................52
5.2
RECOMMENDATIONS ...................................................................................................................................52
REFERENCES ..........................................................................................................................................................53 APPENDIX A
COMPRESSOR DERIVATIONS .............................................................................................54
A.1
OUTLET RELATIVE MACH NUMBER .............................................................................................................54
A.2
OUTLET RELATIVE STAGNATION TEMPERATURE .........................................................................................55
APPENDIX B
BURNER AND TURBINE DERIVATIONS ............................................................................56
B.1
CONSERVATION OF ANGULAR MOMENTUM .................................................................................................56
B.2
CONSERVATION OF ENERGY (FIRST LAW OF THERMODYNAMICS)................................................................56
B.3
EQUATION OF STATE ...................................................................................................................................58
B.4
CONSERVATION OF MASS ............................................................................................................................58
B.5
CONSERVATION OF LINEAR MOMENTUM .....................................................................................................58
B.6
RELATIVE STAGNATION TEMPERATURE EQUATION .....................................................................................59
B.7
RELATIVE STAGNATION TEMPERATURE EQUATION .....................................................................................60
B.8
ABSOLUTE STAGNATION TEMPERATURE EQUATION ....................................................................................62
B.9
RELATIVE MACH NUMBER EQUATION .........................................................................................................63
B.10
ABSOLUTE STAGNATION PRESSURE EQUATION ...........................................................................................63
B.11
ENTROPY EQUATION ...................................................................................................................................64
B.12
BURNER ABSOLUTE STAGNATION TEMPERATURE DISTRIBUTION.................................................................64
B.13
BURNER SPECIFIC WORK .............................................................................................................................65
B.14
TURBINE SPECIFIC WORK ............................................................................................................................66
APPENDIX C
TO DETERMINE PERPENDICULAR (ONE-DIMENSIONAL) FLOW AREA
BETWEEN THE VANES .........................................................................................................................................67 APPENDIX D
CURRENT ENGINE DATA......................................................................................................68
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APPENDIX E
COMPLETE RESULTS FOR CASE 1.....................................................................................72
E.1
INPUT PARAMETERS ....................................................................................................................................72
E.2
OUTPUT VALUES .........................................................................................................................................74
APPENDIX F
COMPLETE RESULTS FOR CASE 2.....................................................................................82
F.1
INPUT PARAMETERS ....................................................................................................................................82
F.2
OUTPUT VALUES .........................................................................................................................................84
APPENDIX G
COMPLETE RESULTS FOR CASE 3.....................................................................................93
G.1
INPUT PARAMETERS ....................................................................................................................................93
G.2
OUTPUT VALUE ...........................................................................................................................................95
APPENDIX H
COMPLETE RESULTS FOR CASE 4...................................................................................100
H.1
INPUT PARAMETERS ..................................................................................................................................100
H.2
OUTPUT VALUES .......................................................................................................................................102
APPENDIX I
SAMPLE ROTOR FOR CASE 1 WITH CALCULATION PROGRAM ...........................108
APPENDIX J
SAMPLE ROTOR FOR CASE 2 WITH CALCULATION PROGRAM ...........................110
APPENDIX K
SAMPLE ROTOR FOR CASE 3 WITH CALCULATION PROGRAM ...........................112
APPENDIX L
SAMPLE ROTOR FOR CASE 4 WITH CALCULATION PROGRAM ...........................114
VITA .........................................................................................................................................................................116
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Table of figures Figure 1-1: Williams International FJ44 turbofan engine, small gas turbine engine (from [1]). .................. 1 Figure 1-2: Pratt & Whitney J58 turbojet engine, large gas turbine engine (from [2]). ............................... 1 Figure 1-3: Years spent in the small gas turbine engine business (from [1])................................................ 3 Figure 1-4: Turboshaft engine (from [2]). .................................................................................................... 4 Figure 1-5: Auxiliary power unit (from [3]). ................................................................................................ 4 Figure 1-6: APU with exhaust vent at the rear of the aircraft (from [4])...................................................... 5 Figure 1-7: The new rotor with the combined components will look something like this compressor impeller (from [5]). ............................................................................................................................... 5 Figure 2-1: Garrett Black Box (from [1]). .................................................................................................... 7 Figure 2-2: GTC43/44 first stage backward curved centrifugal compressor (from [1]). .............................. 8 Figure 2-3: Closed gas turbine engine cycle (from [6]).............................................................................. 10 Figure 2-4: Closed cycle T-s diagram (from [6])........................................................................................ 10 Figure 2-5: T-s diagram for an ideal jet propulsion cycle along with a turbojet engine schematic (from [6]). ..................................................................................................................................................... 11 Figure 2-6: APU centrifugal compressor rotor with inducer vanes (from [3]). .......................................... 11 Figure 2-7: Combustion chambers (from [3])............................................................................................. 12 Figure 2-8: Fuel igniter (from [3]). ............................................................................................................. 13 Figure 2-9: APU turbines (from [3])........................................................................................................... 13 Figure 3-1: Cylindrical coordinate system (from [5])................................................................................. 15 Figure 3-2: Shape of rotor with velocity triangle (from [5])....................................................................... 16 Figure 3-3: Burner and turbine control volume between two vanes across a small step change (from [9]). ............................................................................................................................................................ 22 Figure 3-4: Convergent-divergent nozzle with supersonic exit (from [1]). ................................................ 31 Figure 3-5: Variation of specific rupture strength with service temperature (from [5]). ............................ 38 Figure 4-1: Constant area combustion chamber (from [10]). ..................................................................... 40 Figure 4-2: Constant area flow through a duct with heat addition (from [9])............................................. 41 Figure 4-3: Flow through a duct with variable area (from [9])................................................................... 42 Figure 4-4: Relative Mach number, stagnation temperature (K) and pressure (Pa) according to location in the rotor (Case 1). ............................................................................................................................... 47 Figure 4-5: Variation of the absolute tangential velocity (m/s), rotor speed (m/s) and flow curvature (deg) (Case 1). .............................................................................................................................................. 48 Figure D-1: PSFC and specific power comparison between APU cases and current engines.................... 71 Figure E-1: Case 1 relative Mach number. ................................................................................................. 76 vii
Figure E-2: Case 1 relative stagnation temperature (K).............................................................................. 76 Figure E-3: Case 1 relative stagnation pressure (Pa). ................................................................................. 76 Figure E-4: Case 1 stagnation temperature (K). ......................................................................................... 76 Figure E-5: Case 1 stagnation pressure (Case 1). ....................................................................................... 77 Figure E-6: Case 1 temperature (Case 1). ................................................................................................... 77 Figure E-7: Case 1 pressure (Case 1).......................................................................................................... 77 Figure E-8: Case 1 density (Case 1)............................................................................................................ 77 Figure E-9: Case 1 flow curvature (Case 1)................................................................................................ 77 Figure E-10: Case 1 rotor speed (Case 1). .................................................................................................. 77 Figure E-11: Case 1 specific heat (Case 1). ................................................................................................ 78 Figure E-12: Case 1 specific heat ratio (Case 1)......................................................................................... 78 Figure E-13: Case 1 tangential velocity (Case 1). ...................................................................................... 78 Figure E-14: Case 1 To-s diagram (Case 1). ............................................................................................... 78 Figure E-15: Case 1 Po-v diagram (Case 1). ............................................................................................... 78 Figure E-16: Variation of specific power takeoff with compressor pressure ratio (Case 1)....................... 79 Figure E-17: Variation of PSFC with compressor pressure ratio (Case 1). ................................................ 80 Figure E-18: Variation of compressor radius ratio and pressure ratio (Case 1).......................................... 80 Figure E-19: Variation of rotor radius ratio with compressor pressure ratio (Case 1)................................ 81 Figure E-20: Variation of disk thickness with compressor pressure ratio (Case 1). ................................... 81 Figure F-1: Relative Mach number (Case 2). ............................................................................................. 86 Figure F-2: Relative stagnation temperature (Case 2). ............................................................................... 86 Figure F-3: Relative stagnation pressure (Case 2). ..................................................................................... 86 Figure F-4: Stagnation temperature (Case 2). ............................................................................................. 86 Figure F-5: Stagnation pressure (Case 2).................................................................................................... 87 Figure F-6: Temperature (Case 2)............................................................................................................... 87 Figure F-7: Pressure (Case 2)...................................................................................................................... 87 Figure F-8: Density (Case 2)....................................................................................................................... 87 Figure F-9: Flow curvature (Case 2)........................................................................................................... 87 Figure F-10: Rotor speed (Case 2).............................................................................................................. 87 Figure F-11: Specific heat (Case 2). ........................................................................................................... 88 Figure F-12: Specific heat ratio (Case 2). ................................................................................................... 88 Figure F-13: Tangential velocity (Case 2). ................................................................................................. 88 Figure F-14: To-s diagram (Case 2). ........................................................................................................... 88 Figure F-15: Beginning of To-s diagram (Case 2). ..................................................................................... 88
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Figure F-16: End of To-s diagram (Case 2)................................................................................................. 88 Figure F-17: Po-s diagram (Case 2)............................................................................................................. 89 Figure F-18: Beginning of Po-s diagram (Case 2)....................................................................................... 89 Figure F-19: Variation of specific power takeoff with compressor pressure ratio (Case 2). ...................... 90 Figure F-20: Variation of PSFC with compressor pressure ratio (Case 2). ................................................ 90 Figure F-21: Variation of compressor radius ratio and pressure ratio (Case 2). ......................................... 91 Figure F-22: Variation of rotor radius ratio with compressor pressure ratio (Case 2)................................ 91 Figure F-23: Variation of disk thickness with compressor pressure ratio (Case 2). ................................... 92 Figure G-1: Relative Mach number (Case 3).............................................................................................. 97 Figure G-2: Relative stagnation temperature (Case 3)................................................................................ 97 Figure G-3: Relative stagnation pressure (Case 3). .................................................................................... 97 Figure G-4: Stagnation temperature (Case 3). ............................................................................................ 97 Figure G-5: Stagnation pressure (Case 3). .................................................................................................. 98 Figure G-6: Temperature (Case 3). ............................................................................................................. 98 Figure G-7: Pressure (Case 3)..................................................................................................................... 98 Figure G-8: Density (Case 3). ..................................................................................................................... 98 Figure G-9: Flow curvature (Case 3). ......................................................................................................... 98 Figure G-10: Rotor speed (Case 3). ............................................................................................................ 98 Figure G-11: Specific heat (Case 3)............................................................................................................ 99 Figure G-12: Specific heat ratio (Case 3). .................................................................................................. 99 Figure G-13: Tangential velocity (Case 3). ................................................................................................ 99 Figure G-14: To-s diagram (Case 3)............................................................................................................ 99 Figure G-15: Po-v diagram (Case 3). .......................................................................................................... 99 Figure H-1: Relative Mach number (Case 4)............................................................................................ 104 Figure H-2: Relative stagnation temperature (Case 4).............................................................................. 104 Figure H-3: Relative stagnation pressure (Case 4). .................................................................................. 104 Figure H-4: Stagnation temperature (Case 4). .......................................................................................... 104 Figure H-5: Stagnation pressure (Case 4). ................................................................................................ 105 Figure H-6: Temperature (Case 4). ........................................................................................................... 105 Figure H-7: Pressure (Case 4)................................................................................................................... 105 Figure H-8: Density (Case 4). ................................................................................................................... 105 Figure H-9: Flow curvature (Case 4). ....................................................................................................... 105 Figure H-10: Rotor speed (Case 4). .......................................................................................................... 105 Figure H-11: Specific heat (Case 4).......................................................................................................... 106
ix
Figure H-12: Specific heat ratio (Case 4). ................................................................................................ 106 Figure H-13: Tangential velocity (Case 4). .............................................................................................. 106 Figure H-14: To-s diagram (Case 4).......................................................................................................... 106 Figure H-15: Beginning of To-s diagram (Case 4).................................................................................... 106 Figure H-16: End of To-s diagram (Case 4). ............................................................................................. 106 Figure H-17: Po-s diagram (Case 4). ......................................................................................................... 107 Figure I-1: Sample rotor for Case 1 with side view (starting at station 3)................................................ 109 Figure J-1: Sample rotor for Case 2 with side view (starting at station 3)................................................ 111 Figure K-1: Sample rotor for Case 3 with side view (starting at station 3) .............................................. 113 Figure L-1: Sample rotor for Case 4 with side view (starting at station 3)............................................... 115
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List of tables Table 3-1: Ambient air equation input parameters. .................................................................................... 17 Table 3-2: Compressor equation input parameters. .................................................................................... 18 Table 3-3: Burner equation input parameters. ............................................................................................ 26 Table 3-4: Turbine equation input parameters. ........................................................................................... 32 Table 3-5: Burner exit flow variables. ........................................................................................................ 35 Table 3-6: Turbine exit flow variables........................................................................................................ 36 Table 4-1: Comparison of burner equations to simple flow example (drag and heat addtion)................... 41 Table 4-2: Comparison of burner equations to simple flow example (heat addition)................................. 42 Table 4-3: Comparison of turbine equations to simple flow example (variable area). ............................... 43 Table 4-4: Model Center input parameters with range limits. .................................................................... 44 Table 4-5: Model Center fixed input values. .............................................................................................. 44 Table 4-6: Model Center output constraints. .............................................................................................. 45 Table 4-7: Overall rotor and other properties (Case 1). .............................................................................. 46 Table 4-8: Overall rotor and other properties (Case 2). .............................................................................. 49 Table 4-9: Overall rotor and other properties (Case 3). .............................................................................. 50 Table 4-10: Overall rotor and other properties (Case 4). ............................................................................ 50 Table D-1: Airplane turboprop engine data. ............................................................................................... 68 Table D-2: Helicopter turboshaft engine data............................................................................................. 68 Table D-3: Aircraft (turboprop) and helicopter (turboshaft) dual-purpose engine data.............................. 69 Table D-4: Four-stroke gasoline generator engine data. ............................................................................. 69 Table D-5: Diesel generator engine data..................................................................................................... 70 Table E-1: Air and diffuser input parameter values (Case 1). .................................................................... 72 Table E-2: Compressor input parameter values (Case 1)............................................................................ 72 Table E-3: Burner input parameter values (Case 1).................................................................................... 73 Table E-4: Turbine input parameter values (Case 1). ................................................................................. 73 Table E-5: Air diffuser output values (Case 1). .......................................................................................... 74 Table E-6: Compressor output values (Case 1). ......................................................................................... 74 Table E-7: Burner output value (Case 1). ................................................................................................... 75 Table E-8: Turbine output value (Case 1)................................................................................................... 75 Table E-9: Rotor overall properties (Case 1). ............................................................................................. 76 Table E-10: Data to show Case 1 configuration is the optimum (Case 1 highlighted below). ................... 79 Table F-1: Air and diffuser input parameter values (Case 2)...................................................................... 82 Table F-2: Compressor input parameter values (Case 2)............................................................................ 82 xi
Table F-3: Burner input parameter values (Case 2). ................................................................................... 83 Table F-4: Turbine and stress input parameter values (Case 2).................................................................. 83 Table F-5: Air diffuser output values (Case 2). .......................................................................................... 84 Table F-6: Compressor output values (Case 2)........................................................................................... 84 Table F-7: Burner output value (Case 2). ................................................................................................... 85 Table F-8: Turbine output value (Case 2). .................................................................................................. 85 Table F-9: Rotor overall properties (Case 2). ............................................................................................. 86 Table F-10: Data to show Case 2 configuration is the optimum (Case 2 highlighted below). ................... 89 Table G-1: Air and diffuser input parameter values (Case 3). .................................................................... 93 Table G-2: Compressor input parameter values (Case 3). .......................................................................... 93 Table G-3: Burner input parameter values (Case 3). .................................................................................. 94 Table G-4: Turbine input parameter values (Case 3).................................................................................. 94 Table G-5: Air diffuser output values (Case 3)........................................................................................... 95 Table G-6: Compressor output values (Case 3). ......................................................................................... 95 Table G-7: Burner output value (Case 3).................................................................................................... 96 Table G-8: Turbine output value (Case 3). ................................................................................................. 96 Table G-9: Rotor overall properties (Case 3).............................................................................................. 97 Table H-1: Air and diffuser input parameter values (Case 4). .................................................................. 100 Table H-2: Compressor input parameter values (Case 4). ........................................................................ 100 Table H-3: Burner input parameter values (Case 4). ................................................................................ 101 Table H-4: Turbine and stress input parameter values (Case 4). .............................................................. 101 Table H-5: Air diffuser output values (Case 4)......................................................................................... 102 Table H-6: Compressor output values (Case 4). ....................................................................................... 102 Table H-7: Burner output value (Case 4).................................................................................................. 103 Table H-8: Turbine output value (Case 4). ............................................................................................... 103 Table H-9: Rotor overall properties (Case 4)............................................................................................ 104
xii
Nomenclature Variables
Definition
Mrel
Relative Mach number
τrel
Relative stagnation temperature ratio
πrel
Relative stagnation pressure ratio
τ
Stagnation temperature ratio
π
Stagnation pressure ratio
Torel
Relative stagnation temperature
Porel
Relative stagnation pressure
To
Stagnation temperature
Po
Stagnation pressure
W
Relative velocity
T
Temperature
P
Pressure
ρ
Density
s
Entropy
v
Specific volume
m
Mass flow rate
mf
Fuel mass flow rate
f
Fuel-to-air ratio
hHV
Fuel heating value
CD
Drag coefficient
M
Absolute Mach number
C
Absolute velocity
U
Blade speed
Ω
Impeller rotation speed
R
Gas constant
Cp
Specific heat
γ
Ratio of specific heats
Q
Heat addition per unit seconds
W
Work per unit seconds
PTO
Power takeoff per unit seconds
xiii
ηTH
Thermal efficiency
A
Area perpendicular to flow between two vanes
r
Impeller radius
b
Vane height
β
Relative flow and blade angle
α
Absolute flow angle
Nb
Number of blades
Subscripts
Definition
Engine components d
Diffuser
c
Compressor
b
Burner (combustion chamber)
t
Turbine
Station (location) numbering 0
Ambient air (freestream)
1
Diffuser entry
2
Diffuser exit/Compressor entry
2t
Compressor entry at the blade tip
3
Compressor exit/Burner entry
4
Burner exit/Turbine entry
4.5
Location close to sonic point
5
Turbine exit
Cylindrical coordinate system r
Radial
θ
Tangential
z
Axial
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Chapter 1 1.1
Introduction
About small gas turbine engines From the beginning, gas turbine engine manufactures considered large and small engines as two
separate categories with each having different applications. Both had their own unique set of problems and challenges. With the introduction of large gas turbine engines in the 1940s, military aircrafts followed by civilian ones, began using them in place of piston engines. Since then, the power and size of these engines grew significantly compared to piston engines.
Figure 1-1: Williams International FJ44 turbofan engine, small gas turbine engine (from [1]).
Figure 1-2: Pratt & Whitney J58 turbojet engine, large gas turbine engine (from [2]).
1
The usage of piston engines continued for low power applications. For this reason, the evolution of small gas turbine engines occurred slowly. Over time, this engine was the power plant of choice for a variety of applications such as: a) Remotely Piloted Vehicles (RPV) and Unmanned Aerial Vehicles (UAV) b) Decoy, tactical and strategic missiles c) Military trainer aircraft d) Special purpose aircraft such as Vertical Takeoff and Landing (VTOL) aircraft e) Helicopters They provided greater operational capabilities in terms of speed, payload, altitude and reliability than piston engines. Small gas turbine engines were quite different mechanically from their larger engine counterparts. There were factors such as manufacturing limitations and mechanical design problems. This prevented direct scaling of large engine design and performance. For example, internal engine pressures were about the same for small and large engines [1]. Therefore, it was necessary that the casing of small engines be approximately as thick as large engine casings. As a result, small engines paid an inherent structural weight penalty. Another example was the difficulty that came about during the development of smaller and lighter fuel controls that had the same amount of reliability like larger engines [1]. Small engine fuel controls had critical accuracy problems because of the lower rates of fuel flow. Gradually, these scaling issues declined due to aggressive efforts in technology development. The advances produced by these efforts allowed the small engine to overcome its problems related to size and attain outstanding performance. The military turned to the gas turbine engine manufacturers to develop small gas turbine engines. This attracted manufacturers to the potential of military contracts and a profitable market once they were developed. By late 1950s, the military had plenty of success with these engines such that civilian aircraft started using them. Piston engine makers saw a need to get into the small gas turbine engine business to maintain their market position and profitability level. Established large engine producers seized the opportunity to expand their business by applying their technical expertise to the development of small gas turbine engines [1]. These incentives and potentials led to an array of companies that wanted to enter the small gas turbine engine business. Next is a chart that shows the North American companies that developed and built small gas turbine engines from the early 1940s through the present:
2
Figure 1-3: Years spent in the small gas turbine engine business (from [1]).
After early efforts by Westinghouse, it phased out of the small gas turbine engine market in the I950s. Other US companies also became active in studying, developing, and manufacturing these engines for aircraft propulsion in the 1940s. These companies included Fredric Flader, Boeing, Fairchild, and West Engineering. The military sponsored much of their work, and this led to engines that powered both piloted and unmanned aircraft. Each of these companies eventually phased out of the small gas turbine engine business. One company, Williams International, began developing small gas turbine engines using its own funds with the philosophy that once it had successfully developed an engine, there would be a market for it [1]. Another relevant activity underway during the 1940s was small gas turbine component and nonaircraft research and development. In 1943, Garrett began work on Project A [1]. This project consisted of a two-stage compressor for aircraft cabin pressurization, which later led to turbine environmental control systems, jet engine starters, and auxiliary power units (APU). This made Garrett the first company to begin developing APUs.
3
1.2
Auxiliary power unit (APU) and purpose of research An APU is essentially a small gas turbine engine. It is similar in construction and purpose to a
turboshaft engine, seen in Figure 1-4. A turboshaft engine differs from a turboprop engine primarily in the function of the engine shaft. Instead of driving a propeller, the turboshaft engine connects to a transmission system or gearbox to drive a mechanical load. Therefore, shaft power is the desired output.
Figure 1-4: Turboshaft engine (from [2]).
Like the turboshaft engine, an APU consists of three primary components. They are the compressor, a combustion chamber (burner) and a turbine section. Figure 1-5 shows an example of an APU. In commercial and military aircraft, shaft power from APUs generate electrical power that are used for equipments such as lights, onboard computers, televisions, refrigerators, microwave ovens, and coffee pots. In addition, compressed air supplied by the APU goes for aircraft air-conditioning, heating, and ventilation. Another use of the shaft power is to run pumps.
Figure 1-5: Auxiliary power unit (from [3]).
Figure 1-6 shows a typical location of an APU on modern jetliners. The opening at the aircraft rear indicates the APU exhaust vent.
4
Figure 1-6: APU with exhaust vent at the rear of the aircraft (from [4]).
The purpose of this research is to investigate the possibility of combining the three main components of an APU into a single centrifugal impeller, similar to the compressor design seen in Figure 1-7. The idea of having a power producing turbomachine with only one rotating component suggests that the engine could be lighter, cheaper, and smaller. This in turn could allow it to produce high specific power takeoffs (power takeoff per unit mass flow rate of air). Power takeoff is the amount of mechanical power extracted from the shaft to run equipment such as a generator or hydraulic pump. A numerical simulation of the rotor is to take place in this investigation. Chapter 3 in this thesis shows the derivation of the equations for the analysis. The analysis could apply equally to APUs, turboshaft engines, and so on.
Figure 1-7: The new rotor with the combined components will look something like this compressor impeller (from [5]).
5
Chapter 2 2.1
Literature review
History of the APU
2.1.1 Project A In the spring of 1943, Garrett (today known as AlliedSignal) started to design and develop a twostage compressor for a cabin air compressor. The company called this classified program Project A. Each stage was a centrifugal compressor rotor. The following are the specifications of each rotor: a) Mass flow rate of 45 lb/min at a pressure ratio of 1.75 b) 7.25 inch diameter c) 8 vanes (blades) d) 30 degree backward curvature (measured from the tangent of the outer diameter) e) Shrouded cast aluminum impellers f) Adiabatic efficiency of 78% at the design point Although the unit was just a laboratory development tool, Project A demonstrated early on that high efficiencies over broad operating ranges were characteristics of the backward curved compressor rotor design. This knowledge and experience became an important consideration for aircraft cabin air conditioning equipment. It was also the foundation for Garrett’s first small gas turbine design called the Black Box [1].
2.1.2 The Black Box Boeing wanted a lightweight, compact, self-powered unit that could furnish AC and DC current to its Model 377 Stratocruiser aircraft’s electrical systems. In addition, it must provide airflow for cabin pressurization and air conditioning. It should also supply hot air for the wing anti-icing system. Boeing needed a complete unit in 18 months. Garrett accepted the job and decided to make a small gas turbine engine. Preliminary work started in the spring of 1945. As the design progressed, the unit was nicknamed the Black Box due in part to the secrecy of the project and the fact it had so many gadgets made it look like a magical black box. The engine consisted of a three stage, backward curved centrifugal compressor, a burner, and a single stage axial turbine. A geared power takeoff shaft was to run a blower and a generator/alternator. Compressor bleed air routed through a cooling turbine would generate 40 hp back into the shaft. In addition, the wing anti-icing system would receive a portion of the exhaust gas. Also in this Black Box were primary and secondary
6
heat exchangers, automatic controls, regulators, and air ducts [11]. Figure 2-1 shows the Black Box as it was being assembled.
Figure 2-1: Garrett Black Box (from [1]).
Component testing began by mid 1946 and showed excellent overall compressor efficiency in the neighborhood of 81 to 82%. The high compressor efficiency was not surprising as the technology flowed directly from Project A. The Black Box pressure ratio was three as opposed to 1.75 in Project A. A threestage compressor achieved this ratio. The burner also performed well in tests. This was an important accomplishment for Garrett, as the company had never before built a burner. The turbine wheel component testing did not occur due to the unavailability of a suitable test rig with the capacity to absorb its power. Therefore, turbine testing could only occur until the machine was ready to run. An external power source drove the Black Box after assembly late in the fall of 1946. However, it could not generate sufficient power to run by itself. After a month of trying to get the Black Box to self-run, engineers found the untested turbine component to be the problem. With an efficiency of less than 70%, the turbine engine was on the borderline of being self-supporting. By that time, there was
7
insufficient time left to redesign the turbine and meet the contract deadline. Subsequently, Garrett had to cancel the Black Box program at the end of December 1946. The complexity of the unit, low turbine efficiency, and tight development schedule killed the Black Box project [1]. Despite the cost of the program to Garrett and the problems that it caused with Boeing, there were some important lessons learned, particularly what not to do. Further work on axial turbines discontinued at Garrett in favor of the radial inflow turbine. The highly successful backward curved centrifugal compressor continued in future Garrett projects. The knowledge gained from building a successful combustor was part of the technology base gained from the program. These efforts produced here carried on in future Garrett engines especially in the GTC43/44, the company’s first successful gas turbine engine [1].
2.1.3 The GTC43/44 While the Black Box program was still running, the Navy was looking for a 35 hp gas turbine starter. That was the power needed to start the 5525 hp Allison XT40 turboprop engines in the Navy sponsored Convair XPSY-1 flying boat. Agreeing to develop such a unit, Garrett received a contract in early 1947 by the Navy for the starter. The engine that Garrett ultimately designed was the GTC43/44. The project started after the termination of the Black Box program. The design work took place between March and April 1947. The GTC43/44 contained a two-stage backward curved centrifugal compressor (first stage seen in Figure 2-2) with an overall pressure ratio of three. From the compressor, two outlets connected with elbows led to two independent tubular steel combustion chambers. This engine would have a single stage radial inflow turbine. The unit was to deliver 43 lb of air at 44 lb per square inch absolute pressure, hence its name GTC43/44 [1].
Figure 2-2: GTC43/44 first stage backward curved centrifugal compressor (from [1]).
On July 1, 1947, turbine wheel tests showed 82 to 84% efficiency. Garrett conducted the first self-sustaining test run of the GTC43/44 on August 23, 1947. On June 2, 1948, the engine passed its 200hour Navy endurance test and it was the first small gas turbine engine to pass such a test. Garrett began
8
production of the starter in 1948. Its first flight service was on April 18, 1950 in the Convair XPSY-1 flying boat. Two GTC43/44s provided compressed air for starting the main engines and for driving alternators that powered the XPSY electrical systems. In an effort to further its applications, the North American A2J used a mobile ground power version. The first commercial use of the GTC43/44 was in a ground vehicle for starting the Lockheed Electra. However, the GTC43/44 was not without problems [1]. Automatic fuel controls, designed to provide fully automatic starting and overload protection, proved unreliable in service. The twin combustor design also proved to be a problem. The combustor-turbine coupling became extremely hot and it was difficult to find a suitable fireproof enclosure. The radial inflow turbine also had difficulties such as cracks on the turbine rims. Considerable engineering effort went into solving such field service, packaging, and design problems. The GTC43/44 was however a commercial success and more than 500 units were manufactured between 1949 and early 1950s for a variety of applications. It was Garrett's first successful gas turbine engine. It was also the start of a major new product line, the gas turbine auxiliary power unit (APU), which Garrett dominated the world markets through the 1990s. The GTC43/44 also provided a technology base for future Garrett prime propulsion engines.
2.2
Ideal Brayton Cycle and ideal jet propulsion cycle George Brayton first proposed the Brayton cycle for use in the piston engine that he developed
around 1870 [6]. Today gas turbine engines use it when both the compression and expansion processes take place in rotating machinery. Ambient air, drawn into a compressor, rises in both temperature and pressure [7]. Then burning of fuel occurs when the air proceeds into a combustion chamber (burner). The resulting high-temperature gas then expands in a turbine, and exits the engine. In an APU, this expansion process produces shaft power. When the exhaust gas simply leaves the engine, this process is called an open cycle. Gas turbine engines usually operate on an open cycle. Figure 2-3 shows a closed cycle called the Brayton cycle. This is when a constant-pressure heat rejection process replaces the exhaust air from the open cycle.
9
Figure 2-3: Closed gas turbine engine cycle (from
Figure 2-4: Closed cycle T-s diagram (from [6]).
[6]).
Figure 2-4 shows the temperature-entropy (T-s) diagram for a closed cycle. For an ideal Brayton cycle, the following processes happen: a) Isentropic compression (2-3) b) Constant pressure heat addition or combustion (3-4) c) Isentropic expansion (4-5) d) Constant pressure heat rejection (5-1) Figure 2-4 shows the maximum temperature occurring at the end of the combustion process. Material constraints contribute to this temperature limitation. Aircraft gas turbine engines operate on an open cycle called a jet propulsion cycle. The ideal jet propulsion cycle differs from the ideal Brayton cycle simply that the gases do not expand to the ambient pressure in the turbine [6]. Instead, it expands in the turbine to produce just sufficient power to drive the compressor and, if any, auxiliary equipment. The equipment could be a small generator or hydraulic pump. Figure 2-5 shows a turbojet engine and its ideal T-s diagram. Ambient air pressure rises slightly as it decelerates in the diffuser. Air, compressed in the compressor, mixes and burns with jet fuel in the combustion chamber at constant pressure. This high pressure-temperature gas then partially expands in the turbine to produce enough power to run the compressor. For a turbojet, the gas exiting the turbine expands to ambient pressure in the nozzle to produce thrust. The ideal T-s diagram for an APU will be similar to the one below.
10
Figure 2-5: T-s diagram for an ideal jet propulsion cycle along with a turbojet engine schematic (from [6]).
2.3
How current APUs work
Figure 2-6: APU centrifugal compressor rotor with inducer vanes (from [3]).
Air drawn into the engine first goes through a centrifugal compressor rotor. Curved vanes at the compressor intake area, called inducers, guide the air into the compressor. Rotors without inducers are usually very noisy due to flow separation [5]. As the air passes through the compressor, it accelerates outward at high speed and slows down in a ring of stationary vanes called the diffuser. This causes the air pressure to rise. Immediately after the compressor section, an air bleed system is usually present. This releases a portion of the airflow in the engine. Since this bleed air is very energetic, it can pressurize aircraft cabins or drive small cold turbines to develop shaft horsepower. Valves or venturis control this air bleed to within pre-determined limits [3].
11
Figure 2-7: Combustion chambers (from [3]).
The diffuser sends this air to the combustion chamber. The chamber causes it to heat and expand [3]. Combustion chambers vary in design but they all work in the same way. A metal liner inside the engine holds a flame in place by injecting air through a number of holes and orifices. One or more nozzles then spray fuel into the chamber where it burns continuously once ignited. With about a quarter of the air burned through the APU, the rest mixes with the combustion exhaust to lower its temperature so that it can pass through the turbine. Two basic types of combustion chambers exist. They are the can type or the annular type [3]. The can type is mounted on one side of the engine. Heat resistant ducting guides the combustion gases from the combustion chamber on to the turbine nozzle. In some cases, there are two combustion chambers on either side of the APU. It has the advantage of being easy to remove from the APU. An annular combustion chamber placed around the axis of the engine takes the form of a cylinder. It usually guides the exhaust gases directly onto the turbine nozzle. This chamber design allows the APU to maintain a small size. A mechanical or electronic governing system controls the amount of fuel supplied to the combustion chamber. The system must ensure that the engine starts and accelerates smoothly without getting too hot [3]. It must also keep the engine running at constant speed regardless of load. Fuel pumps normally consist of gear pumps or small piston pumps operated by a rotating plate arrangement. The fuel pump usually receives power from a separate electric motor.
12
Figure 2-8: Fuel igniter (from [3]).
The ignition of APUs is similar to that of larger engines. High-energy ignition is the most common ignition. A capacitor, charged to a high voltage (about 3,000V), is discharged into a special sparkplug [3]. The charge comes from a DC inverter, which steps up a battery supply. The sparkplug extrudes into the combustion chamber and is close to the fuel nozzle. A cold engine is quite difficult to light. The energy from the discharged spark is as much as several joules. It occurs across the surface of the plug at a rate of one to two sparks per second. Some models of engines are equipped with automotive type ignition. Here a trembler induction coil provides a very high voltage (about 20,000 to 30,000V) but with a low energy spark [3].
Figure 2-9: APU turbines (from [3]).
The hot gases generated by the combustion process drive one or more turbine wheels that create shaft power. A single shaft connects the turbine, compressor and an external load (via a gearbox)
13
together. A second mechanically independent turbine can also drive the load. Thus, this engine is equipped with two shafts. In most APUs, the compressor uses about two thirds of the mechanical power developed [3]. There are two types of turbines found in APUs. They are the inflow radial (IFR) and axial turbine. The design of the IFR turbine is similar to a centrifugal compressor rotor but is made of heat resistant metal. A nozzle ring directs hot gases from the combustion chamber inwards and tangentially on to the radial blades of the turbine. The gases flow inward and then along the axis of the wheel and out through an exhaust duct. For axial turbines, a disc is fitted with aerofoil cross-sectional blades around its circumference. A ring of similar static blades that form a nozzle directs hot gases onto it. The turbine disc and nozzle are also made of heat resistant metal. Axial turbines can be put together to form multiple stages. Small engines generally employ a maximum of two turbine stages [3]. Compressor bleed air keep the turbine and nozzle assembly cool by allowing it to flow around the components. Twin-shaft APUs are less common than the single-shaft ones. Both normally drive a load via a reduction gearbox. The same gearbox may also drive engine accessories such as fuel and oil pumps. A typical load is an electrical generator or a mechanical pump. A single-shaft engine generally cannot accept any kind of load until it has started and accelerated to operating speed. Most aircraft APUs are of single-shaft designs. Twin-shaft APUs are especially useful for starting larger engines and are known as gas turbine starters (GTS). Most of the twin-shaft APUs work as a GTS unit [3]. Lubrication of APU bearings occur in a similar way to larger propulsion engines. That is, by spraying small oil jets onto them. A pressure pump with a relief valve pressurizes the system feeding the jets. Oil normally returns to a reservoir under gravity or collected by a second larger capacity pump. The larger capacity pump is required as the oil picks up a lot of air and can become foamy. The oil circulating around an APU usually becomes hot such that it passes through some sort of cooling device like a fancooled radiator. Oil pumps are generally gear types. However, compressor air can also pressurize the lubricating oil. On some models, a separate electric motor circulates the oil around the engine. Oil seals keep the oil around the bearing assemblies so that it would not enter the combustion process. Carbon seals are common in APUs. A ring or disc of carbon is spring loaded against a highly polished rotating surface through which oil cannot escape. APU lubricating oils are synthetic and thinner than the ones used in piston engines. APUs are often started by electric motors. A heavy-duty motor can accelerate the APU to light up speed and assist the engine until it becomes self-sustaining. Most APUs self sustain at about 25 to 30% of their rated speed [3]. Self-sustaining speed is the point where the compressor begins to develop significant gauge pressure. When this happens, the mechanical load on the starter motor reduces and its power automatically cuts off.
14
Chapter 3 3.1
Formulas used for the APU
General information As mentioned before, the new APU combines a compressor, burner and turbine into a single
centrifugal impeller (rotor). The rotor consists of a number of blades (usually curved), also called vanes, arranged in a regular pattern around a rotating shaft, as seen in Figure 1-7. First, it is essential to become familiar with the variables and their accompanying subscripts for this research in the Nomenclature section. The subscripts describe the following [8]: a) Rotor components b) Location within the APU (station number) c) Coordinate system for the velocities This rotor will use the cylindrical coordinate system for convenience. There are no axial velocity components (z-direction) within the rotor since it is radial in design. Figure 3-1 shows the absolute velocity in this coordinate system.
Figure 3-1: Cylindrical coordinate system (from [5]).
A velocity triangle graphically relates the velocities C, W and U. Figure 3-2 shows the general shape of the rotor along with the velocity triangle:
15
Figure 3-2: Shape of rotor with velocity triangle (from [5]).
Figure 3-2 indicates a backward leaning configuration. This means the angle β here is positive. The angle of the relative velocity is the same as the blade angle. Equations in this chapter are valid for any configuration of velocity triangles. Figure 3-1 and Figure 3-2 give the following relationships for the velocities: Cθ = U-Wθ Cr = Wr Cz = Wz
(1)
C2 = Cr2+Cθ2+Cz2 W2 = Wr2+Wθ2+Wz2 The following sections show the equations needed to analyze the turbomachinery cycle of this new rotor. Each portion of the rotor has its own set of equations. The entire analysis in this study ignores the effects of gravity and the gas is continuous (motion of individual molecules does not have to be considered). In addition, the viscosity of the flow, magnetic and electrical effects are also negligible.
16
3.2
Ambient air and diffuser
Table 3-1: Ambient air equation input parameters.
Input
Description
M0
Freestream Mach number
T0
Freestream temperature
P0
Freestream pressure
γ0
Specific heat ratio
s0
Freestream entropy
R
Air gas constant
τd
Diffuser stagnation temperature ratio
πd
Diffuser stagnation pressure ratio
Like most gas turbine engines, this APU has a diffuser at the inlet. The diffuser assumptions here are: a) Steady flow b) Calorically perfect Ambient air first passes through the diffuser before entering the compressor. The equations used to determine ambient air and diffuser flow properties are: a) Specific heat of ambient air:
⎛⎜ γ 0 ⎞ ⋅R ⎜ γ0 − 1 ⎝ ⎠
Cp0
(2)
b) Ratio of To0 (stagnation temperature at station 0) to T0, τr: τr
1+
⎛ γ1 − 1 ⎞ 2 ⎜ ⋅M ⎝ 2 ⎠ 0
(3)
c) Ratio of Po0 (stagnation pressure at station 0) to P0, πr: γ0
πr
τr
γ 0− 1
(4)
d) Ambient air density ρ0
P0 R⋅ T0
e) Diffuser exit stagnation temperature:
17
(5)
To2
τd ⋅ τr⋅ T0
(6)
Po2
πd ⋅ πr⋅ P0
(7)
f) Diffuser exit stagnation pressure:
3.3
Compressor
Table 3-2: Compressor equation input parameters.
Input
Description
M2rel
Inlet relative Mach number
β2t
Inlet tip (blade edge at inlet outer diameter) flow angle
β3
Outlet blade angle
ec
Polytropic efficiency
ζc
Inlet hub-to-tip ratio
U3/(γ0*R*To2)^(1/2)
Allowable outlet tip speed ratio
Cθ2t/(γ0*R*To2)^(1/2)
Inlet swirl parameter
Wr3/U3
Outlet flow coefficient
The first portion of the rotor is the centrifugal compressor similar to the one in Figure 1-7. Assumptions for the compressor are: a) Steady-flow adiabatic compression b) Calorically perfect The Hill and Peterson textbook [5] provided all the following equations necessary to determine the compressor properties except two that needed derivation as shown in Appendix A: a) Inlet tip temperature: 2 ⎡ Cz2t γ 0 − 1 ⎡⎢ ⎛ ⎞ ⎢ ⎜ T2t To2⋅ 1 − ⋅ + ⎢ 2 ⎢ ⎜ γ ⋅ R⋅ T 0 o2 ⎣ ⎣⎝ ⎠
⎛⎜ Cθ2t ⎞ ⎜ γ 0⋅ R⋅ To2 ⎝ ⎠
2⎤ ⎤
⎥⎥ ⎥⎥ ⎦⎦
(8)
For Equation (8): 2⎤ ⎡ Cθ2t γ0 − 1 ⎛ ⎞ ⎢ ⎥ ⋅M 2 cos ( β 2t) ⋅ 1 − ⋅⎜ ⎢ ⎥ 2rel 2 ⎜ γ ⋅ R⋅ T ⎣ ⎝ 0 o2 ⎠ ⎦ 2
Cz2t γ 0⋅ R⋅ To2
1+
γ0 − 1 2
(
( ))2
⋅ M 2rel⋅ cos β 2t
b) Inlet tip pressure:
18
(9)
Po2
P2t
γ0
(10)
γ 0− 1
⎛ To2 ⎞ ⎜T ⎝ 2t ⎠ c) Inlet tip density:
P2t
ρ 2t
(11)
R⋅ T2t
d) Inlet relative stagnation temperature: To2rel
⎛
γ0 − 1
⎝
2
T2t⋅ ⎜ 1 +
2⎞
⋅ M 2rel
⎠
(12)
e) Inlet relative stagnation pressure: γ0
⎛ To2rel ⎞ Po2rel P2t⋅ ⎜ ⎝ T2t ⎠
γ 0− 1
(13)
f) Stagnation temperature ratio:
τc
(
⎛
)⎜
1 + γ0 − 1 ⋅⎜
⎝
⎡ 2⎢ W r3 U3 ⎞ ⎢ ⋅ 1− ⋅ tan ( β 3) − ⎢ U3 γ 0⋅ R⋅ To2 ⎠ ⎢ ⎢ ⎣
Cz2t ⎛⎜ Cθ2t ⎞ Cθ2t ⎤ + ⋅ tan ( β 2t) ⋅ ⎥ ⎜ γ 0⋅ R⋅ To2 γ 0⋅ R⋅ To2 γ 0⋅ R⋅ To2 ⎥ ⎝ ⎠ ⎥ (14) 2 U3 ⎛⎜ ⎞ ⎥ ⎥ ⎜ γ 0⋅ R⋅ To2 ⎝ ⎠ ⎦
g) Adiabatic efficiency: ec
ηc =
τc − 1
(15)
τc − 1
h) Stagnation pressure ratio: γ0
πc
i)
⎡⎣ 1 + η c( τc − 1) ⎤⎦
γ 0− 1
(16)
Absolute outlet Mach number: a
M3 1−
γ0 − 1 2
19
⋅a
(17)
For Equation (17):
a
j)
2 W r3 ⎛ ⎞ ⋅ tan ( β 3) + 2 ⎜1 − U3 ⎞ ⎝ ⎠ ⋅
U3 ⎛⎜ ⎜ γ 0⋅ R⋅ To2 ⎝ ⎠
⎛ W r3 ⎞ ⎜ U ⎝ 3⎠
2
(18)
τc
Relative outlet Mach number (Equation 1 in Appendix A): U3
Wr3 U3
( )
M 3rel
cos β 3
γ 0⋅ R⋅ T o2
⋅
τc 1+
γ0 − 1 2
(19) 2
⋅ M3
k) Outlet absolute stagnation temperature:
l)
T o3 = τ c⋅ τ d⋅ τ r⋅ T 0
(20)
P o3 = π c⋅ π d⋅ π r⋅ P 0
(21)
Outlet absolute stagnation pressure:
m) Outlet relative stagnation temperature (Equation 2 in Appendix A): To3rel
⎡
γ0 − 1
⎢ ⎢ ⎣
γ0 − 1 ⎛ 2⎞ 2⋅ ⎜ 1 + ⋅ M3 2 ⎝ ⎠
To3⋅ ⎢ 1 −
⎤ 2⎞ ⎥
⋅ ⎛ M 3 − M 3rel ⎝ 2
⎠⎥ ⎥ ⎦
(22)
n) Outlet relative stagnation pressure: Po3
Po3rel
γ0
⎛ ⎞ ⎜T ⎝ o3rel ⎠ To3
γ 0− 1
(23)
o) Outlet temperature: To3rel
T3 1+
γ0 − 1 2
2
⋅ M 3rel
p) Outlet pressure:
20
(24)
Po3rel
P3
γ0
⎛ To3rel ⎞ ⎜ T ⎝ 3 ⎠
(25)
γ 0− 1
q) Outlet relative velocity:
(
)
2⋅ Cp0⋅ To3rel − T3
W3
(26)
r) Outlet density: P3
ρ3 =
(27)
R⋅ T 3
s) Dimensionless impeller rotation: 1
m3 Po2
t)
⋅
⎛
Ω 1
=
2⋅ π ⋅ γ c⋅ ⎜
⎜ ⎝
( γ c⋅ R⋅ To2) 4
Cθ2t γ c⋅ R⋅ T o2
+
⌠ ⎮ 2⎮ Cz2t ⎞ ⎮ ⋅ tan β 2t ⋅ ⎮ γ c⋅ R⋅ T o2 ⎠ ⎮ ⌡ζ
1
⎡ ⎢1 − ⎢ ⎣
( )
γ c − 1 ⎡⎛ ⎢ ⋅ ⎜ ⎢⎜ 2 ⎣⎝
⎞
Cz2t γ c⋅ R⋅ T o2
⎠
2
+
⎛ ⎜ ⎜ ⎝
⎤⎤ Cθ2t ⎞ 2 ⎥⎥ ⋅(2 − y ) ⎥⎥ γ c⋅ R⋅ T o2 ⎠ ⎦⎦ 2
γ c− 1
⋅
⎛ ⎜ ⎜ ⎝
⎞
Cz2t γ c⋅ R⋅ T o2
2
⎛
− 2⋅ ⎜
⎜ ⎝
⎠
Cθ2t γ c⋅ R⋅ T o2
⎞
2
(
2
)
⋅ y − 1 ⋅ y dy
(28)
⎠
c
Radius ratio: U3 γ 0⋅ R⋅ T o2
r3 Cθ2t
r2t
γ 0⋅ R⋅ To2
+
Cz2t γ 0⋅ R⋅ To2
(29)
( )
⋅ tan β 2t
u) Outlet blade height to radius ratio: 1
⎛ γ0 + 1 ⎞ ⎜ ⎝ 2 ⎠
b3
γ 0− 1
2 ⎛ r3 ⎞ ⋅ ⎛ 1 − ζc ⎞ ⋅ ⎜ ⎝ ⎠ r ⎝ 2t ⎠
−2
r3
1
⎡ ⎢ 2⋅ 1 + ⎢ ⎣
⎛ 1 + ηc ⎞ ⎛ ⎜ ⋅ ( γ 0 − 1) ⋅ ⎜ ⎜ ⎝ 2 ⎠ ⎝
U3 W r3 ⎞ ⎛ ⎞ ⎥⎤ ⋅⎜ 1 − ⋅ tan ( β 3) ⎥ U3 γ 0⋅ R⋅ To2 ⎝ ⎠⎦ ⎠ 2
(30)
γ 0− 1
v) Outlet rotor speed: U3
U3 ⎛⎜ ⎞ ⋅ γ 0⋅ R⋅ To2 ⎜ γ 0⋅ R⋅ To2 ⎝ ⎠
w) Inlet rotor tip speed:
21
(31)
⎛ r3 ⎞ U 2t = U 3⋅ ⎜ ⎜ r2t ⎝ ⎠
−1
(32)
x) Mass flow rate to outlet area ratio: m3 A3
ρ 3⋅ W 3
(33)
y) Outlet entropy: s3
( )
( )
s 0 + Cp0⋅ ln τc − R⋅ ln πc
(34)
z) Specific power: Wc = −C p0 ⋅ (To3 − To 2 ) m3
3.4
(35)
Burner and turbine After the compressor, the rotor vanes extend to include a burner followed by a turbine. It is
necessary to select the governing equations for both components. The best way is to choose the generalized quasi-one-dimensional compressible flow equations. In general, these equations are to take into account the following effects: a) Flow area change b) Heat exchange c) Work done by or on the flow d) Drag force on the flow e) Mass addition (fuel) into the flow
Figure 3-3: Burner and turbine control volume between two vanes across a small step change (from [9]).
22
First, define a control volume over a differentially short portion of the flow as seen in Figure 3-3. The assumptions here are steady flow and that the added fuel does not alter the gas properties significantly. Then select the governing equations based on the following principles: a) Conservation of angular momentum b) Conservation of energy (first law of thermodynamics) c) Equation of state d) Conservation of mass e) Conservation of linear momentum f) Relative stagnation temperature equation g) Relative stagnation pressure equation h) Absolute stagnation temperature equation i)
Relative Mach number equation
j)
Absolute stagnation pressure equation
k) Entropy equation The textbook by Oosthuizen and Carscallen [9] provides some of the equations while the others required derivation, as seen in Appendix B. The equations here assumed constant specific heats. By assuming the change in Cp is very small across the differential step size, it is variable using the following formula [6]: 28.11⋅
kJ kmol⋅ K
−2
+ 0.1967⋅ 10 ⋅
Cp =
kJ kmol⋅ K
−5
2
⋅ T + 0.4802⋅ 10 ⋅
kJ kmol⋅ K
28.97
2
3
−9
⋅ T − 1.966⋅ 10 ⋅
kJ kmol⋅ K
4
⋅T
3
(36)
kg kmol
Therefore, the burner and turbine equations are thermally perfect. The specific heat ratio is then: γ =
Cp Cp − R
(37)
3.4.1 Burner equations For the new rotor, the flow in the burner is subsonic. This allows the combustion process to take place since it is difficult to place a flame in the flow to ignite the fuel if the velocities are too fast. Equations 1 through 9 and 11 from Appendix B describe the flow through the burner vanes. The drag coefficient seen in the conservation of linear momentum equation represents the flame holder located only at the beginning of the burner. The equations for this component are: a) Conservation of energy and angular momentum combination:
23
⎛ 2⋅ h ⋅ η + U2 − W2 ⎞ 2 d( m) W d( W) d( T ) U ⋅ d( U) ⎜ HV b −1 ⋅ − ⋅ − =− ⎜ 2⋅ Cp ⋅ T Cp ⋅ T W Cp ⋅ T T ⎝ ⎠ m
(38)
b) Equation of state: d( ρ ) ρ
d( T )
+
−
d( P)
=0
(39)
P
T
c) Conservation of mass: d( W)
d( ρ )
+
ρ
W
d( m)
−
=−
d( A )
m
(40)
A
d) Conservation of linear momentum: d( W)
d( m)
+
W
P
+
2
m
ρ ⋅W
⋅
d( P) P
( )
1 = − ⋅ d CD 2
(41)
e) Relative stagnation temperature equation:
( )
d To
−
To
(
)
T orel d T orel U⋅ W⋅ sin( β ) d( W) U⋅ d( U) − W⋅ d( U ⋅ sin( β ) ) ⋅ + ⋅ = T orel W To Cp ⋅ T o Cp ⋅ T o
(42)
f) Relative stagnation pressure equation:
(
)
d Porel Porel
2
+
γ ⋅ M rel 2
⋅
(
)
d T orel T orel
2
2
+ γ ⋅ M rel ⋅
d( m)
=−
γ ⋅ M rel
m
2
( )
⋅ d CD
(43)
g) Absolute stagnation temperature equation:
( )
d To To
−
T d( T ) W⋅ ( W − U ⋅ sin( β ) ) d( W) U ⋅ d( U ) − W⋅ d( U⋅ sin( β ) ) ⋅ − ⋅ = To T W Cp ⋅ T o Cp ⋅ T o
(44)
h) Relative Mach number equation: M rel =
i)
W
Absolute stagnation pressure equation:
⎛ To ⎞ P o = P orel⋅ ⎜ ⎜ T orel ⎝ ⎠ j)
(45)
γ ⋅ R⋅ T
γ γ −1
(46)
Entropy equation: s
⎛ To ⎞ ⎛ Po ⎞ s 3 + Cp ⋅ ln⎜ − R⋅ ln⎜ ⎝ To3 ⎠ ⎝ Po3 ⎠
Notice that Equations (38) through (44) are differential equations that require a numerical solution.
24
(47)
3.4.2 Burner input parameters and method of solving equations In Equations (38) through (47), the specified variables are A, β, U, To CD, hHV, and ηb. The outline below shows how to deal with these variables: a) Specify the radius ratio r4/r3 and the number of iteration steps, nb. b) Next, consider the radius variation along the burner flow to be r = r3+δr (δr is the difference between r as it varies along the burner and r3). At the inlet, δr (or δr3) is zero when r = r3. The equation for r/r3 is: r
δr
= 1+
r3
(48)
r3
c) At station 4, δr/r3 is δr4/r3 = (r4/r3)-1. It follows that the small step change is d(δr/r3) = (δr4/r3)/nb or: r4
⎛ δr ⎞ d⎜ ⎝ r3 ⎠
r3
−1
(49)
nb
d) It is now possible to vary the quantity δr/r3 starting with zero in steps of d(δr/r3) from index i = 0 to nb as follows: δr r3
⎛ δr ⎞
i⋅ d ⎜
(50)
⎝ r3 ⎠
This also allows the variation of r/r3 from one to r4/r3. e) Vary the flow area as the ratio A/A3 using the second order polynomial below: A A3
⎛ δr ⎞ + Y2⋅ ⎛ δr ⎞ ⎜r ⎝ r3 ⎠ ⎝ 3⎠
2
= 1 + Y1⋅ ⎜
(51)
The variables Y1 and Y2 are specified coefficients. f) Obtain the variation of angle β using the following polynomial:
⎛ δr ⎞ + S2⋅ ⎛ δr ⎞ β = β 3 + S1⋅ ⎜ ⎜r ⎝ r3 ⎠ ⎝ 3⎠
2
(52)
The variables S1 and S2 are specified coefficients. The initial value of β is β3. g) Define the change in rotor speed using:
⎛ r⎞ U = U 3⋅ ⎜ ⎝ r3 ⎠
(53)
h) Assuming a linear variation of To (stagnation temperature) with initial value To3 and final value To4 (maximum stagnation temperature in burner) gives:
25
T o = T o3 +
T o4 − T o3 r4
−1
⎛ δr ⎞
⋅⎜
⎝ r3 ⎠
(54)
r3
Appendix B shows the derivation of Equation (54). i)
Specify a drag coefficient, CD for the flame holder along with the fuel heating value, hHV and burner
efficiency, ηb [10]. The second-order polynomials in Equations (51) and (52) are chosen for convenience; other variations with r are possible. Table 3-3 summarizes the input parameters for the burner: Table 3-3: Burner equation input parameters.
Input
Description
r4/r3
Burner radius ratio
Y1, Y2 A/A3 second order polynomial coefficients S1, S2
β second order polynomial coefficients
To4
Maximum burner stagnation temperature
CD
Flame holder drag coefficient
hHV
Fuel heating value
ηb
Burner efficiency
nb
Number of iteration steps
With the input parameters established, it is necessary to show how to solve Equations (38) through (47). For Equations (38) through (44), they give the following form: ⎡ 2 W ⎢ −1 − 0 ⎢ Cp ⋅ T ⎢ 0 −1 ⎢ 1 ⎢ 0 1 0 ⎢ P 1 ⎢ 0 2 ⎢ ρ⋅W ⎢ U⋅ W ⋅ sin ( β ) ⎢ 0 0 ⎢ Cp ⋅ To ⎢ ⎢ ⎢ 0 0 0 ⎢ ⎢ T W ⋅ ( W − U⋅ sin ( β ) ) 0 − ⎢− To Cp ⋅ To ⎣
2
0
0
1
0
0
2⋅ h HV⋅ η b + U − W 2⋅ Cp ⋅ T
0
0
1
0
0
−1
0
0
0
1
0
0
0 −
Torel To 2
0 0
γ ⋅ M rel 2 0
1 0
2
γ ⋅ M rel 0
Inverting the matrix in Equation (55) gives:
26
2
⎤ ⎛⎜ ⎥ −1 ⎜ ⎥ ⎜ ⎥ ⎜ ⎥⎜ ⎥ ⎜ ⎥ ⎥⎜ ⎜ ⎥ ⎜ ⎥ ⋅⎜ ⎥ ⎜ ⎥ ⎥⎜ ⎜ ⎥ ⎥⎜ ⎜ ⎥ ⎜ ⎥⎜ ⎥ ⎜ ⎦⎝
d ( T)
⎞
T
d ( m)
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
m
⎠
d( W ) W d ( P) P d(ρ )
(
ρ
)
d Torel Torel
(
)
d Porel Porel
U⋅ d ( U) ⎛ − ⎜ Cp ⋅ T ⎜ ⎜ 0 ⎜ d (A) ⎜ − A ⎜ ⎜ 1 − ⋅ d ( CD) ⎜ 2 ⎜ ⎜ d ( To ) U⋅ d( U) − W ⋅ d ( U⋅ sin ( β ) ) ⎜− T + Cp ⋅ To o ⎜ ⎜ 2 γ ⋅ M rel ⎜ − ⋅ d ( CD) ⎜ 2 ⎜ ⎜ − d ( To ) + U⋅ d( U) − W ⋅ d ( U⋅ sin ( β ) ) ⎜ To Cp ⋅ To ⎝
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ (55) ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
d ( T)
⎞
T
d ( m)
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
m
⎠
d( W ) W d ( P) P d(ρ )
(
ρ
)
d Torel Torel
(
)
d Porel Porel
⎡ 2 W ⎢ −1 − 0 Cp ⋅ T ⎢ ⎢ 0 −1 ⎢ 1 ⎢ 0 1 0 ⎢ P 1 ⎢ 0 2 ⎢ ρ⋅W ⎢ U⋅ W ⋅ sin ( β ) ⎢ 0 0 ⎢ Cp ⋅ To ⎢ ⎢ ⎢ 0 0 0 ⎢ ⎢ T W ⋅ ( W − U⋅ sin ( β ) ) − 0 ⎢− To Cp ⋅ To ⎣
2
2⋅ h HV⋅ η b + U − W
0
0
0
1
0
0
0
1
0
0
−1
0
0
0
1
0
0
1
γ ⋅ M rel
0
0
0 −
Torel To
2
2⋅ Cp ⋅ T
2
γ ⋅ M rel
0
2
0
0
2
⎤ − 1⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
−1
U⋅ d ( U) ⎛ ⎞ − ⎜ Cp ⋅ T ⎜ ⎜ 0 ⎜ d (A) ⎜ − A ⎜ ⎜ 1 − ⋅ d ( CD) ⎜ 2 ⋅⎜ (56) d T ⎜ ( o) U⋅ d ( U) − W ⋅ d ( U⋅ sin ( β ) ) − + ⎜ T Cp ⋅ To o ⎜ ⎜ 2 γ ⋅ M rel ⎜ − ⋅ d ( CD) ⎜ 2 ⎜ ⎜ − d( To) + U⋅ d ( U) − W ⋅ d ( U⋅ sin ( β ) ) ⎜ To Cp ⋅ To ⎝ ⎠
Initial flow values for the burner are the compressor exit properties. Solving Equation (56) numerically from r/r3 = 1 to r4/r3 in steps of d(δr/r3) gives the following flow properties: T
W
i
i
T
i− 1
W
i− 1
+
⎛ d ( T) ⎞ ⋅ T ⎜ T ⎝ ⎠ i−1
+
⎛ d(W ) ⎞ ⋅ W ⎜ ⎝ W ⎠ i−1
P
P
+
⎛ d ( P) ⎞ ⋅ P ⎜ ⎝ P ⎠ i−1
ρi
ρ i− 1 +
⎛ d( ρ ) ⎞ ⋅ ρ ⎜ i− 1 ⎝ ρ ⎠
i
i− 1
Porel + i− 1
Porel i
To
i
To
i− 1
+
⎛ d ( Porel) ⎞ ⋅ Porel ⎜ P i− 1 ⎝ orel ⎠ ⎛ d ( To ) ⎞ ⋅ To ⎜ T ⎝ o ⎠ i−1
⎛ m ⎞ ⎛ m ⎞ + ⎛ d ( m) ⎞ ⋅ ⎛ m ⎞ ⎜ ⎟⎜ ⎟ ⎜A ⎟ ⎜A ⎟ ⎝ 3 ⎠ i ⎝ 3 ⎠ i−1 ⎝ m ⎠ ⎝ A 3 ⎠ i−1 W M rel
i
i
γ i⋅ R⋅ T
i γi
Po i
s
i
⎛ To ⎞ i Porel ⋅ ⎜ i ⎜ Torel i⎠ ⎝
γ i−1
⎛ To i ⎞ ⎛ Poi ⎞ ⎜ ⎜ s 3 + Cp ⋅ ln − R⋅ ln ⎜ ⎜P i ⎝ To3 ⎠ ⎝ o3 ⎠
27
(57)
The subscripts i-1 and i refer to the index before and after the differential control volume seen in Figure 3-3. All the variables on the right hand side of Equation (56) are at index i-1 (except the constants) and it follows that:
d( A )
−
A
⎛⎡ A ⎤⎞ − ⎛⎡ A ⎤⎞ ⎜⎢A ⎥ ⎜⎢ ⎥ ⎝ ⎣ 3⎦ ⎠ i ⎝ ⎣ A3 ⎦ ⎠ i− 1 ⎛⎡ A ⎤⎞ ⎜⎢A ⎥ ⎝ ⎣ 3⎦ ⎠ i− 1
( )
d ( U⋅ sin ( β ) )
(
U ⋅ sin β i − U ⋅ sin β i− 1 i i− 1
)
(58)
U −U
d ( U)
i− 1
i
( )
To − To i i− 1
d To
At the exit, the burner flow variables in Equation (57) will use the subscript four. Knowing that the amount of fuel added is mf = m4-m3 gives the burner fuel-to-air ratio defined as f = mf/m3 or: m4
f
m3
−1
(59)
The ratio m4/m3 = (m4/A3)/(m3/A3). In the event f is an input, then the To distribution would require calculation. Next, using the definition of angular momentum [12] from Appendix B and Cθ = UW*sin(β), the specific power of the burner is: ⎡m ⎤ Wb = − ⎢ 4 ⋅ U 4 ⋅ ( U 4 − W4 ⋅ sin(β 4 )) − U 3 ⋅ ( U 3 − W3 ⋅ sin(β 3 ))⎥ m3 ⎣ m3 ⎦
(60)
Appendix B shows the derivation of Equation (60).
3.4.3 Turbine equations To produce as much power as possible, the flow will have to exit at high relative Mach numbers to help spin the rotor. Like the burner, Equations 1 through 8 along with 10 and 12 in Appendix B describe the flow through the turbine but without the heat addition, mass addition and drag force terms. They are: a) Conservation of energy and angular momentum combination: −
W
2
Cp ⋅ T
⋅
d(W ) W
−
d ( T)
−
T
U⋅ d ( U) Cp ⋅ T
(61)
b) Equation of state: d(ρ ) ρ
+
d ( T) T
−
d ( P)
28
P
0
(62)
c) Conservation of mass: d(W ) W
+
d(ρ )
−
ρ
d( A )
(63)
A
d) Conservation of linear momentum: d(W ) W
P
+
ρ⋅W
2
⋅
d ( P) P
0
(64)
e) Relative stagnation temperature equation:
( )
d To To
−
(
)
Torel d Torel U⋅ W ⋅ sin ( β ) d ( W ) ⋅ + ⋅ Cp ⋅ To W To Torel
U⋅ d ( U) − W ⋅ d ( U⋅ sin ( β ) ) Cp ⋅ To
(65)
f) Relative stagnation pressure equation:
(
)
d Porel Porel
2
+
γ ⋅ M rel 2
⋅
(
)
d Torel Torel
(66)
0
g) Absolute stagnation temperature equation:
( )
d To To
−
W ⋅ ( W − U⋅ sin ( β ) ) d ( W ) T d ( T) ⋅ − ⋅ To T Cp ⋅ To W
U⋅ d ( U) − W ⋅ d ( U⋅ sin ( β ) ) Cp ⋅ To
(67)
h) Relative Mach number equation: M rel =
i)
W
(68)
γ ⋅ R⋅ T
Absolute stagnation pressure equation: γ
⎛ To ⎞ Po Po4⋅ ⎜ ⎝ To4 ⎠ j)
γ −1
(69)
Entropy equation s
⎛ To ⎞ ⎛ Po ⎞ s 4 + Cp ⋅ ln⎜ − R⋅ ln⎜ ⎝ To4 ⎠ ⎝ Po4 ⎠
(70)
Notice that Equations (61) through (67) are differential equations that require a numerical solution.
3.4.4 Turbine input parameters The turbine flow will initially be subsonic. It can then proceed to a supersonic flow region. If the subsonic flow approaches the sonic point and needs to go supersonic, the calculations must terminate and cannot continuously cross the sonic point. The chosen termination point is when Mrel = 0.99 (station number 4.5). In Equations (61) through (70), the specified variables are A, β, and U.
29
3.4.4.1
Subsonic turbine Assuming first that the sonic point does not occur, perform the following steps to obtain the input
parameters: a) Specify the radius ratio r5/r4 and the number of iterations steps, nt. b) Next, consider the radius along the turbine flow to be r = r4+δr (δr is the difference between r as it varies along the burner and r4). At the inlet, δr (or δr4) is zero when r = r4. The variation of r/r4 is: r
1+
r4
δr
(71)
r4
c) At station 5, δr/r4 is δr5/r4 = (r5/r4)-1. It follows that the small step change is d(δr/r4) = (δr5/r4)/nt or: r5
⎛ δr ⎞ d⎜ ⎝ r4 ⎠
r4
−1
(72)
nt
d) It is now possible to vary the quantity δr/r4 starting with zero in steps of d(δr/r4) from index i = 0 to nt as follows:
⎛ δr ⎞
δr
i⋅ d ⎜
(73)
⎝ r4 ⎠
r4
This then allows the variation of r/r4. e) For the flow to accelerate, decrease the area as the ratio A/A4 using the second order polynomial below: A A4
⎛ δr ⎞ + K2⋅ ⎛ δr ⎞ ⎜r ⎝ r4 ⎠ ⎝ 4⎠
2
1 + K1⋅ ⎜
(74)
The variables K1 and K2 are specified coefficients. f) Obtain the variation of angle β using the following polynomial: β
⎛ δr ⎞ + B2⋅ ⎛ δr ⎞ ⎜r ⎝ r4 ⎠ ⎝ 4⎠
2
β 4 + B1⋅ ⎜
(75)
The variables B1 and B2 are specified coefficients. The initial value of β is β4. g) Define the change in rotor speed using: U
r U4⋅ ⎛⎜ ⎞ ⎝ r4 ⎠
(76)
The second-order polynomials in Equations (74) and (75) are chosen for convenience; other variations with r are possible. They are variable using any type of functions.
30
3.4.4.2
Supersonic turbine If the turbine reaches station 4.5, it can only go supersonic if it satisfies the condition P0/Po4.5rel <
0.528. Otherwise, the flow stops at station 4.5 and r5/r4 is shorter than the one specified in 3.4.4.1. Figure 3-4 from the Anderson textbook [11] provides the basis for this condition:
Figure 3-4: Convergent-divergent nozzle with supersonic exit (from [1]).
For the input parameters, repeat the steps in Section 3.4.4.1 by changing the subscripts 4 to 4.5. This means r5/r4 in Step a) becomes r5/r4.5. However, r5/r4.5 needs to be calculated. First, obtain the ratio r4.5/r4, which is the final value of r/r4. Now, the ratio r5/r4.5 is:
⎛ r5 ⎞ ⎛ r4.5 ⎞ ⎜ r ⋅⎜ r r4.5 ⎝ 4⎠ ⎝ 4 ⎠ r5
31
−1
(77)
This ensures that by the end of the turbine calculations, the radius ratio is the specified r5/r4. For the flow to accelerate, the area will have to increase. Replace Equation (74) in Step e) with the supersonic area ratio A/A4.5 polynomial:
⎛ δr ⎞ + KK2⋅ ⎛ δr ⎞ 1 + KK1⋅ ⎜ ⎜r A 4.5 ⎝ r4.5 ⎠ ⎝ 4.5 ⎠ A
2
(78)
The variables KK1 and KK2 are specified coefficients. The decreasing flow area in the subsonic region and increasing flow area in the supersonic region means the turbine resembles a convergent-divergent nozzle. The overall turbine area ratio is then: A5 A4
⎛ A 4.5 ⎞ ⎛ A 5 ⎞ ⎜ A ⎜A ⎝ 4 ⎠ ⎝ 4.5 ⎠
(79)
The ratio A4.5/A4 is the value of Equation (74) at station 4.5.
3.4.5 Method of solving turbine equations Table 3-4 summarizes the input parameters for both the subsonic and supersonic turbine. Table 3-4: Turbine equation input parameters.
Input
Description
r5/r4
Turbine radius ratio
K1,K2
A/A4 second order polynomial coefficients
KK1,KK2 A/A4.5 second order polynomial coefficients B1,B2
β second order polynomial coefficients
nt
number of iteration steps
With the input parameters established, it is necessary to show how to solve Equations (61) through (70). For Equations (61) through (67), they give the following form:
32
2 ⎡ W ⎢ −1 0 − Cp ⋅ T ⎢ ⎢ 1 −1 0 ⎢ 0 1 ⎢ 0 ⎢ P 1 ⎢ 0 2 ρ⋅W ⎢ ⎢ U⋅ W ⋅ sin ( β ) ⎢ 0 0 Cp ⋅ To ⎢ ⎢ ⎢ 0 0 ⎢ 0 ⎢ ⎢ T W ⋅ ( W − U⋅ sin ( β ) ) 0 ⎢− T − Cp ⋅ To o ⎣
0
0
0
1
0
0
1
0
0
0
0
0
0 −
Torel To
0
2
0
γ ⋅ M rel
0
2 0
1 0
⎛ ⎤⎜ 0⎥ ⎜ ⎥⎜ ⎥⎜ 0 ⎥⎜ 0⎥ ⎜ ⎥⎜ 0⎥ ⎜ ⎥⎜ ⎥ ⋅⎜ 1⎥ ⎜ ⎥⎜ ⎥⎜ ⎥⎜ 0⎥ ⎜ ⎥⎜ ⎥⎜ 1⎥ ⎜ ⎦⎜ ⎝
d ( T) T d( W ) W d ( P) P d(ρ )
(
ρ
)
d Torel Torel
(
)
d Porel Porel
( )
d To To
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
U⋅ d ( U) ⎛ − ⎜ Cp ⋅ T ⎜ ⎜ 0 ⎜ d( A ) ⎜ − A ⎜ ⎜ 0 ⎜ ⎜ U⋅ d ( U) − W ⋅ d ( U⋅ sin ( β ) ) Cp ⋅ To ⎜ ⎜ 0 ⎜ ⎜ U⋅ d ( U) − W ⋅ d ( U⋅ sin ( β ) ) ⎜ Cp ⋅ To ⎝
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
(80)
⎠
Inverting the matrix in Equation (80) gives:
⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
d To
( )
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
To
⎠
d ( T) T d(W ) W d ( P) P d( ρ )
(
ρ
)
d Torel Torel
(
)
d Porel Porel
2 ⎡ W ⎢ −1 − 0 Cp ⋅ T ⎢ ⎢ 1 0 −1 ⎢ 1 0 ⎢ 0 ⎢ P 1 ⎢ 0 2 ⎢ ρ⋅W ⎢ U⋅ W ⋅ sin ( β ) ⎢ 0 0 Cp ⋅ To ⎢ ⎢ ⎢ ⎢ 0 0 0 ⎢ ⎢ T W ⋅ ( W − U⋅ sin ( β ) ) 0 ⎢− T − Cp ⋅ To ⎣ o
0
0
0
1
0
0
1
0
0
0
0
0
0 −
Torel To
0
2
0
γ ⋅ M rel 2
0
0
1 0
⎤ 0⎥ ⎥ ⎥ 0 ⎥ 0⎥ ⎥ 0⎥ ⎥ ⎥ 1⎥ ⎥ ⎥ ⎥ 0⎥ ⎥ ⎥ 1⎥ ⎦
−1
U⋅ d ( U) ⎛ − ⎜ Cp ⋅ T ⎜ ⎜ 0 ⎜ d( A ) ⎜ − A ⎜ ⎜ 0 ⋅ ⎜ ⎜ U⋅ d( U) − W ⋅ d( U⋅ sin ( β ) ) Cp ⋅ To ⎜ ⎜ 0 ⎜ ⎜ U⋅ d( U) − W ⋅ d( U⋅ sin ( β ) ) ⎜ Cp ⋅ To ⎝
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
(81)
⎠
Initial flow values for the turbine are the burner exit properties. For the supersonic flow, the initial values are the ones at station 4.5 except for T, W, P, ρ, and Mrel. Modify them according to the following steps: a) If P0/Po4.5rel < 0.528, then Mrel = 1.01. b) Calculate the others using:
33
To4.5rel
T
γ 4.5 − 1
1+
2
2
⋅ M rel
M rel⋅ γ 4.5⋅ R⋅ T
W
Po4.5rel
P
(82)
γ 4.5
γ 4.5 − 1 ⎛ 2⎞ ⎜1 + ⋅ M rel 2 ⎝ ⎠ ρ
γ 4.5− 1
P R⋅ T
Solving Equation (81) numerically across the subsonic and supersonic flows until the turbine radius ratio is the specified r5/r4 gives the following flow properties: T
W
T
i− 1
W
i− 1
i
i
+
⎛ d ( T) ⎞ ⋅ T ⎜ T ⎝ ⎠ i−1
+
⎛ d(W ) ⎞ ⋅ W ⎜ W ⎝ ⎠ i−1
P
P
+
⎛ d ( P) ⎞ ⋅ P ⎜ P ⎝ ⎠ i−1
ρi
ρ i− 1 +
⎛ d( ρ ) ⎞ ⋅ ρ ⎜ i− 1 ⎝ ρ ⎠
i− 1
i
Porel + i− 1
Porel i
To
i
To
i− 1
+
⎛ d ( Porel) ⎞ ⋅ Porel ⎜ P i− 1 ⎝ orel ⎠ ⎛ d ( To ) ⎞ ⋅ To ⎜ T ⎝ o ⎠ i−1 W
M rel
i
(83)
i
γ i⋅ R⋅ T
i γi
Po i
s
i
⎛ To i ⎞ ⎜ Po4 ⋅ ⎜ To4 ⎝ ⎠
γ i−1
⎛ To i ⎞ ⎛ ⎜ ⎜ s 4 + Cp ⋅ ln − R⋅ ln ⎜ i ⎜ To4 ⎝ ⎠ ⎝
Po i
⎞
Po4 ⎠
All the variables on the right hand side of Equation (81) are at index i-1 (except the constants) and it follows that: 34
subsonic:
d( A) A
⎛ A ⎞ −⎛ A ⎞ ⎜A ⎜ ⎝ 4 ⎠ i ⎝ A 4 ⎠ i−1 ⎛ ⎞ ⎜A ⎝ 4 ⎠ i−1 A
, supersonic:
( )
d ( U⋅ sin ( β ) )
d(A)
⎛ A ⎞ −⎛ A ⎞ ⎜A ⎜ ⎝ 4.5 ⎠ i ⎝ A4.5 ⎠ i−1 ⎛ A ⎞ ⎜A ⎝ 4.5 ⎠ i−1
A
(
U ⋅ sin β i − U ⋅ sin β i− 1 i i− 1 d ( U)
(84)
)
U −U i
i− 1
At the exit, the turbine flow variables in Equation (83) will carry the subscript five. Using the definition of angular momentum [12] from Appendix B and Cθ = U-W*sin(β), the specific power of the turbine is: Wt = −(1 + f ) ⋅ [U 5 ⋅ ( U 5 − W5 ⋅ sin(β 5 )) − U 4 ⋅ ( U 4 − W4 ⋅ sin(β 4 ))] m3
Appendix B shows the derivation of Equation (85).
3.4.6 Burner and turbine output summary In summary, the burner exit variables are: Table 3-5: Burner exit flow variables.
Output
Description
M4rel
Exit relative Mach number
τbrel
Relative stagnation temperature ratio (To4rel/To3rel)
πbrel
Relative stagnation pressure ratio (Po4rel/Po3rel)
τb
Absolute stagnation temperature ratio (To4/To3)
πb
Absolute stagnation pressure ratio (Po4/Po3)
To4rel
Relative stagnation temperature
Po4rel
Relative stagnation pressure
Po4
Absolute stagnation pressure
W4
Outlet relative velocity
T4
Outlet temperature
P4
Outlet pressure
ρ4
Outlet density
s4
Outlet entropy
m4/m3
Mass flow rate ratio
A4/A3
Area ratio
35
(85)
Output
Description
β4
Outlet flow angle
U4
Outlet rotor speed
f
Fuel-to-air ratio
Cp4
Outlet specific heat
γ4
Outlet specific heat ratio
Wb/m3
Burner specific power
For the turbine: Table 3-6: Turbine exit flow variables.
Output
Description
M5rel
Exit relative Mach number
τtrel
Relative stagnation temperature ratio (To5rel/To4rel)
πtrel
Relative stagnation pressure ratio (Po5rel/Po4rel)
τt
Absolute stagnation temperature ratio (To5/To5)
πt
Absolute stagnation pressure ratio (Po5/Po4)
To5rel
Relative stagnation temperature
Po5rel
Relative stagnation pressure
To5
Absolute stagnation pressure
Po5
Absolute stagnation pressure
W5
Outlet relative velocity
T5
Outlet temperature
P5
Outlet pressure
ρ5
Outlet density
s5
Outlet entropy
A5/A4
Area ratio
β5
Outlet flow angle
U5
Outlet rotor speed
Cp5
Outlet specific heat
γ5
Outlet specific heat ratio
Wt/m3
Turbine specific power
36
The output variables above are the important ones. It is still possible to obtain other output properties not mentioned in this chapter by using a combination of variables seen in Table 3-5 and Table 3-6. Plotting the burner and turbine results show how they vary throughout the flow.
3.5
Overall APU properties With the flow properties now known at each component, it is necessary to determine the overall
performance of the rotor. This is important when comparing the new APU to other engines currently in service. Here are the important parameters that describe the overall performance: a) Rotor specific power takeoff: PTO W t Wb Wc = + + mc mc mc mc
(86)
b) Power takeoff coefficient:
CTO
⎛ PTO ⎞ ⎜ m ⎝ c ⎠
(87)
Cpc ⋅ T0
c) Power specific fuel consumption (PSFC): mf
f
PTO
⎛ PTO ⎞ ⎜ m ⎝ c ⎠
(88)
d) Thermal efficiency: η TH
CTO f⋅ hHV
(89)
Cpc ⋅ T 0
e) Rotor radius-to-compressor inlet tip ratio: r5 r2t
⎛ r3 ⎞ ⎛ r4 ⎞ ⎛ r5 ⎞ ⎜ r ⋅⎜ r ⋅⎜ r ⎝ 2t ⎠ ⎝ 3 ⎠ ⎝ 4 ⎠
(90)
In most cases, the specific power takeoff and PSFC are the two parameters used when comparing the rotor with other engines. In order to limit the impeller size, it is important to introduce the concept of centrifugal stress. To make this analysis as simple as possible, the assumptions are: a) The rotor bottom is a relatively flat disk. b) The disk thickness is tapered in such a way that its centrifugal stress is uniform everywhere. First, calculate the ratio r5/r2h:
37
r5 r2h
ζc
−1
⎛ r3 ⎞ ⎛ r4 ⎞ ⎛ r5 ⎞ ⋅⎜ ⋅⎜ ⎝ r2t ⎠ ⎝ r3 ⎠ ⎝ r4 ⎠
⋅⎜
(91)
The disk hub-to-rim thickness ratio, z2h/z5 from the Hill and Petersen textbook [5] is:
z2h z5
⎡ ⎢ ⎢ exp⎢ ⎢ ⎢ ⎣
−2 ⎤ ⎛ r5 ⎞ ⎥ 1−⎜ ⎥ r2h ⎝ ⎠ ⎥ 2 ⎛ σ ⎞ ⎥ ⋅⎜ 2 ρ ⎥ U5 ⎝ material ⎠ ⎦
Use Figure 3-5 to select an appropriate value of σ/ρmaterial (also called specific rupture strength):
Figure 3-5: Variation of specific rupture strength with service temperature (from [5]).
It is thicker at the hub than at the rim.
38
(92)
Chapter 4
Results of analysis
A MathCAD program created to carry out the new rotor analysis consisted of an input parameters section and an equations/results section. In the equations/results section, the program performed the analysis according to the following steps: a) Air and diffuser b) Compressor c) Burner d) Turbine e) Overall APU properties The turbine portion evaluates only the subsonic flow if the supersonic region does not occur. Before continuing with the rotor analysis, it was important to validate the burner and turbine equations by comparing them with simple flow problems. The purpose was to provide confidence in the usage of the burner and turbine equations.
4.1
Simple one-dimensional flow A simple flow involves no curvature (β does not change) and rotation along with constant
specific heats. For convenience, the equations in this section will use relative frame variables (absolute and relative frames are the same for simple flows). Simple cases for the burner are: a) Flow in a constant area duct with drag and heat addition. b) Flow in a constant area duct with only heat addition. For both cases above, mf is extremely small compared to m3. As for the turbine, the chosen simple case is the variable area duct flow.
39
4.1.1 Burner 4.1.1.1
Constant area flow with drag and heat addition
Figure 4-1: Constant area combustion chamber (from [10]).
Consider a constant area duct with flame holders at the beginning that contribute a drag force to the flow as seen in Figure 4-1. The equation for M4rel in terms of the upstream variables and a prescribed
τbrel [10] is: 2⋅ χ
M 4rel
1
(
)
1 − 2⋅ γ b ⋅ χ + ⎡ 1 − 2⋅ γ b + 1 ⋅ χ ⎤ ⎣ ⎦
(93)
2
where: γc − 1 2⎛ 2⎞ ⋅ M 3rel γ c M 3rel ⋅ ⎜ 1 + 2 ⎝ ⎠ ⋅τ ⋅ 2 brel γb C ⎡ ⎛ ⎞ ⎤ D ⎢ 1 + γ c⋅ M3rel2⋅ ⎜ 1 − ⎥ 2 ⎠⎦ ⎣ ⎝
χ
(94)
Next, the equation for πbrel [10] is: γb
CD ⎞ ⎛ γb − 1 ⎛ 2⎞ 1 + γ c⋅ M 3rel ⋅ ⎜ 1 − ⎜1 + ⋅ M 4rel 2 ⎠ ⎝ 2 ⎝ ⎠ ⋅
γ b− 1
2
πbrel
2
1 + γ b ⋅ M 4rel
γc − 1 ⎛ 2⎞ ⎜1 + ⋅ M 3rel 2 ⎝ ⎠
γc
(95)
γ c− 1
In this section, the input parameters for the burner equations in the program are for a simple case. This allows the comparison of the quantities M4rel and πbrel in Equations (93) and (95) with its respective values in the program. Table 4-1 summarizes this comparison: 40
Table 4-1: Comparison of burner equations to simple flow example (drag and heat addtion).
Burner
To4
input
(K)
πbrel
M4rel
τbrel
Burner
Equation
Burner
Equation
equations
(93)
equations
(95)
M3rel = 0.2 T3 = 500K
600
1.190476
0.231116
0.230973
0.952213
0.952305
900
1.785714
0.297083
0.294143
0.93304
0.934004
1200
2.380952
0.362688
0.355036
0.911921
0.914513
P3 = 500kPa CD = 1.5 hHV = 18,000 BTU/lbm
ηb = 0.98 R = 0.287 kJ/(kg*K)
γ0 = γ4 = 1.4 r4/r3 = 2 nb = 1000
4.1.1.2
Constant area flow with only heat addition
Figure 4-2: Constant area flow through a duct with heat addition (from [9]).
Consider the heat addition flow thorough the control volume shown in Figure 4-2. The same equations in Section 4.1.1.1 are applicable for the above control volume but with CD = 0. Once again, the input parameters for the burner equations are such that it is a simple case for this section. Table 4-2 summarizes the program results and the ones from Equations (93) and (95):
41
Table 4-2: Comparison of burner equations to simple flow example (heat addition).
Burner
To4
input
(K)
πbrel
M4rel
τbrel
Burner
Equation
Burner
Equation
equations
(93)
equations
(95)
M3rel = 0.2 T3 = 500K
600
1.190476
0.221122
0.220526
0.994463
0.994625
900
1.785714
0.283147
0.27976
0.976184
0.97725
1200
2.380952
0.343898
0.336013
0.956187
0.958871
P3 = 500kPa CD = 0 hHV = 18,000 BTU/lbm
ηb = 0.98 R = 0.287 kJ/(kg*K)
γ0 = γ4 = 1.4 r4/r3 = 2 nb = 1000
4.1.2 Variable area flow
Figure 4-3: Flow through a duct with variable area (from [9]).
Consider the flow shown in Figure 4-3. At any two points in the flow, the area ratio [9] is:
42
γ t+ 1
A5 A4
⎛ ⎜1+ M ⎛ 4rel ⎞ ⎜ ⋅ ⎜M ⎝ 5rel ⎠ ⎜ 1 + ⎜ ⎝
γt − 1 2 γt − 1 2
2⎞
(
2⋅ γ t − 1
)
⋅ M 5rel
(96)
⎟ ⎟ 2
⋅ M 4rel
⎠
Equation (96) also works for convergent-divergent ducts. This type of nozzle can generate a supersonic flow. The turbine input parameters in the program are such that both a subsonic and supersonic region exists for a simple case. That means the turbine is a convergent-divergent nozzle just like in Figure 4-3. Equation (96) uses the values of M4rel and M5rel from the turbine calculations to determine A5/A4. Table 4-3 summarizes the results: Table 4-3: Comparison of turbine equations to simple flow example (variable area).
Turbine input
A5/A4
Μ4rel
Μ5rel
Turbine equations
Equation (96)
T4 = 500K P4 = 500kPa
0.5
2.013896
1.273793
1.274185
0.7
2.426491
2.249443
2.250246
0.9
2.583947
2.826604
2.827483
Cpt = 1.008123 kJ/(kg*K) R = 0.287005 kJ/(kg*K)
γ4 = 1.4 r5/r4 = 2 nt = 8000
4.2
Single rotor APU results
4.2.1 Model Center and input/output constraints In order to implement the MathCAD program for the new rotor, it was preferable to optimize it using Model Center (created by Phoenix Integration Inc.). Unlike other optimizers, Model Center has a specially created plug-in that easily wraps Mathcad programs into it. With the wrapping completed, Model Center needed a range of values for all or just a selected number of input parameters to perform the optimization. For the compressor input parameters, they were obtainable using typical values from the Hill and Petersen textbook. As for the burner and turbine, the 43
range of values was as wide as possible but within reasonable limits. The objective of the optimizer in this study was to maximize the specific power takeoff. Model Center then chose the best values for the input parameters when the optimizing process ended. Table 4-4 summarizes these input parameters: Table 4-4: Model Center input parameters with range limits.
Range of values
Input parameters
Lower limit Upper limit M2rel
0.3
0.9
β2t (deg)
10
50
ζc
0.15
0.4
U3/(γ0*R*To2)^(1/2)
0.45
2.5
Cθ2t/(γ0*R*To2)^(1/2)
0
0.4
Wr3/U3
0.1
0.6
Y1
-15
15
S1
0
50
r4/r3
1
4
K1
-50
-1
K2
1
1.5
KK1
1
50
B1
0
50
r5/r4
1
4
σ/ρmaterial (kPa/kg/m3)
15
30
The input parameters that have fixed values were: Table 4-5: Model Center fixed input values.
Input parameters
Values
Input parameters
Values
Input parameters
Values
M0
0
τd
0.99
hHV (BTU/lbm)
18000
T0 (K)
300
πd
0.99
nb
2000
P0 (kPa)
101.325
Y2
0
CD
1.5
γ0
1.398
S2
0
KK2
0
s0 (kJ/(kg*K))
1.70203
To4 (K)
1200
B2
0
R (kJ/(kg*K))
0.287005
ηb
0.98
nt
8000
44
Setting a limit on σ/ρmaterial allowed the selection of an appropriate material from Figure 3-5 at the end of the optimizer run. To obtain the best results, it was preferable to set constraints on some of the output variables. This eventually helped speed up the optimization process. Table 4-6 shows the chosen variables along with their given constraints: Table 4-6: Model Center output constraints.
Output variables M3rel
Constraints
maximum of 0.7 between 1.1 and
πc
30
r3/r2t
at least 1.1
Output variables
β4 (deg)
Output
Constraints
variables
maximum of 90
β5
deg
Constraints maximum of 90 deg
A5/A4.5
maximum of 4
z2h/z5
between 1 and 3
M4rel
maximum of 0.8
|(P0-P5)/P5|
less than 0.005
The limit placed on A5/A4.5 ensured the relative Mach number did not become too large in the event the flow goes supersonic. In addition, the limit |(P0-P5)/P5| (for backpressure matching consideration) being less than 0.005 means the rotor flow exits close to atmospheric pressure. Limits placed on the compressor, burner and turbine size prevented them from becoming too small or big. With the information in Table 4-4 and Table 4-6, Model Center found the maximum specific power takeoff for the cases mentioned in Section 4.2.2. There were three available optimizing methods in Model Center: a) Method of feasible directions (MFD). b) Sequential linear programming (SLP). c) Sequential quadratic programming (SQP). Of the three, SQP was the newest algorithm. Before starting the optimizer, it required initial values for the variables listed in Table 4-4 along with the constraints mentioned in Table 4-6. MFD was found not to reach the maximum point for the given constraints and limitations. For this reason, it was the quickest among the three. However, the results from this run served as the initial values for the SLP method. According to Model Center, SLP had the most efficient algorithm. To make sure SLP found a true optimum, it was necessary to restart the calculation using initial values from the previous run. Usually, it took SLP as much as two times to find the true optimum point. The downside with using the SLP method was it took between one and a half to two hours to complete a run. The entire optimization procedure
45
needed repeating using different initial values to make sure the true optimum point did indeed occur. SQP was capable of reaching a maximum point close to the one achieved by SLP but required many repeated runs.
4.2.2 Results from Model Center The Model Center analysis consisted of four different cases: a) Without the stress (z2h/z5) and |(P0-P5)/P5| constraints. b) With the stress constraint but without the |(P0-P5)/P5| constraint. c) Without the stress constraint but with the |(P0-P5)/P5| constraints. d) With the stress and |(P0-P5)/P5| constraints. The rest of the limitation in Table 4-4 and Table 4-6 stayed the same.
4.2.2.1
Case 1: Without the stress and |(P0-P5)/P5| constraints. This case was necessary to investigate how large the rotor will get without the limitation of
material (stress) or whether the flow exited at atmospheric pressure. Model Center then provided the following results (complete results in Appendix E): Table 4-7: Overall rotor and other properties (Case 1).
PTO/m3
mf/PTO
(W/kg/s)
(kg/s/W)
412218.499688
πc 6.061594
ηTH
r5/r2h
z2h/z5
5.256221E-8
0.454406
65.390339
1.748837E+4
β5
Wc/m3
Wb/m3
Wt/m3
(deg)
(W/kg/s)
(W/kg/s)
(W/kg/s)
89.433445
-228368.78268
124301.705278 516285.577093
The optimizer stopped when A5/A4.5 reached a value of 4.012, which limited the turbine size. Figure E-16 through Figure E-20 (all created by only varying U3/(γ0*R*To2)1/2) shows that this configuration was the optimum point based on the given constraints and limitations in Section 4.2.1. The optimum occurred when the compressor pressure ratio was just enough to satisfy the P0/Po4.5rel < 0.528 condition and allowed the flow to go past the sonic point into the supersonic region (refer to Figure 4-4). In Figure D-1, Case 1 clearly competed very well with the other gas turbine engines. The high specific power takeoff resulted due to both the burner and turbine producing a large amount of specific power. Figure E-14 showed how this configuration compared to the Brayton cycle.
46
1500
3
⎡ r3 ⎤ ⎡ r4 ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ r2t ⎦ ⎣ r2t ⎦
2 Mrel
To 500
1
0
⎡ r3 ⎤ ⎡ r4 ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ r2t ⎦ ⎣ r2t ⎦
1000
0
10
20
0
30
0
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
10
20
30
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
r
r
6 1 .10
⎡ r3 ⎤ ⎡ r4 ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ r2t ⎦ ⎣ r2t ⎦
P o 5 .105
0
0
10
20
30
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
Figure 4-4: Relative Mach number, stagnation temperature (K) and pressure (Pa) according to location in the rotor (Case 1).
The plots in Figure 4-4 show that the rotor went supersonic which in turn caused the stagnation temperature and pressure to drop considerably in the turbine. There was an unavoidable stagnation pressure drop in the burner that was consistent with the concept of burning at finite relative Mach numbers. This would occur in the burner for all the other cases.
47
500
1000
⎡ r3 ⎤ ⎡ r4 ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ r2t ⎦ ⎣ r2t ⎦
0 Cθ
U
⎡ r3 ⎤ ⎡ r4 ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ r2t ⎦ ⎣ r2t ⎦
500
500
1000
0
10
20
0
30
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
0
10
20
30
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
r
r
100
β deg
⎡ r3 ⎤ ⎡ r4 ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ r2t ⎦ ⎣ r2t ⎦
50
0
0
10
20
30
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
Figure 4-5: Variation of the absolute tangential velocity (m/s), rotor speed (m/s) and flow curvature (deg) (Case 1).
Figure 4-5 indicates that U and β achieved high values that contributed to a large negative drop in the value of Cθ for the turbine. This permitted it to achieve a high specific power value. The drop in Cθ in the burner made it act like a turbine and produced specific power. The burner would act like a turbine in all the subsequent cases. However, Table 4-7 showed a huge value for z2h/z5 making this rotor unrealistic. A r5/r2h value of 65.390339, which resulted in U5 being 765.698776 m/s, made the rotor become as strong as possible to withstand the amount of stress associated with this configuration.
4.2.2.2
Case 2: With the stress constraint but without the |(P0-P5)/P5| constraint The analysis now included material limitation but still without taking into account the
backpressure. Table 4-8 shows some of the results from Model Center with the complete set located in Appendix F:
48
Table 4-8: Overall rotor and other properties (Case 2).
PTO/m3
mf/PTO
(W/kg/s)
(kg/s/W)
127884.924331
ηTH
r5/r2h
z2h/z5
2.053457E-7
0.116314
3.211841
2.932721
β5
Wc/m3
Wb/m3
Wt/m3
(deg)
(W/kg/s)
(W/kg/s)
(W/kg/s)
81.378184
-16268.094656
81713.818746
62439.200241
πc 1.183161
Comparing the results in Appendix E and Appendix F showed similar patterns in the flow characteristics. This would be the situation for all the subsequent cases. The exception here was there was no supersonic flow region, as seen in Figure F-1. The compressor pressure ratio was not high enough to satisfy the condition P0/Po4.5rel < 0.528 and caused the flow to cutoff at station 4.5. From Case 1, it was known that the compressor pressure ratio had to be around six and higher to go past the sonic point. In this case, the optimizer halted when r3/r2t came close to its minimum value. Figure F-19 through Figure F-23 (all created by only varying M2rel) indicated that Case 2 was indeed an optimum. In fact, all the points in Figure F-19 through Figure F-23 had their flow end at station 4.5. The plots also showed that more specific power takeoff was possible but only when r3/r2t became smaller than one. Like in Case 1, Figure F-14 showed how this case compared to the Brayton cycle (to view the compressor and turbine temperature change in Figure F-14, see Figure F-15 and Figure F-16). In Figure F-13, the size limitation affected the range at which Cθ could drop in the burner and turbine. The value of U5 was definitely a lot smaller than in Case 1. This made the burner and turbine each produce specific power that was not as high as in the previous case. Therefore, the stress limitation clearly prevented the rotor from achieving a high specific power takeoff. When placed into Figure D-1, this case was close to the other gas turbine engines but could not compete very well in terms of specific power takeoff and PSFC. Unlike Case 1, the value of z2h/z5 here was more realistic. This was due to a smaller rotor size and exit rotor speed. Figure F-23 showed the disk thickness would continue to become larger with increasing compressor pressure ratio and specific power takeoff.
4.2.2.3
Case 3: Without the stress constraint but with the |(P0-P5)/P5| constraint It was now necessary to determine the rotor characteristics using the backpressure as a constraint
but without any stress limitations. This gave the following selected results in Table 4-9 (complete results in Appendix G):
49
Table 4-9: Overall rotor and other properties (Case 3).
PTO/m3
mf/PTO
(W/kg/s)
(kg/s/W)
132248.596648
ηTH
r5/r2h
z2h/z5
1.57612E-7
0.15154
19.533881
1.292149E+4
β5
Wc/m3
Wb/m3
Wt/m3
(deg)
(W/kg/s)
(W/kg/s)
(W/kg/s)
88.76084
-436026.78374
310579.14829
257696.232095
πc 17.4036
There was an obvious improvement in the results compared to Case 2. From Figure G-1, the flow managed to get into the supersonic region. This meant it satisfied the P0/Po4.5rel < 0.528 condition and at the same time had an exit pressure close to atmospheric. To accomplish this, the compressor pressure ratio had to be very large. However, this large compressor had a negative effect on the specific power takeoff. Table 4-9 showed that the burner and turbine together produced a significant amount of specific power. This was due to the large Cθ drop seen in Figure G-12 with high β5 and U5 values. Nevertheless, the compressor power demand took up most of this specific power leaving a specific power takeoff a little more than in Case 2. In Figure D-1, Case 3 and Case 2 were close together but compared not very well to the other gas turbine engines. With no material limitation, the rotor size behaved similar to Case 1 and the disk thickness went as large as possible. The plots in Appendix G showed that the major portion of the rotor was in fact the compressor. Therefore, a z2h/z5 value of 1.292149E+4 made the manufacturing of this rotor impossible.
4.2.2.4
Case 4: With the stress and |(P0-P5)/P5| constraints In this case, Model Center used all the constraints and limitations mentioned in Table 4-4 and
Table 4-6 to give the selected results in Table 4-10: Table 4-10: Overall rotor and other properties (Case 4).
PTO/m3
mf/PTO
(W/kg/s)
(kg/s/W)
17641.820934
πc 1.572254
ηTH
r5/r2h
z2h/z5
1.357239E-6
0.017598
3.495073
2.489291
β5
Wc/m3
Wb/m3
Wt/m3
(deg)
(W/kg/s)
(W/kg/s)
(W/kg/s)
84.535271
-45803.940205
19140.007112
44305.754028
50
When completely constrained, this case produced the lowest specific power takeoff and highest PSFC among the four cases. Part of this reason was that the flow ended at M5rel equal to 0.476262371. Figure H-13 showed there was not much of a drop in Cθ between the burner inlet and turbine exit to significantly power the compressor and produce specific power takeoff at the same time. Like in Case 3, the compressor absorbed most of the specific power generated. Placing this case into Figure D-1 showed that it was far from the gas turbine engine points in terms of both specific power takeoff and PSFC. Appendix H shows the rest of the results for this case.
4.2.3 Rotor material and size The specific strength for this rotor (including all the other cases) ended at its highest given limit of 30 kPa/kg/m3. This once again indicated that the rotor required the strongest material possible. Using Figure 3-5 (with To4 as the service temperature), a possible material was molybdenum alloy stainless steel. Next, it was important to illustrate how the rotor for each case would look like. Appendix I through Appendix L shows how this took place. All four cases had eight vanes and r2t = 2 inches. Appendix I showed Case 1 had an unusual blade height distribution. It was very small at the compressor outlet but took on a very large value at the rotor exit. In Case 2, the compressor exit blade height started out at 0.934 inches but when it reached the burner outlet, the height was 9.738 inches. Both the first two cases obviously had unusual rotor designs. This also occurred in Cases 3 and 4. The value of b5 equal to 1.159 inches in Case 3 made this rotor seem reasonable. However, both the ratio b5/b3 = 53.131 and z2h/z5 = 1.292149E+4 in Case 3 made this rotor unrealistic. As for Case 4, b4 = 0.743 inches and b5 = 4.312 inches. Each rotor design had geometries that made their manufacturing not practical. A general observation was the compressor always had the largest size compared to the other components.
51
Chapter 5 5.1
Conclusion
Summary The idea of this study was to combine the compressor, burner and turbine of a gas turbine engine
into a single radial rotor and simulate it mathematically according to the principles of quasi-onedimensional flow. The simulations consisted of four different cases with each producing a unique set of results. An optimizer maximized the specific power takeoff for each case using a set of design constraints placed on the input parameters and output variables. The results from the first case indicated that with no restrictions on the type of material and exit backpressure matching, the rotor size became as large as possible with high supersonic exit relative Mach numbers. This allowed a large specific power takeoff that was able to compete very well with current gas turbine engines in service. Stress analysis indicated this rotor had an unrealistic disk thickness distribution. With the rotor fully constrained, as in the last case, it was unable to achieve supersonic exit velocities and produced a very low specific power takeoff. This made it compare poorly with current gas turbine engine performance. Constraining the rotor size prevented a large absolute tangential velocity change. This in turn affected the specific powers produced by the burner and turbine components. Each case also had the disadvantage of having large compressors (compared to the other components). However, all the gas turbine engines (including the four cases) compared badly with the gasoline and diesel power generators. These generators usually do not have any weight limitations since they are ground-based. This allows them to have extra components such as regenerators, intercoolers and so on giving them high efficiencies. Each case investigated had a sample rotor drawn to help visualize their shapes. The pictures indicated that all four rotors had an unusual combination of blade heights and disk thicknesses making their construction difficult. Therefore, the results in general suggest that the single radial rotor concept may not be such a good idea, at least for large PTO/m3.
5.2
Recommendations With the study now complete, some recommendations for continuation of this research are:
a) Optimize the PSFC, not just the specific power takeoff. b) Consider secondary flows to analyze the rotor with loss terms such as friction. c) Provide constraints for the blade height variation to make rotor shape more practical. d) Perform a combined burner and turbine analysis. e) Provide more degrees of freedom to the analysis by extending the rotor geometry to allow flow in the axial direction (z-direction). 52
References [1]
Fleming, William A., and Richard A. Leyes II. The History of North American Small Gas Turbine Aircraft Engines. Reston: AIAA, 1999.
[2]
Turboprop, Turboshaft, Ramjet, Scramjet and Turbojet/Ramjet; [http://www.aircraftenginedesign.com/abe_right4.html].
[3]
How it Works: Small Gas Turbine Engine (APU); [http://www.users.globalnet.co.uk/~spurr/sec.htm].
[4]
Auxiliary Power Unit: The APU described; 1999; [http://www.b737.org.uk/apu.htm]
[5]
Hill, Philip G., and Carl R. Peterson. Mechanics and Thermodynamics of Propulsion. 2nd ed. Reading: Addison-Wesley, 1992.
[6]
Boles, Michael A., and Yunus A. Cengel. Thermodynamics: An Engineering Approach. Boston: McGraw-Hill, 1998.
[7]
Bloch, Heinz P. A Practical Guide to Compressor Technology. New York: McGraw-Hill, 1995.
[8]
Daley, Daniel H., William H. Heiser, and Jack D. Mattingly. Aircraft Engine Design. Washington: AIAA, 1987
[9]
Carscallen, William E., and Patrick H. Oosthuizen. Compressible Fluid Flow. New York: McGraw, 1997.
[10] Oates, Gordon C. Aerothermodynamics of Gas Turbine and Rocket Propulsion. 3rd ed. Reston: AIAA, 1984. [11] Anderson Jr., John D. Modern Compressible Flow: With Historical Perspective. New York: McGraw, 1982 [12] Dixon, S. L. Fluid Mechanics and Thermodynamics of Turbomachinery. 4th ed. Boston: BH, 1998.
53
Appendix A A.1 M 3rel
M 3rel
Compressor Derivations
Outlet relative Mach number W3 γ 0⋅ R⋅ T3
W r3
( )
cos β 3
1
⋅
<----- W 3
γ 0⋅ R⋅ T3
W r3
( )
cos β 3
Wr3
M 3rel
U3
( )
cos β 3
⋅
U3 γ 0⋅ R⋅ T3
Wr3
M 3rel
U3
( )
cos β 3
U3
⋅
γ 0⋅ R⋅ To3
1+
γ0 − 1
2
To3
<----- T3
1+
γ0 − 1
2
⋅ M3
Wr3
M 3rel
U3
( )
cos β 3
U3
⋅
γ 0⋅ R⋅ To2⋅
1+
γ0 − 1
2
To3 To2 2
⋅ M3
Wr3
M 3rel
U3
( )
cos β 3
⋅
U3 γ 0⋅ R⋅ To2⋅ τc
1+
U3
( )
cos β 3
γ0 − 1
2
To3 To2
2
⋅ M3
U3
Wr3
M 3rel
<----- τc
γ 0⋅ R⋅ T o2
⋅
<-----Equation 1
τc
1+
γ0 − 1
2
2
⋅ M3
54
2
2
⋅ M3
A.2
Outlet relative stagnation temperature 2
2
C3
since Cp0⋅ T3
and Cp0⋅ T3 Cp0⋅ To3 − 2 2
Cp0⋅ To3rel −
Cp0⋅ To3 − 2
2
2
C3
Cp0⋅ To3 − + 2
To3 −
:
C3
2
To3rel
2
2
W3
2
Cp0⋅ To3rel
Cp0⋅ To3rel −
W3
W3 2
2
C3 − W 3 2⋅ Cp0
2 2 ⎛ C3 − W 3 ⎞ ⎜ To3rel To3⋅ 1 − ⎜ 2⋅ Cp0⋅ To3 ⎝ ⎠ 2 2 ⎡ W 3 ⎞ ⎥⎤ γ 0⋅ R⋅ T3 ⎛⎜ C3 ⎢ To3rel To3⋅ 1 − ⋅ − ⎢ 2⋅ Cp0⋅ To3 ⎜ γ 0⋅ R⋅ T3 γ 0⋅ R⋅ T3 ⎥ ⎣ ⎝ ⎠⎦ 2
γ 0⋅ R ⎡ 2 2 ⎤ To3rel To3⋅ ⎢ 1 − ⋅ ⎛ M 3 − M 3rel ⎞ ⎥ ⎠ To3 ⎝ ⎢ ⎥ ⋅ 2 ⋅ C p0 T ⎢ ⎥ 3 ⎣ ⎦
To3rel
To3rel
(
⎡
To3⋅ ⎢ 1 −
⎢ ⎢ ⎣
)
γ 0⋅ γ 0 − 1
⎛
γ0 − 1
⎝
2
2⋅ γ 0⋅ ⎜ 1 +
⋅ M3
⎡
γ0 − 1
⎢ ⎢ ⎣
γ0 − 1 ⎛ 2⋅ ⎜ 1 + ⋅ M3 2 ⎝ ⎠
To3⋅ ⎢ 1 −
<-----M 3rel
⎤ 2⎞ ⎥
⋅ ⎛ M 3 − M 3rel 2
⎝ 2⎞ ⎠
2
⎠⎥ ⎥ ⎦
⎤ 2⎞ ⎥
⋅ ⎛ M 3 − M 3rel
⎝ 2⎞
W3
2
⎠⎥ ⎥ ⎦
55
<-----
γ 0⋅ R⋅ T3
R
γ0 − 1
Cp0
γ0
<-----Equation 2
and
2
2
and M 3
To3 T3
C3
γ 0⋅ R⋅ T3
1+
γ0 − 1 2
2
⋅ M3
Appendix B B.1
Burner and turbine derivations
Conservation of angular momentum
−W = ( m ⋅ U ⋅ C θ ) out − ( m ⋅ U ⋅ C θ ) in
B.2
Conservation of energy (first law of thermodynamics)
Q − W = ( m ⋅ h o ) out − ( m ⋅ h o ) in 2
h+
since h o
C
2
:
⎡ ⎛ ⎡ ⎛ C 2 ⎞⎟⎤ C 2 ⎞⎟⎤ ⎥ ⎥ − ⎢m ⋅ ⎜ h + Q − W = ⎢m ⋅ ⎜ h + 2 ⎟⎥ 2 ⎟⎥ ⎢ ⎜⎝ ⎢ ⎜⎝ ⎠ ⎠ ⎣ ⎦ out ⎣ ⎦ in
(
)out − (m⋅ U⋅ Cθ )in
(
)out
Q + m⋅ U⋅ Cθ
Q + m⋅ U⋅ Cθ
⎡ ⎛
Q + ⎢m⋅ ⎜ U⋅ Cθ −
⎣ ⎝
⎛
C
⎝
2
− ⎜ m⋅
2 ⎞⎤ 2 ⎞⎤ ⎡ ⎛ ⎡ ⎛ ⎢m⋅ ⎜ h + C ⎥ − ⎢m⋅ ⎜ h + C ⎥ 2 ⎠⎦ out ⎣ ⎝ 2 ⎠⎦ in ⎣ ⎝
2⎞
(
− m⋅ U⋅ Cθ
⎠ out
)in
⎛
C
⎝
2
2⎞
+ ⎜ m⋅
⎠ in
( m⋅ h )
out
<-----place conservation of angular momentum here
− ( m⋅ h )
in
2 ⎞⎤ ⎡ ⎛ ⎥ − ⎢m⋅ ⎜ U⋅ Cθ − C ⎥ ( m⋅ h ) − ( m⋅ h ) out in 2 ⎠⎦ out ⎣ ⎝ 2 ⎠⎦ in 2 ⎞⎤
C
2
C : for U⋅ Cθ − 2 2
2
U⋅ Cθ −
C
U⋅ Cθ −
C
U⋅ Cθ −
C
2
U⋅ Cθ −
2
2
(
C
2 2
( (
C
U⋅ Cθ −
C
U
2
2
2
) )
U⋅ U − W θ −
U⋅ Cθ −
2
)
U⋅ U − W θ −
2
U⋅ Cθ −
2
U − U⋅ W θ − 2
−
2 2
<----- C
2
U⋅ U − W θ −
2
2
Cr + Cθ
W
2
(
Wr + U − Wθ
)2
2 2
2
2
Cr + Cθ
U − W θ and W r
<-----Cθ
W r + U − 2⋅ U⋅ W θ + W θ
2
2
2 2
2
W + U − 2⋅ U⋅ W θ
<----- W
2 W 2
2
2
−
U
2
+ U⋅ W θ
2
2
56
2
2
Wr + Wθ
2
Cr (both from velocity triangle)
2
2
U
C place U⋅ Cθ − 2
−
2
W
2
into the first law equation:
2
⎡ ⎛ U2 W 2 ⎞⎤ ⎡ ⎛ U2 W 2 ⎞⎤ ⎢ ⎜ ⎥ ⎥ Q + m⋅ − − ⎢m⋅ ⎜ − ( m⋅ h ) − ( m⋅ h ) out in 2 ⎠⎦ out ⎣ ⎝ 2 2 ⎠⎦ in ⎣ ⎝ 2 2⎞
⎛
U
⎝
2
⎛
U
⎝
2
Q + ⎜ m⋅
Q + ⎜ m⋅
2⎞
⎛
U
⎠ out ⎝
2
− ⎜ m⋅
2⎞
W
⎠ in ⎝
2
⎠ out
W
2⎞
2⎞
⎛
U
⎠ out ⎝
2
− ⎜ m⋅
2⎞
⎛
− ⎜ m⋅
⎡⎛
− ⎢ ⎜ m⋅
⎠ in ⎢⎣ ⎝
2
2⎞
⎛
W
⎝
2
⎠ in
W
2⎞
+ ⎜ m⋅
⎛
− ⎜ m⋅
(m⋅ Cp⋅ T)out − (m⋅ Cp⋅ T)in
⎤ ⎥ 2 ⎠ in ⎥ ⎦
⎠ out ⎝
<----- h
Cp ⋅ T
(m⋅ Cp⋅ T)out − (m⋅ Cp⋅ T)in
convert equation above into differential form: 2⎞
⎛
U
⎝
2
d ( Q) + d ⎜ m⋅
⎠
⎛ U2 ⎞
m
⎛ U2 ⎞ ⎝ 2 ⎠
h HV⋅ η b ⋅
d ( m)
h HV⋅ η b ⋅
d ( m)
m
m
2⎞
⎝
2
⎠
U
+
⎝ 2 ⎠
+ d⎜
W
2
d ( Q) + m⋅ d ⎜
d ( Q)
⎛
− d ⎜ m⋅
2 2
U
+
2
⋅
⎛ W2 ⎞
⋅ d ( m) − m⋅ d ⎜
⎝ 2 ⎠
⎝ 2 ⎠ 2
2
⋅
− d⎜
⎝ 2 ⎠
m
⎛ U2 ⎞
U
⎛ W2 ⎞
d ( m)
+ d⎜
+
(
d m⋅ Cp ⋅ T
2
+
d ( m) m
U
2 −
⋅
W 2
d ( m) m 2
⋅
−
) −
W 2
W 2 2
⋅
⎝ 2 ⎠
d ( m) m
⋅ d ( m)
m −
m⋅ Cp ⋅ d ( T) + Cp ⋅ T⋅ d ( m) d ( m) Cp ⋅ d ( T) + Cp ⋅ T⋅ m
d ( m)
⎛ W2 ⎞
− d⎜
2
W 2
2
⋅
<-----divide by m
d ( m) Cp ⋅ d ( T) + Cp ⋅ T⋅ m
d ( m) m 2⎞
⎛U d ( m) − Cp ⋅ T⋅ + d⎜ m ⎝ 2
⎠
⎛ W2 ⎞
− d⎜
⎝ 2 ⎠
− Cp ⋅ d ( T)
<----- d ( Q)
0
2 2 ⎛ ⎛ 2⎞ ⎛ 2⎞ ⎞ ⎜ hHV⋅ η b + U − W − Cp⋅ T ⋅ d ( m) + d⎜ U − d ⎜ W − Cp⋅ d ( T) 0 2 2 ⎝ ⎠ m ⎝ 2 ⎠ ⎝ 2 ⎠
⎛ 2⋅ h HV⋅ η b U2 W 2 ⎞ d( m) ⎜ + − − Cp ⋅ T ⋅ + U⋅ d ( U) − W ⋅ d ( W ) − Cp ⋅ d ( T) 0 2 2 2 ⎝ ⎠ m ⎛ 2⋅ h ⋅ η + U2 − W 2 ⎞ U⋅ d ( U) W ⋅ d( W ) d ( T) ⎜ HV b d ( m) −1 ⋅ + − − ⎜ 2⋅ Cp ⋅ T Cp ⋅ T Cp ⋅ T T ⎝ ⎠ m ⎛ 2⋅ h ⋅ η + U2 − W 2 ⎞ 2 W d( W ) d ( T) ⎜ HV b d ( m) −1 ⋅ − ⋅ − ⎜ 2⋅ Cp ⋅ T T Cp ⋅ T W ⎝ ⎠ m
57
−
0
U⋅ d ( U) Cp ⋅ T
<-----divide by Cp ⋅ T
<-----Equation 1
h HV⋅ η b ⋅ d ( m)
B.3
Equation of state
d( ρ )
d ( T)
+
ρ
B.4
T
<-----Equation 2
0
P
Conservation of mass
d( W )
d( ρ )
+
W d( W )
ρ d( ρ )
+
W
B.5
ρ
+
d(A)
d ( m)
A
m
−
d ( m)
−
m
d( A )
<-----Equation 3
A
Conservation of linear momentum P
−
d ( P)
−
ρ ⋅W
2
⋅
d ( P)
d( W )
P
W
( )
d FD
for
+
d ( m) m
+
( )
d FD 2
ρ ⋅W ⋅A
:
2
ρ ⋅W ⋅A 1
( )
d FD
2 1
2
ρ ⋅W ⋅A
2
( )
d FD
1
( )
⋅ d FD 2
⋅ρ ⋅W ⋅A
( )
d FD
⋅
2 1
2
ρ ⋅W ⋅A
2
( )
d FD
1
2
d FD
( )
1
2
2
P ρ ⋅W
2
⋅
<----- d CD
( )
d FD 1 2
ρ ⋅W ⋅A −
( )
( )
⋅ d CD 2
ρ ⋅W ⋅A
place
2
⋅ρ ⋅W ⋅A
2
⋅ρ ⋅W ⋅A
( )
⋅ d CD into the linear momentum equation:
d ( P)
d( W )
P
W
+
d ( m) m
+
1 2
( )
⋅ d CD
58
d( W )
d ( m)
+
W
B.6
P
+
m
ρ ⋅W
⋅
2
d ( P) P
( )
1 − ⋅ d CD 2
Relative stagnation temperature equation 2
2
C Cp ⋅ To − 2
W Cp ⋅ Torel − 2 W
2
2
C
Cp ⋅ Torel − + 2 2
Cp ⋅ To
Cp ⋅ Torel +
2
2
C −W 2
2
C −W 2
2
: 2
2
2
2
2 2
<-----C
2
2
C −W
2
(
Wr + U − Wθ
2
2
)2 − W r2 − W θ 2
2
2
C −W
2
2
U − 2⋅ U⋅ W θ + W θ − W θ
2
2
2
2
Cr + Cθ and W
<-----Cθ
2
2
Wr + Wθ
U − W θ and W r
2
2
2
2
U
2
2
2
C −W
2
2
U
2
2 2
C −W 2
2
− U⋅ W θ − U⋅ W ⋅ sin ( β )
<-----W θ
W ⋅ sin ( β ) (from velocity triangle)
2
− U⋅ W ⋅ sin ( β ) into Cp ⋅ To 2
U
2
Cp ⋅ Torel +
2
Cp ⋅ To
U Cp ⋅ Torel + − U⋅ W ⋅ sin ( β ) 2
convert equation above into differential form:
59
C −W 2
2
:
2
Cr (both from velocity triangle)
2
C −W
place
2
Cr + Cθ − W r − W θ
2
W : Cp ⋅ Torel − 2
2
Cp ⋅ To
C −W
2
C and Cp ⋅ T Cp ⋅ To − 2
since Cp ⋅ T
for
<-----Equation 4
Cp ⋅ d Torel +
( )
Cp ⋅ d Torel + U⋅ d ( U) − U⋅ sin ( β ) ⋅ d ( W ) − W ⋅ d ( U⋅ sin ( β ) )
Cp ⋅ d To
( )
(
)
(
)
(
)
( 2) − d(U⋅W⋅ sin (β ))
( )
Cp ⋅ d To
d U 2
Cp ⋅ d To − Cp ⋅ d Torel + U⋅ sin ( β ) ⋅ d ( W )
(
U⋅ d ( U) − W ⋅ d ( U⋅ sin ( β ) )
)
d Torel d( W ) Cp ⋅ d To − Cp ⋅ Torel⋅ + U⋅ W ⋅ sin ( β ) ⋅ Torel W
( )
( )
d To
−
To
B.7
(
)
Torel d Torel U⋅ W ⋅ sin ( β ) d ( W ) ⋅ + ⋅ To Torel Cp ⋅ To W
U⋅ d ( U) − W ⋅ d ( U⋅ sin ( β ) )
U⋅ d ( U) − W ⋅ d ( U⋅ sin ( β ) )
<-----divide by Cp ⋅ To
Cp ⋅ To
Relative stagnation temperature equation 2
(
γ ⋅ M rel
)
d Porel
d ( P)
Porel
P
d ⎛ M rel
+
γ−1
1+
2
2⎞
⎝ ⋅
2 2
⋅ M rel
⎠
2
M rel
from the the conservation of linear momentum equation: d( W ) W d( W ) W d( W ) W
+
+
+
d ( m) m
+
P ρ⋅W
d ( m) m
+
2
⋅
d ( m) m
+
1 − ⋅ d CD 2
P
γ⋅P γ⋅ρ ⋅W
( )
d ( P)
2
1 2
⋅
d ( P)
⋅
d ( P)
( )
1 − ⋅ d CD 2
P
γ ⋅ M rel
( )
1 − ⋅ d CD 2
P
2
<----- M rel 2
2 d(W )
γ ⋅ M rel ⋅
W
2 d ( m)
+ γ ⋅ M rel ⋅
m
+
d ( P) P
−
γ ⋅ M rel 2
2
d ( P) P
2 d(W )
−γ ⋅ M rel ⋅
W
2 d ( m)
− γ ⋅ M rel ⋅
m
−
γ ⋅ M rel 2
place
P
2 d( W )
−γ ⋅ M rel ⋅
W
2 d ( m)
− γ ⋅ M rel ⋅
m
−
2
γ⋅P
( )
⋅ d CD
( )
⋅ d CD
2
d ( P)
ρ⋅W
γ ⋅ M rel 2
( )
⋅ d CD into the
60
(
)
d Porel Porel
equation:
<-----Equation 5
2
(
)
2
d Porel
2 d(W )
−γ ⋅ M rel ⋅
Porel
W
d ⎛ M rel
2 d ( m)
− γ ⋅ M rel ⋅
−
m
γ ⋅ M rel 2
γ ⋅ M rel
( )
d ⎛ M rel
⎝ ⋅
2
⋅ d CD + 1+
γ−1 2
2
⋅ M rel
2⎞
⎠
2
M rel
2⎞
⎝ from definition of
⎠:
2
M rel d ⎛ M rel
2⎞
⎝
⎠
2⋅
2
d(W ) W
M rel
d ( T) T
⎞ ⎛ d⎛ M 2 ⎞ rel ⎠ d ( T) ⎝ ⎜ ⋅ + 2 2 ⎜ T M rel ⎝ ⎠
d( W )
1
W
place
−
⎞ ⎛ d⎛ M 2 ⎞ d ( Porel) d ( T) rel ⎠ ⎝ ⎜ into the equation: ⋅ + 2 Porel 2 ⎜ T ⎝ Mrel ⎠ 1
d(W ) W
2
(
)
2
d Porel
−
Porel
γ ⋅ M rel 2
γ ⋅ M rel
2 2 ⎞ ⎛ d⎛ M 2 ⎞ γ ⋅ M rel d ⎛ M rel ⎞ rel ⎠ 2 d ( T) 2 d ( m) ⎝ ⎝ ⎠ ⎜ ⋅ + − γ ⋅ M rel ⋅ − ⋅ d ( CD) + ⋅ m ⎜ M 2 2 2 T γ−1 2 M rel 1+ ⋅ M rel rel ⎝ ⎠ 2
by definition, the relative stagnation temperature is:
(
γ−1
)
d Torel
d ( T)
Torel
T
(
+ 1+
)
d ( T)
d Torel
T
Torel
place
2
γ−1 2
γ−1 2
− 1+
(
)
d ( T)
d Torel
T
Torel
2
d ⎛ M rel
⋅ M rel
2
⋅ M rel 2
2
− 1+
⎠
2
M rel 2
d ⎛ M rel
⋅ M rel
γ−1 2
2⎞
⎝ ⋅ 2
2
2
M rel
⋅ M rel
γ−1
⎠
d ⎛ M rel
⋅ M rel
γ−1
2⎞
⎝ ⋅
⎝ ⋅ 2
⋅ M rel
2⎞
⎠ into the d ( Porel) equation: Porel
2
M rel
61
2 2
(
γ ⋅ M rel γ−1 2 ⎛ 2 2 ⎞ 2 2 ⋅ M rel γ ⋅ M rel ⎜ d ⎛ M rel ⎞ d ( Torel) d ⎛ M rel ⎞ γ M ⋅ d ⎛ M rel ⎞ 2 d ( m ) rel 2 ⎝ ⎠ + ⎝ ⎠ − γ ⋅ M 2⋅ ⎝ ⎠ − ⋅⎜ − ⋅ − ⋅ d C + ⋅ ( ) ⎟ rel m D 2 2 Torel 2 2 2 γ−1 γ−1 2 2 M rel ⎜ M rel M rel 1+ ⋅ M rel 1+ ⋅ M rel 2 2 ⎝ ⎠
)
2
d Porel Porel
2
(
γ ⋅ M rel
2 2 2 γ ⋅ M rel ⎛⎜ d ( Torel) d ⎛ M rel ⎞ ⎞ d ⎛ M rel ⎞ γ ⋅ M rel 1 2 2 d ( m) ⎝ ⎠ ⎝ ⎠ − ⋅ + ⋅ − γ ⋅ M rel ⋅ − ⋅ d ( CD) + ⋅ ⎟ ⎜ T 2 2 2 2 m γ − 1 γ − 1 2 2 M rel M rel 1+ ⋅ M rel 1+ ⋅ M rel ⎜ orel 2 2 ⎝ ⎠
)
2
d Porel Porel
2
2
(
)
2
d Porel
−
Porel
(
)
−
Porel
(
)
Porel
B.8
⋅
2
(
)
d Torel
γ ⋅ M rel
⋅
2
2
+
γ ⋅ M rel
⋅
2
1+
(
)
)
Torel
⎝ ⋅
2
2
⋅ Mrel
2
− γ ⋅ M rel ⋅
m
−
⎠ − γ ⋅ M 2⋅ d ( m) − γ ⋅ Mrel ⋅ d C + ( D) rel
+ γ ⋅ M rel ⋅
m
γ ⋅ M rel
d ⎛ Mrel
⎝ ⋅
2
1+
γ−1 2
2
⋅ Mrel
( )
⋅ d CD
2
−
2
m
2
2 d ( m)
γ ⋅ M rel
2
Mrel
2 d ( m)
Torel
(
γ−1
2⎞
2
d Torel
d Torel
d ⎛ Mrel
2
−
Torel
2
d Porel
d Porel
γ ⋅ Mrel
γ ⋅ M rel
γ ⋅ M rel
( )
⋅ d CD
2
<-----Equation 6
Absolute stagnation temperature equation 2
Cp ⋅ To
C Cp ⋅ T + 2
2
Cp ⋅ To
Cp ⋅ T +
Cr + Cθ
Cp ⋅ T +
Cp ⋅ T +
Cp ⋅ T +
Cp ⋅ To
Cp ⋅ T +
)2
2
2
Cr + Cθ
2
<-----Cθ
W r + U − 2⋅ U⋅ W θ + W θ
2
U − W θ and W r
Cr (both from velocity triangle)
2
2 2
Cp ⋅ To
(
Wr + U − Wθ
2
Cp ⋅ To
2
<----- C
2 2
Cp ⋅ To
2
2
W + U − 2⋅ U⋅ W θ
<----- W
2
2 W + U − 2⋅ U⋅ W ⋅ sin ( β ) 2
2
2
<-----W θ
62
2
Wr + Wθ
2
W ⋅ sin ( β ) (from velocity triangle)
2⎞
2
Mrel
⎠
2
2
W U Cp ⋅ T + + − U⋅ W ⋅ sin ( β ) 2 2
Cp ⋅ To
convert equation above into differential form:
( 2) + d(U2) − d(U⋅W⋅ sin (β ))
( )
Cp ⋅ d ( T) +
( )
Cp ⋅ d ( T) + W ⋅ d ( W ) + U⋅ d ( U) − U⋅ sin ( β ) ⋅ d ( W ) − W ⋅ d ( U⋅ sin ( β ) )
( )
Cp ⋅ d ( T) + ( W − U⋅ sin ( β ) ) ⋅ d ( W ) + U⋅ d ( U) − W ⋅ d ( U⋅ sin ( β ) )
( )
U⋅ d ( U) − W ⋅ d ( U⋅ sin ( β ) )
Cp ⋅ d To
Cp ⋅ d To Cp ⋅ d To
d W 2
2
Cp ⋅ d To − Cp ⋅ d ( T) − ( W − U⋅ sin ( β ) ) ⋅ d ( W )
( )
d ( T) d(W ) Cp ⋅ d To − Cp ⋅ T⋅ − W ⋅ ( W − U⋅ sin ( β ) ) ⋅ T W
( )
d To
−
To
B.9
T d ( T) W ⋅ ( W − U⋅ sin ( β ) ) d ( W ) ⋅ − ⋅ To T Cp ⋅ To W
U⋅ d ( U) − W ⋅ d ( U⋅ sin ( β ) ) U⋅ d ( U) − W ⋅ d ( U⋅ sin ( β ) ) Cp ⋅ To
Relative Mach number equation W
M rel
<-----Equation 8
γ ⋅ R⋅ T
B.10 Absolute stagnation pressure equation burner: γ
since
Po P
⎛ To ⎞ ⎜ ⎝ T ⎠
γ
γ −1
and
Porel P
⎛ Torel ⎞ ⎜ ⎝ T ⎠
γ −1
:
γ Po P
⎛ To ⎞ ⎜ ⎝ T ⎠
γ −1
γ
Porel P
⎛ Torel ⎞ ⎜ ⎝ T ⎠
γ −1
63
<-----divide by Cp ⋅ To
<-----Equation 7
γ
Po
To
γ −1 γ
Porel Torel
γ −1 γ
⎛ To ⎞ Po Porel⋅ ⎜ ⎝ Torel ⎠
γ −1
<-----Equation 9
turbine: γ
⎛ To ⎞ Po Po4⋅ ⎜ ⎝ To4 ⎠
γ −1
<-----Equation 10
B.11 Entropy equation burner:
s
⎛ To ⎞ ⎛ Po ⎞ − R⋅ ln⎜ s 3 + Cp ⋅ ln⎜ ⎝ To3 ⎠ ⎝ Po3 ⎠
<-----Equation 11
turbine:
s
⎛ To ⎞ ⎛ Po ⎞ − R⋅ ln⎜ s 4 + Cp ⋅ ln⎜ ⎝ To4 ⎠ ⎝ Po4 ⎠
<-----Equation 12
B.12 Burner absolute stagnation temperature distribution the To distribution is linear: To
⎛ δr ⎞
a + b⋅ ⎜
at the inlet,
<-----
⎝ r3 ⎠ r r3
1, which means
δr
r
r3
r3
δr3
r3
−1
0 and To
To3:
64
To3 a
⎛ δr3 ⎞
a + b⋅ ⎜
⎝ r3 ⎠
To3
at the outlet,
r
r4
r3
r3
To3 + b ⋅ ⎜
To4
To3 + b ⋅ ⎜
r4
r3
r3
− 1 and To
To4:
⎝ r3 ⎠ ⎛ r4 ⎝ r3
−1
⎞ ⎠
To4 − To3 r4 r3
place a
−1
To3 and b
To4 − To3 r4 r3
To
δr4
⎛ δr4 ⎞
To4
b
, which means
To3 +
To4 − To3 r4 r3
−1
in the To distribution equation:
−1
⎛ δr ⎞
⋅⎜
<-----Equation 13
⎝ r3 ⎠
B.13 Burner specific work −W b = ( m ⋅ U ⋅ C θ ) 4 − ( m ⋅ U ⋅ C θ ) 3
<-----from the definition of angular momentum
⎡m ⎤ − W b = m 3 ⋅ ⎢ 4 ⋅ U 4 ⋅ ( U 4 − Wθ4 ) − U 3 ⋅ ( U 3 − Wθ3 ) ⎥ ⎣ m3 ⎦
<-----Cθ
⎡m ⎤ − Wb = m 3 ⋅ ⎢ 4 ⋅ U 4 ⋅ ( U 4 − W4 ⋅ sin(β 4 )) − U 3 ⋅ ( U 3 − W3 ⋅ sin(β 3 ))⎥ ⎣ m3 ⎦ ⎡m ⎤ Wb = − ⎢ 4 ⋅ U 4 ⋅ ( U 4 − W4 ⋅ sin(β 4 )) − U 3 ⋅ ( U 3 − W3 ⋅ sin(β 3 ))⎥ m3 ⎣ m3 ⎦
65
U − Wθ
<-----W θ
W ⋅ sin ( β )
<-----Equation 14
B.14 Turbine specific work − W t = m 4 ⋅ [U 5 ⋅ C θ5 − U 4 ⋅ C θ4 ]
<-----from the definition of angular momentum
− W t = ( m 3 + m f ) ⋅ [U 5 ⋅ ( U 5 − Wθ5 ) − U 4 ⋅ ( U 4 − Wθ4 )] − W t = m 3 ⋅ (1 +
<-----Cθ
U − Wθ
mf ) ⋅ [U 5 ⋅ ( U 5 − W5 ⋅ sin(β 5 )) − U 4 ⋅ ( U 4 − W4 ⋅ sin(β 4 ))] m3
− W t = m 3 ⋅ (1 + f ) ⋅ [U 5 ⋅ ( U 5 − W5 ⋅ sin(β 5 )) − U 4 ⋅ ( U 4 − W4 ⋅ sin(β 4 ))] Wt = −(1 + f ) ⋅ [U 5 ⋅ ( U 5 − W5 ⋅ sin(β 5 )) − U 4 ⋅ ( U 4 − W4 ⋅ sin(β 4 ))] m3
66
<-----W θ
W ⋅ sin ( β )
<-----f = m f / m 3
<-----Equation 15
Appendix C
To determine perpendicular (one-dimensional) flow area between the vanes
mass flow rate associated with the circular area between two vanes: ⌠ →→ ⎮ ( ) ⎮ ρ ⋅ C⋅ n ⋅ 2⋅ π⋅ b dr ⌡ m
m
m
m
m
Nb 2⋅ π⋅ r⋅ b ⋅ ρ ⋅ C⋅ cos ( α ) Nb 2⋅ π⋅ r⋅ b ⋅ ρ ⋅ Cr
<----- Cr C⋅ cos ( α )
Nb 2⋅ π⋅ r⋅ b ⋅ ρ ⋅ W r
<----- W r Cr
Nb 2⋅ π⋅ r⋅ b ⋅ ρ ⋅ W ⋅ cos ( β ) Nb
<----- W r W ⋅ cos ( β )
mass flow rateassociated with area perpendicular to flow between two vane
m
⌠ →→ ⎮ ⎮ ρ ⋅ W ⋅ n dA ⌡
m
ρ ⋅W⋅A
since the mass flow rates are the same:
ρ ⋅W⋅A
A
2⋅ π⋅ r⋅ b ⋅ ρ ⋅ W ⋅ cos ( β ) Nb
2⋅ π⋅ r⋅ b ⋅ cos ( β ) Nb
67
Appendix D
Current engine data
Table D-1: Airplane turboprop engine data.
PTO
mf/PTO
m3
(shp)
(lbm/h/shp)
(lbm/s)
TPE 331-5
710
0.602
7.75
TPE 331-T76
577
0.6
6.17
Klimov Corporation
TV7-117
2466
0.397
17.53
NK
NK-12MV
14795
0.501
143
OEDB
TVD-20-01
1380
0.506
11.9
P&WC
PT6A-27
680
0.633
6.8
AE 2100J
4591
0.41
16.33
Allison T56-A15
4591
0.536
32.4
Turbomeca
Bastan VIC
798
0.773
10
Walter
M602B
2012
0.498
16.6
Honeywell
TPE 331-5
710
0.602
7.75
Company
Model
Honeywell
Rolls Royce
Table D-2: Helicopter turboshaft engine data.
PTO
mf/PTO
m3
(shp)
(lbm/h/shp)
(lbm/s)
T58 (GE-10)
1400
0.6
13.7
CT58-110
1250
0.64
12.7
T700-401C
1800
0.459
10
Ivchenko Prog. ZMKB
D-136
10000
0.436
79.4
JSC 'Aviadvigatel
D-25V
5500
0.639
57.8
TV2-117
1500
0.606
18.5
TV3-117
2190
0.507
19.84
LHTEC
CTS-800-4
1362
0.465
7.8
MTR
MTR 390
1285
0.46
7.05
PZL Rzeszow
GTD-350
394
0.84
4.83
Company
General Electric
Klimov Corporation
Model
68
Company
PTO
mf/PTO
m3
(shp)
(lbm/h/shp)
(lbm/s)
Gazelle
1400
0.68
17
GEM-42
1000
0.65
7.52
Gnome (H-1400)
1250
0.608
13.7
Turbomeca RM 322
2241
0.442
12.69
Makila (1A2)
1845
0.551
12.1
Turmo (IIIC3)
1480
0.603
13
Model
Rolls Royce
Turbomeca
Table D-3: Aircraft (turboprop) and helicopter (turboshaft) dual-purpose engine data.
PTO
mf/PTO
m3
(shp)
(lbm/h/shp)
(lbm/s)
T64 (GE-413)
3925
0.47
29.4
LTC1, T53 (T5313B, L-13B)
1400
0.58
10.5
LTC4, T55 (GA-714)
4868
0.503
29.08
LTS/LTP 101 (750B-1)
550
0.577
5.1
TVD-1500/RD 600 (1500 S)
1300
0.454
8.8
Company
Model
General Electric
Honeywell
Table D-4: Four-stroke gasoline generator engine data.
Generator
cylinders
model
(hp)
(ft3/min)
5ERKM
11.5
19
0.78
Onan
Microquiet 4000
9.5
19
0.71
Kohler
7ER
16
24
0.94
CME 5500
12.9
17.2
0.95
CMM 7000
14
18.9
1.22
10CCE
13
35
5.6
12CCE
17
35
5.6
Company
Kohler
Onan
Kohler
1
2
Power Air intake
Fuel
Number of
consumption (gal/hr)
4
69
Table D-5: Diesel generator engine data.
Company
Generac
Number of Generator Power Air intake cylinders
3
Kohler
Fuel consumption
model
(hp)
(ft3/min)
GR8
11
22
0.67
10EOR/Z
17.7
36
0.97
GR25
31
87
1.4
GR50
58
94
2.6
GR70
85
150
3.5
GR85
93
178
3.8
15EOR/Z
26.1
54
1.4
20EOR/Z
36.1
70
1.67
GR125
144
283
5.7
GR160
175
283
7.4
GR190
206
283
8.6
GR210
220
283
9.8
(gal/hr)
Generac 4
Kohler
Generac
6
70
Figure D-1: PSFC and specific power comparison between APU cases and current engines.
71
Appendix E E.1
Complete results for Case 1
Input parameters
Table E-1: Air and diffuser input parameter values (Case 1).
Input
Values
M0
0
T0 (K)
300
P0 (kPa)
101.325
γ0
1.398
s0 (kJ/(kg*K)) 1.70203 R (kJ/(kg*K))
0.287
τd
0.99
πd
0.99
Table E-2: Compressor input parameter values (Case 1).
Input
Values
β2t (deg)
10
β3 (deg)
0
ec
0.905
M2rel
0.5
ζc
0.4
U3/(γ0*R*To2)^(1/2)
1.384337
Cθ2t/(γ0*R*To2)^(1/2)
0
Wr3/U3
0.251381
72
Table E-3: Burner input parameter values (Case 1).
Input
Values
Y1
0.692596
Y2
0
S1
3.323625
S2
0
To4 (K)
1200
ηb
0.98
CD
1.5
hHV (BTU/lbm)
18000
r4/r3
1.349041
nb
2000
Table E-4: Turbine input parameter values (Case 1).
Input
Values
K1
-10.024377
K2
1.350002
KK1
16.430168
KK2
0
B1
2.142985
B2
0
r5/r4
1.187721
nt
8000
σ/ρmaterial (kPa/kg/m3)
30
73
E.2
Output values
Table E-5: Air diffuser output values (Case 1).
Output
Values
Cp0 (kJ/(kg*K))
1.00812309
τr
1
πr
1
ρ0 (kg/m3)
1.176808766
To2 (K)
297
Po2 (kPa)
100.31175
Table E-6: Compressor output values (Case 1).
Output
Values
Output
Values
T2t (K)
283.3293979
Po3 (kPa) 2
608.0490906
P2t (kPa)
85.00958077
m3/A3 (kg/(m *s))
252.0767384
3
ρ2t (kg/m )
1.045410296
W3 (m/s)
120.1298177
U2t (m/s)
29.27415519
T3 (K)
403.1068889
To2rel (kPa)
297.4250355
P3 (kPa)
242.7681092
Po2rel (kPa)
100.8169058
ρ3 (kg/m )
2.098369441
M3rel
0.29870493
(m3/ Po2)1/2*Ω/(γ0*R* To2)1/4
0.106466006
M3
1.22522507
r3/r2t
16.32427858
τc
1.762722794
b3/r3
0.000639812
πc
6.061593887
U3 (m/s)
477.8794646
To3rel (K)
410.2643349
ηc
0.878835559
Po3rel (kPa)
258.2498426
s3 (kJ/(kg*K))
1.756319109
To3 (K)
523.5286698
Wc/m3 (W/kg/s)
-228368.7827
3
74
Table E-7: Burner output value (Case 1).
Output
Values
Output
Values
M4rel
0.799978645
P4 (kPa)
128.1908667
τbrel
3.126086403
ρ4 (kg/m3)
0.386067689
πbrel
0.743665982
s4 (kJ/(kg*K))
3.131401637
τb
2.29213808
m4/m3
1.021667114
πb
0.241091813
A4/A3
1.2417444
To4rel (K)
1282.521759
β4 (deg)
66.467768
Po4rel (kPa)
192.0516227
Cp4 (kJ/(kg*K))
1.165537217
To4 (K)
1200
γ4
1.326686938
Po4 (kPa)
146.5956577
U4 (m/s)
644.6789908
W4 (m/s)
530.8233787
f
0.021667114
T4 (K)
1156.33955
Wb/m3 (W/kg/s)
124301.7053
Table E-8: Turbine output value (Case 1).
Output
Values
Output
Values
M5rel
2.630817867
P5 (kPa)
7.205425768
τtrel
1.061557783
ρ5 (kg/m3)
0.04096135
πtrel
0.817183228
s5 (kJ/(kg*K))
3.131401637
τt
0.628156809
A5/A4
3.863164912
πt
0.179957283
A5/A4.5
4.012205168
To5rel (K)
1361.470956
r5/r4
1.187721
Po5rel (kPa)
156.941365
β5 (deg)
89.433445
To5 (K)
753.7881702
Cp5 (kJ/(kg*K))
1.058624556
Po5 (kPa)
26.3809563
γ5
1.371951433
W5 (m/s)
1292.700693
U5 (m/s)
765.6987756
T5 (K)
613.1774811
Wt/m3 (W/kg/s)
516285.5771
75
Table E-9: Rotor overall properties (Case 1).
Output
Values
PTO/m3 (W/kg/s) 412218.4997 CTO
1.362989975
mf/PTO (kg/s/W)
5.25622E-08
ηTH
0.45440614
r5/r2h
65.39033925
z2h/z5
17488.37489
1500
3
⎡ r3 ⎤ ⎡ r4 ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ r2t ⎦ ⎣ r2t ⎦
2 Mrel
T orel 500
1
0
⎡ r3 ⎤ ⎡ r4 ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ r2t ⎦ ⎣ r2t ⎦
1000
0
10
20
0
30
0
10
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
20
30
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
Figure E-1: Case 1 relative Mach number.
Figure E-2: Case 1 relative stagnation temperature (K).
3 .10
5
1500
⎡ r3 ⎤ ⎡ r4 ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ r2t ⎦ ⎣ r2t ⎦
P orel2 .105
⎡ r3 ⎤ ⎡ r4 ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ r2t ⎦ ⎣ r2t ⎦
1000 To 500
1 .10
5 0
10
20
0
30
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
0
10
20
30
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
Figure E-3: Case 1 relative stagnation pressure (Pa).
Figure E-4: Case 1 stagnation temperature (K).
76
1 .10
6
1500
⎡ r3 ⎤ ⎡ r4 ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ r2t ⎦ ⎣ r2t ⎦
P o 5 .105
⎡ r3 ⎤ ⎡ r4 ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ r2t ⎦ ⎣ r2t ⎦
1000 T 500
0
0
10
20
0
30
0
10
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
Figure E-6: Case 1 temperature (Case 1).
3 .10
5
3
⎡ r3 ⎤ ⎡ r4 ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ r2t ⎦ ⎣ r2t ⎦
2 .10
5
P
⎡ r3 ⎤ ⎡ r4 ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ r2t ⎦ ⎣ r2t ⎦
2
ρ
5 1 .10
1
0
0
10
20
0
30
0
10
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
30
Figure E-8: Case 1 density (Case 1).
100
1000
⎡ r3 ⎤ ⎡ r4 ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ r2t ⎦ ⎣ r2t ⎦
50
0
20
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
Figure E-7: Case 1 pressure (Case 1).
β
30
r
Figure E-5: Case 1 stagnation pressure (Case 1).
deg
20
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
r
0
10
20
U
500
0
30
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
⎡ r3 ⎤ ⎡ r4 ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ r2t ⎦ ⎣ r2t ⎦
0
10
20
30
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
Figure E-9: Case 1 flow curvature (Case 1).
Figure E-10: Case 1 rotor speed (Case 1).
77
1200
1.4
⎡ r3 ⎤ ⎡ r4 ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ r2t ⎦ ⎣ r2t ⎦
Cp 1100
1000
0
10
20
γ
1.35
1.3
30
⎡ r3 ⎤ ⎡ r4 ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ r2t ⎦ ⎣ r2t ⎦
0
10
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
20
30
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
r
r
Figure E-11: Case 1 specific heat (Case 1).
Figure E-12: Case 1 specific heat ratio (Case 1).
500
1500
⎡ r3 ⎤ ⎡ r4 ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ r2t ⎦ ⎣ r2t ⎦
0 Cθ
s3
s4
1000 To
500
1000
500
0
10
20
0 1500
30
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
2000
2500
3000
3500
s
Figure E-14: Case 1 To-s diagram (Case 1).
Figure E-13: Case 1 tangential velocity (Case 1). 1 .10
6
P o 5 .105
1 1 ρ3 ρ4
0
0
10
20
30
1
ρ
Figure E-15: Case 1 Po-v diagram (Case 1).
78
Table E-10: Data to show Case 1 configuration is the optimum (Case 1 highlighted below).
U3/(γ0*R*To2)1/2
πc
r3/r2t
0.5
1.35193
5.896064
89465.712 2.69E-07
0.7
1.761836
8.254489
0.9 1
PTO/m3
mf/PTO
r5/r2h
z2h/z5
A5/A4.5
21.16218
2.776372
1
88795.587 2.64E-07 29.233796
7.033892
1
2.430948 10.612915 69279.005 3.28E-07 37.102533 23.187477
1
2.901004 11.792128 53339.868 4.18E-07 40.966537
46.20024
1
1.05
3.178761 12.381734
43761.26
5.04E-07 42.881601 66.682409
1
1.1
3.489095
12.97134
32332.3
6.76E-07
97.638476
1
6.061594 16.324279
412218.5
5.26E-08 65.390339
1.75E+04
4.012205
6.130212 16.391057 416331.58 5.27E-08 65.657836
1.89E+04
4.033592
1.384337 1.39
44.785
450000 400000
PTO/m3 (W/kg/s)
350000 300000 250000 200000 150000 100000 50000 0 0
1
2
3
4
5
6
πc
Figure E-16: Variation of specific power takeoff with compressor pressure ratio (Case 1).
79
7
8.00E-07 7.00E-07
mf /PTO (kg/s/W)
6.00E-07 5.00E-07 4.00E-07 3.00E-07 2.00E-07 1.00E-07 0.00E+00 0
1
2
3
4
5
6
7
Ď&#x20AC;c
Figure E-17: Variation of PSFC with compressor pressure ratio (Case 1).
18 16 14
r3 /r2t
12 10 8 6 4 2 0 0
1
2
3
4
5
6
Ď&#x20AC;c
Figure E-18: Variation of compressor radius ratio and pressure ratio (Case 1).
80
7
70
60
r5 /r2h
50
40
30
20
10
0 0
1
2
3
4
5
6
7
Ď&#x20AC;c
Figure E-19: Variation of rotor radius ratio with compressor pressure ratio (Case 1).
20000
15000
z2h/z5
10000
5000
0 0
1
2
3
4
5
6
-5000 Ď&#x20AC;c
Figure E-20: Variation of disk thickness with compressor pressure ratio (Case 1).
81
7
Appendix F F.1
Complete results for Case 2
Input parameters
Table F-1: Air and diffuser input parameter values (Case 2).
Input
Values
M0
0
T0 (K)
300
P0 (kPa)
101.325
γ0
1.398
s0 (kJ/(kg*K)) 1.70203 R (kJ/(kg*K))
0.287
τd
0.99
πd
0.99
Table F-2: Compressor input parameter values (Case 2).
Input
Values
β2t (deg)
19.585342
β3 (deg)
0
ec
0.905
M2rel
0.64997
ζc
0.398559
U3/(γ0*R*To2)^(1/2)
0.614369
Cθ2t/(γ0*R*To2)^(1/2)
0.398209
Wr3/U3
0.536858
82
Table F-3: Burner input parameter values (Case 2).
Input
Values
Y1
3.99312
Y2
0
S1
5.537262
S2
0
To4 (K)
1200
ηb
0.98
CD
1.5
hHV (BTU/lbm)
18000
r4/r3
1.255677
nb
2000
Table F-4: Turbine and stress input parameter values (Case 2).
Input
Values
K1
-41.638544
K2
1.287858
KK1
13.058003
KK2
0
B1
1.147403
B2
0
r5/r4
1.995009
nt
8000
σ/ρmaterial (kPa/kg/m ) -41.638544 3
83
F.2
Output values
Table F-5: Air diffuser output values (Case 2).
Output
Values
Cp0 (kJ/(kg*K))
1.00812309
τr
1
πr
1
ρ0 (kg/m3)
1.176808766
To2 (K)
297
Po2 (kPa)
100.31175
Table F-6: Compressor output values (Case 2).
Output
Values
Output
Values
T2t (K)
267.6547712
Po3 (kPa) 2
118.6849691
P2t (kPa)
69.60638697
m3/A3 (kg/(m *s))
118.0581585
ρ2t (kg/m )
0.906117753
W3 (m/s)
113.8584493
U2t (m/s)
208.863337
T3 (K)
284.3989835
To2rel (kPa)
290.1564381
P3 (kPa)
84.63465782
Po2rel (kPa)
92.42506928
ρ3 (kg/m )
1.036885354
M3rel
0.337056672
(m3/ Po2)1/2*Ω/(γ0*R* To2)1/4
0.827809062
M3
0.712587056
r3/r2t
1.015415114
τc
1.054333375
b3/r3
0.460075489
πc
1.183161186
U3 (m/s)
212.082989
To3rel (K)
290.8286281
ηc
0.902709389
Po3rel (kPa)
91.54868673
s3 (kJ/(kg*K))
1.707097155
To3 (K)
313.1370122
Wc/m3 (W/kg/s)
-16268.09466
3
3
84
Table F-7: Burner output value (Case 2).
Output
Values
Output
Values
M4rel
0.603615629
P4 (kPa)
49.41581802
τbrel
4.344274537
ρ4 (kg/m3)
0.144872207
πbrel
0.683366737
s4 (kJ/(kg*K))
3.524437467
τb
3.832188317
m4/m3
1.026260618
πb
0.427190894
A4/A3
2.020948942
To4rel (K)
1263.439404
β4 (deg)
81.116531
Po4rel (kPa)
62.56132736
Cp4 (kJ/(kg*K))
1.171053612
To4 (K)
1200
γ4
1.324648437
Po4 (kPa)
50.70113808
U4 (m/s)
266.3077314
W4 (m/s)
405.5837271
f
0.026260618
T4 (K)
1187.544785
Wb/m3 (W/kg/s)
81713.81875
Table F-8: Turbine output value (Case 2).
Output
Values
Output
Values
M5rel
0.987347437
P5 (kPa)
34.23796475
τtrel
1.00019199
ρ5 (kg/m3)
0.109959882
πtrel
0.999932428
s5 (kJ/(kg*K))
3.524437467
τt
0.956413596
A5/A4
0.834297496
πt
0.836135196
A5/A4.5
1
To5rel (K)
1263.681971
r5/r4
1.003980036
Po5rel (kPa)
62.55709997
β5 (deg)
81.378184
To5 (K)
1147.696315
Cp5 (kJ/(kg*K))
1.152564501
Po5 (kPa)
42.39300603
γ5
1.331583213
W5 (m/s)
635.9900172
U5 (m/s)
267.3676458
T5 (K)
1085.68245
Wt/m3 (W/kg/s)
62439.20024
85
Table F-9: Rotor overall properties (Case 2).
Output
Values
PTO/m3 (W/kg/s) 127884.9243 CTO
0.422848247
mf/PTO (kg/s/W)
2.05346E-07
ηTH
0.116314054
r5/r2h
3.211840863
z2h/z5
2.932720727
1500
1
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
Mrel 0.5
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
1000 T orel
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
500
0
0.9
1
1.1
1.2
0
1.3
0.9
1
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
1.1
1.2
1.3
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
r
r
Figure F-1: Relative Mach number (Case 2).
Figure F-2: Relative stagnation temperature (Case 2).
1 .10
5
1500
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
P orel8 .104
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
1000 To
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
500
6 .10
4 0.9
1
1.1
1.2
0 0.9
1.3
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
1
1.1
1.2
1.3
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
Figure F-3: Relative stagnation pressure (Case 2).
Figure F-4: Stagnation temperature (Case 2).
86
1.5 .10
5
1500
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
1 .10
5
Po
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
1000 T
5 .10
4
500
0 0.9
1
1.1
1.2
0
1.3
0.9
1
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
1.1
1.2
1.3
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
r
r
Figure F-5: Stagnation pressure (Case 2).
Figure F-6: Temperature (Case 2).
1 .10
5
P
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
1.5
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
5 .10
4
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
1
ρ
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
0.5
0 0.9
1
1.1
1.2
0
1.3
0.9
1
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
Figure F-7: Pressure (Case 2).
β
1.3
300
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
50
0
1.2
Figure F-8: Density (Case 2).
100
deg
1.1
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
0.9
1
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
1.1
1.2
U
250
200
1.3
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
0.9
1
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
1.1
1.2
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
Figure F-9: Flow curvature (Case 2).
Figure F-10: Rotor speed (Case 2).
87
1.3
1200
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
Cp 1100
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
1.4
γ
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
1.35
1000 0.9
1
1.1
1.2
1.3
1.3
0.9
1
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
1.2
1.3
r
Figure F-11: Specific heat (Case 2).
Figure F-12: Specific heat ratio (Case 2).
500
Cθ
1.1
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
r
1500
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
0
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
s3
s4
1000 To 500
500
0.9
1
1.1
1.2
0 1000
1.3
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
2000
3000
4000
s
Figure F-14: To-s diagram (Case 2).
Figure F-13: Tangential velocity (Case 2). 350 s3
s4 T o 1200
To 300
1700
1750
3400
s
3600 s
Figure F-15: Beginning of To-s diagram (Case 2).
Figure F-16: End of To-s diagram (Case 2).
88
1.5 .10
5 1 ρ4
1 ρ3
1 .10
5
1.2 .10
5
Po
1 ρ3
Po 4 5 .10
1 .10
5
0
0
5
10
1
2
1
1
ρ
ρ
Figure F-17: Po-s diagram (Case 2).
Figure F-18: Beginning of Po-s diagram (Case 2).
Table F-10: Data to show Case 2 configuration is the optimum (Case 2 highlighted below).
M2rel
πc
r3/r2t
PTO/m3
mf/PTO
r5/r2h
z2h/z5
A5/A4.5
0.3
1.245687 1.237661 122769.28 2.11E-07 3.917246
3.050059
1
0.35
1.236315 1.198915 122186.45 2.13E-07 3.794145
3.03363
1
0.4
1.227077 1.162841 121858.95 2.14E-07 3.679526
3.016945
1
0.45
1.217979 1.129212 121724.06 2.14E-07 3.572674
3.000041
1
0.5
1.209031 1.097826 121739.93 2.15E-07 3.472944
2.982956
1
0.55
1.200241 1.068503 121877.15 2.15E-07 3.379761
2.965726
1
0.6
1.191614 1.041077 122114.24 2.15E-07 3.292602 2.95E+00
1
0.64997 1.183161 1.015415 122430.27 2.14E-07 3.211045 2.93E+00
1
0.7
1.174873 0.991339 122827.23 2.14E-07 3.134521 2.91E+00
1
0.8
1.158849 0.947586 123789.87 2.13E-07 2.995435
2.878672
1
0.9
1.143564 0.908966 124950.45 2.12E-07 2.872641
2.844053
1
1.129034
2.809908
1
1
0.87477
126283.75 2.10E-07 2.763883
89
127000
PTO/m3 (W/kg/s)
126000
125000
124000
123000
122000
121000 1.12
1.14
1.16
1.18
1.2
1.22
1.24
1.26
Ď&#x20AC;c
Figure F-19: Variation of specific power takeoff with compressor pressure ratio (Case 2).
2.15E-07 2.15E-07 2.14E-07
mf /PTO (kg/s/W)
2.14E-07 2.13E-07 2.13E-07 2.12E-07 2.12E-07 2.11E-07 2.11E-07 2.10E-07 1.12
1.14
1.16
1.18
1.2
1.22
1.24
Ď&#x20AC;c
Figure F-20: Variation of PSFC with compressor pressure ratio (Case 2).
90
1.26
1.4
1.2
1
r3 /r2t
0.8
0.6
0.4
0.2
0 1.12
1.14
1.16
1.18
1.2
1.22
1.24
1.26
Ď&#x20AC;c
Figure F-21: Variation of compressor radius ratio and pressure ratio (Case 2).
4.5 4 3.5
r5 /r2h
3 2.5 2 1.5 1 0.5 0 1.12
1.14
1.16
1.18
1.2
1.22
1.24
Ď&#x20AC;c
Figure F-22: Variation of rotor radius ratio with compressor pressure ratio (Case 2).
91
1.26
3.1
3.05
3
z2h/z5
2.95
2.9
2.85
2.8
2.75 1.12
1.14
1.16
1.18
1.2
1.22
1.24
Ď&#x20AC;c
Figure F-23: Variation of disk thickness with compressor pressure ratio (Case 2).
92
1.26
Appendix G G.1
Complete results for Case 3
Input parameters
Table G-1: Air and diffuser input parameter values (Case 3).
Input
Values
M0
0
T0 (K)
300
P0 (kPa)
101.325
γ0
1.398
s0 (kJ/(kg*K)) 1.70203 R (kJ/(kg*K))
0.287
τd
0.99
πd
0.99
Table G-2: Compressor input parameter values (Case 3).
Input
Values
β2t (deg)
10.158584
β3 (deg)
0
ec
0.905
M2rel
0.368845
ζc
0.4
U3/(γ0*R*To2)^(1/2)
1.928568
Cθ2t/(γ0*R*To2)^(1/2)
0.215862
Wr3/U3
0.203182
93
Table G-3: Burner input parameter values (Case 3).
Input
Values
Y1
1.335502
Y2
0
S1
21.227624
S2
0
To4 (K)
1200
ηb
0.98
CD
1.5
hHV (BTU/lbm)
18000
r4/r3
1.069184
nb
2000
Table G-4: Turbine input parameter values (Case 3).
Input
Values
K1
-3.524361
K2
1.350092
KK1
4.724029
KK2
0
B1
1.349654
B2
0
r5/r4
1.06018
nt
8000
σ/ρmaterial (kPa/kg/m3)
30
94
G.2
Output value
Table G-5: Air diffuser output values (Case 3).
Output
Values
Cp0 (kJ/(kg*K))
1.00812309
τr
1
πr
1
ρ0 (kg/m3)
1.176808766
To2 (K)
297
Po2 (kPa)
100.31175
Table G-6: Compressor output values (Case 3).
Output
Values
Output
Values
T2t (K)
286.7249039
Po3 (kPa) 2
1745.785.622
P2t (kPa)
88.64231542
m3/A3 (kg/(m *s))
437.905847
3
ρ2t (kg/m )
1.07717487
W3 (m/s)
135.2685171
U2t (m/s)
96.5817412
T3 (K)
500.6121768
To2rel (kPa)
294.4874876
P3 (kPa)
465.1305732
Po2rel (kPa)
97.36252913
ρ3 (kg/m )
3.237307959
M3rel
0.301819786
(m3/ Po2)1/2*Ω/(γ0*R* To2)1/4
0.326260951
M3
1.515817267
r3/r2t
6.893129974
τc
2.456274202
b3/r3
0.001581712
πc
17.40360049
U3 (m/s)
665.7504952
To3rel (K)
509.6872451
ηc
0.861980802
Po3rel (kPa)
495.4286911
s3 (kJ/(kg*K))
1.788094816
To3 (K)
729.513438
Wc/m3 (W/kg/s)
-436026.7837
3
95
Table G-7: Burner output value (Case 3).
Output
Values
Output
Values
M4rel
0.799710092
P4 (kPa)
253.6339543
τbrel
2.529142176
ρ4 (kg/m3)
0.75986289
πbrel
0.766859421
s4 (kJ/(kg*K))
2.889971791
τb
1.644932002
m4/m3
1.020843961
πb
0.162667148
A4/A3
1.09239537
To4rel (K)
1289.071508
β4 (deg)
84.145266
Po4rel (kPa)
379.9241592
Cp4 (kJ/(kg*K))
1.166692558
To4 (K)
1200
γ4
1.326257883
Po4 (kPa)
283.9819685
U4 (m/s)
711.8097775
W4 (m/s)
532.0416408
f
0.020843961
T4 (K)
1162.809722
Wb/m3 (W/kg/s)
310579.1483
Table G-8: Turbine output value (Case 3).
Output
Values
Output
Values
M5rel
1.51580327
P5 (kPa)
100.820663
τtrel
1.021361317
ρ5 (kg/m3)
0.369539251
πtrel
0.97683752
s5 (kJ/(kg*K))
2.889971791
τt
0.823512255
A5/A4
1.192265289
πt
0.466709557
A5/A4.5
1.235197655
To5rel (K)
1316.607773
r5/r4
1.06018
Po5rel (kPa)
371.1241734
β5 (deg)
88.76084
To5 (K)
988.2147063
Cp5 (kJ/(kg*K))
1.126352764
Po5 (kPa)
132.5370987
γ5
1.341938124
W5 (m/s)
917.1864066
U5 (m/s)
754.6464899
T5 (K)
950.6213979
Wt/m3 (W/kg/s)
257696.2321
96
Table G-9: Rotor overall properties (Case 3).
Output
Values
PTO/m3 (W/kg/s) 132248.5966 CTO
0.437276618
mf/PTO (kg/s/W)
1.57612E-07
ηTH
0.151540461
r5/r2h
19.53388085
z2h/z5
12921.49295
1500
2
⎡ r⎡3 r⎤4 ⎤ ⎢⎢⎥⎥ ⎣ r⎣2tr⎦2t ⎦
1.5 Mrel
1
⎡ r⎡3 r⎤4 ⎤ ⎢⎢⎥⎥ ⎣ r⎣2tr⎦2t ⎦
1000 T orel 500
0.5 0
0
5
0
10
0
5
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
10
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
Figure G-1: Relative Mach number (Case 3).
Figure G-2: Relative stagnation temperature (Case 3).
6 .10
5
1500
⎡ r3⎡ ⎤r4 ⎤ ⎢ ⎢⎥ ⎥ ⎣ r2t⎣ ⎦r2t ⎦
4 .10
5
P orel
⎡ r⎡3 r⎤4 ⎤ ⎢⎢⎥⎥ ⎣ r⎣2tr⎦2t ⎦
1000 To
5 2 .10
500
0
0
5
0
10
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
0
5
10
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
Figure G-3: Relative stagnation pressure (Case 3).
Figure G-4: Stagnation temperature (Case 3).
97
2 .10
6
1500
⎡ r3⎡ ⎤r4 ⎤ ⎢ ⎢⎥ ⎥ ⎣ r2t⎣ ⎦r2t ⎦
P o 1 .106
⎡ r⎡3 r⎤4 ⎤ ⎢⎢⎥⎥ ⎣ r⎣2tr⎦2t ⎦
1000 T 500
0
0
5
0
10
0
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
5
10
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
r
r
Figure G-5: Stagnation pressure (Case 3).
Figure G-6: Temperature (Case 3).
6 .10
5
4
⎡ r3⎡ ⎤r4 ⎤ ⎢ ⎢⎥ ⎥ ⎣ r2t⎣ ⎦r2t ⎦
4 .10
5
P
ρ
⎡ r⎡3 r⎤4 ⎤ ⎢⎢⎥⎥ ⎣ r⎣2tr⎦2t ⎦
2
5 2 .10
0
0
5
0
10
0
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
Figure G-7: Pressure (Case 3).
β
1000
⎡ r⎡3 r⎤4 ⎤ ⎢⎢⎥⎥ ⎣ r⎣2tr⎦2t ⎦
50
0
10
Figure G-8: Density (Case 3).
100
deg
5
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
0
5
U
500
0
10
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
⎡ r⎡3 r⎤4 ⎤ ⎢⎢⎥⎥ ⎣ r⎣2tr⎦2t ⎦
0
5
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
Figure G-9: Flow curvature (Case 3).
Figure G-10: Rotor speed (Case 3).
98
10
1200
1.4
⎡ r⎡3 r⎤4 ⎤ ⎢⎢⎥⎥ ⎣ r⎣2tr⎦2t ⎦
Cp 1100
1000
0
γ
5
1.35
1.3
10
⎡ r⎡3 r⎤4 ⎤ ⎢⎢⎥⎥ ⎣ r⎣2tr⎦2t ⎦
0
5
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
10
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
r
r
Figure G-11: Specific heat (Case 3).
Figure G-12: Specific heat ratio (Case 3).
1000
1500
⎡ r⎡3 r⎤4 ⎤ ⎢⎢⎥⎥ ⎣ r⎣2tr⎦2t ⎦
500 Cθ
s3
s4
1000 To
0
500
500
0
5
0 1500
10
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
2000
2500
3000
s
Figure G-14: To-s diagram (Case 3).
Figure G-13: Tangential velocity (Case 3). 2 .10
6 1 ρ4
1 ρ3
P o 1 .106
0
0
1
2 1
ρ
Figure G-15: Po-v diagram (Case 3).
99
3
Appendix H H.1
Complete results for Case 4
Input parameters
Table H-1: Air and diffuser input parameter values (Case 4).
Input
Values
M0
0
T0 (K)
300
P0 (kPa)
101.325
γ0
1.398
s0 (kJ/(kg*K)) 1.70203 R (kJ/(kg*K))
0.287
τd
0.99
πd
0.99
Table H-2: Compressor input parameter values (Case 4).
Input
Values
β2t (deg)
49.999979
β3 (deg)
0
ec
0.905
M2rel
0.672518
ζc
0.399999
U3/(γ0*R*To2)^(1/2)
0.619976
Cθ2t/(γ0*R*To2)^(1/2)
0
Wr3/U3
0.2
100
Table H-3: Burner input parameter values (Case 4).
Input
Values
Y1
-1.434396
Y2
0
S1
6.332654
S2
0
To4 (K)
1200
ηb
0.98
CD
1.5
hHV (BTU/lbm)
18000
r4/r3
1.1
nb
2000
Table H-4: Turbine and stress input parameter values (Case 4).
Input
Values
K1
-7.855414
K2
1.196379
KK1
1.7
KK2
0
B1
22.766825
B2
0
r5/r4
1.036995
nt
8000
σ/ρmaterial (kPa/kg/m ) 3
30
101
H.2
Output values
Table H-5: Air diffuser output values (Case 4).
Output
Values
Cp0 (kJ/(kg*K))
1.00812309
τr
1
πr
1
ρ0 (kg/m3)
1.176808766
To2 (K)
297
Po2 (kPa)
100.31175
Table H-6: Compressor output values (Case 4).
Output
Values
Output
Values
T2t (K)
286.3513274
Po3 (kPa) 2
157.7155002
P2t (kPa)
88.23730292
m3/A3 (kg/(m *s))
57.39563798
3
ρ2t (kg/m )
1.07365206
W3 (m/s)
42.80371022
U2t (m/s)
174.624696
T3 (K)
318.808737
To2rel (kPa)
312.1240383
P3 (kPa)
122.6922554
Po2rel (kPa)
119.4319393
ρ3 (kg/m )
1.34090334
M3rel
0.119679011
(m3/ Po2)1/2*Ω/(γ0*R* To2)1/4
0.604641308
M3
0.610245611
r3/r2t
1.225591545
τc
1.152979356
b3/r3
0.31377556
πc
1.572253502
U3 (m/s)
214.0185511
To3rel (K)
319.7174343
ηc
0.898764101
Po3rel (kPa)
123.9250334
s3 (kJ/(kg*K))
1.715663037
To3 (K)
342.4348687
Wc/m3 (W/kg/s)
-45803.94021
3
102
Table H-7: Burner output value (Case 4).
Output
Values
Output
Values
M4rel
0.314921598
P4 (kPa)
110.0527876
τbrel
3.751173898
ρ4 (kg/m3)
0.324832124
πbrel
0.947802185
s4 (kJ/(kg*K))
3.266250485
τb
3.504316032
m4/m3
1.023944159
πb
0.746469231
A4/A3
0.8565604
To4rel (K)
1199.315694
β4 (deg)
36.283435
Po4rel (kPa)
117.4564174
Cp4 (kJ/(kg*K))
1.169595183
To4 (K)
1200
γ4
1.325184899
Po4 (kPa)
117.7297682
U4 (m/s)
235.4204062
W4 (m/s)
210.9026514
f
0.023944159
T4 (K)
1179.215682
Wb/m3 (W/kg/s)
19140.00711
Table H-8: Turbine output value (Case 4).
Output
Values
Output
Values
Output
Values
M5rel
0.476262371
To5 (K)
1162.957271
A5/A4
0.711062276
τtrel
1.00149048
Po5 (kPa)
103.6519401
r5/r4
1.036995
πtrel
0.99985241
W5 (m/s)
316.1401293
β5 (deg)
84.535271
τt
0.969131059
T5 (K)
1157.254069
Cp5 (kJ/(kg*K))
1.165700874
πt
0.880422528
P5 (kPa)
101.3265771
γ5
1.326626093
To5rel (K)
1201.103251
ρ5 (kg/m3)
0.304751534
U5 (m/s)
244.1286954
Po5rel (kPa)
117.439082
s5 (kJ/(kg*K))
3.266250485
Wt/m3 (W/kg/s)
44305.75403
103
Table H-9: Rotor overall properties (Case 4).
Output
Values
PTO/m3 (W/kg/s) 17641.82093 CTO
0.058332232
mf/PTO (kg/s/W)
1.35724E-06
ηTH
0.017597931
r5/r2h
3.495072575
z2h/z5
2.489291269
1500
1
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
Mrel 0.5
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
1000 T orel
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
500
0
1
1.2
0
1.4
1
1.2
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
1.4
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
r
r
Figure H-1: Relative Mach number (Case 4).
Figure H-2: Relative stagnation temperature (Case 4).
1.25 .10
5
1500
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
P orel 1.2 .105
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
1000 To
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
500
1.15 .10
5 1
1.2
0
1.4
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
1
1.2
1.4
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
r
r
Figure H-3: Relative stagnation pressure (Case 4).
Figure H-4: Stagnation temperature (Case 4).
104
2 .10
5
1500
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
P o 1.5 .105
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
1000 T
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
500
1 .10
5 1
1.2
0
1.4
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
5
1.5
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
1.2 .10
5
P
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
1
ρ
5 1 .10
1
1.2
0
1.4
1
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
1.2
1.4
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
Figure H-7: Pressure (Case 4).
Figure H-8: Density (Case 4).
100
250
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
50
0
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
0.5
4
β
1.4
Figure H-6: Temperature (Case 4).
1.4 .10
deg
1.2 r
Figure H-5: Stagnation pressure (Case 4).
8 .10
1
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
r
1
1.2
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
U
200
150
1.4
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
1
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
1.2
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
Figure H-9: Flow curvature (Case 4).
Figure H-10: Rotor speed (Case 4).
105
1.4
1200
1.4
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
Cp 1100
1000
1
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
1.2
γ
1.35
1.3
1.4
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
1
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
1.2
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
1.4
⎡ ⎤ ⎢r ⎥ ⎣ 2t ⎦
r
r
Figure H-11: Specific heat (Case 4).
Figure H-12: Specific heat ratio (Case 4).
400
1500
⎡ r3 ⎤ ⎢ ⎥ ⎣ r2t ⎦
200 Cθ
⎡ r4 ⎤ ⎢ ⎥ ⎣ r2t ⎦
s3
s4
1000 To
0
200
500
1
1.2
0 1500
1.4
⎡ r⎤ ⎢r ⎥ ⎣ 2t ⎦
2000
2500
3000
3500
s
Figure H-14: To-s diagram (Case 4).
Figure H-13: Tangential velocity (Case 4).
1300 s3
s4
400 T o 1200
To
1100 1600
1700
1800
3200
s
3300 s
Figure H-15: Beginning of To-s diagram (Case 4).
Figure H-16: End of To-s diagram (Case 4).
106
2 .10
5
P o 1.5 .105
1 .10
1 ρ4
1 ρ3
5 0
1
2
3
1
ρ
Figure H-17: Po-s diagram (Case 4).
107
4
Appendix I
Sample rotor for Case 1 with calculation program
⎡ A 4⎤ ⎢ A ⎥ := 1.241744 ⎣ 3⎦
⎡ A4 ⎤ ⎢ A ⎥ := 1.03858 ⎣ 4.5⎦
⎡ b3⎤ −4 ⎢ r ⎥ := 6.39812× 10 ⎣ 3⎦
⎡ r3 ⎤ ⎢ r ⎥ := 16.324279 ⎣ 2t ⎦
β 3 := 0⋅ deg
β 4 := 66.467768deg ⋅
ζc := 0.4
r2t := 2⋅ in
⎡ r5⎤ r5 := r4⋅ ⎢ ⎥ ⎣ r4⎦
d 2h := 2⋅ r2h
⎡ b3⎤ ⎥ ⎣ r3 ⎦
b 3 := r3⋅ ⎢
⎛ ⎡ A4 ⎤ ⎞ A 4.5 := A 4⋅ ⎜ ⎢ ⎥ ⎝ ⎣ A 4.5⎦ ⎠
A 3 :=
d 2t := 2⋅ r2t
( )
2⋅ π⋅ r3⋅ b 3⋅ cos β 3
−1
b 4.5 :=
⎡ r4⎤ ⎢ r ⎥ := 1.349041 ⎣ 3⎦
β 4.5 := 66.922988deg ⋅
r2h := r2t⋅ ζc
Nb
⎡ A5 ⎤ ⎢ A ⎥ := 4.012205 ⎣ 4.5⎦ ⎡ r4.5⎤ ⎢ r ⎥ := 1.003707 ⎣ 4⎦ β 5 := 89.433445deg ⋅
⎡ r3 ⎤ r3 := r2t⋅ ⎢ ⎥ ⎣ r2t ⎦ d 3 := 2⋅ r3
⎡ A 4⎤ ⎥ ⎣ A 3⎦
d 5 := 2⋅ r5
Nb ⋅ A 4
b 4 :=
( )
2⋅ π⋅ r4⋅ cos β 4
⎡ A5 ⎤ ⎥ ⎣ A 4.5⎦
A 5 := A 4.5⋅ ⎢
( )
⎡ r4.5⎤ r4.5 := r4⋅ ⎢ ⎥ ⎣ r4 ⎦
d 4.5 := 2⋅ r4.5
A 4 := A 3⋅ ⎢
2⋅ π⋅ r4.5⋅ cos β 4.5
Nb := 8
⎡ r4⎤ r4 := r3⋅ ⎢ ⎥ ⎣ r3⎦
d 4 := 2⋅ r4
Nb ⋅ A 4.5
⎡ r5⎤ ⎢ r ⎥ := 1.187721 ⎣ 4⎦
b 5 :=
Nb ⋅ A 5
( )
2⋅ π⋅ r5⋅ cos β 5
r2h = 0.8in
r3 = 32.649in
r4 = 44.044in
r4.5 = 44.208in
r5 = 52.312in
d 2h = 1.6in
d 2t = 4 in
d 3 = 65.297in
d 4 = 88.088in
d 4.5 = 88.415in
d 5 = 104.625in
b 3 = 0.021in
b 4 = 0.048in
b 4.5 = 0.047in
b 5 = 6.325in
2
A 4 = 0.665in
A 3 = 0.536in
2
2
A 4.5 = 0.64in
108
2
A 5 = 2.569in
89.4°= β5 67.0° = β4.5
Ø 104.624 = d5
Ø 88.416 = d4.5 Ø 4.000 = d2t
Ø 1.600 = d2h
Ø 88.088 = d4
Ø 65.298 = d3
r θ
b4 = .048
.047 = b4.5
b3 = .021 b5 = 6.325
Figure I-1: Sample rotor for Case 1 with side view (starting at station 3).
109
Appendix J
Sample rotor for Case 2 with calculation program
⎡ A 4⎤ ⎢ A ⎥ := 2.020949 ⎣ 3⎦
⎡ A 5⎤ ⎢ A ⎥ := 0.834297 ⎣ 4⎦
⎡ r3 ⎤ ⎢ ⎥ := 1.015415 ⎣ r2t ⎦
⎡ b3⎤ ⎢ r ⎥ := 0.460075 ⎣ 3⎦
⎡ r4⎤ ⎢ r ⎥ := 1.255677 ⎣ 3⎦
⎡ r5⎤ ⎢ r ⎥ := 1.00398 ⎣ 4⎦
β 3 := 0⋅ deg
β 4 := 81.116531deg ⋅
β 5 := 81.378184deg ⋅
Nb := 8
ζc := 0.4
r2t := 2⋅ in
d 2h := 2⋅ r2h
r2h := r2t⋅ ζc
d 2t := 2⋅ r2t
⎡ b3⎤ ⎥ ⎣ r3 ⎦
b 3 := r3⋅ ⎢
⎡ A 5⎤ ⎥ ⎣ A 4⎦
A 3 :=
d 3 := 2⋅ r3
( )
2⋅ π⋅ r3⋅ b 3⋅ cos β 3 Nb
⎡ r3 ⎤ r3 := r2t⋅ ⎢ ⎥ ⎣ r2t ⎦ d 4 := 2⋅ r4
⎡ r4⎤ r4 := r3⋅ ⎢ ⎥ ⎣ r3⎦ d 5 := 2⋅ r5
⎡ A 4⎤ ⎥ ⎣ A 3⎦
A 4 := A 3⋅ ⎢
b 4 :=
Nb ⋅ A 4
( )
2⋅ π⋅ r4⋅ cos β 4
Nb ⋅ A 5
A 5 := A 4⋅ ⎢
b 5 :=
r2h = 0.8in
r3 = 2.031in
r4 = 2.55in
r5 = 2.56in
d 2h = 1.6in
d 2t = 4 in
d 3 = 4.062in
d 4 = 5.1in
b 3 = 0.934in
b 4 = 9.738in
b 5 = 8.336in
2
A 4 = 3.012in
A 3 = 1.49in
⎡ r5⎤ r5 := r4⋅ ⎢ ⎥ ⎣ r4⎦
( )
2⋅ π⋅ r5⋅ cos β 5
2
2
A 5 = 2.513in
110
d 5 = 5.12in
81.3°=β 5 Ø 5.120 = d5 Ø 5.100 = d4 Ø 4.062 = d3 Ø 4.000 = d2t Ø 1.594 = d2h
r θ
9.738 =b4
8.336 =b5
b3 = .934
Figure J-1: Sample rotor for Case 2 with side view (starting at station 3).
111
Appendix K
Sample rotor for Case 3 with calculation program
⎡ A 4⎤ ⎢ A ⎥ := 1.092395 ⎣ 3⎦
⎡ A4 ⎤ ⎢ A ⎥ := 1.036009 ⎣ 4.5⎦
⎡ r3 ⎤ ⎢ r ⎥ := 6.89313 ⎣ 2t ⎦ β 3 := 0⋅ deg
⎡ b3⎤ −3 ⎢ r ⎥ := 1.581712× 10 ⎣ 3⎦ β 4 := 84.145266deg ⋅
ζc := 0.4
r2t := 2⋅ in
⎡ r5⎤ r5 := r4⋅ ⎢ ⎥ ⎣ r4⎦ ⎡ b3⎤ ⎥ ⎣ r3 ⎦
b 3 := r3⋅ ⎢
⎛ ⎡ A4 ⎤ ⎞ A 4.5 := A 4⋅ ⎜ ⎢ ⎥ ⎝ ⎣ A 4.5⎦ ⎠
A 3 :=
( )
2⋅ π⋅ r3⋅ b 3⋅ cos β 3
−1
b 4.5 :=
d 3 := 2⋅ r3
⎡ r4.5⎤ ⎢ r ⎥ := 1.0099 ⎣ 4⎦
⎡ r4⎤ r4 := r3⋅ ⎢ ⎥ ⎣ r3⎦ d 4 := 2⋅ r4
⎡ A 4⎤ ⎥ ⎣ A 3⎦
d 5 := 2⋅ r5
( )
2⋅ π⋅ r4⋅ cos β 4
⎡ A5 ⎤ ⎥ ⎣ A 4.5⎦
( )
⎡ r4.5⎤ r4.5 := r4⋅ ⎢ ⎥ ⎣ r4 ⎦
Nb ⋅ A 4
b 4 :=
A 5 := A 4.5⋅ ⎢
2⋅ π⋅ r4.5⋅ cos β 4.5
Nb := 8
d 4.5 := 2⋅ r4.5
A 4 := A 3⋅ ⎢
Nb ⋅ A 4.5
⎡ r5⎤ ⎢ r ⎥ := 1.06018 ⎣ 4⎦
β 5 := 88.76084deg ⋅
⎡ r3 ⎤ r3 := r2t⋅ ⎢ ⎥ ⎣ r2t ⎦
d 2t := 2⋅ r2t
Nb
⎡ r4⎤ ⎢ r ⎥ := 1.069184 ⎣ 3⎦
β 4.5 := 84.910798deg ⋅
r2h := r2t⋅ ζc
d 2h := 2⋅ r2h
⎡ A5 ⎤ ⎢ A ⎥ := 1.235198 ⎣ 4.5⎦
b 5 :=
Nb ⋅ A 5
( )
2⋅ π⋅ r5⋅ cos β 5
r2h = 0.8in
r3 = 13.786in
r4 = 14.74in
r4.5 = 14.886in
r5 = 15.627in
d 2h = 1.6in
d 2t = 4 in
d 3 = 27.573in
d 4 = 29.48in
d 4.5 = 29.772in
d 5 = 31.254in
b 3 = 0.022in
b 4 = 0.218in
b 4.5 = 0.24in
b 5 = 1.159in
2
A 4 = 0.258in
A 3 = 0.236in
2
2
A 4.5 = 0.249in
112
2
A 5 = 0.308in
84.1° = β 4 88.8° = β 5 Ø 29.772 = d4.5 Ø 31.254 = d5
Ø 29.480 = d4 Ø 27.572 = d3 Ø 4.000 = d2t Ø 1.600 = d2h
r b4 = .218 θ
.240 = b4.5
b3 = .022 1.159 = b5
Figure K-1: Sample rotor for Case 3 with side view (starting at station 3)
113
Appendix L
Sample rotor for Case 4 with calculation program
⎡ A 4⎤ ⎢ A ⎥ := 0.85656 ⎣ 3⎦
⎡ A 5⎤ ⎢ A ⎥ := 0.711062 ⎣ 4⎦
⎡ r3 ⎤ ⎢ r ⎥ := 1.225592 ⎣ 2t ⎦
⎡ b3⎤ ⎢ r ⎥ := 0.313776 ⎣ 3⎦
⎡ r4⎤ ⎢ r ⎥ := 1.1 ⎣ 3⎦
β 3 := 0⋅ deg
β 4 := 36.283435deg ⋅
β 5 := 84.535271deg ⋅
ζc := 0.4
r2t := 2⋅ in
d 2h := 2⋅ r2h
r2h := r2t⋅ ζc
d 2t := 2⋅ r2t
⎡ b3⎤ ⎥ ⎣ r3 ⎦
b 3 := r3⋅ ⎢
⎡ A 5⎤ ⎥ ⎣ A 4⎦
A 5 := A 4⋅ ⎢
A 3 :=
b 5 :=
d 3 := 2⋅ r3
( )
2⋅ π⋅ r3⋅ b 3⋅ cos β 3 Nb
⎡ r3 ⎤ r3 := r2t⋅ ⎢ ⎥ ⎣ r2t ⎦
⎡ r5⎤ r5 := r4⋅ ⎢ ⎥ ⎣ r4⎦
d 5 := 2⋅ r5
⎡ A 4⎤ ⎥ ⎣ A 3⎦
A 4 := A 3⋅ ⎢
b 4 :=
Nb ⋅ A 4
( )
2⋅ π⋅ r4⋅ cos β 4
Nb ⋅ A 5
( )
2⋅ π⋅ r5⋅ cos β 5
r4 = 2.696in
r5 = 2.796in
d 2h = 1.6in
d 2t = 4 in
d 3 = 4.902in
b 3 = 0.769in
b 4 = 0.743in
b 5 = 4.312in
2
Nb := 8
⎡ r4⎤ r4 := r3⋅ ⎢ ⎥ ⎣ r3⎦
d 4 := 2⋅ r4
r3 = 2.451in
A 3 = 1.481in
⎡ r5⎤ ⎢ r ⎥ := 1.036995 ⎣ 4⎦
2
A 4 = 1.268in
d 4 = 5.393in
2
A 5 = 0.902in
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d 5 = 5.592in
β 4 = 36° 85° = β 5 Ø 5.592 = d5 Ø 5.392 = d4 Ø 4.902 = d3 Ø 1.600 = d2t Ø 4.000 = d2h
r
θ
4.312= b4
.743 = b5
b3 = .769
Figure L-1: Sample rotor for Case 4 with side view (starting at station 3).
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Vita Manoharan Thiagarajan was born to Malaysian parents in Baton Rouge, Louisiana on August 23, 1977. His early education up until high school was in Kuala Lumpur, Malaysia. After completing high school at Cochrane Road Secondary School (Malaysia), he enrolled in McNeese State University at Lake Charles, Louisiana from 1995 to 1996. He then transferred to Louisiana State University and completed his B.S. in Mechanical Engineering in Fall of 2000. His father, R. Thiagarajan and mother, G. Easwari are Malaysian Government employees. They both served as an agricultural officer and teacher, respectively. In the Fall of 2001, he enrolled at the Mechanical Engineering Department of Virginia Tech as a M.S. graduate student and completed his defense in Summer II. He plans to continue with his studies by pursuing a PhD degree in Aerospace Engineering.
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