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 Oâ€™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 Oâ€™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 authorâ€™s graduate committee composed of Dr. Walter Oâ€™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 authorâ€™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

v

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

x

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

xiv

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 Garrettâ€™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 aircraftâ€™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 companyâ€™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 (Î˛ 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 CÎ¸ 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€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€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€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€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€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€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€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€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€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

114

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).

115

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.

116

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