[PANTONE STIRLING GEET SPAD] Thesis_Simple Gas Turbine Engine Design

A Design Study of Single-Rotor Turbomachinery Cycles

by

Manoharan Thiagarajan

A thesis submitted to the faculty of the Virginia Polytechnic Institute and State University in partial

fulfillment of the requirements for the degree of

Master of Science

in

Mechanical Engineering

Committee

Dr. Peter King, Chairman

Dr. Walter O’Brien, Committee Member

Dr. Clint Dancey, Committee Member

August 12, 2004

Blacksburg, Virginia

Keywords: Auxiliary power unit, single radial rotor, specific power takeoff, compressor, burner, turbine

A Design Study of Single-Rotor Turbomachinery Cycles by

Manoharan Thiagarajan

Dr. Peter King, Chairman

Dr. Walter O’Brien, Committee Member

Dr. Clint Dancey, Committee Member

(ABSTRACT)

Gas turbine engines provide thrust for aircraft engines and supply shaft power for various

applications. They consist of three main components. That is, a compressor followed by a combustion

chamber (burner) and a turbine. Both turbine and compressor components are either axial or centrifugal

(radial) in design. The combustion chamber is stationary on the engine casing. The type of engine that is

of interest here is the gas turbine auxiliary power unit (APU). A typical APU has a centrifugal

compressor, burner and an axial turbine. APUs generate mechanical shaft power to drive equipments such

as small generators and hydraulic pumps. In airplanes, they provide cabin pressurization and ventilation.

They can also supply electrical power to certain airplane systems such as navigation. In comparison to

thrust engines, APUs are usually much smaller in design.

The purpose of this research was to investigate the possibility of combining the three components

of an APU into a single centrifugal rotor. To do this, a set of equations were chosen that would describe

the new turbomachinery cycle. They either were provided or derived using quasi-one-dimensional

compressible flow equations. A MathCAD program developed for the analysis obtained best design

points for various cases with the help of an optimizer called Model Center. These results were then

compared to current machine specifications (gas turbine engine, gasoline and diesel generators). The

result of interest was maximum specific power takeoff. The results showed high specific powers in the

event there was no restriction to the material and did not exhaust at atmospheric pressure. This caused the

rotor to become very large and have a disk thickness that was unrealistic. With the restrictions fully in

place, they severely limited the performance of the rotor. Sample rotor shapes showed all of them to have

unusual designs. They had a combination of unreasonable blade height variations and very large disk

thicknesses. Indications from this study showed that the single radial rotor turbomachinery design might

not be a good idea. Recommendations for continuation of research include secondary flow consideration,

blade height constraints and extending the flow geometry to include the axial direction.

Acknowledgement The author wishes to express his sincere gratitude to Dr. Peter King, major professor, for

contributing valuable time, advice, and assistance to the research and to the preparation of this

manuscript. Sincere thanks are due to the members of the author’s graduate committee composed of Dr.

Walter O’Brien, and Dr. Clint Dancey for their advice and constructive criticism. The author also is

grateful to Phoenix Integration for allowing him to use Model Center for the purpose of optimization to

help in the completion of this research project.

Very special thanks are due to the author’s parents for their understanding, patience, and

encouragement throughout the course of this study. Heartiest thanks are also due to Rene Villanueva, An

Song Nguyen, and Kevin Duffy for all their encouragement. Special appreciation goes out to Ms. Lisa

Stables for all her assistance during this research.

To all turbolabbers, warp speed ahead. Space is the final frontier.

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Table of contents

TABLE OF FIGURES ............................................................................................................................................VII

LIST OF TABLES.................................................................................................................................................... XI

NOMENCLATURE .............................................................................................................................................. XIII

CHAPTER 1 INTRODUCTION.........................................................................................................................1

1.1 ABOUT SMALL GAS TURBINE ENGINES ..........................................................................................................1 1.2 AUXILIARY POWER UNIT (APU) AND PURPOSE OF RESEARCH .......................................................................4

CHAPTER 2 LITERATURE REVIEW.............................................................................................................6

2.1 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 RESULTS OF ANALYSIS.........................................................................................................39

4.1 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 Variable area flow ........................................................................................................................42 4.2 SINGLE ROTOR APU RESULTS .....................................................................................................................43

4.2.1 Model Center and input/output constraints ..................................................................................43 4.2.2 Results from Model Center............................................................................................................46

4.2.2.1 Case 1: Without the stress and |(P0-P5)/P5| constraints. ................................................................................. 46 4.2.2.2 Case 2: With the stress constraint but without the |(P0-P5)/P5| constraint...................................................... 48 4.2.2.3 Case 3: Without the stress constraint but with the |(P0-P5)/P5| constraint...................................................... 49 4.2.2.4 Case 4: With the stress and |(P0-P5)/P5| constraints ....................................................................................... 50

4.2.3 Rotor material and size .................................................................................................................51

CHAPTER 5 CONCLUSION............................................................................................................................52

5.1 SUMMARY...................................................................................................................................................52 5.2 RECOMMENDATIONS...................................................................................................................................52

REFERENCES ..........................................................................................................................................................53

APPENDIX A COMPRESSOR DERIVATIONS .............................................................................................54

A.1 OUTLET RELATIVE MACH NUMBER.............................................................................................................54 A.2 OUTLET RELATIVE STAGNATION TEMPERATURE .........................................................................................55

APPENDIX B BURNER AND TURBINE DERIVATIONS............................................................................56

B.1 CONSERVATION OF ANGULAR MOMENTUM .................................................................................................56 B.2 CONSERVATION OF ENERGY (FIRST LAW OF THERMODYNAMICS)................................................................56 B.3 EQUATION OF STATE ...................................................................................................................................58 B.4 CONSERVATION OF MASS ............................................................................................................................58 B.5 CONSERVATION OF LINEAR MOMENTUM .....................................................................................................58 B.6 RELATIVE STAGNATION TEMPERATURE EQUATION .....................................................................................59 B.7 RELATIVE STAGNATION TEMPERATURE EQUATION .....................................................................................60 B.8 ABSOLUTE STAGNATION TEMPERATURE EQUATION ....................................................................................62 B.9 RELATIVE MACH NUMBER EQUATION.........................................................................................................63 B.10 ABSOLUTE STAGNATION PRESSURE EQUATION ...........................................................................................63 B.11 ENTROPY EQUATION ...................................................................................................................................64 B.12 BURNER ABSOLUTE STAGNATION TEMPERATURE DISTRIBUTION.................................................................64 B.13 BURNER SPECIFIC WORK .............................................................................................................................65 B.14 TURBINE SPECIFIC WORK ............................................................................................................................66

APPENDIX C TO DETERMINE PERPENDICULAR (ONE-DIMENSIONAL) FLOW AREA

BETWEEN THE VANES .........................................................................................................................................67

APPENDIX D CURRENT ENGINE DATA......................................................................................................68

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APPENDIX E COMPLETE RESULTS FOR CASE 1.....................................................................................72

E.1 INPUT PARAMETERS ....................................................................................................................................72 E.2 OUTPUT VALUES .........................................................................................................................................74

APPENDIX F COMPLETE RESULTS FOR CASE 2.....................................................................................82

F.1 INPUT PARAMETERS ....................................................................................................................................82 F.2 OUTPUT VALUES .........................................................................................................................................84

APPENDIX G COMPLETE RESULTS FOR CASE 3.....................................................................................93

G.1 INPUT PARAMETERS ....................................................................................................................................93 G.2 OUTPUT VALUE...........................................................................................................................................95

APPENDIX H COMPLETE RESULTS FOR CASE 4...................................................................................100

H.1 INPUT PARAMETERS ..................................................................................................................................100 H.2 OUTPUT VALUES .......................................................................................................................................102

APPENDIX I SAMPLE ROTOR FOR CASE 1 WITH CALCULATION PROGRAM ...........................108

APPENDIX J SAMPLE ROTOR FOR CASE 2 WITH CALCULATION PROGRAM ...........................110

APPENDIX K SAMPLE ROTOR FOR CASE 3 WITH CALCULATION PROGRAM ...........................112

APPENDIX L SAMPLE ROTOR FOR CASE 4 WITH CALCULATION PROGRAM ...........................114

VITA.........................................................................................................................................................................116

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Table of figures Figure 1-1: Williams International FJ44 turbofan engine, small gas turbine engine (from [1]). .................. 1 Figure 1-2: Pratt & Whitney J58 turbojet engine, large gas turbine engine (from [2]). ............................... 1 Figure 1-3: Years spent in the small gas turbine engine business (from [1])................................................ 3 Figure 1-4: Turboshaft engine (from [2]). .................................................................................................... 4 Figure 1-5: Auxiliary power unit (from [3]). ................................................................................................ 4 Figure 1-6: APU with exhaust vent at the rear of the aircraft (from [4]). ..................................................... 5 Figure 1-7: The new rotor with the combined components will look something like this compressor

impeller (from [5]). ............................................................................................................................... 5 Figure 2-1: Garrett Black Box (from [1]). .................................................................................................... 7 Figure 2-2: GTC43/44 first stage backward curved centrifugal compressor (from [1]). .............................. 8 Figure 2-3: Closed gas turbine engine cycle (from [6]). ............................................................................. 10 Figure 2-4: Closed cycle T-s diagram (from [6])........................................................................................ 10 Figure 2-5: T-s diagram for an ideal jet propulsion cycle along with a turbojet engine schematic (from

[6]). ..................................................................................................................................................... 11 Figure 2-6: APU centrifugal compressor rotor with inducer vanes (from [3]). .......................................... 11 Figure 2-7: Combustion chambers (from [3])............................................................................................. 12 Figure 2-8: Fuel igniter (from [3]). ............................................................................................................. 13 Figure 2-9: APU turbines (from [3])........................................................................................................... 13 Figure 3-1: Cylindrical coordinate system (from [5])................................................................................. 15 Figure 3-2: Shape of rotor with velocity triangle (from [5])....................................................................... 16 Figure 3-3: Burner and turbine control volume between two vanes across a small step change (from [9]).

............................................................................................................................................................ 22

Figure 3-4: Convergent-divergent nozzle with supersonic exit (from [1]). ................................................ 31 Figure 3-5: Variation of specific rupture strength with service temperature (from [5]). ............................ 38 Figure 4-1: Constant area combustion chamber (from [10]). ..................................................................... 40 Figure 4-2: Constant area flow through a duct with heat addition (from [9])............................................. 41 Figure 4-3: Flow through a duct with variable area (from [9]). .................................................................. 42 Figure 4-4: Relative Mach number, stagnation temperature (K) and pressure (Pa) according to location in

the rotor (Case 1). ............................................................................................................................... 47 Figure 4-5: Variation of the absolute tangential velocity (m/s), rotor speed (m/s) and flow curvature (deg)

(Case 1). .............................................................................................................................................. 48 Figure D-1: PSFC and specific power comparison between APU cases and current engines. ................... 71 Figure E-1: Case 1 relative Mach number. ................................................................................................. 76

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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

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Figure H-12: Specific heat ratio (Case 4). ................................................................................................ 106 Figure H-13: Tangential velocity (Case 4). .............................................................................................. 106 Figure H-14: To-s diagram (Case 4).......................................................................................................... 106 Figure H-15: Beginning of To-s diagram (Case 4).................................................................................... 106 Figure H-16: End of To-s diagram (Case 4). ............................................................................................. 106 Figure H-17: Po-s diagram (Case 4). ......................................................................................................... 107 Figure I-1: Sample rotor for Case 1 with side view (starting at station 3). ............................................... 109 Figure J-1: Sample rotor for Case 2 with side view (starting at station 3)................................................ 111 Figure K-1: Sample rotor for Case 3 with side view (starting at station 3) .............................................. 113 Figure L-1: Sample rotor for Case 4 with side view (starting at station 3)............................................... 115

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List of tables Table 3-1: Ambient air equation input parameters. .................................................................................... 17 Table 3-2: Compressor equation input parameters. .................................................................................... 18 Table 3-3: Burner equation input parameters. ............................................................................................ 26 Table 3-4: Turbine equation input parameters. ........................................................................................... 32 Table 3-5: Burner exit flow variables. ........................................................................................................ 35 Table 3-6: Turbine exit flow variables........................................................................................................ 36 Table 4-1: Comparison of burner equations to simple flow example (drag and heat addtion)................... 41 Table 4-2: Comparison of burner equations to simple flow example (heat addition)................................. 42 Table 4-3: Comparison of turbine equations to simple flow example (variable area). ............................... 43 Table 4-4: Model Center input parameters with range limits. .................................................................... 44 Table 4-5: Model Center fixed input values. .............................................................................................. 44 Table 4-6: Model Center output constraints. .............................................................................................. 45 Table 4-7: Overall rotor and other properties (Case 1). .............................................................................. 46 Table 4-8: Overall rotor and other properties (Case 2). .............................................................................. 49 Table 4-9: Overall rotor and other properties (Case 3). .............................................................................. 50 Table 4-10: Overall rotor and other properties (Case 4). ............................................................................ 50 Table D-1: Airplane turboprop engine data. ............................................................................................... 68 Table D-2: Helicopter turboshaft engine data............................................................................................. 68 Table D-3: Aircraft (turboprop) and helicopter (turboshaft) dual-purpose engine data.............................. 69 Table D-4: Four-stroke gasoline generator engine data. ............................................................................. 69 Table D-5: Diesel generator engine data..................................................................................................... 70 Table E-1: Air and diffuser input parameter values (Case 1). .................................................................... 72 Table E-2: Compressor input parameter values (Case 1)............................................................................ 72 Table E-3: Burner input parameter values (Case 1).................................................................................... 73 Table E-4: Turbine input parameter values (Case 1). ................................................................................. 73 Table E-5: Air diffuser output values (Case 1). .......................................................................................... 74 Table E-6: Compressor output values (Case 1). ......................................................................................... 74 Table E-7: Burner output value (Case 1). ................................................................................................... 75 Table E-8: Turbine output value (Case 1)................................................................................................... 75 Table E-9: Rotor overall properties (Case 1). ............................................................................................. 76 Table E-10: Data to show Case 1 configuration is the optimum (Case 1 highlighted below). ................... 79 Table F-1: Air and diffuser input parameter values (Case 2)...................................................................... 82 Table F-2: Compressor input parameter values (Case 2)............................................................................ 82

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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 Introduction

1.1 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 non-

aircraft 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 Literature review

2.1 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 two-

stage compressor for a cabin air compressor. The company called this classified program Project A. Each

stage was a centrifugal compressor rotor. The following are the specifications of each rotor:

a) Mass flow rate of 45 lb/min at a pressure ratio of 1.75

b) 7.25 inch diameter

c) 8 vanes (blades)

d) 30 degree backward curvature (measured from the tangent of the outer diameter)

e) Shrouded cast aluminum impellers

f) Adiabatic efficiency of 78% at the design point

Although the unit was just a laboratory development tool, Project A demonstrated early on that

high efficiencies over broad operating ranges were characteristics of the backward curved compressor

rotor design. This knowledge and experience became an important consideration for aircraft cabin air

conditioning equipment. It was also the foundation for Garrett’s first small gas turbine design called the

Black Box [1].

2.1.2 The Black Box

Boeing wanted a lightweight, compact, self-powered unit that could furnish AC and DC current

to its Model 377 Stratocruiser aircraft’s electrical systems. In addition, it must provide airflow for cabin

pressurization and air conditioning. It should also supply hot air for the wing anti-icing system. Boeing

needed a complete unit in 18 months. Garrett accepted the job and decided to make a small gas turbine

engine. Preliminary work started in the spring of 1945.

As the design progressed, the unit was nicknamed the Black Box due in part to the secrecy of the

project and the fact it had so many gadgets made it look like a magical black box. The engine consisted of

a three stage, backward curved centrifugal compressor, a burner, and a single stage axial turbine. A

geared power takeoff shaft was to run a blower and a generator/alternator. Compressor bleed air routed

through a cooling turbine would generate 40 hp back into the shaft. In addition, the wing anti-icing

system would receive a portion of the exhaust gas. Also in this Black Box were primary and secondary

6

heat exchangers, automatic controls, regulators, and air ducts [11]. Figure 2-1 shows the Black Box as it

was being assembled.

Figure 2-1: Garrett Black Box (from [1]).

Component testing began by mid 1946 and showed excellent overall compressor efficiency in the

neighborhood of 81 to 82%. The high compressor efficiency was not surprising as the technology flowed

directly from Project A. The Black Box pressure ratio was three as opposed to 1.75 in Project A. A three-

stage compressor achieved this ratio. The burner also performed well in tests. This was an important

accomplishment for Garrett, as the company had never before built a burner.

The turbine wheel component testing did not occur due to the unavailability of a suitable test rig

with the capacity to absorb its power. Therefore, turbine testing could only occur until the machine was

ready to run. An external power source drove the Black Box after assembly late in the fall of 1946.

However, it could not generate sufficient power to run by itself. After a month of trying to get the Black

Box to self-run, engineers found the untested turbine component to be the problem. With an efficiency of

less than 70%, the turbine engine was on the borderline of being self-supporting. By that time, there was

7

insufficient time left to redesign the turbine and meet the contract deadline. Subsequently, Garrett had to

cancel the Black Box program at the end of December 1946. The complexity of the unit, low turbine

efficiency, and tight development schedule killed the Black Box project [1].

Despite the cost of the program to Garrett and the problems that it caused with Boeing, there were

some important lessons learned, particularly what not to do. Further work on axial turbines discontinued

at Garrett in favor of the radial inflow turbine. The highly successful backward curved centrifugal

compressor continued in future Garrett projects. The knowledge gained from building a successful

combustor was part of the technology base gained from the program. These efforts produced here carried

on in future Garrett engines especially in the GTC43/44, the company’s first successful gas turbine engine

[1].

2.1.3 The GTC43/44

While the Black Box program was still running, the Navy was looking for a 35 hp gas turbine

starter. That was the power needed to start the 5525 hp Allison XT40 turboprop engines in the Navy

sponsored Convair XPSY-1 flying boat. Agreeing to develop such a unit, Garrett received a contract in

early 1947 by the Navy for the starter.

The engine that Garrett ultimately designed was the GTC43/44. The project started after the

termination of the Black Box program. The design work took place between March and April 1947. The

GTC43/44 contained a two-stage backward curved centrifugal compressor (first stage seen in Figure 2-2)

with an overall pressure ratio of three. From the compressor, two outlets connected with elbows led to

two independent tubular steel combustion chambers. This engine would have a single stage radial inflow

turbine. The unit was to deliver 43 lb of air at 44 lb per square inch absolute pressure, hence its name

GTC43/44 [1].

Figure 2-2: GTC43/44 first stage backward curved centrifugal compressor (from [1]).

On July 1, 1947, turbine wheel tests showed 82 to 84% efficiency. Garrett conducted the first

self-sustaining test run of the GTC43/44 on August 23, 1947. On June 2, 1948, the engine passed its 200-

hour 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

[6]).

Figure 2-4: Closed cycle T-s diagram (from [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 fan-

cooled 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 Formulas used for the APU

3.1 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

C2 = Cr2+Cθ

2+Cz2

W2 = Wr2+Wθ

2+Wz2

(1)

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:

Cp0γ0

γ0 1−

⎛⎜⎜⎝

⎞

⎠R⋅

(2)

b) Ratio of To0 (stagnation temperature at station 0) to T0, τr:

τr 1γ1 1−

2

⎛⎜⎝

⎞

⎠M0

2⋅+

(3)

c) Ratio of Po0 (stagnation pressure at station 0) to P0, πr:

πr τr

γ0

γ0 1−

(4)

d) Ambient air density

ρ0P0

R T0⋅ (5)

e) Diffuser exit stagnation temperature:

17

To2 τd τr⋅ T0⋅ (6)

f) Diffuser exit stagnation pressure:

Po2 πd πr⋅ P0⋅ (7)

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:

T2t To2 1γ0 1−

2

Cz2t

γ0 R⋅ To2⋅

⎛⎜⎜⎝

⎞

⎠

2 Cθ2t

γ0 R⋅ To2⋅

⎛⎜⎜⎝

⎞

⎠

2

+⎡⎢⎢⎣

⎤⎥⎥⎦

⋅−⎡⎢⎢⎣

⎤⎥⎥⎦

⋅

(8)

For Equation (8):

Cz2t

γ0 R⋅ To2⋅

cos β2t( )21

γ0 1−

2

Cθ2t

γ0 R⋅ To2⋅

⎛⎜⎜⎝

⎞

⎠

2

⋅−⎡⎢⎢⎣

⎤⎥⎥⎦

⋅ M2rel2

⋅

1γ0 1−

2M2rel cos β2t( )⋅( )2

⋅+

(9)

b) Inlet tip pressure:

18

P2tPo2

To2T2t

⎛⎜⎝

⎞

⎠

γ0

γ0 1−

(10)

c) Inlet tip density:

ρ 2tP2t

R T2t⋅

(11)

d) Inlet relative stagnation temperature:

To2rel T2t 1γ0 1−

2M2rel

2⋅+

⎛⎜⎝

⎞

⎠⋅

(12)

e) Inlet relative stagnation pressure:

Po2rel P2tTo2rel

T2t

⎛⎜⎝

⎞

⎠

γ0

γ0 1−

⋅

(13)

f) Stagnation temperature ratio:

τc 1 γ0 1−( )U3

γ0 R⋅ To2⋅

⎛⎜⎜⎝

⎞

⎠

2

⋅ 1Wr3U3

tan β3( )⋅−

Cθ2t

γ0 R⋅ To2⋅

Cz2t

γ0 R⋅ To2⋅tan β2t( )⋅+

⎛⎜⎜⎝

⎞

⎠

Cθ2t

γ0 R⋅ To2⋅⋅

U3

γ0 R⋅ To2⋅

⎛⎜⎜⎝

⎞

⎠

2−

⎡⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎦

⋅+

(14)

g) Adiabatic efficiency:

ηcτc

ec 1−

τc 1−=

(15)

h) Stagnation pressure ratio:

πc 1 ηc τc 1−( )+⎡⎣ ⎤⎦

γ0

γ0 1−

(16)

i) Absolute outlet Mach number:

M3a

1γ0 1−

2a⋅−

(17)

19

For Equation (17):

aU3

γ0 R⋅ To2⋅

⎛⎜⎜⎝

⎞

⎠

2 1Wr3U3

tan β3( )⋅−⎛⎜⎝

⎞

⎠

2 Wr3U3

⎛⎜⎝

⎞

⎠

2

+

τc⋅

(18)

j) Relative outlet Mach number (Equation 1 in Appendix A):

M3rel

Wr3

U3

cos β3( )

U3

γ0 R⋅ To2⋅

τc

1γ0 1−

2M3

2⋅+

⋅

(19)

k) Outlet absolute stagnation temperature:

To3 τc τd⋅ τr⋅ T0⋅= (20)

l) Outlet absolute stagnation pressure:

Po3 πc πd⋅ πr⋅ P0⋅= (21)

m) Outlet relative stagnation temperature (Equation 2 in Appendix A):

To3rel To3 1γ0 1−

2 1γ0 1−

2M3

2⋅+

⎛⎜⎝

⎞

⎠⋅

M32 M3rel

2−⎛

⎝⎞⎠⋅−

⎡⎢⎢⎢⎣

⎤⎥⎥⎥⎦

⋅

(22)

n) Outlet relative stagnation pressure:

Po3relPo3

To3To3rel

⎛⎜⎝

⎞

⎠

γ0

γ0 1−

(23)

o) Outlet temperature:

T3To3rel

1γ0 1−

2M3rel

2⋅+

(24)

p) Outlet pressure:

20

P3Po3rel

To3relT3

⎛⎜⎝

⎞

⎠

γ0

γ0 1−

(25)

q) Outlet relative velocity:

W3 2 Cp0⋅ To3rel T3−( )⋅ (26)

r) Outlet density:

ρ3P3

R T3⋅=

(27)

s) Dimensionless impeller rotation:

m3

Po2

Ω

γc R⋅ To2⋅( )14

⋅ 2 π⋅ γc⋅Cθ2t

γc R⋅ To2⋅

Cz2t

γc R⋅ To2⋅tan β2t( )⋅+

⎛⎜⎜⎝

⎞

⎠

2

⋅

ζc

1

y1γc 1−

2

Cz2t

γc R⋅ To2⋅

⎛⎜⎜⎝

⎞

⎠

2 Cθ2t

γc R⋅ To2⋅

⎛⎜⎜⎝

⎞

⎠

2

2 y2−( )⋅+⎡⎢⎢⎣

⎤⎥⎥⎦

⋅−⎡⎢⎢⎣

⎤⎥⎥⎦

1γc 1−

Cz2t

γc R⋅ To2⋅

⎛⎜⎜⎝

⎞

⎠

2

2Cθ2t

γc R⋅ To2⋅

⎛⎜⎜⎝

⎞

⎠

2

⋅ y2 1−( )⋅−⋅ y⋅

⌠⎮⎮⎮⎮⎮⌡

d⋅= (28)

t) Radius ratio:

r3r2t

U3

γ0 R⋅ To2⋅

Cθ2t

γ0 R⋅ To2⋅

Cz2t

γ0 R⋅ To2⋅tan β2t( )⋅+

(29)

u) Outlet blade height to radius ratio:

b3r3

γ0 1+

2

⎛⎜⎝

⎞

⎠

1

γ0 1−

1 ζc2

−⎛⎝

⎞⎠⋅

r3r2t

⎛⎜⎝

⎞

⎠

2−

⋅

2 11 ηc+

2

⎛⎜⎝

⎞

⎠γ0 1−( )⋅

U3

γ0 R⋅ To2⋅

⎛⎜⎜⎝

⎞

⎠

2

⋅ 1Wr3U3

tan β3( )⋅−⎛⎜⎝

⎞

⎠⋅+

⎡⎢⎢⎣

⎤⎥⎥⎦

1

γ0 1−

⋅

(30)

v) Outlet rotor speed:

U3U3

γ0 R⋅ To2⋅

⎛⎜⎜⎝

⎞

⎠γ0 R⋅ To2⋅⋅

(31)

w) Inlet rotor tip speed:

21

U2t U3r3

r2t

⎛⎜⎜⎝

⎞

⎠

1−

⋅=

(32)

x) Mass flow rate to outlet area ratio:

m3A3

ρ 3 W3⋅

(33)

y) Outlet entropy:

s3 s0 Cp0 ln τc( )⋅+ R ln πc( )⋅− (34)

z) Specific power:

)TT(C

m 2o3o0p3

c −⋅−=W

(35)

3.4 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]:

Cp

28.11kJ

kmol K⋅⋅ 0.1967 10 2−⋅

kJ

kmol K2⋅⋅ T⋅+ 0.4802 10 5−⋅

kJ

kmol K3⋅⋅ T2⋅+ 1.966 10 9−⋅

kJ

kmol K4⋅⋅ T3⋅−

28.97kg

kmol

=

(36)

Therefore, the burner and turbine equations are thermally perfect. The specific heat ratio is then:

γCp

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 hHV⋅ ηb⋅ U2+ W2−

2 Cp⋅ T⋅1−

⎛⎜⎜⎝

⎞

⎠

d m( )

m⋅

W2

Cp T⋅

d W( )

W⋅−

d T( )

T−

U d U( )⋅

Cp T⋅−=

(38)

b) Equation of state:

d ρ( )

ρ

d T( )

T+

d P( )

P− 0=

(39)

c) Conservation of mass:

d W( )

W

d ρ( )

ρ+

d m( )

m−

d A( )

A−=

(40)

d) Conservation of linear momentum:

d W( )

W

d m( )

m+

P

ρ W2⋅

d P( )

P⋅+

1

2− d CD( )⋅=

(41)

e) Relative stagnation temperature equation:

d To( )To

Torel

To

d Torel( )Torel

⋅−U W⋅ sin β( )⋅

Cp To⋅

d W( )

W⋅+

U d U( )⋅ W d U sin β( )⋅( )⋅−

Cp To⋅=

(42)

f) Relative stagnation pressure equation:

d Porel( )Porel

γ Mrel2⋅

2

d Torel( )Torel

⋅+ γ Mrel2⋅

d m( )

m⋅+

γ Mrel2⋅

2− d CD( )⋅=

(43)

g) Absolute stagnation temperature equation:

d To( )To

T

To

d T( )

T⋅−

W W U sin β( )⋅−( )⋅

Cp To⋅

d W( )

W⋅−

U d U( )⋅ W d U sin β( )⋅( )⋅−

Cp To⋅=

(44)

h) Relative Mach number equation:

MrelW

γ R⋅ T⋅=

(45)

i) Absolute stagnation pressure equation:

Po PorelTo

Torel

⎛⎜⎜⎝

⎞

⎠

γγ 1−

⋅=

(46)

j) Entropy equation:

s s3 Cp lnToTo3

⎛⎜⎝

⎞

⎠⋅+ R ln

PoPo3

⎛⎜⎝

⎞

⎠⋅−

(47)

Notice that Equations (38) through (44) are differential equations that require a numerical solution.

24

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

r31

δr

r3+=

(48)

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:

dδrr3

⎛⎜⎝

⎞

⎠

r4r3

1−

nb

(49)

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:

δrr3

i dδrr3

⎛⎜⎝

⎞

⎠⋅

(50)

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

A31 Y1

δr

r3

⎛⎜⎝

⎞

⎠⋅+ Y2

δr

r3

⎛⎜⎝

⎞

⎠

2⋅+=

(51)

The variables Y1 and Y2 are specified coefficients.

f) Obtain the variation of angle β using the following polynomial:

β β3 S1δr

r3

⎛⎜⎝

⎞

⎠⋅+ S2

δr

r3

⎛⎜⎝

⎞

⎠

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:

U U3r

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

To To3To4 To3−

r4

r31−

δr

r3

⎛⎜⎝

⎞

⎠⋅+=

(54)

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:

1−

1

0

0

0

0

TTo

−

W2

Cp T⋅−

0

1

1

U W⋅ sin β( )⋅

Cp To⋅

0

W W U sin β( )⋅−( )⋅

Cp To⋅−

0

1−

0

P

ρ W2⋅

0

0

0

0

1

1

0

0

0

0

0

0

0

0

TorelTo

−

γ Mrel2

⋅

2

0

0

0

0

0

0

1

0

2 hHV⋅ ηb⋅ U2+ W2

−

2 Cp⋅ T⋅1−

0

1−

1

0

γ Mrel2

⋅

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

d T( )T

d W( )W

d P( )P

d ρ( )ρ

d Torel( )Torel

d Porel( )Porel

d m( )m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⋅

U d U( )⋅

Cp T⋅−

0

d A( )A

−

12

− d CD( )⋅

d To( )To

−U d U( )⋅ W d U sin β( )⋅(⋅−

Cp To⋅

)+

γ Mrel2

⋅

2− d CD( )⋅

d To( )To

−U d U( )⋅ W d U sin β( )⋅( )⋅−

Cp To⋅+

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

(55)

Inverting the matrix in Equation (55) gives:

26

d T( )T

d W( )W

d P( )P

d ρ( )ρ

d Torel( )Torel

d Porel( )Porel

d m( )m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

1−

1

0

0

0

0

TTo

−

W2

Cp T⋅−

0

1

1

U W⋅ sin β( )⋅

Cp To⋅

0

W W U sin β( )⋅−( )⋅

Cp To⋅−

0

1−

0

P

ρ W2⋅

0

0

0

0

1

1

0

0

0

0

0

0

0

0

TorelTo

−

γ Mrel2

⋅

2

0

0

0

0

0

0

1

0

2 hHV⋅ ηb⋅ U2+ W2−

2 Cp⋅ T⋅1−

0

1−

1

0

γ Mrel2⋅

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1− U d U( )⋅

Cp T⋅−

0

d A( )A

−

12

− d CD( )⋅

d To( )To

−U d U( )⋅ W d U sin β( )⋅(⋅−

Cp To⋅

)+

γ Mrel2

⋅

2− d CD( )⋅

d To( )To

−U d U( )⋅ W d U sin β( )⋅( )⋅−

Cp To⋅+

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞

⎠

⋅ (56)

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:

Ti Ti 1−d T( )

T⎛⎜⎝

⎞⎠

Ti 1−⋅+

Wi Wi 1−d W( )

W⎛⎜⎝

⎞⎠

Wi 1−⋅+

Pi Pi 1−d P( )

P⎛⎜⎝

⎞⎠

Pi 1−⋅+

ρi ρi 1−d ρ( )

ρ⎛⎜⎝

⎞⎠

ρi 1−⋅+

PoreliPoreli 1−

d Porel( )Porel

⎛⎜⎝

⎞

⎠Poreli 1−

⋅+

ToiToi 1−

d To( )To

⎛⎜⎝

⎞

⎠Toi 1−

⋅+

mA3

⎛⎜⎝

⎞⎟⎠ i

mA3

⎛⎜⎝

⎞⎟⎠ i 1−

d m( )m

⎛⎜⎝

⎞⎟⎠

mA3

⎛⎜⎝

⎞⎟⎠ i 1−

⋅+

Mreli

Wi

γi R⋅ Ti⋅

PoiPoreli

Toi

Toreli

⎛⎜⎜⎝

⎞

⎠

γ i

γ i 1−

⋅

si s3 Cpiln

Toi

To3

⎛⎜⎜⎝

⎞

⎠⋅+ R ln

Poi

Po3

⎛⎜⎜⎝

⎞

⎠⋅−

(57)

27

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

AA3

⎡⎢⎣

⎤⎥⎦

⎛⎜⎝

⎞⎠ i

AA3

⎡⎢⎣

⎤⎥⎦

⎛⎜⎝

⎞⎠i 1−

−

AA3

⎡⎢⎣

⎤⎥⎦

⎛⎜⎝

⎞⎠ i 1−

−

d U sin β( )⋅( ) Ui sin βi( )⋅ Ui 1− sin βi 1−( )⋅−

d U( ) Ui Ui 1−−

d To( ) Toi

Toi 1−−

(58)

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:

fm4m3

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 = U-

W*sin(β), the specific power of the burner is:

⎥⎦

⎤⎢⎣

⎡β⋅−⋅−β⋅−⋅⋅−= ))sin(WU(U))sin(WU(U

mm

m 333344443

4

3

bW

(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:

W2

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( )

P− 0

(62)

28

c) Conservation of mass:

d W( )W

d ρ( )ρ

+d A( )

A−

(63)

d) Conservation of linear momentum:

d W( )W

P

ρ W2⋅

d P( )P

⋅+ 0

(64)

e) Relative stagnation temperature equation:

d To( )To

TorelTo

d Torel( )Torel

⋅−U W⋅ sin β( )⋅

Cp To⋅

d W( )W

⋅+U d U( )⋅ W d U sin β( )⋅( )⋅−

Cp To⋅ (65)

f) Relative stagnation pressure equation:

d Porel( )Porel

γ Mrel2

⋅

2

d Torel( )Torel

⋅+ 0

(66)

g) Absolute stagnation temperature equation:

d To( )To

TTo

d T( )T

⋅−W W U sin β( )⋅−( )⋅

Cp To⋅

d W( )W

⋅−U d U( )⋅ W d U sin β( )⋅( )⋅−

Cp To⋅ (67)

h) Relative Mach number equation:

MrelW

γ R⋅ T⋅=

(68)

i) Absolute stagnation pressure equation:

Po Po4ToTo4

⎛⎜⎝

⎞

⎠

γ

γ 1−

⋅

(69)

j) Entropy equation

s s4 Cp lnToTo4

⎛⎜⎝

⎞

⎠⋅+ R ln

PoPo4

⎛⎜⎝

⎞

⎠⋅−

(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:

rr4

1δrr4

+

(71)

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:

dδrr4

⎛⎜⎝

⎞

⎠

r5r4

1−

nt

(72)

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:

δrr4

i dδrr4

⎛⎜⎝

⎞

⎠⋅

(73)

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:

AA4

1 K1δrr4

⎛⎜⎝

⎞

⎠⋅+ K2

δrr4

⎛⎜⎝

⎞

⎠

2⋅+

(74)

The variables K1 and K2 are specified coefficients.

f) Obtain the variation of angle β using the following polynomial:

β β4 B1δrr4

⎛⎜⎝

⎞

⎠⋅+ B2

δrr4

⎛⎜⎝

⎞

⎠

2⋅+

(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 U4rr4

⎛⎜⎝

⎞⎠

⋅

(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:

r5r4.5

r5r4

⎛⎜⎝

⎞

⎠

r4.5r4

⎛⎜⎝

⎞

⎠

1−

⋅

(77)

31

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:

AA4.5

1 KK1δr

r4.5

⎛⎜⎝

⎞

⎠⋅+ KK2

δrr4.5

⎛⎜⎝

⎞

⎠

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:

A5A4

A4.5A4

⎛⎜⎝

⎞

⎠

A5A4.5

⎛⎜⎝

⎞

⎠ (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

1−

1

0

0

0

0

TTo

−

W2

Cp T⋅−

0

1

1

U W⋅ sin β( )⋅

Cp To⋅

0

W W U sin β( )⋅−( )⋅

Cp To⋅−

0

1−

0

P

ρ W2⋅

0

0

0

0

1

1

0

0

0

0

0

0

0

0

TorelTo

−

γ Mrel2⋅

2

0

0

0

0

0

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 T( )T

d W( )W

d P( )P

d ρ( )ρ

d Torel( )Torel

d Porel( )Porel

d To( )To

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

1−

1

0

0

0

0

TTo

−

W2

Cp T⋅−

0

1

1

U W⋅ sin β( )⋅

Cp To⋅

0

W W U sin β( )⋅−( )⋅

Cp To⋅−

0

1−

0

P

ρ W2⋅

0

0

0

0

1

1

0

0

0

0

0

0

0

0

TorelTo

−

γ Mrel2

⋅

2

0

0

0

0

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

TTo4.5rel

1γ4.5 1−

2Mrel

2⋅+

W Mrel γ4.5 R⋅ T⋅⋅

PPo4.5rel

1γ4.5 1−

2Mrel

2⋅+

⎛⎜⎝

⎞

⎠

γ4.5

γ4.5 1−

ρP

R T⋅

(82)

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:

Ti Ti 1−d T( )

T⎛⎜⎝

⎞⎠

Ti 1−⋅+

Wi Wi 1−d W( )

W⎛⎜⎝

⎞⎠

Wi 1−⋅+

Pi Pi 1−d P( )

P⎛⎜⎝

⎞⎠

Pi 1−⋅+

ρi ρi 1−d ρ( )

ρ⎛⎜⎝

⎞⎠

ρi 1−⋅+

PoreliPoreli 1−

d Porel( )Porel

⎛⎜⎝

⎞

⎠Poreli 1−

⋅+

ToiToi 1−

d To( )To

⎛⎜⎝

⎞

⎠Toi 1−

⋅+

Mreli

Wi

γi R⋅ Ti⋅

PoiPo4

Toi

To4

⎛⎜⎜⎝

⎞

⎠

γ i

γ i 1−

⋅

si s4 Cpiln

Toi

To4

⎛⎜⎜⎝

⎞

⎠⋅+ R ln

Poi

Po4

⎛⎜⎜⎝

⎞

⎠⋅−

(83)

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

AA4

⎛⎜⎝

⎞⎠i

AA4

⎛⎜⎝

⎞⎠i 1−

−

AA4

⎛⎜⎝

⎞⎠i 1−

, supersonic: d A( )A

AA4.5

⎛⎜⎝

⎞⎠i

AA4.5

⎛⎜⎝

⎞⎠i 1−

−

AA4.5

⎛⎜⎝

⎞⎠i 1−

d U sin β( )⋅( ) Ui sin βi( )⋅ Ui 1− sin βi 1−( )⋅−

d U( ) Ui Ui 1−−

(84)

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:

[ ]))sin(WU(U))sin(WU(U)f1(m 44445555

3

t β⋅−⋅−β⋅−⋅⋅+−=W

(85)

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

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:

cc

cb

ct

cTO

mmmmP WWW

++=

(86)

b) Power takeoff coefficient:

CTO

PTOmc

⎛⎜⎝

⎞

⎠Cpc T0⋅

(87)

c) Power specific fuel consumption (PSFC):

mfPTO

fPTOmc

⎛⎜⎝

⎞

⎠

(88)

d) Thermal efficiency:

ηTHCTO

f hHV⋅

Cpc T0⋅

(89)

e) Rotor radius-to-compressor inlet tip ratio:

r5r2t

r3r2t

⎛⎜⎝

⎞

⎠

r4r3

⎛⎜⎝

⎞

⎠⋅

r5r4

⎛⎜⎝

⎞

⎠⋅

(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

r5r2h

ζc1− r3

r2t

⎛⎜⎝

⎞

⎠⋅

r4r3

⎛⎜⎝

⎞

⎠⋅

r5r4

⎛⎜⎝

⎞

⎠⋅

(91)

The disk hub-to-rim thickness ratio, z2h/z5 from the Hill and Petersen textbook [5] is:

z2hz5

exp

1r5r2h

⎛⎜⎝

⎞

⎠

2−

−

2

U52

σ

ρ material

⎛⎜⎝

⎞

⎠⋅

⎡⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎦

(92)

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

Chapter 4 Results of analysis A MathCAD program created to carry out the new rotor analysis consisted of an input parameters

section and an equations/results section. In the equations/results section, the program performed the

analysis according to the following steps:

a) Air and diffuser

b) Compressor

c) Burner

d) Turbine

e) Overall APU properties

The turbine portion evaluates only the subsonic flow if the supersonic region does not occur.

Before continuing with the rotor analysis, it was important to validate the burner and turbine

equations by comparing them with simple flow problems. The purpose was to provide confidence in the

usage of the burner and turbine equations.

4.1 Simple one-dimensional flow

A simple flow involves no curvature (β does not change) and rotation along with constant

specific heats. For convenience, the equations in this section will use relative frame variables (absolute

and relative frames are the same for simple flows). Simple cases for the burner are:

a) Flow in a constant area duct with drag and heat addition.

b) Flow in a constant area duct with only heat addition.

For both cases above, mf is extremely small compared to m3. As for the turbine, the chosen simple case is

the variable area duct flow.

39

4.1.1 Burner

4.1.1.1 Constant area flow with drag and heat addition

Figure 4-1: Constant area combustion chamber (from [10]).

Consider a constant area duct with flame holders at the beginning that contribute a drag force to

the flow as seen in Figure 4-1. The equation for M4rel in terms of the upstream variables and a prescribed

τbrel [10] is:

M4rel2 χ⋅

1 2 γb⋅ χ⋅− 1 2 γb 1+( )⋅ χ⋅−⎡⎣ ⎤⎦

1

2+

(93)

where:

χγcγb

M3rel2 1

γc 1−

2M3rel

2⋅+

⎛⎜⎝

⎞

⎠⋅

1 γc M3rel2

⋅ 1CD2

−⎛⎜⎝

⎞⎠

⋅+⎡⎢⎣

⎤⎥⎦

2⋅ τbrel⋅

(94)

Next, the equation for πbrel [10] is:

πbrel

1 γc M3rel2

⋅ 1CD2

−⎛⎜⎝

⎞⎠

⋅+

1 γb M4rel2

⋅+

1γb 1−

2M4rel

2⋅+

⎛⎜⎝

⎞

⎠

γb

γb 1−

1γc 1−

2M3rel

2⋅+

⎛⎜⎝

⎞

⎠

γc

γc 1−

⋅

(95)

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).

M4rel πbrelBurner

input

To4

(K) τbrel Burner

equations

Equation

(93)

Burner

equations

Equation

(95)

600 1.190476 0.231116 0.230973 0.952213 0.952305

900 1.785714 0.297083 0.294143 0.93304 0.934004

M3rel = 0.2

T3 = 500K

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

1200 2.380952 0.362688 0.355036 0.911921 0.914513

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).

M4rel πbrelBurner

input

To4

(K) τbrel Burner

equations

Equation

(93)

Burner

equations

Equation

(95)

600 1.190476 0.221122 0.220526 0.994463 0.994625

900 1.785714 0.283147 0.27976 0.976184 0.97725

M3rel = 0.2

T3 = 500K

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

1200 2.380952 0.343898 0.336013 0.956187 0.958871

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

A5A4

M4relM5rel

⎛⎜⎝

⎞

⎠

1γt 1−

2M5rel

2⋅+

1γt 1−

2M4rel

2⋅+

⎛⎜⎜⎜⎜⎝

⎞

⎟⎟

⎠

γ t 1+

2 γ t 1−( )⋅

⋅

(96)

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).

A5/A4Turbine

input Μ4rel Μ5rel Turbine

equations Equation (96)

0.5 2.013896 1.273793 1.274185

0.7 2.426491 2.249443 2.250246

T4 = 500K

P4 = 500kPa

Cpt = 1.008123 kJ/(kg*K)

R = 0.287005 kJ/(kg*K)

γ4 = 1.4

r5/r4 = 2

nt = 8000 0.9 2.583947 2.826604 2.827483

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 Constraints

Output

variables Constraints

Output

variables Constraints

M3rel maximum of 0.7 β4 (deg) maximum of 90

deg β5

maximum of 90

deg

πcbetween 1.1 and

30 A5/A4.5 maximum of 4 z2h/z5 between 1 and 3

r3/r2t at least 1.1 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

(W/kg/s)

mf/PTO

(kg/s/W) ηTH r5/r2h z2h/z5

412218.499688 5.256221E-8 0.454406 65.390339 1.748837E+4

πcβ5

(deg)

Wc/m3

(W/kg/s)

Wb/m3

(W/kg/s)

Wt/m3

(W/kg/s)

6.061594 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

0 10 20 300

1

2

3

Mrel

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

0 10 20 300

500

1000

1500

To

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

0 10 20 300

5 .105

1 .106

Po

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

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

0 10 20 301000

500

0

500

Cθ

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

0 10 20 300

500

1000

U

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

0 10 20 300

50

100

βdeg

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

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

(W/kg/s)

mf/PTO

(kg/s/W) ηTH r5/r2h z2h/z5

127884.924331 2.053457E-7 0.116314 3.211841 2.932721

πcβ5

(deg)

Wc/m3

(W/kg/s)

Wb/m3

(W/kg/s)

Wt/m3

(W/kg/s)

1.183161 81.378184 -16268.094656 81713.818746 62439.200241

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

(W/kg/s)

mf/PTO

(kg/s/W) ηTH r5/r2h z2h/z5

132248.596648 1.57612E-7 0.15154 19.533881 1.292149E+4

πcβ5

(deg)

Wc/m3

(W/kg/s)

Wb/m3

(W/kg/s)

Wt/m3

(W/kg/s)

17.4036 88.76084 -436026.78374 310579.14829 257696.232095

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

(W/kg/s)

mf/PTO

(kg/s/W) ηTH r5/r2h z2h/z5

17641.820934 1.357239E-6 0.017598 3.495073 2.489291

πcβ5

(deg)

Wc/m3

(W/kg/s)

Wb/m3

(W/kg/s)

Wt/m3

(W/kg/s)

1.572254 84.535271 -45803.940205 19140.007112 44305.754028

50

When completely constrained, this case produced the lowest specific power takeoff and highest PSFC

among the four cases. Part of this reason was that the flow ended at M5rel equal to 0.476262371. Figure

H-13 showed there was not much of a drop in Cθ between the burner inlet and turbine exit to significantly

power the compressor and produce specific power takeoff at the same time. Like in Case 3, the

compressor absorbed most of the specific power generated. Placing this case into Figure D-1 showed that

it was far from the gas turbine engine points in terms of both specific power takeoff and PSFC. Appendix

H shows the rest of the results for this case.

4.2.3 Rotor material and size

The specific strength for this rotor (including all the other cases) ended at its highest given limit

of 30 kPa/kg/m3. This once again indicated that the rotor required the strongest material possible. Using

Figure 3-5 (with To4 as the service temperature), a possible material was molybdenum alloy stainless

steel.

Next, it was important to illustrate how the rotor for each case would look like. Appendix I

through Appendix L shows how this took place. All four cases had eight vanes and r2t = 2 inches.

Appendix I showed Case 1 had an unusual blade height distribution. It was very small at the compressor

outlet but took on a very large value at the rotor exit. In Case 2, the compressor exit blade height started

out at 0.934 inches but when it reached the burner outlet, the height was 9.738 inches. Both the first two

cases obviously had unusual rotor designs. This also occurred in Cases 3 and 4. The value of b5 equal to

1.159 inches in Case 3 made this rotor seem reasonable. However, both the ratio b5/b3 = 53.131 and z2h/z5

= 1.292149E+4 in Case 3 made this rotor unrealistic. As for Case 4, b4 = 0.743 inches and b5 = 4.312

inches. Each rotor design had geometries that made their manufacturing not practical. A general

observation was the compressor always had the largest size compared to the other components.

51

Chapter 5 Conclusion

5.1 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-one-

dimensional 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 Compressor Derivations

A.1 Outlet relative Mach number

M3relW3

γ0 R⋅ T3⋅

M3relWr3

cos β3( )1

γ0 R⋅ T3⋅⋅ <-----W3

Wr3cos β3( )

M3rel

Wr3

U3

cos β3( )U3

γ0 R⋅ T3⋅⋅

M3rel

Wr3

U3

cos β3( )U3

γ0 R⋅ To3⋅

1γ0 1−

2M3

2⋅+

⋅ <-----T3To3

1γ0 1−

2M3

2⋅+

M3rel

Wr3

U3

cos β3( )U3

γ0 R⋅ To2⋅To3To2

⋅

1γ0 1−

2M3

2⋅+

⋅

M3rel

Wr3

U3

cos β3( )U3

γ0 R⋅ To2⋅ τc⋅

1γ0 1−

2M3

2⋅+

⋅ <-----τcTo3To2

M3rel

Wr3

U3

cos β3( )

U3

γ0 R⋅ To2⋅

τc

1γ0 1−

2M3

2⋅+

⋅ <-----Equation 1

54

A.2 Outlet relative stagnation temperature

since Cp0 T3⋅ Cp0 To3⋅C3

2

2− and Cp0 T3⋅ Cp0 To3rel⋅

W32

2− :

Cp0 To3rel⋅W3

2

2− Cp0 To3⋅

C32

2−

Cp0 To3rel⋅ Cp0 To3⋅C3

2

2−

W32

2+

To3rel To3C3

2 W32

−

2 Cp0⋅−

To3rel To3 1C3

2 W32−

2 Cp0⋅ To3⋅−

⎛⎜⎜⎝

⎞

⎠⋅

To3rel To3 1γ0 R⋅ T3⋅

2 Cp0⋅ To3⋅

C32

γ0 R⋅ T3⋅

W32

γ0 R⋅ T3⋅−

⎛⎜⎜⎝

⎞

⎠⋅−

⎡⎢⎢⎣

⎤⎥⎥⎦

⋅

To3rel To3 1γ0 R⋅

2 Cp0⋅To3T3

⋅

M32 M3rel

2−⎛

⎝⎞⎠⋅−

⎡⎢⎢⎢⎣

⎤⎥⎥⎥⎦

⋅ <-----M3rel2 W3

2

γ0 R⋅ T3⋅ and M3

2 C32

γ0 R⋅ T3⋅

To3rel To3 1γ0 γ0 1−( )⋅

2 γ0⋅ 1γ0 1−

2M3

2⋅+⎛⎜⎝

⎞

⎠⋅

M32 M3rel

2−⎛

⎝⎞⎠⋅−

⎡⎢⎢⎢⎣

⎤⎥⎥⎥⎦

⋅ <-----R

Cp0

γ0 1−

γ0 and

To3T3

1γ0 1−

2M3

2⋅+

To3rel To3 1γ0 1−

2 1γ0 1−

2M3

2⋅+

⎛⎜⎝

⎞

⎠⋅

M32 M3rel

2−⎛⎝

⎞⎠⋅−

⎡⎢⎢⎢⎣

⎤⎥⎥⎥⎦

⋅ <-----Equation 2

55

Appendix B Burner and turbine derivations

B.1 Conservation of angular momentum

inout )CUm()CUm( θθ ⋅⋅−⋅⋅=−W

B.2 Conservation of energy (first law of thermodynamics)

inoouto )hm()hm(Q ⋅−⋅=− W

since ho hC2

2+ :

in

2

out

2

2Chm

2ChmQ

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⋅−

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⋅=− W

Q m U⋅ Cθ⋅( )out

+ m U⋅ Cθ⋅( )in

− m hC2

2+

⎛⎜⎝

⎞

⎠⋅

⎡⎢⎣

⎤⎥⎦out

m hC2

2+

⎛⎜⎝

⎞

⎠⋅

⎡⎢⎣

⎤⎥⎦in

− <-----place conservation of angular momentum here

Q m U⋅ Cθ⋅( )out

+ mC2

2⋅

⎛⎜⎝

⎞

⎠out− m U⋅ Cθ⋅( )

in− m

C2

2⋅

⎛⎜⎝

⎞

⎠in+ m h⋅( )out m h⋅( )in−

Q m U Cθ⋅C2

2−

⎛⎜⎝

⎞

⎠⋅

⎡⎢⎣

⎤⎥⎦out

+ m U Cθ⋅C2

2−

⎛⎜⎝

⎞

⎠⋅

⎡⎢⎣

⎤⎥⎦ in

− m h⋅( )out m h⋅( )in−

for U Cθ⋅C2

2− :

U Cθ⋅C2

2− U Cθ⋅

Cr2 Cθ

2+

2− <-----C2 Cr

2 Cθ2

+

U Cθ⋅C2

2− U U Wθ−( )⋅

Wr2 U Wθ−( )2

+

2− <-----Cθ U Wθ− and Wr Cr (both from velocity triangle)

U Cθ⋅C2

2− U U Wθ−( )⋅

Wr2 U2

+ 2 U⋅ Wθ⋅− Wθ2

+

2−

U Cθ⋅C2

2− U U Wθ−( )⋅

W2 U2+ 2 U⋅ Wθ⋅−

2− <-----W2 Wr

2 Wθ2

+

U Cθ⋅C2

2− U2 U Wθ⋅−

W2

2−

U2

2− U Wθ⋅+

U Cθ⋅C2

2−

U2

2W2

2−

56

<-----Equation 12 hHV⋅ ηb⋅ U2

+ W2−

2 Cp⋅ T⋅1−

⎛⎜⎜⎝

⎞

⎠

d m( )m

⋅W2

Cp T⋅d W( )

W⋅−

d T( )T

−U d U( )⋅

Cp T⋅−

<-----divide by Cp T⋅ 2 hHV⋅ ηb⋅ U2

+ W2−

2 Cp⋅ T⋅1−

⎛⎜⎜⎝

⎞

⎠

d m( )m

⋅U d U( )⋅

Cp T⋅+

W d W( )⋅

Cp T⋅−

d T( )T

− 0

2 hHV⋅ ηb⋅

2U2

2+

W2

2− Cp T⋅−

⎛⎜⎝

⎞

⎠d m( )

m⋅ U d U( )⋅+ W d W( )⋅− Cp d T( )⋅− 0

hHV ηb⋅U2

2+

W2

2− Cp T⋅−

⎛⎜⎝

⎞

⎠d m( )

m⋅ d

U2

2

⎛⎜⎝

⎞

⎠+ d

W2

2

⎛⎜⎝

⎞

⎠− Cp d T( )⋅− 0

hHV ηb⋅d m( )

m⋅

U2

2d m( )

m⋅+

W2

2d m( )

m⋅− Cp T⋅

d m( )m

⋅− dU2

2

⎛⎜⎝

⎞

⎠+ d

W2

2

⎛⎜⎝

⎞

⎠− Cp d T( )⋅− 0

<-----d Q( ) hHV ηb⋅ d m( )⋅ hHV ηb⋅d m( )

m⋅ d

U2

2

⎛⎜⎝

⎞

⎠+

U2

2d m( )

m⋅+ d

W2

2

⎛⎜⎝

⎞

⎠−

W2

2d m( )

m⋅− Cp d T( )⋅ Cp T⋅

d m( )m

⋅+

<-----divide by m d Q( )m

dU2

2

⎛⎜⎝

⎞

⎠+

U2

2d m( )

m⋅+ d

W2

2

⎛⎜⎝

⎞

⎠−

W2

2d m( )

m⋅− Cp d T( )⋅ Cp T⋅

d m( )m

⋅+

d Q( ) m dU2

2

⎛⎜⎝

⎞

⎠⋅+

U2

2d m( )⋅+ m d

W2

2

⎛⎜⎝

⎞

⎠⋅−

W2

2d m( )⋅− m Cp⋅ d T( )⋅ Cp T⋅ d m( )⋅+

d Q( ) d mU2

2⋅

⎛⎜⎝

⎞

⎠+ d m

W2

2⋅

⎛⎜⎝

⎞

⎠− d m Cp⋅ T⋅( )

convert equation above into differential form:

Q mU2

2⋅

⎛⎜⎝

⎞

⎠out+ m

U2

2⋅

⎛⎜⎝

⎞

⎠ in− m

W2

2⋅

⎛⎜⎝

⎞

⎠outm

W2

2⋅

⎛⎜⎝

⎞

⎠ in−

⎡⎢⎢⎣

⎤⎥⎥⎦

− m Cp⋅ T⋅( )out

m Cp⋅ T⋅( )in

−

<-----h Cp T⋅ Q mU2

2⋅

⎛⎜⎝

⎞

⎠out+ m

U2

2⋅

⎛⎜⎝

⎞

⎠ in− m

W2

2⋅

⎛⎜⎝

⎞

⎠out− m

W2

2⋅

⎛⎜⎝

⎞

⎠in+ m Cp⋅ T⋅( )

outm Cp⋅ T⋅( )

in−

Q mU2

2W2

2−

⎛⎜⎝

⎞

⎠⋅

⎡⎢⎣

⎤⎥⎦out

+ mU2

2W2

2−

⎛⎜⎝

⎞

⎠⋅

⎡⎢⎣

⎤⎥⎦ in

− m h⋅( )out m h⋅( )in−

place U Cθ⋅C2

2−

U2

2W2

2− into the first law equation:

57

B.3 Equation of state

d ρ( )ρ

d T( )T

+d P( )

P− 0 <-----Equation 2

B.4 Conservation of mass

d W( )W

d ρ( )ρ

+d A( )

A+

d m( )m

d W( )W

d ρ( )ρ

+d m( )

m−

d A( )A

− <-----Equation 3

B.5 Conservation of linear momentum

P

ρ W2⋅

−d P( )

P⋅

d W( )W

d m( )m

+d FD( )

ρ W2⋅ A⋅

+

for d FD( )

ρ W2⋅ A⋅

:

d FD( )ρ W2

⋅ A⋅

12

d FD( )⋅

12

ρ⋅ W2⋅ A⋅

d FD( )ρ W2

⋅ A⋅

12

d FD( )12

ρ⋅ W2⋅ A⋅

⋅

d FD( )ρ W2

⋅ A⋅

12

d CD( )⋅ <-----d CD( )d FD( )

12

ρ⋅ W2⋅ A⋅

place d FD( )

ρ W2⋅ A⋅

12

d CD( )⋅ into the linear momentum equation:

P

ρ W2⋅

−d P( )

P⋅

d W( )W

d m( )m

+12

d CD( )⋅+

58

d W( )W

d m( )m

+P

ρ W2⋅

d P( )P

⋅+12

− d CD( )⋅ <-----Equation 4

B.6 Relative stagnation temperature equation

since Cp T⋅ Cp To⋅C2

2− and Cp T⋅ Cp Torel⋅

W2

2− :

Cp To⋅C2

2− Cp Torel⋅

W2

2−

Cp To⋅ Cp Torel⋅W2

2−

C2

2+

Cp To⋅ Cp Torel⋅C2 W2

−

2+

for C2 W2

−

2:

C2 W2−

2

Cr2 Cθ

2+ Wr

2− Wθ

2−

2<-----C2 Cr

2 Cθ2

+ and W2 Wr2 Wθ

2+

C2 W2−

2

Wr2 U Wθ−( )2

+ Wr2

− Wθ2

−

2<-----Cθ U Wθ− and Wr Cr (both from velocity triangle)

C2 W2−

2

U2 2 U⋅ Wθ⋅− Wθ2

+ Wθ2

−

2

C2 W2−

2U2

2U Wθ⋅−

C2 W2−

2U2

2U W⋅ sin β( )⋅− <-----Wθ W sin β( )⋅ (from velocity triangle)

place C2 W2

−

2U2

2U W⋅ sin β( )⋅− into Cp To⋅ Cp Torel⋅

C2 W2−

2+ :

Cp To⋅ Cp Torel⋅U2

2+ U W⋅ sin β( )⋅−

convert equation above into differential form:

59

Cp d To( )⋅ Cp d Torel( )⋅d U2( )

2+ d U W⋅ sin β( )⋅( )−

Cp d To( )⋅ Cp d Torel( )⋅ U d U( )⋅+ U sin β( )⋅ d W( )⋅− W d U sin β( )⋅( )⋅−

Cp d To( )⋅ Cp d Torel( )⋅− U sin β( )⋅ d W( )⋅+ U d U( )⋅ W d U sin β( )⋅( )⋅−

Cp d To( )⋅ Cp Torel⋅d Torel( )

Torel⋅− U W⋅ sin β( )⋅

d W( )W

⋅+ U d U( )⋅ W d U sin β( )⋅( )⋅−

d To( )To

TorelTo

d Torel( )Torel

⋅−U W⋅ sin β( )⋅

Cp To⋅

d W( )W

⋅+U d U( )⋅ W d U sin β( )⋅( )⋅−

Cp To⋅<-----divide by Cp To⋅ <-----Equation 5

B.7 Relative stagnation temperature equation

d Porel( )Porel

d P( )P

γ Mrel2⋅

2

1γ 1−

2Mrel

2⋅+

d Mrel2⎛

⎝⎞⎠

Mrel2

⋅+

from the the conservation of linear momentum equation:

d W( )W

d m( )m

+P

ρ W2⋅

d P( )P

⋅+12

− d CD( )⋅

d W( )W

d m( )m

+γ P⋅

γ ρ⋅ W2⋅

d P( )P

⋅+12

− d CD( )⋅

d W( )W

d m( )m

+1

γ Mrel2

⋅

d P( )P

⋅+12

− d CD( )⋅ <-----Mrel2 ρ W2

⋅

γ P⋅

γ Mrel2

⋅d W( )

W⋅ γ Mrel

2⋅

d m( )m

⋅+d P( )

P+

γ Mrel2

⋅

2− d CD( )⋅

d P( )P

γ− Mrel2

⋅d W( )

W⋅ γ Mrel

2⋅

d m( )m

⋅−γ Mrel

2⋅

2d CD( )⋅−

place d P( )

Pγ− Mrel

2⋅

d W( )W

⋅ γ Mrel2

⋅d m( )

m⋅−

γ Mrel2

⋅

2d CD( )⋅− into the

d Porel( )Porel

equation:

60

d Porel( )Porel

γ− Mrel2

⋅d W( )

W⋅ γ Mrel

2⋅

d m( )m

⋅−γ Mrel

2⋅

2d CD( )⋅−

γ Mrel2⋅

2

1γ 1−

2Mrel

2⋅+

d Mrel2⎛

⎝⎞⎠

Mrel2

⋅+

from definition of d Mrel

2⎛⎝

⎞⎠

Mrel2

:

d Mrel2⎛

⎝⎞⎠

Mrel2

2d W( )

W⋅

d T( )T

−

d W( )W

12

d Mrel2⎛

⎝⎞⎠

Mrel2

d T( )T

+⎛⎜⎜⎝

⎞

⎠

⋅

place d W( )

W12

d Mrel2⎛

⎝⎞⎠

Mrel2

d T( )T

+⎛⎜⎜⎝

⎞

⎠

⋅ into the d Porel( )

Porel equation:

d Porel( )Porel

γ Mrel2

⋅

2−

d Mrel2⎛

⎝⎞⎠

Mrel2

d T( )T

+⎛⎜⎜⎝

⎞

⎠

⋅ γ Mrel2

⋅d m( )

m⋅−

γ Mrel2

⋅

2d CD( )⋅−

γ Mrel2⋅

2

1γ 1−

2Mrel

2⋅+

d Mrel2⎛

⎝⎞⎠

Mrel2

⋅+

by definition, the relative stagnation temperature is:

d Torel( )Torel

d T( )T

γ 1−

2Mrel

2⋅

1γ 1−

2Mrel

2⋅+

d Mrel2⎛

⎝⎞⎠

Mrel2

⋅+

d T( )T

d Torel( )Torel

γ 1−

2Mrel

2⋅

1γ 1−

2Mrel

2⋅+

d Mrel2⎛

⎝⎞⎠

Mrel2

⋅−

place d T( )

T

d Torel( )Torel

γ 1−

2Mrel

2⋅

1γ 1−

2Mrel

2⋅+

d Mrel2⎛

⎝⎞⎠

Mrel2

⋅− into the d Porel( )

Porel equation:

61

2

d Porel( )Porel

γ Mrel2⋅

2−

d Mrel2⎛

⎝⎞⎠

Mrel2

d Torel( )Torel

+

γ 1−

2Mrel

2⋅

1γ 1−

2Mrel

2⋅+

d Mrel2⎛

⎝⎞⎠

Mrel2

⋅−

⎛⎜⎜⎜⎝

⎞

⎟

⎠

⋅ γ Mrel2⋅

d m( )m

⋅−γ Mrel

2⋅

2d CD( )⋅−

γ Mrel2⋅

2

1γ 1−

2Mrel

2⋅+

d Mrel2⎛

⎝⎞⎠

Mrel2

⋅+

d Porel( )Porel

γ Mrel2

⋅

2−

d Torel( )Torel

1

1γ 1−

2Mrel

2⋅+

d Mrel2⎛

⎝⎞⎠

Mrel2

⋅+⎛⎜⎜⎜⎝

⎞

⎟

⎠

⋅ γ Mrel2

⋅d m( )

m⋅−

γ Mrel2

⋅

2d CD( )⋅−

γ Mrel2⋅

2

1γ 1−

2Mrel

2⋅+

d Mrel2⎛

⎝⎞⎠

Mrel2

⋅+

d Porel( )Porel

γ Mrel2

⋅

2−

d Torel( )Torel

⋅

γ Mrel2⋅

2

1γ 1−

2Mrel

2⋅+

d Mrel2⎛

⎝⎞⎠

Mrel2

⋅− γ Mrel2

⋅d m( )

m⋅−

γ Mrel2

⋅

2d CD( )⋅−

γ Mrel2⋅

2

1γ 1−

2Mrel

2⋅+

d Mrel2⎛

⎝⎞⎠

Mrel2

⋅+

d Porel( )Porel

γ Mrel2

⋅

2−

d Torel( )Torel

⋅ γ Mrel2

⋅d m( )

m⋅−

γ Mrel2

⋅

2d CD( )⋅−

d Porel( )Porel

γ Mrel2

⋅

2

d Torel( )Torel

⋅+ γ Mrel2

⋅d m( )

m⋅+

γ Mrel2

⋅

2− d CD( )⋅ <-----Equation 6

B.8 Absolute stagnation temperature equation

Cp To⋅ Cp T⋅C2

2+

Cp To⋅ Cp T⋅Cr

2 Cθ2

+

2+ <-----C2 Cr

2 Cθ2

+

Cp To⋅ Cp T⋅Wr

2 U Wθ−( )2+

2+ <-----Cθ U Wθ− and Wr Cr (both from velocity triangle)

Cp To⋅ Cp T⋅Wr

2 U2+ 2 U⋅ Wθ⋅− Wθ

2+

2+

Cp To⋅ Cp T⋅W2 U2

+ 2 U⋅ Wθ⋅−

2+ <-----W2 Wr

2 Wθ2

+

Cp To⋅ Cp T⋅W2 U2

+ 2 U⋅ W⋅ sin β( )⋅−

2+ <-----Wθ W sin β( )⋅ (from velocity triangle)

62

Cp To⋅ Cp T⋅W2

2U2

2++ U W⋅ sin β( )⋅−

convert equation above into differential form:

Cp d To( )⋅ Cp d T( )⋅d W2( )

2+

d U2( )2

+ d U W⋅ sin β( )⋅( )−

Cp d To( )⋅ Cp d T( )⋅ W d W( )⋅+ U d U( )⋅+ U sin β( )⋅ d W( )⋅− W d U sin β( )⋅( )⋅−

Cp d To( )⋅ Cp d T( )⋅ W U sin β( )⋅−( ) d W( )⋅+ U d U( )⋅+ W d U sin β( )⋅( )⋅−

Cp d To( )⋅ Cp d T( )⋅− W U sin β( )⋅−( ) d W( )⋅− U d U( )⋅ W d U sin β( )⋅( )⋅−

Cp d To( )⋅ Cp T⋅d T( )

T⋅− W W U sin β( )⋅−( )⋅

d W( )W

⋅− U d U( )⋅ W d U sin β( )⋅( )⋅−

d To( )To

TTo

d T( )T

⋅−W W U sin β( )⋅−( )⋅

Cp To⋅

d W( )W

⋅−U d U( )⋅ W d U sin β( )⋅( )⋅−

Cp To⋅<-----divide by Cp To⋅ <-----Equation 7

B.9 Relative Mach number equation

MrelW

γ R⋅ T⋅<-----Equation 8

B.10 Absolute stagnation pressure equation

burner:

since PoP

ToT

⎛⎜⎝

⎞⎠

γ

γ 1−

and Porel

P

TorelT

⎛⎜⎝

⎞⎠

γ

γ 1−

:

Po

P

Porel

P

ToT

⎛⎜⎝

⎞⎠

γ

γ 1−

TorelT

⎛⎜⎝

⎞⎠

γ

γ 1−

63

PoPorel

To

γ

γ 1−

Torel

γ

γ 1−

Po PorelTo

Torel

⎛⎜⎝

⎞

⎠

γ

γ 1−

⋅ <-----Equation 9

turbine:

Po Po4ToTo4

⎛⎜⎝

⎞

⎠

γ

γ 1−

⋅ <-----Equation 10

B.11 Entropy equation

burner:

s s3 Cp lnToTo3

⎛⎜⎝

⎞

⎠⋅+ R ln

PoPo3

⎛⎜⎝

⎞

⎠⋅− <-----Equation 11

turbine:

s s4 Cp lnToTo4

⎛⎜⎝

⎞

⎠⋅+ R ln

PoPo4

⎛⎜⎝

⎞

⎠⋅− <-----Equation 12

B.12 Burner absolute stagnation temperature distribution

the To distribution is linear:

To a bδrr3

⎛⎜⎝

⎞

⎠⋅+ <-----

δrr3

rr3

1−

at the inlet, rr3

1, which means δr3r3

0 and To To3:

64

To3 a bδr3r3

⎛⎜⎝

⎞

⎠⋅+

a To3

at the outlet, rr3

r4r3

, which means δr4r3

r4r3

1− and To To4:

To4 To3 bδr4r3

⎛⎜⎝

⎞

⎠⋅+

To4 To3 br4r3

1−⎛⎜⎝

⎞

⎠⋅+

bTo4 To3−

r4r3

1−

place a To3 and bTo4 To3−

r4r3

1−

in the To distribution equation:

To To3To4 To3−

r4r3

1−

δrr3

⎛⎜⎝

⎞

⎠⋅+ <-----Equation 13

B.13 Burner specific work

34b )CUm()CUm( θθ ⋅⋅−⋅⋅=−W <-----from the definition of angular momentum

⎥⎦

⎤⎢⎣

⎡−⋅−−⋅⋅⋅=− θθ )WU(U)WU(U

mm

m 33344434

3bW

<-----Cθ U Wθ−

⎥⎦

⎤⎢⎣

⎡β⋅−⋅−β⋅−⋅⋅⋅=− ))sin(WU(U))sin(WU(U

mm

m 3333444434

3bW

<-----Wθ W sin β( )⋅

⎥⎦

⎤⎢⎣

⎡β⋅−⋅−β⋅−⋅⋅−= ))sin(WU(U))sin(WU(U

mm

m 3333444434

3bW

<-----Equation 14

65

B.14 Turbine specific work

[ ]44554t CUCUm θθ ⋅−⋅⋅=− W <-----from the definition of angular momentum

[ ])WU(U)WU(U)mm( 444555f3t θθ −⋅−−⋅⋅+=− W <-----Cθ U Wθ−

[ ]))sin(WU(U))sin(WU(U)mm

1(m 444455553f

3t β⋅−⋅−β⋅−⋅⋅+⋅=− W

<-----Wθ W sin β( )⋅

[ ]))sin(WU(U))sin(WU(U)f1(m 444455553t β⋅−⋅−β⋅−⋅⋅+⋅=− W <-----f = m f / m3

[ ]))sin(WU(U))sin(WU(U)f1(m 44445555

3t β⋅−⋅−β⋅−⋅⋅+−=

W

<-----Equation 15

66

Appendix C To determine perpendicular (one-dimensional) flow

area between the vanes

mass flow rate associated with the circular area between two vanes:

m

rρ C→⋅ n

→⋅ 2 π⋅ b⋅( )⋅

⌠⎮⎮⌡

d

Nb

m2 π⋅ r⋅ b⋅ ρ⋅ C⋅ cos α( )⋅

Nb

m2 π⋅ r⋅ b⋅ ρ⋅ Cr⋅

Nb<----- Cr C cos α( )⋅

m2 π⋅ r⋅ b⋅ ρ⋅ Wr⋅

Nb<----- Wr Cr

m2 π⋅ r⋅ b⋅ ρ⋅ W⋅ cos β( )⋅

Nb<----- Wr W cos β( )⋅

mass flow rate associated with area perpendicular to flow between two vane

m Aρ W→

⋅ n→⋅

⌠⎮⎮⌡

d

m ρ W⋅ A⋅

since the mass flow rates are the same:

ρ W⋅ A⋅2 π⋅ r⋅ b⋅ ρ⋅ W⋅ cos β( )⋅

Nb

A2 π⋅ r⋅ b⋅ cos β( )⋅

Nb

67

Appendix D Current engine data

Table D-1: Airplane turboprop engine data.

Company Model PTO

(shp)

mf/PTO

(lbm/h/shp)

m3

(lbm/s)

TPE 331-5 710 0.602 7.75 Honeywell

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 Rolls Royce

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

Table D-2: Helicopter turboshaft engine data.

Company Model PTO

(shp)

mf/PTO

(lbm/h/shp)

m3

(lbm/s)

T58 (GE-10) 1400 0.6 13.7

CT58-110 1250 0.64 12.7 General Electric

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 Klimov Corporation

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

68

Company Model PTO

(shp)

mf/PTO

(lbm/h/shp)

m3

(lbm/s)

Gazelle 1400 0.68 17

GEM-42 1000 0.65 7.52

Gnome (H-1400) 1250 0.608 13.7 Rolls Royce

Turbomeca RM 322 2241 0.442 12.69

Makila (1A2) 1845 0.551 12.1 Turbomeca

Turmo (IIIC3) 1480 0.603 13

Table D-3: Aircraft (turboprop) and helicopter (turboshaft) dual-purpose engine data.

Company Model PTO

(shp)

mf/PTO

(lbm/h/shp)

m3

(lbm/s)

General Electric 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 Honeywell

TVD-1500/RD 600 (1500 S) 1300 0.454 8.8

Table D-4: Four-stroke gasoline generator engine data.

Company Number of

cylinders

Generator

model

Power

(hp)

Air intake

(ft3/min)

Fuel

consumption

(gal/hr)

Kohler 5ERKM 11.5 19 0.78

Onan 1

Microquiet 4000 9.5 19 0.71

Kohler 7ER 16 24 0.94

CME 5500 12.9 17.2 0.95 Onan

2

CMM 7000 14 18.9 1.22

10CCE 13 35 5.6 Kohler 4

12CCE 17 35 5.6

69

Table D-5: Diesel generator engine data.

Company Number of

cylinders

Generator

model

Power

(hp)

Air intake

(ft3/min)

Fuel

consumption

(gal/hr)

Generac GR8 11 22 0.67

Kohler 3

10EOR/Z 17.7 36 0.97

GR25 31 87 1.4

GR50 58 94 2.6

GR70 85 150 3.5 Generac

GR85 93 178 3.8

15EOR/Z 26.1 54 1.4 Kohler

4

20EOR/Z 36.1 70 1.67

GR125 144 283 5.7

GR160 175 283 7.4

GR190 206 283 8.6 Generac 6

GR210 220 283 9.8

70

Figure D-1: PSFC and specific power comparison between APU cases and current engines.

71

Appendix E Complete results for Case 1

E.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) 608.0490906

P2t (kPa) 85.00958077 m3/A3 (kg/(m2*s)) 252.0767384

ρ2t (kg/m3) 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/m3) 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

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

0 10 20 300

1

2

3

Mrel

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure E-1: Case 1 relative Mach number.

0 10 20 300

500

1000

1500

Torel

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure E-2: Case 1 relative stagnation temperature

(K).

0 10 20 301 .105

2 .105

3 .105

Porel

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure E-3: Case 1 relative stagnation pressure (Pa).

0 10 20 300

500

1000

1500

To

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure E-4: Case 1 stagnation temperature (K).

76

0 10 20 300

5 .105

1 .106

Po

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure E-5: Case 1 stagnation pressure (Case 1).

0 10 20 300

500

1000

1500

T

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure E-6: Case 1 temperature (Case 1).

0 10 20 300

1 .105

2 .105

3 .105

P

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure E-7: Case 1 pressure (Case 1).

0 10 20 300

1

2

3

ρ

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure E-8: Case 1 density (Case 1).

0 10 20 300

50

100

βdeg

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure E-9: Case 1 flow curvature (Case 1).

0 10 20 300

500

1000

U

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure E-10: Case 1 rotor speed (Case 1).

77

0 10 20 301000

1100

1200

Cp

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure E-11: Case 1 specific heat (Case 1).

0 10 20 301.3

1.35

1.4

γ

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure E-12: Case 1 specific heat ratio (Case 1).

0 10 20 301000

500

0

500

Cθ

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure E-13: Case 1 tangential velocity (Case 1).

1500 2000 2500 3000 35000

500

1000

1500

To

s3 s4

s

Figure E-14: Case 1 To-s diagram (Case 1).

0 10 20 300

5 .105

1 .106

Po

1ρ 3

1ρ 4

1ρ

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 PTO/m3 mf/PTO r5/r2h z2h/z5 A5/A4.5

0.5 1.35193 5.896064 89465.712 2.69E-07 21.16218 2.776372 1

0.7 1.761836 8.254489 88795.587 2.64E-07 29.233796 7.033892 1

0.9 2.430948 10.612915 69279.005 3.28E-07 37.102533 23.187477 1

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 44.785 97.638476 1

1.384337 6.061594 16.324279 412218.5 5.26E-08 65.390339 1.75E+04 4.012205

1.39 6.130212 16.391057 416331.58 5.27E-08 65.657836 1.89E+04 4.033592

0

50000

100000

150000

200000

250000

300000

350000

400000

450000

0 1 2 3 4 5 6

π c

PTO

/m3 (

W/k

g/s)

7

Figure E-16: Variation of specific power takeoff with compressor pressure ratio (Case 1).

79

0.00E+00

1.00E-07

2.00E-07

3.00E-07

4.00E-07

5.00E-07

6.00E-07

7.00E-07

8.00E-07

0 1 2 3 4 5 6 7

π c

mf /P

TO

(kg/

s/W

)

Figure E-17: Variation of PSFC with compressor pressure ratio (Case 1).

0

2

4

6

8

10

12

14

16

18

0 1 2 3 4 5 6 7

π c

r3/r

2t

Figure E-18: Variation of compressor radius ratio and pressure ratio (Case 1).

80

0

10

20

30

40

50

60

70

0 1 2 3 4 5 6 7

π c

r5/r

2h

Figure E-19: Variation of rotor radius ratio with compressor pressure ratio (Case 1).

-5000

0

5000

10000

15000

20000

0 1 2 3 4 5 6 7

π c

z2h /

z5

Figure E-20: Variation of disk thickness with compressor pressure ratio (Case 1).

81

Appendix F Complete results for Case 2

F.1 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/m3) -41.638544

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) 118.6849691

P2t (kPa) 69.60638697 m3/A3 (kg/(m2*s)) 118.0581585

ρ2t (kg/m3) 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/m3) 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

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

0.9 1 1.1 1.2 1.30

0.5

1

Mrel

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure F-1: Relative Mach number (Case 2).

0.9 1 1.1 1.2 1.30

500

1000

1500

Torel

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure F-2: Relative stagnation temperature (Case 2).

0.9 1 1.1 1.2 1.36 .104

8 .104

1 .105

Porel

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure F-3: Relative stagnation pressure (Case 2).

0.9 1 1.1 1.2 1.30

500

1000

1500

To

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure F-4: Stagnation temperature (Case 2).

86

0.9 1 1.1 1.2 1.30

5 .104

1 .105

1.5 .105

Po

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure F-5: Stagnation pressure (Case 2).

0.9 1 1.1 1.2 1.30

500

1000

1500

T

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure F-6: Temperature (Case 2).

0.9 1 1.1 1.2 1.30

5 .104

1 .105

P

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure F-7: Pressure (Case 2).

0.9 1 1.1 1.2 1.30

0.5

1

1.5

ρ

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure F-8: Density (Case 2).

0.9 1 1.1 1.2 1.30

50

100

βdeg

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure F-9: Flow curvature (Case 2).

0.9 1 1.1 1.2 1.3200

250

300

U

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure F-10: Rotor speed (Case 2).

87

0.9 1 1.1 1.2 1.31000

1100

1200

Cp

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure F-11: Specific heat (Case 2).

0.9 1 1.1 1.2 1.31.3

1.35

1.4

γ

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure F-12: Specific heat ratio (Case 2).

0.9 1 1.1 1.2 1.3500

0

500

Cθ

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure F-13: Tangential velocity (Case 2).

1000 2000 3000 40000

500

1000

1500

To

s3 s4

s

Figure F-14: To-s diagram (Case 2).

1700 1750

300

350

To

s3

s

Figure F-15: Beginning of To-s diagram (Case 2).

3400 3600

1200To

s4

s

Figure F-16: End of To-s diagram (Case 2).

88

0 50

5 .104

1 .105

1.5 .105

10

Po

1ρ 3

1ρ 4

1ρ

Figure F-17: Po-s diagram (Case 2).

1 2

1 .105

1.2 .105

Po

1ρ 3

1ρ

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 1.129034 0.87477 126283.75 2.10E-07 2.763883 2.809908 1

89

121000

122000

123000

124000

125000

126000

127000

1.12 1.14 1.16 1.18 1.2 1.22 1.24 1.26

π c

PTO

/m3 (

W/k

g/s)

Figure F-19: Variation of specific power takeoff with compressor pressure ratio (Case 2).

2.10E-07

2.11E-07

2.11E-07

2.12E-07

2.12E-07

2.13E-07

2.13E-07

2.14E-07

2.14E-07

2.15E-07

2.15E-07

1.12 1.14 1.16 1.18 1.2 1.22 1.24 1.26

π c

mf /P

TO

(kg/

s/W

)

Figure F-20: Variation of PSFC with compressor pressure ratio (Case 2).

90

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.12 1.14 1.16 1.18 1.2 1.22 1.24 1.26

π c

r3/r

2t

Figure F-21: Variation of compressor radius ratio and pressure ratio (Case 2).

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

1.12 1.14 1.16 1.18 1.2 1.22 1.24 1.26

π c

r5/r

2h

Figure F-22: Variation of rotor radius ratio with compressor pressure ratio (Case 2).

91

2.75

2.8

2.85

2.9

2.95

3

3.05

3.1

1.12 1.14 1.16 1.18 1.2 1.22 1.24 1.26

π c

z2h /

z5

Figure F-23: Variation of disk thickness with compressor pressure ratio (Case 2).

92

Appendix G Complete results for Case 3

G.1 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) 1745.785.622

P2t (kPa) 88.64231542 m3/A3 (kg/(m2*s)) 437.905847

ρ2t (kg/m3) 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/m3) 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

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

0 50

0.5

1

1.5

2

10

Mrel

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure G-1: Relative Mach number (Case 3).

0 50

500

1000

1500

10

Torel

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure G-2: Relative stagnation temperature (Case

3).

0 50

2 .105

4 .105

6 .105

10

Porel

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure G-3: Relative stagnation pressure (Case 3).

0 50

500

1000

1500

10

To

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure G-4: Stagnation temperature (Case 3).

97

0 50

1 .106

2 .106

10

Po

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure G-5: Stagnation pressure (Case 3).

0 50

500

1000

1500

10

T

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure G-6: Temperature (Case 3).

0 50

2 .105

4 .105

6 .105

10

P

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure G-7: Pressure (Case 3).

0 50

2

4

10

ρ

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure G-8: Density (Case 3).

0 50

50

100

10

βdeg

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure G-9: Flow curvature (Case 3).

0 50

500

1000

10

U

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure G-10: Rotor speed (Case 3).

98

0 51000

1100

1200

10

Cp

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure G-11: Specific heat (Case 3).

0 51.3

1.35

1.4

10

γ

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure G-12: Specific heat ratio (Case 3).

0 5 10500

0

500

1000

Cθ

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure G-13: Tangential velocity (Case 3).

1500 2000 2500 30000

500

1000

1500

To

s3 s4

s

Figure G-14: To-s diagram (Case 3).

0 1 2 30

1 .106

2 .106

Po

1ρ 3

1ρ 4

1ρ

Figure G-15: Po-v diagram (Case 3).

99

Appendix H Complete results for Case 4

H.1 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/m3) 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) 157.7155002

P2t (kPa) 88.23730292 m3/A3 (kg/(m2*s)) 57.39563798

ρ2t (kg/m3) 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/m3) 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

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

1 1.2 1.40

0.5

1

Mrel

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure H-1: Relative Mach number (Case 4).

1 1.2 1.40

500

1000

1500

Torel

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure H-2: Relative stagnation temperature (Case

4).

1 1.2 1.41.15 .105

1.2 .105

1.25 .105

Porel

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure H-3: Relative stagnation pressure (Case 4).

1 1.2 1.40

500

1000

1500

To

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure H-4: Stagnation temperature (Case 4).

104

1 1.2 1.41 .105

1.5 .105

2 .105

Po

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure H-5: Stagnation pressure (Case 4).

1 1.2 1.40

500

1000

1500

T

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure H-6: Temperature (Case 4).

1 1.2 1.48 .104

1 .105

1.2 .105

1.4 .105

P

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure H-7: Pressure (Case 4).

1 1.2 1.40

0.5

1

1.5

ρ

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure H-8: Density (Case 4).

1 1.2 1.40

50

100

βdeg

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure H-9: Flow curvature (Case 4).

1 1.2 1.4150

200

250

U

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure H-10: Rotor speed (Case 4).

105

1 1.2 1.41000

1100

1200

Cp

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure H-11: Specific heat (Case 4).

1 1.2 1.41.3

1.35

1.4

γ

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure H-12: Specific heat ratio (Case 4).

1 1.2 1.4200

0

200

400

Cθ

r3r2t

⎡⎢⎣

⎤⎥⎦

r4r2t

⎡⎢⎣

⎤⎥⎦

rr2t

⎡⎢⎣

⎤⎥⎦

Figure H-13: Tangential velocity (Case 4).

1500 2000 2500 3000 35000

500

1000

1500

To

s3 s4

s

Figure H-14: To-s diagram (Case 4).

1600 1700 1800

400

To

s3

s

Figure H-15: Beginning of To-s diagram (Case 4).

3200 3300

1100

1200

1300

To

s4

s

Figure H-16: End of To-s diagram (Case 4).

106

0 1 2 3 41 .105

1.5 .105

2 .105

Po

1ρ 3

1ρ 4

1ρ

Figure H-17: Po-s diagram (Case 4).

107

Appendix I Sample rotor for Case 1 with calculation program

A32 π⋅ r3⋅ b3⋅ cos β3( )⋅

Nb:= A4 A3

A4A3

⎡⎢⎣

⎤⎥⎦

⋅:= b4Nb A4⋅

2 π⋅ r4⋅ cos β4( )⋅:=

A4.5 A4A4

A4.5

⎡⎢⎣

⎤⎥⎦

⎛⎜⎝

⎞

⎠

1−

⋅:= b4.5Nb A4.5⋅

2 π⋅ r4.5⋅ cos β4.5( )⋅:= A5 A4.5

A5A4.5

⎡⎢⎣

⎤⎥⎦

⋅:= b5Nb A5⋅

2 π⋅ r5⋅ cos β5( )⋅:=

r2h 0.8in= r3 32.649in= r4 44.044in= r4.5 44.208in= r5 52.312in=

d2h 1.6in= d2t 4 in= d3 65.297in= d4 88.088in= d4.5 88.415in=

d5 104.625in= b3 0.021in= b4 0.048in= b4.5 0.047in= b5 6.325in=

A3 0.536in2= A4 0.665in2

= A4.5 0.64in2= A5 2.569in2

=

A4A3

⎡⎢⎣

⎤⎥⎦

1.241744:=A4

A4.5

⎡⎢⎣

⎤⎥⎦

1.03858:=A5

A4.5

⎡⎢⎣

⎤⎥⎦

4.012205:=

b3r3

⎡⎢⎣

⎤⎥⎦

6.39812 10 4−×:=

r3r2t

⎡⎢⎣

⎤⎥⎦

16.324279:=r4r3

⎡⎢⎣

⎤⎥⎦

1.349041:=r4.5r4

⎡⎢⎣

⎤⎥⎦

1.003707:=r5r4

⎡⎢⎣

⎤⎥⎦

1.187721:=

β3 0 deg⋅:= β4 66.467768deg⋅:= β4.5 66.922988deg⋅:= β5 89.433445deg⋅:= Nb 8:=

ζc 0.4:= r2t 2 in⋅:= r2h r2t ζc⋅:= r3 r2tr3r2t

⎡⎢⎣

⎤⎥⎦

⋅:= r4 r3r4r3

⎡⎢⎣

⎤⎥⎦

⋅:= r4.5 r4r4.5r4

⎡⎢⎣

⎤⎥⎦

⋅:=

r5 r4r5r4

⎡⎢⎣

⎤⎥⎦

⋅:= d2h 2 r2h⋅:= d2t 2 r2t⋅:= d3 2 r3⋅:= d4 2 r4⋅:= d4.5 2 r4.5⋅:= d5 2 r5⋅:=

b3 r3b3r3

⎡⎢⎣

⎤⎥⎦

⋅:=

108

67.0°

89.4°

Ø 104.624

Ø 88.416

Ø 88.088

Ø 65.298

Ø 4.000

Ø 1.600

6.325

.047.048

.021

= d2h

= d2t

= d3

= d4

= d4.5

= d5

= β5

= β4.5

b3 =

b4 = = b4.5

b5 =

r

θ

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

b3 r3b3r3

⎡⎢⎣

⎤⎥⎦

⋅:= A32 π⋅ r3⋅ b3⋅ cos β3( )⋅

Nb:= A4 A3

A4A3

⎡⎢⎣

⎤⎥⎦

⋅:= b4Nb A4⋅

2 π⋅ r4⋅ cos β4( )⋅:=

A5 A4A5A4

⎡⎢⎣

⎤⎥⎦

⋅:= b5Nb A5⋅

2 π⋅ r5⋅ cos β5( )⋅:=

r2h 0.8in= r3 2.031in= r4 2.55in= r5 2.56in=

d2h 1.6in= d2t 4 in= d3 4.062in= d4 5.1in= d5 5.12in=

b3 0.934in= b4 9.738in= b5 8.336in=

A3 1.49in2= A4 3.012in2

= A5 2.513in2=

A4A3

⎡⎢⎣

⎤⎥⎦

2.020949:=A5A4

⎡⎢⎣

⎤⎥⎦

0.834297:=

r3r2t

⎡⎢⎣

⎤⎥⎦

1.015415:=b3r3

⎡⎢⎣

⎤⎥⎦

0.460075:=r4r3

⎡⎢⎣

⎤⎥⎦

1.255677:=r5r4

⎡⎢⎣

⎤⎥⎦

1.00398:=

β3 0 deg⋅:= β4 81.116531deg⋅:= β5 81.378184deg⋅:= Nb 8:=

ζc 0.4:= r2t 2 in⋅:= r2h r2t ζc⋅:= r3 r2tr3r2t

⎡⎢⎣

⎤⎥⎦

⋅:= r4 r3r4r3

⎡⎢⎣

⎤⎥⎦

⋅:= r5 r4r5r4

⎡⎢⎣

⎤⎥⎦

⋅:=

d2h 2 r2h⋅:= d2t 2 r2t⋅:= d3 2 r3⋅:= d4 2 r4⋅:= d5 2 r5⋅:=

110

Ø 5.120

Ø 5.100

81.3°

Ø 4.062

Ø 4.000

Ø 1.594

.934

9.738

8.336

= d5

= d4

= d3

= d2t

= d2h

b3 =

=b4

=b5

=β5

r

θ

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

A32 π⋅ r3⋅ b3⋅ cos β3( )⋅

Nb:= A4 A3

A4A3

⎡⎢⎣

⎤⎥⎦

⋅:= b4Nb A4⋅

2 π⋅ r4⋅ cos β4( )⋅:=

A4.5 A4A4

A4.5

⎡⎢⎣

⎤⎥⎦

⎛⎜⎝

⎞

⎠

1−

⋅:= b4.5Nb A4.5⋅

2 π⋅ r4.5⋅ cos β4.5( )⋅:= A5 A4.5

A5A4.5

⎡⎢⎣

⎤⎥⎦

⋅:= b5Nb A5⋅

2 π⋅ r5⋅ cos β5( )⋅:=

r2h 0.8in= r3 13.786in= r4 14.74in= r4.5 14.886in= r5 15.627in=

d2h 1.6in= d2t 4 in= d3 27.573in= d4 29.48in= d4.5 29.772in=

d5 31.254in= b3 0.022in= b4 0.218in= b4.5 0.24in= b5 1.159in=

A3 0.236in2= A4 0.258in2

= A4.5 0.249in2= A5 0.308in2

=

A4A3

⎡⎢⎣

⎤⎥⎦

1.092395:=A4

A4.5

⎡⎢⎣

⎤⎥⎦

1.036009:=A5

A4.5

⎡⎢⎣

⎤⎥⎦

1.235198:=

r3r2t

⎡⎢⎣

⎤⎥⎦

6.89313:=b3r3

⎡⎢⎣

⎤⎥⎦

1.581712 10 3−×:=

r4r3

⎡⎢⎣

⎤⎥⎦

1.069184:=r4.5r4

⎡⎢⎣

⎤⎥⎦

1.0099:=r5r4

⎡⎢⎣

⎤⎥⎦

1.06018:=

β3 0 deg⋅:= β4 84.145266deg⋅:= β4.5 84.910798deg⋅:= β5 88.76084deg⋅:= Nb 8:=

ζc 0.4:= r2t 2 in⋅:= r2h r2t ζc⋅:= r3 r2tr3r2t

⎡⎢⎣

⎤⎥⎦

⋅:= r4 r3r4r3

⎡⎢⎣

⎤⎥⎦

⋅:= r4.5 r4r4.5r4

⎡⎢⎣

⎤⎥⎦

⋅:=

r5 r4r5r4

⎡⎢⎣

⎤⎥⎦

⋅:= d2h 2 r2h⋅:= d2t 2 r2t⋅:= d3 2 r3⋅:= d4 2 r4⋅:= d4.5 2 r4.5⋅:= d5 2 r5⋅:=

b3 r3b3r3

⎡⎢⎣

⎤⎥⎦

⋅:=

112

88.8°84.1°

Ø 31.254

Ø 29.772

Ø 29.480

Ø 27.572

Ø 4.000

Ø 1.600

.022

.218 .240

1.159

= d5

= d4.5

= d4

= d3

= d2t

= d2h

= b5

= b4.5b4 =

b3 =

= β4= β5

r

θ

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

b3 r3b3r3

⎡⎢⎣

⎤⎥⎦

⋅:= A32 π⋅ r3⋅ b3⋅ cos β3( )⋅

Nb:= A4 A3

A4A3

⎡⎢⎣

⎤⎥⎦

⋅:= b4Nb A4⋅

2 π⋅ r4⋅ cos β4( )⋅:=

A5 A4A5A4

⎡⎢⎣

⎤⎥⎦

⋅:= b5Nb A5⋅

2 π⋅ r5⋅ cos β5( )⋅:=

r3 2.451in= r4 2.696in= r5 2.796in=

d2h 1.6in= d2t 4 in= d3 4.902in= d4 5.393in= d5 5.592in=

b3 0.769in= b4 0.743in= b5 4.312in=

A3 1.481in2= A4 1.268in2

= A5 0.902in2=

A4A3

⎡⎢⎣

⎤⎥⎦

0.85656:=A5A4

⎡⎢⎣

⎤⎥⎦

0.711062:=

r3r2t

⎡⎢⎣

⎤⎥⎦

1.225592:=b3r3

⎡⎢⎣

⎤⎥⎦

0.313776:=r4r3

⎡⎢⎣

⎤⎥⎦

1.1:=r5r4

⎡⎢⎣

⎤⎥⎦

1.036995:=

β3 0 deg⋅:= β4 36.283435deg⋅:= β5 84.535271deg⋅:= Nb 8:=

ζc 0.4:= r2t 2 in⋅:= r2h r2t ζc⋅:= r3 r2tr3r2t

⎡⎢⎣

⎤⎥⎦

⋅:= r4 r3r4r3

⎡⎢⎣

⎤⎥⎦

⋅:= r5 r4r5r4

⎡⎢⎣

⎤⎥⎦

⋅:=

d2h 2 r2h⋅:= d2t 2 r2t⋅:= d3 2 r3⋅:= d4 2 r4⋅:= d5 2 r5⋅:=

114

36°

85°Ø 5.592

Ø 5.392

Ø 4.902

Ø 1.600

Ø 4.000

.769

.743

4.312

= d5

= d4

= d3

= d2t

= d2h

b3 =

= b4

= b5

β4 =

= β5

r

θ

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