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This is to certify that the
dissertation entitled

PERFORMANCE PREDICTION AND PRELIMINARY DESIGN
OF WAVE ROTORS ENHANCING GAS TURBINE CYCLES

presented by

PEZHMAN AKBARI

has been accepted towards fulfillment
of the requirements for the

Ph.D. degree in Mechanical Engineering

 

 

A

 

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Ma‘ﬁr‘Profe'ssor’s Signature

8/23/2004

 

Date

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PERFORMANCE PREDICTION AND PRELIMINARY DESIGN OF
WAVE ROTORS ENHANCING GAS TURBINE CYCLES

By

Pezhman Akbari

A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

2004

ABSTRACT

PERFORMANCE PREDICTION AND PRELIMINARY DESIGN OF WAVE ROTORS
ENHANCING GAS TURBINE CYCLES

By

Pezhman Akbari

Wave rotors as a family of unsteady-ﬂow devices have shown unique capabilities to
enhance the performance and operating characteristics of a variety of engines and
machinery utilizing thermodynamic cycles. The wave rotor is an unsteady-ﬂow machine
that utilizes compression and expansion waves to exchange energy between fluids with
different pressures. The ﬁrst part of this study presents several thermodynamic cycle
analyses proving the performance improvement of small gas turbines (microturbines) by
implementing various advantageous four-port wave rotor topping cycles. General
performance maps are generated showing the design space and optima for baseline and
topped engines. In the second part of this work, a one-dimensional analytical gasdynamic
model of the hi gh-pressure phase is suggested to calculate flow characteristics inside the
wave rotor channels. Useful design parameters such as port widths and rotor size are
determined by computing transit times of the waves traveling inside the channels.
Reasonable agreement has been found comparing the predicted results of the analytical
design procedure with the available numerical data. Using the commercial software
FLUENT, some CFD simulations are performed showing that commonly available CFD
software can be utilized for wave rotor simulations and to support their design. Overall,
the predicted results are in good agreement with established wave rotor theories. Finally,

several innovative wave rotor concepts and designs including have been studied.

 

To My Family

iii

ACKNOWLEDGMENTS

I would like ﬁrst to express my gratitude to my supervisor Prof. Norbert Miiller for
providing an opportunity to make the present research a true learning process. He
generously allowed me to start this work in order to learn and satisfy my own curiosity.
His sense of guidance, unending patience, refreshing humor, and friendly suggestions
have made my research experience both fruitful and enjoyable. His encouragement and
timely advice have always helped me to overcome the difﬁculties I faced during my
research. I consider myself lucky to have Prof. Norbert Miiller as my supervisor.

I have had a privilege to learn a lot from my Ph.D. committee members. As a former
student of Prof. Indrek Wichman, I admire his inspiring work and his continuous support.
Prof. Abraham Engeda deserves my special thanks as one of the most supportive advisors
through my research, offering very helpful discussions and suggestions. I am also grateful
to Prof. Farhad Jaberi on my Ph.D. committee for his valuable suggestions. He has
always been a valuable mentor and an excellent model for me. Special thanks to Prof.
Charles MacCluer for serving on my Ph.D. committee. His helpful advice and feedback
have been very beneﬁcial for me. I am also very thankful to Prof. Razi Nalim at the
Purdue School of Engineering and Technology at IUPUI. As an acting member of my
Ph.D. committee and top specialist in his field, he has provided me with continuous and
valuable guidance in improving this work.

I would particularly like to thank Prof. Jenasz Piechna at Warsaw University of
Technology in Poland with whom I have had many constructive discussions. I appreciate

his unique talents and brightness and enjoy his friendship.

iv

Thanks to Prof. Manoochehr Koochesfahani for assisting me to begin my Ph.D. at
MSU. I am honored to had a chance to work in his laboratory in the early stages of my
Ph.D. program. I am also very grateful to my former supervisor Prof. Kaveh Ghorbanian
at Sharif University of Technology in Iran who ﬁrst inspired me to work on such a
challenging topic. Working with him was a valuable and enjoyable undertaking.

This work would have never been completed without the support of several
colleagues in the MSU Turbomachinery Laboratory. I am thankful for the collaboration
and friendship of Florin Iancu whose assistance in numerical simulations performed in
this work is appreciated. His patience during our collaboration is appreciated. I would
like to thank Amir A. Kharazi for a productive collaboration, leading to our several
technical publications. The materials presented in Chapter 6 are the result of my
collaboration with the above friends at the MSU wave rotor team. I am also very thankful
to Azadeh Behinfar as a real friend and a great source of support. Several of the 3D
sketches presented here are results of her excellent skills. Thanks for keeping me happy
during those difﬁcult moments.

Finally and most importantly, I certainly remain indebted to my parents and all other
family members for their continuing love, encouragement, and endless support through

this journey. I could never pay them back what they have given me.

TABLE OF CONTENTS

 

 

 

 

 

 

LIST OF TABLES VIII
LIST OF FIGURES IX
CHAPTER 1: INTRODUCTION 1
1.1 Unsteady-Flow Devices ......................................................................................................... l
1.2 Wave Rotors .......................................................................................................................... 4
1.3 Wave Rotor Applications ...................................................................................................... 6
1.3.1 Gas Turbine Applications ............................................................................................... 7
CHAPTER 2: HISTORICAL REVIEW OF WAVE ROTOR TECHNOLOGY ..... 14
CHAPTER 3: WAVE ROTOR THEORY 48
3.1 Energy Exchange with Waves ............................................................................................. 48
3.2 Charging Process in Wave Rotors ....................................................................................... 53
3.3 Principles of F our-Port Wave Rotor Operation ................................................................... 57
3.3.1 Through-Flow versus Reverse-Flow Wave Rotors ...................................................... 57
3.3.2 How Does it Work Inside? ........................................................................................... 59
3.4 Gasdynamic Equations ........................................................................................................ 63
3.4.1 Moving Normal Shock Wave Relations ....................................................................... 63
3.4.2 Expansion Wave Relations ........................................................................................... 66
CHAPTER 4: THERMODYNAMIC ANALYSIS 70
4.1 Microturbines ...................................................................................................................... 70
4.2 Gas Turbine without Recuperation ...................................................................................... 72
4.2.1 Implementation Cases .................................................................................................. 72
4.2.2 Thermodynamic Calculations ...................................................................................... 76
4.2.3 Predicted Performance Results ..................................................................................... 86
4.2.4 Comparison Between Adding a Second Compressor Stage with Wave-Rotor-Topping
............................................................................................................................................. 105
4.2.5 Substituting the Compressor in the C-60 Engine with the C-30 Compressor plus Wave
Rotor .................................................................................................................................... 111
4.2.6 Effect of Compressor Inlet Temperature .................................................................... 112
4.3 Gas Turbine with Recuperation ......................................................................................... 117
4.3.1 Thermodynamic Calculations .................................................................................... 119
4.3.2 Predicted Performance Results ................................................................................... 122
4.4 Turbojet Engines Topped with Wave Rotors .................................................................... 128
4.4.1 Thermodynamic Calculations .................................................................................... 130
4.4.2 Predicted Performance Results ................................................................................... 134
CHAPTER 5: PRELIMINARY WAVE ROTOR DESIGN 138
5.1 Analytical Approach .......................................................................................................... 138
5.1.1 Charging Process with a Single Shock Wave ............................................................ 140
5.1.2 Charging Process with Two Shock Waves ................................................................. 146
5.1.3 Validation ................................................................................................................... 163
5.2 Numerical Simulation ........................................................................................................ 165
5.2.1 Approach .................................................................................................................... 167
5.2.2 Numerical Results ...................................................................................................... 172

vi

CHAPTER 6: INNOVATIVE WAVE ROTOR DESIGNS AND APPLICATIONS

 

 

 

 

186

6.1 Radial Wave Rotor Concept .............................................................................................. 186
6.1.1 Radial-Flow Wave Rotor with Straight Channels ...................................................... 186
6.1.2 Radial-Flow Wave Rotor with Curved Channels ....................................................... 190
6.1.3 Radial-Flow Wave Rotor Concept ............................................................................. 193

6.2 Condensing Wave Rotor .................................................................................................... 197
6.2.1 How Does a Condensing Wave Rotor Work? ............................................................ 198
6.2.2 Performance Evaluation ............................................................................................. 204

6.3 Wave Rotors for Ultra—Micro Gas Turbines ...................................................................... 206
6.3.1 Performance Enhancement of UuGT Using Wave Rotors ......................................... 207
6.3.2 Conceptual Designs of Wave Rotor Implementation into UuGT .............................. 209
CONCLUSION 213
REFERENCES 217
APPENDICES 233
Appendix A: Shock Waves and Diffusers ............................................................................... 234
Appendix B: Temperature-Entropy Diagram Construction ..................................................... 239

vii

LIST OF TABLES

Table 1: Baseline engine data, assuming T 0 =300 K, Cpa,,=1.005 kJ/kgK, Cpgas=1.148 kJ/kgK, ya,

=1.4,yga,=l.33 ...................................................................................................................... 77
Table 2: Performance comparison between baseline engines and ﬁve cases of wave-rotor-topping
with a wave rotor pressure ratio of 1.8 ................................................................................ 101

Table 3: Performance comparison between adding a conventional second compressor stage and
wave-rotor-topping Case A and E with a wave rotor pressure ratio of 1.8; baseline engine C-
30 ......................................................................................................................................... 108

Table 4: Performance comparison between adding a conventional second compressor stage and
wave-rotor-topping Case A and E with a wave rotor pressure ratio of 1.8; baseline engine C-

60 ......................................................................................................................................... 109
Table 5: Ambient temperature effect on performance - comparison between baseline engines and
Cases A and B of wave-rotor—topping with PRW =1.8; baseline engine C-3O ..................... 114
Table 6: Ambient temperature effect on performance - comparison between baseline engines and
Cases A and B of wave-rotor-topping with PRW =1.8; baseline engine 060 ..................... 114
Table 7: Performance comparison between baseline engines and two cases of wave-rotor-topping
with a wave rotor pressure ratio of 1.8 ................................................................................ 128
Table 8: Turbojet baseline engine data, assuming T 0 =300 K, Cpai,=1.005 kJ/kgK, Cpgas =1 . 148
kJ/kgK, yai,=1.4 , ygm=1.33 ................................................................................................. 130
Table 9: Performance comparison between baseline turbojet engine and ﬁve cases of wave-rotor-
topping with a wave rotor pressure ratio of 1.8 ................................................................... 135
Table 10: Enhanced engine data, assuming PRW = pmpﬂ =1.8 for Case ................................... 143
Table 11: Flow properties inside rotor channels for L=20 cm ..................................................... 152
Table 12: Wave rotor design parameters for L=20 cm, N=30, nm,0,=10000 rpm, and m, =03 l 8
kg/s, ..................................................................................................................................... 152

Table 13: Inlet and outlet port data of University of Tokyo wave rotor, taken ﬁ'om Ref. [215] 163
Table 14: Comparison between the analytical model and the numerical data of the Tokyo wave
rotor ..................................................................................................................................... 165

viii

LIST OF FIGURES

Figure 1: Comparison of pressure gain (local pressure ratio) of moving shock and steady-flow

isentropic diffuser for y=1.4 .................................................................................................... 2
Figure 2: Shock wave, diffuser, and compressor isentropic efﬁciencies as functions of pressure
gain .......................................................................................................................................... 3
Figure 3: Schematic conﬁguration of a typical wave rotor .............................................................. 4
Figure 4: Schematic conﬁguration of a simple gas turbine engine .................................................. 7
Figure 5: Schematic temperature-entropy diagram for a gas turbine with low and high pressure
ratios ........................................................................................................................................ 9
Figure 6: Schematic of a gas turbine topped by a four-port wave rotor ........................................ 10
Figure 7: Schematic example of the physical implementation of a wave rotor in a gas turbine
(exploded view, piping not shown) ....................................................................................... 10
Figure 8: Schematic T-s diagrams for a gas turbine with and without a wave rotor ..................... 12
Figure 9: Wave rotor as a topping stage for the locomotive gas turbine, taken from Ref. [17].... 16
Figure 10: The Comprex®, taken from Ref. [34] .......................................................................... 17
Figure 11: Photograph of the CAL Wave Supercharger, taken from Ref. [5] ............................... 21
Figure 12: Schematic of a double wave rotor cycle, taken from Ref. [98] .................................... 22
Figure 13: The experimental divider test rig at Imperial College, taken from Ref. [5] ................. 23
Figure 14: The Pearson rotor (left) and rear and front stator plates (right), taken from Ref. [124]
............................................................................................................................................... 24
Figure 15: Ideal wave diagram of the Klapproth rotor, taken from Ref. [126] .............................. 28
Figure 16: Ideal wave diagram of the GPC rotor, taken from Ref. [109] ...................................... 30
Figure 17: Schematic of the MSNW experimental set up, taken from Ref. [21] ........................... 33
Figure 18: Conceptual design of a turbofan engine incorporating a wave rotor, taken from Ref.
[138] ...................................................................................................................................... 35
Figure 19: Four-port wave rotor of NASA .................................................................................... 39
Figure 20: Temperature distribution of partition exit ﬂow, taken from Ref. [204] ....................... 43
Figure 21: Rotary Wave Ejector Pulse Detonation Engine, taken from Ref. [210] ....................... 44
Figure 22: Wave Rotor PDE the ‘CVC’ Engine, taken from Ref. [211] ....................................... 45
Figure 23: University of Tokyo single-channel test rig, taken from Ref. [214] ............................ 46

Figure 24: Historical perspective of wave rotor technology. Red: gas turbine application, Green:
IC engine supercharging, Blue: refrigeration cycle, Pink: pressure divider and equalizer,
Purple: wave superheater, Orange: internal combustion wave rotors, Black: general

applications ........................................................................................................................... 47
Figure 25: Flow in a shock tube after the diaphragm is ruptured .................................................. 49
Figure 26: Closed tube containing stationary gas .......................................................................... 49
Figure 27: Left end opening to gas at higher pressure ................................................................... 50
Figure 28: Leﬁ end opening to gas at lower pressure .................................................................... 51
Figure 29: Opened tube containing ﬂowing gas ............................................................................ 52
Figure 30: Left end closing to generate an expansion wave .......................................................... 52
Figure 31: Right end closing to generate a shock wave ................................................................. 53
Figure 32: Wave diagram of charging process with two primary shock waves ............................ 54
Figure 33: Wave diagram of charging process using single shock wave and expansion wave 56
Figure 34: Schematic of a gas turbine topped by a reverse-ﬂow four-port wave rotor ................. 58
Figure 35: Wave diagrams for through-ﬂow (left) and reverse-ﬂow (right) four-port wave rotors,

taken form Ref. [168] ............................................................................................................ 61
Figure 36: Transformation from a moving shock wave to a stationary shock wave ..................... 64
Figure 37: Illustration of the characteristic lines in the x-t plane ................................................... 67

ix

Figure 38: Leﬁ end closing to generate an expansion wave .......................................................... 67
Figure 39 : Schematic T-s diagrams for a baseline cycle and ﬁve wave-rotor-topped cycles ....... 73

Figure 40: Pressure gain ratio across the wave rotor versus temperature ratio across it ................ 83
Figure 41: Pressure gain ratio across the wave rotor versus wave rotor compression ratio .......... 84
Figure 42: T-s diagrams for the C-30 and C-60 wave-rotor—topped engines for Case A
implementation ...................................................................................................................... 87
Figure 43: Thermal efﬁciency, speciﬁc work, and SF C for the wave-rotor-topped engines versus
the wave rotor pressure ratio and overall pressure ratio, Case A consideration .................... 89

Figure 44: Relative values of thermal efﬁciency, speciﬁc work, and SF C for the wave-rotor-
topped engines versus the wave rotor pressure ratio and overall pressure ratio, Case A

consideration ......................................................................................................................... 90
Figure 45: T-s diagrams for the C-30 and C-60 wave-rotor-topped engines for Case B
implementation ...................................................................................................................... 92

Figure 46: Thermal efﬁciency, speciﬁc work, and SF C for the wave-rotor-topped C-30 engine
versus the wave rotor pressure ratio and compressor pressure ratio, Case B consideration 94
Figure 47: Relative values of thermal efficiency, speciﬁc work, and SF C for the wave-rotor-
topped C-30 engine versus the wave rotor pressure ratio and compressor pressure ratio,
Case B consideration ............................................................................................................. 94
Figure 48: Thermal efﬁciency, speciﬁc work, and SF C for the wave-rotor-topped C-30 engine
versus the wave rotor pressure ratio and overall pressure ratio for Case A (solid) and versus
wave rotor pressure and compressor pressure ratio for Case B (dashed) .............................. 97
Figure 49: Thermal efﬁciency, speciﬁc work, and SF C for the wave-rotor—topped C-6O engine
versus the wave rotor pressure ratio and overall pressure ratio for Case A (solid) and versus

wave rotor pressure and compressor pressure ratio for Case B (dashed) .............................. 97
Figure 50: T-s diagrams for the C-30 and C-60 wave-rotor-topped engines for Case C
implementation ...................................................................................................................... 98
Figure 51: T-s diagrams for the C-30 and C-60 wave-rotor-topped engines for Case D
implementation ...................................................................................................................... 99
Figure 52: T-s diagrams for the C-30 and C-60 wave-rotor-topped engines for Case E
implementation .................................................................................................................... 100

Figure 53: Performance map for wave-rotor-topping of the C-30 engine, Cases A, B, and D... 102
Figure 54: Performance map for wave-rotor-topping of the C-60 engine, Cases A, B, and D... 102
Figure 55: Performance map for wave-rotor—topping of the C-30 engine, Cases C and E ......... 103
Figure 56: Performance map for wave-rotor-topping of the 060 engine, Cases C and E ......... 103
Figure 57: T-s diagrams for baseline cycle, conventional cycle with two-stage compressor
(double compression work) and wave-rotor-topped cycle Case E with PRW =1.8, 030
engine .................................................................................................................................. 109
Figure 58: Thermal efﬁciency, speciﬁc work, and speciﬁc fuel consumption of the wave-rotor-
topped C-60 engine using the C-30 compressor versus wave rotor compression ratio and
compressor pressure ratio for an overall pressure ratio ﬁxed at 4.8 (yellow points for
original 060 baseline engine, orange points for wave-rotor-topped C—60 engine with C-3O

compressor) ......................................................................................................................... 1 12
Figure 59: Effect of ambient temperature: absolute and relative changes of thermal efﬁciency,
speciﬁc work, and SF C versus wave rotor pressure ratio for Case A ................................. 115
Figure 60: Effect of ambient temperature: absolute and relative changes of thermal efﬁciency,
speciﬁc work, and SF C versus wave rotor pressure ratio for Case B ................................. 116
Figure 61: Schematic of a recuperated gas turbine topped by a four-port wave rotor ................. 117
Figure 62: T-s diagrams for recuperated baseline and wave-rotor-topped cycles, Case A .......... 118
Figure 63: T-s diagrams for recuperated baseline and wave-rotor-topped cycles, Case B .......... 118
Figure 64: Thermal efﬁciency, speciﬁc work, and SF C of the enhanced C-30 recuperated engine
versus the wave rotor pressure ratio and overall pressure ratio for Case A ........................ 124

Figure 65: Relative values of thermal efﬁciency, speciﬁc work, and SF C of the enhanced C-30
recuperated engine versus the wave rotor pressure ratio and overall pressure ratio for Case B

............................................................................................................................................. 124
Figure 66: Thermal efﬁciency, speciﬁc work, and SF C of the enhanced C-3O recuperated engine
versus the wave rotor pressure ratio and compressor pressure ratio for Case B ................. 126

Figure 67: Relative values of thermal efﬁciency, speciﬁc work, and SF C of the C-30 recuperated
engine topped with a wave rotor versus the wave rotor pressure ratio and compressor

pressure ratio for Case B ..................................................................................................... 126
Figure 68: Schematic of a turbojet engine topped by a four-port wave rotor .............................. 129
Figure 69 : Schematic T-s diagrams for a baseline turbojet cycle and ﬁve different wave-rotor-

topped cycles ....................................................................................................................... 129
Figure 70: Performance maps for wave-rotor-topping of the C-30 turbojet engine, Cases A, B,

and D ................................................................................................................................... 136
Figure 71: Performance map for wave-rotor—topping of the C-30 turbojet engine, Cases C and E

............................................................................................................................................. 137
Figure 72: Wave diagram of high-pressure part with only one shock wave ................................ 141
Figure 73: Wave diagram of high-pressure part with two primary shock waves ........................ 147
Figure 74: Preliminary port designs of the charging process, corresponding to the wave diagram

sketched in Figure 73 .......................................................................................................... 153
Figure 75: Outlet port width comparison before and after rotor shortening ................................ 154
Figure 76: Variation of 175 and HR versus PRW for a constant value of 17comb ............................. 157
Figure 77: Shock strengths 175 , 17R , entropy production As/R , and exit velocity u; versus

PR w and [Lamb ..................................................................................................................... 158
Figure 78: Variation of Lg“, /L versus PRW and 11%,, ................................................................. 161
Figure 79: Variation of dimensionless values of (WT- WWJ/ WT versus PRW and 17mm), ............ 162
Figure 80: University of Tokyo wave rotor, taken from Ref. [215]. ........................................... 164
Figure 81: Non-dimensional total temperature contour of the Tokyo wave rotor, taken from Ref.

[215] .................................................................................................................................... 168
Figure 82: Non-dimensional total pressure contour of the Tokyo wave rotor, taken from Ref.

[215] .................................................................................................................................... 169
Figure 83: Generated mesh .......................................................................................................... 171
Figure 84: Finer mesh at port interface and channels, inlet high-pressure port ........................... 171
Figure 85: Interface and gap between stationary ports and moving channels ............................. 172
Figure 86: Total pressure contours, the effect of high-pressure gas inlet port on the lst channel

............................................................................................................................................. 176
Figure 87: Total pressure contours, the effect of high-pressure gas inlet port on the lst and 2nd

channels ............................................................................................................................... 177
Figure 88: Total pressure contours, traveling of compression waves toward end wall ............... 178
Figure 89: Total pressure contours, generation of reﬂected shock wave at two different time steps

............................................................................................................................................. 179
Figure 90: Total pressure contours, air scavenging process at two different time steps .............. 180
Figure 91: Total pressure contours, ingestion of fresh air at two different time steps ................. 181
Figure 92: Total pressure contour, end of the ﬁrst operating cycle ............................................. 182
Figure 93: Axial component of relative velocity vector, gas inlet port and channels .................. 182
Figure 94: Axial component of relative velocity, lower and upper comers of gas inlet port ....... 183
Figure 95: Axial component of relative velocity, air outlet port .................................................. 184
Figure 96: Axial component of relative velocity, gas outlet port ................................................ 184
Figure 97: Axial component of relative velocity, air inlet port .................................................... 185
Figure 98: Mesh for multi cycles ................................................................................................. 185
Figure 99: Radial—ﬂow wave rotor with straight channels ........................................................... 187

xi

Figure 100: Reverse-ﬂow wave disk with straight channels ....................................................... 188
Figure 101: Wave disc with straight channels: contour plots of local pressure (top), local

temperature (middle), and velocity (bottom) at time=8.6e-O4 s .......................................... 189
Figure 102: Radial-ﬂow wave rotor with curved channels .......................................................... 190
Figure 103: Reverse-ﬂow wave disk with curved channels ......................................................... 191
Figure 104: Wave disc with curved channels: contour plots of local pressure (top), local

temperature (middle), and velocity (bottom), at time=1.73e-3 s ......................................... 192
Figure 105: Stack of wave discs with straight channels .............................................................. 193
Figure 106: Stacked radial wave discs and radial compressor ..................................................... 194
Figure 107: Cut-view of a radial wave rotor topping a gas turbine ............................................. 195
Figure 108: Flow through an internal combustion radial wave rotor topping a gas turbine ........ 196
Figure 109: Component parts of a radial wave rotor topping a gas turbine ................................. 197
Figure 110: Schematic of a R718 cycle with direct condensation and evaporation .................... 199
Figure 111: Schematic of a R718 cycle enhanced by a three-port condensing wave rotor

substituting for the condenser and for one compressor stage .............................................. 199
Figure 112: Schematic of a three-port condensing wave rotor .................................................... 200
Figure 113: Regions modeled for each compression and condensation ...................................... 200
Figure 114: Schematic wave and phase-change diagram for the three-port condensing wave rotor

(high-pressure part) ............................................................................................................. 200
Figure 115: Schematic p-h diagram of a R718 baseline cycle and a wave rotor enhanced cycle

............................................................................................................................................. 203
Figure 116: Schematic T-s diagram of a R718 baseline cycle (cooling water cycle not shown) and

a wave rotor enhanced cycle ............................................................................................... 203
Figure 117: Performance map representing maximum performance increase and optimum wave

rotor pressure ratios ............................................................................................................. 205
Figure 118: Efﬁciency trend of compression process. Green: wave rotor efﬁciency, Red:

compressor efﬁciency ......................................................................................................... 209
Figure 119: First conceptual design of an UuGT equipped with a four-port wave rotor ............ 210
Figure 120: Second conceptual design of an UuGT equipped with a four-port wave rotor ........ 211
Figure 121: Third conceptual design of an UuGT equipped with a four-port wave rotor ........... 212

xii

CHAPTER 1: INTRODUCTION

1.1 Unsteady-F low Devices

Oscillatory and pulsatile ﬂuid motion has been neatly utilized in nature, yet is
comparatively poorly studied by engineers despite the invention of cyclically operating
engines and machines. The potential for utilizing unsteady ﬂows has been recognized
since the early twentieth century but neglected as long as substantive improvements could
be made to conceptually simple steady-ﬂow or semi-static devices. Also, the inherent
non-linearity of large—amplitude wave phenomena in compressible ﬂuids and unusual
geometry of non-steady ﬂow devices necessitates detailed calculations, which until
recently were too laborious or expensive or imprecise. By understanding and exploiting
complex unsteady ﬂows, a quantum increase in engine performance is possible. Often it
has been feasible to simplify the hardware of engines, making them less costly, more
responsive, and more durable by employing unsteady-ﬂow processes.

Shock tubes, shock tunnels, pulse combustors, pulse detonation engines, and wave
rotors are a few examples of unsteady-ﬂow devices. The basic concept underlying these
devices is the transfer of energy by pressure waves. By generating compression and
expansion waves in appropriate geometries, wave machines can transfer energy directly
between different ﬂuids without using mechanical components such as pistons or vaned
impellers. In fact, these devices properly represent applications of classical, unsteady,
one-dimensional, compressible ﬂow theory. The major beneﬁt of these unsteady-ﬂow
machines is their potential to generate much greater pressure rises than those obtained in
steady-state ﬂow devices [1, 2]. For instance as illustrated in Figure 1, for the same

change from a given inlet Mach number M, to a given outlet Mach number M2 , the local

(static) pressure ratio p2 /p, across a single shock wave (solid line) moving in a
frictionless channel is always much greater than that obtained by an isentropic
deceleration in a 100% efﬁcient diffuser (dotted line). However, it must be noted that
friction effects always exist, and hence pressure gains are actually less than those

predicted by Figure 1.

 

770,17 =100%

M
Mlﬁ— M2 E, E,

2.5 . (I) <2) 1) (2)

 

Moving shock wave Diffuser

p2/p1
N

1.5 -

 

 

 

 

0 0.2 0.4 0.6 0.8 1
M1

Figure 1: Comparison of pressure gain (local pressure ratio) of moving shock and steady-
ﬂow isentropic diffuser for y=1.4

It is also worthwhile to note that shock wave compression is a relatively efﬁcient
process as indicated in Figure 2, where shock isentropic efﬁciency ”Shock (red) is
compared with compressor isentropic efﬁciency neompmm, (green) and diffuser isentropic
efﬁciency 770mm (blue). Figure 2 shows variations of these parameters as functions of
the pressure gain p; /p, obtained by a moving shock wave in a frictionless channel, by a

compressor with different values of polytropic efﬁciencies, and by a diffuser with

different values of total pressure drop across the diffuser expressed by pg /p,/ ,
respectively. The comparison reveals that for the same pressure gain pg/pl, the ideal
shock compression efﬁciency may far exceed the efﬁciency obtained by a decelerating
diffuser or a compressor. For example, it is seen that for a pressure gain of up to p2 /p,
=2.2, ”Shock is greater than about 93% and thus is greater than those of typical diffusers
and compressors. Therefore a gain in cycle performance can be expected when a
compressor (or a diffuser) is replaced by an unsteady-ﬂow device utilizing shock waves.
Flow friction-effects would lower the efﬁciency of wave devices and reduces their
efﬁciency advantage (not shown in Figure 2), but the relative advantage is expected to

persist. Detailed calculations for developing Figure l and Figure 2 are presented in

 

 

 

 

 

 

Appendix A.
1
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Figure 2: Shock wave, diffuser, and compressor isentropic efﬁciencies as functions of
pressure gain

1.2 Wave Rotors

There are at least two classes of wave machines: rotating wave machines and non-
rotating wave machines. Dynamic pressure exchangers and wave rotors are two types of
rotating wave devices developed to reach the high performance targets of thermodynamic
cycles. In Europe, however, both terms have been often used interchangeably [3].

The essential feature of wave rotors is an array of channels arranged around the axis
of a cylindrical drum. As schematically shown in Figure 3, the drum rotates between two
end plates each of which has a few ports or manifolds, controlling the ﬂuid ﬂow through
the channels. The number of ports and their positions vary for different applications. By
carefully selecting their locations and widths to generate and utilize wave processes, a
signiﬁcant and efﬁcient transfer of energy can be obtained between ﬂows in the

connected ducts.

Stator end plate

Channels Rotating drum

 

Figure 3: Schematic conﬁguration of a typical wave rotor

Through rotation, the channel ends are periodically exposed to the ports located on
the stationary end plates initiating compression and expansion waves within the wave
rotor channels. Thus, pressure is exchanged dynamically between ﬂuids by utilizing

unsteady pressure waves. Therefore, unlike a steady-ﬂow turbomachine which either

compresses or expands the ﬂuid, the wave rotor accomplishes both compression and
expansion within a single component. To minimize leakage, the gap between the end
plates and the rotor has to be very small or the end plates with sealing material could
contact the rotor.

An inverted design with stationary rotor and rotating ports is also possible [4]. Such a
conﬁguration has more than one rotating part and usually doubles interfaces that need to
be sealed between rotating and stationary parts. However, it may be preferred for
laboratory investigations because it easily enables ﬂow measurement in the channels
where the important dynamic interactions take place. However, this arrangement rarely
seems to be convenient for commercial purposes.

The rotor may be gear or belt driven or preferably direct driven by an electrical motor
(not shown). The power required to keep the rotor at a correctly designed speed is
negligible [5, 6]. It only needs to overcome rotor windage and friction in the bearings and
contact sealing if used. Alternatively, rotors can be made self-driving. This conﬁguration,
known as the “free-running rotor”, can drive itself by using the momentum of the ﬂow to
rotate the rotor [7, 8].

In wave rotor machines, two basic ﬂuid-exchange processes usually happen at least
once per revolution of the rotor: the high-pressure process (charging process) and the
low-pressure process (scavenging process). In the high-pressure process, compression
waves transfer the energy directly from a ﬂuid at a higher pressure (driver ﬂuid) to
another ﬂuid at a lower pressure (driven ﬂuid). In the low-pressure process the driver
ﬂuid is scavenged from the rotor channels, using expansion waves. Generation of

expansion waves allows ingestion of a fresh low-pressure ﬂuid into the rotor channels.

There are several important advantages of wave rotor machines. Their rotational
speed is low compared with turbomachines, which results in low material stresses. But
they can respond on the timescale of pressure waves, with no rotor inertial lag. From a
mechanical point of view, their geometries can be simpler than those of turbomachines.
Therefore, they can be manufactured relatively inexpensively. Also, the rotor channels
are less prone to erosion damage than the blades of turbomachines. This is mainly due to
the lower velocity of the working ﬂuid in the channels, which is about one-third of what
is typical within turbomachines [5]. Another important advantage of wave rotors is their
self-cooling capabilities. They are naturally cooled by the fresh cold ﬂuid ingested by the
rotor. Therefore, applied to a heat engine, the rotor channels pass through both cool air
and hot gas ﬂow in the cycle at least once per rotor revolution. As a result, the rotor
material temperature is always maintained between the temperature of the cool air which
is being compressed, and the hot gas which is being expanded.

Despite very attractive features, several challenges have impeded the vast appearance
of commercial wave rotors in some applications though numerous research efforts have
been carried out during the past century. Besides unusual ﬂow complexity and
anticipated off-design problems and uncertainties about the selection of the best wave
rotor conﬁguration for a particular application, the obstacles have been mainly of a
mechanical nature, like sealing and thermal expansion issues, as mentioned throughout
this study. However, due to the recent energy crises, technology improvement, and

economic reasons, new desires for wave rotor technology have been stimulated.

1.3 Wave Rotor Applications

As a combined expansion and compression device, the wave rotor can be used as a

supercharging device for IC engines, a topping component for gas turbines, in

refrigeration cycles, and more. In advanced conﬁgurations, combustion occurs internally
in the wave rotor channels allowing extremely short residence times at high temperature,
hence potentially reducing NOx emissions. A condensing wave rotor may be viewed as a
similarly advanced conﬁguration that enhances the performance of water refrigeration
cycles. Recently, wave rotor technology has been envisioned to enhance the performance
of ultra-micro gas turbines manufactured using microfabrication technologies. In this

study, the focus is on using wave rotor as topping components for gas turbine cycles.

1.3.1 Gas Turbine Applications

Over the last decades, gas turbines have played a major role in a wide size range of
propulsion and power generation systems. A schematic diagram for a simple-cycle,
single-shaﬂ gas turbine is shown in Figure 4. It consists of three components:

compressor, combustion chamber, and turbine.

Combustion Chamber

 

 

Compressor Turbine

Figure 4: Schematic conﬁguration of a simple gas turbine engine
Air enters the compressor at state “0” and is compressed to some higher pressure.
Upon leaving the compressor, the compressed air enters the combustion chamber at state
“1” where fuel is injected in a nearly constant pressure process. At state “4”, the hot gases

enter the turbine where part of their thermal energy is converted into work. In a power

generation application, some of the work produced by the turbine is used to drive the

compressor, and the remainder is available as output work produced by the cycle. The
expanded gases leave the turbine to the surroundings at state “5”.

There are several methods to enhance the performance of gas turbines, including
improvement in aerodynamic designs of turbomachinery components or thermodynamic
cycle enhancement. The aerodynamics of turbomachinery has already yielded very high
component efﬁciencies up to around 90% [9]. Further improvement is possible, but huge
gains seem unlikely. From a thermodynamic point of view, increasing the turbine inlet
temperature is the most efﬁcient way to improve both thermal efﬁciency and output
work. For a given engine, this can be achieved by increasing the cycle pressure ratio
using a larger compressor. This is shown in a schematic temperature-entropy (T-s)
diagram in Figure 5, where an increased pressure ratio has modiﬁed the baseline cycle
from O-lb-4b-5b to 0-1-4-5. The numbers on this diagram correspond to the numbers used
in Figure 4.

According to the ﬁrst law of thermodynamics, the cycle net work (Wm) is equal to
the cycle net heat transfer. Since the cycle with the higher pressure ratio has a greater
heat supply (QH) with the same heat rejected (QL) as the baseline cycle, it has greater

output work and also greater thermal efﬁciency deﬁned by:

Wnet : QH —QL :1_&
QH QH QH

 

’7: (1)

However, the maximum temperature of the gas entering the turbine is usually ﬁxed
by material considerations. Therefore, cycle 0-1-4-5 has an intolerably high turbine inlet
temperature (T4) much greater than that for the baseline engine (T 41,). One innovative
solution for this turbine blade temperature limitation is to cool the burned gases by

extracting energy from the ﬂow before it enters the turbine. This way, the cycle peak

temperature is de-coupled from the turbine inlet temperature. It can be accomplished by
letting the hot gas leaving the combustion chamber compresses the air coming from the
compressor, utilizing shock waves in an appropriate geometry. This is the basic principle

behind the operation of wave rotors.

0-1-4-5 Higher pressure ratio cycle
0-1b-4b-5b Baseline cycle

 

T turbine

Temperature

 

 

Entropy

Figure 5: Schematic temperature-entropy diagram for a gas turbine with low and high
pressure ratios

0 Wave Rotor Topping Cycles

In a wave-rotor-topped cycle, the combustion can take place at a higher temperature
while the turbine inlet temperature can be equal to that of the baseline cycle. Also, a
pressure gain additional to that provided by the compressor is obtained by the wave rotor.
Thus, the wave rotor can increase the overall pressure ratio and peak cycle temperature

beyond the limits of ordinary turbomachinery. As a result, the performance enhancement

is achieved by increasing both the thermal efﬁciency and the output work, hence reducing
the speciﬁc fuel consumption rate considerably. In the following, a four—port wave rotor
integrated into a simple gas turbine cycle is selected to prove these facts and its principal
operation is brieﬂy discussed

In a conventional arrangement, the wave rotor is embedded between the compressor
and turbine “parallel” to the combustion chamber. Figure 6 illustrates how a four-port
wave rotor is used to top a gas turbine cycle. Figure 7 schematically shows how the wave
rotor can be embedded physically into a baseline engine that uses single stage radial

compressor and turbine.

Combustion Chamber

b.)

 

0 Wave Rotor 1‘

 

 

 

Compressor Turbine

Figure 6: Schematic of a gas turbine topped by a four-port wave rotor

\Vm'c Rotor Turhlne

Combustor

 

Compressor

Figure 7: Schematic example of the physical implementation of a wave rotor in a
gas turbine (exploded view, piping not shown)

10

Following the ﬂow path shown in Figure 6, air from the compressor enters the wave
rotor (state 1) and is further compressed inside the wave rotor channels. After the
additional compression of the air in the wave rotor, it discharges into the combustion
chamber (state 2). Here, combustion takes place at a higher pressure and temperature than
in the baseline engine. The hot gas leaving the combustion chamber (state 3) enters the
wave rotor and compresses the air received from the compressor (state 1). To provide the
energy transfer to compress the air, the burned gas expands and is afterward scavenged
toward the turbine (state 4). Due to the pre-expansion in the wave rotor, the burned gas
enters the turbine with a lower temperature than that of the combustor exit. However, the
gas pressure is still higher than the compressor exit pressure by the pressure gain obtained
in the wave rotor. The turbine inlet total pressure is typically 15 to 20% higher than the
air pressure delivered by the compressor [10]. This pressure gain is in contrast to the
untopped engine, where the turbine inlet pressure is always lower than the compressor
discharge pressure due to the pressure loss across the combustion chamber. As a result of
the wave rotor pressure gain, more work can be extracted from the turbine increasing
overall engine thermal efﬁciency and speciﬁc work. Finally, the channels are re-
connected to the compressor outlet, allowing fresh pre-compressed air to ﬂow into the
wave rotor channels and the cycle repeats.

The general advantage of using a wave rotor becomes apparent when comparing the
thermodynamic cycles of baseline and wave—rotor—enhanced engines. Figure 8 shows
schematic T-s diagrams of the baseline engine and the corresponding wave-rotor-topped
engine. The shown wave rotor implementation is the one most commonly discussed in

references, referred to as Case A in this study. It is evident that both gas turbines are

11

operating with the same turbine inlet temperature and compressor pressure ratio. Each
wave rotor investigated in this work has zero shaﬁ work. Therefore, the wave rotor
compression work is equal to the wave rotor expansion work. Thus, the energy increase
from state “lb” to “4b” in the baseline engine and from state “1 ” to “4” in the wave-rotor-
topped engine is the same. This results in the same heat addition for both cycles.
However, the output work of the topped engine is higher than that of the baseline engine

“t”

due to the pressure gain across the wave rotor (p,4>p,4b , where subscript indicates
total values). Therefore, the thermal efﬁciency for the topped engine is higher than that of
the baseline engine. The inherent gas dynamic design of the wave rotor compensates for

the combustor pressure loss from state “2” to “3”, meaning that the compressed air

leaving the wave rotor is at higher pressure than the hot gas entering the wave rotor.

   
   
   

0-1-2-3-4-5
Engine topped with a
wave rotor

Tturbine — —————————————— - — —-.
E
3
‘5
I-
Q)
Q.
E
O
[—
1b=
0 0-1b-4b-5b

Baseline engine

 

 

Entropy

Figure 8: Schematic T-s diagrams for a gas turbine with and without a wave rotor

12

Other advantageous implementation cases for the wave rotor into the given baseline
engine are also possible. Four more advantageous cases have been extensively studied in
this work and their advantages and disadvantages will be discussed in detail in Chapter 4.

The goal of this chapter was to introduce the reader to the concept of the wave rotor
and its application for gas turbine cycles. Next chapter (Chapter 2) provides a succinct
review of past and current research in developing wave rotor technology. Chapter 3
describes the gasdynamic principle behind the operation of wave rotors. In Chapter 4, a
comprehensive and systematic performance analysis of two microturbines known as the
C-30 and C-60 engines which are topped with a four-port wave rotor in various wave-
rotor—topping cycles is presented. Chapter 5 describes an analytical design procedure for
the critical high-pressure phase of four-port wave rotors to predict some of their useful
design parameters. A CF D simulation is also performed showing that commonly
available CFD software FLUENT can be utilized for wave rotor simulations and to
support their design. Finally, Chapter 6 introduces and studies several innovative wave
rotor concepts and designs including radial-ﬂow wave rotors, integrating wave rotors in
ultra-micro gas turbines (UuGT), and water refrigeration systems working with wave

1'0t01’S .

13

 

CHAPTER 2: HISTORICAL REVIEW OF WAVE ROTOR TECHNOLOGY

Recent advances and experiences obtained by the wave rotor community have
renewed interest in this technology. These advances include new computational
capabilities allowing accurate simulation of the ﬂow ﬁeld inside the wave rotor, and
modern experimental measurements and diagnostic techniques. Improvements in
aerodynamic design, sealing technologies, and thermal control methods have been
sought. Recent developments in related unsteady ﬂow and combustion processes of
pulsed detonation engines have also provoked renewed interest. For this reason it is

worthwhile to review the past and current work.1

The Early Work (1906-1940)
The earliest pressure exchanger was suggested by Knauff in 1906 in which he did not

employ the action of pressure waves [11]. The pressure exchanger introduced by him
consisted of a cellular drum that rotates between two end plates containing several ports
through which ﬂows with different pressures enter and leave, exchanging their pressure.
Knauff initially described rotor channels with curved blades and proposed inclined
nozzles in the stator to achieve output shaft power (pressure exchange engine) besides
pressure equalization inside the rotor. Reported by Pearson [12], Knauff in his second
patent in 1906 [13] and Burghard in 1913 [14] proposed a simpler device in which the
pressure exchange takes place in long narrow channel conﬁgurations @ressure
exchanger) known later as the Lebre machine following Lebre’s patent in 1928 [15].
Today, the term “static pressure exchanger” is normally given to this type of device.

Around 1928, Burghard proposed the utilization of pressure waves in another invention

 

' Materials presented in this chapter have been accepted for publication in 2004 International Mechanical
Engineering Conference, ASA/{E Paper [MEC52004-60082, USA, Nov 2004.

14

[16] that was termed the “dynamic pressure exchanger” to distinguish it from the static
pressure exchanger. Here, the term “dynamic” implies the utilization of pressure waves in
both compression and expansion processes taking place inside the rotor channels.
However, difﬁculties mainly related to poor knowledge about unsteady ﬂow processes
limited the dissemination of the dynamic pressure exchanger concept [3] until World War
II when Seippel in Switzerland implemented this concept into real engines, as discussed

below.

The Comprex® Pressure Wave Supercharger (l940-Present)

Brown Boveri Company (BBC), later Asea Brown Boveri (ABB) and now Alston, in
Switzerland has a long history in wave rotor technology. As reported by Meyer [17], their
initial investigations in the beginning of the 19405 were aimed at implementing a wave
rotor as a topping stage for a 1640 kW (2200 hp) locomotive gas turbine plant of British
Railways [18-21]. They expected to obtain a power increase of 80% (2983 kW or 4000
hp) and a 25% efﬁciency increase (from 18% to 22.5%) based on the patents of Seippel
[22-25]. This arrangement is shown in Figure 9. The wave rotor had 30 channels rotating
at 6000 rpm, with two opening ports on each side through which air and gas entered and
left. It had originally shown a pressure ratio up to 3:1 and total efﬁciency of 69% in
previous tests during 1941-1943, which could approximately result in a 83% efﬁciency

for each compression and expansion process [17].

15

 

" Wave Rotor

ﬂ!

 

 

 

 

Figure 9: Wave rotor as a topping stage for the locomotive gas turbine,
taken from Ref. [17]

The ﬁrst wave rotor worked satisfactorily, proving the concept of wave rotor
machines. However, its performance when installed in the engine was far from
expectations, mainly because of its inefﬁcient design and crude integration [20].

Seippel’s work also initiated the notion of using the wave rotor as a pressure wave
supercharger for diesel engines. The extensive practical knowledge accumulated by BBC
during investigations of gas turbine topping cycles was then used to develop pressure
wave superchargers ﬁrst by the ITE Circuit Breaker Company in the US [26-28]. In an
effort jointly sponsored by the US Bureau of Aeronautics and ITE supervised by
Kantrowitz of Cornell University and Berchtold of ITE, the ﬁrst units were successfully
manufactured and tested on vehicle diesel engines between 1947 and 1955. As a result of
this success, a co-operative program with BBC was started in 1955 and BBC decided to
concentrate on the development of pressure wave superchargers for diesel engines, due to
their higher payoff compared to other applications [29]. As a manufacturer of
superchargers, BBC later continued the project in collaboration with the Swiss Federal

Institute of Technology (ETH Zurich). While the ﬁrst prototype was installed in a truck

engine in 1971 [30], the supercharging of passenger car diesel engines was started in
1978 [31, 32] with a ﬁrst successful test on an Opel 2.1 liter diesel engine [32, 33]. This
supercharger was given the trade name Comprex® shown in Figure 10. The port
arrangement indicates the use of two operating cycles per revolution, shortening the rotor
length and reducing thermal loads. The main advantage of the Comprex® compared to a
conventional turbocharger is its rapid response to changes in engine operating conditions
where the turbocharger is less responsive, due to its rotational inertia. Furthermore, as the
efﬁciency of the Comprex® is independent of scale, its light weight and compact size
make this device attractive for supercharging small engines (below about 75 kW or 100

hp) [34, 35].

 

Figure 10: The Comprex®, taken from Ref. [34]

By 1987, the ﬁrst wide application of the Comprex® in passenger cars occurred in the
Mazda 626 Capella [7, 36]. Since then, ABB’s Comprex® pressure wave supercharger
has been commercialized for several passenger car and heavy diesel engines. For
instance, once Mazda produced 150,000 diesel passenger cars equipped with pressure
wave superchargers [37]. The Comprex® has also tested successfully on vehicles such as
the Mercedes-Benz diesel car [8], Peugeot [29], and Ferrari [29].

The progress by BBC/ABB took almost ﬁve decades to accomplish. The successful
development of the Comprex® has been enabled by efforts of numerous researchers.
Besides the above mentioned names, only some more are listed here: Gyarrnathy [6],
Burri [38], Wunsch [39], Croes [40], Summerauer [41], Kollbrunner [42], Jenny [43],
Keller [44], Rebling [45], and Schneider [46]. Further references related to the
development of the Comprex® by BBC [47-60] and other organizations [61-71] until
1990 can be found in the literature. By the end of the 19805, when the Comprex® activity
was transferred to the Mazda company in Japan [3, 72], researchers at ABB turned to the
idea of utilizing wave rotor technology for gas turbine applications [9, 73].

During 19908, a few groups continued the development of pressure wave
superchargers. Nour Eldin and his associates at the University of Wuppertal in Germany
have developed a fast and accurate numerical method for predicting the unsteady-ﬂow
ﬁeld in pressure wave machines, using the theory of characteristics [74-80]. Piechna et al.
at the Warsaw University of Technology in Poland have developed experimentally
validated one-dimensional and two-dimensional numerical codes to analyze the ﬂow ﬁeld
inside the Comprex® [81-87]. Piechna has also proposed a compilation of the pressure

exchanger with the internal combustion wave rotor, presenting the idea of the

18

autonomous pressure wave compressor [88]. Oguri et al. at Sophia University in Japan
have performed measurements on a car gasoline engine supercharged by the pressure
wave supercharger to investigate the feasibility of such integrations [89]. This effort
sought to extend the application from diesel engines to gasoline engines and achieved a
satisfactory increase of thermal efﬁciency of the supercharged engines. Guzzella et a1.
[35, 90-94] at ETH in Switzerland have developed a control-oriented model that
describes the entire engine supercharged by pressure wave devices, with special emphasis
on the modeling of transient exhaust gas recirculation phenomena. The experimentally
validated model has introduced an optimized strategy to operate a supercharged engine
with good drivability. Finally, an investigation of Comprex® supercharging on diesel
emissions has been recently performed in Turkey [95], demonstrating that the Comprex®
has the potential for reducing NOx in diesel engines.

To date, the Comprex® has been recognized the most successful application of wave
rotor technology and represents a practical utilization of the wave rotor concept. The
Comprex® development by BBC/ABB also has established fabrication techniques for
wave rotors in commercial quantities and considered as a matured and reliable machine
for internal combustion engine supercharging. For this application, BBC/ABB has solved
difﬁcult development challenges like sealing against leakages, noise, and the thermal
stress problems. For instance, enclosing the rotor in a pressurized casing and using a rotor
material with a low thermal expansion coefﬁcient over the operating temperature range
has kept Comprex® leakages to an acceptable level [29]. Furthermore, several pockets
have been cast into the end plates to control wave reﬂections and to achieve good off-

design performance when engine speed changes [59].

19

In recent years, Swissauto WENKO AG in Switzerland has developed a modern
version of the pressure wave supercharger [37]. This new generation of Comprex®
known as the Hyprex® is designed for small gasoline engines. It beneﬁts from new
control features, enabling higher pressure ratios at low engine speeds, further reduced
noise levels, and improvement of the compression efﬁciency at medium or high engine
speeds. The Hyprex® has been successfully demonstrated in the SmILE (Small,
Intelligent, Light and Efﬁcient) vehicle which is a modiﬁed Renault Twingo, achieving
very low speciﬁc fuel consumption and low emissions. ETH is collaborating closely in

this effort by developing control systems for the proper operation of the device.

Cornell Aeronautical Laboratory and Cornell University (1948-2001)
Inspired by the cooperation with BBC in the late 19405, work on unsteady-ﬂow

concepts was initiated at Cornell Aeronautical Laboratory (CAL). Among several novel
concepts including development of energy exchangers for gas turbine cycles and various
stationary power applications [96], the CAL Wave Superheater was built in 1958 and
utilized until 1969 [21]. The 2 m diameter wave superheater used heated helium as the
low molecular weight driver gas to provide a steady stream of high-temperature and high-
pressure air for a hypersonic wind tunnel test facility. It compressed and heated air to
more than 4000 K and up to 120 atm for run times as long as 15 seconds. Figure 11 is a
photograph of this device. The CAL Wave Superheater was a landmark demonstration of

the high temperature capabilities of wave rotor devices [21, 96].

20

 

Figure 11: Photograph of the CAL Wave Supercharger, taken from Ref. [5]

Around 1985, Resler, a former member of the CAL Wave Superheater team, resumed
the wave rotor research at Cornell University. His efforts and those of his group led to the
development of new wave rotor concepts and analytic methods for three-port wave rotor
diffusers [97], double wave rotor cycles [98], ﬁve-port wave rotors [98—105], and
supersonic combustor aircraft engines using wave rotors [106]. Five-port wave rotors
have shown signiﬁcant potential for reducing NOx in gas turbine engine applications.
Figure 12 illustrates a double wave rotor in a gas turbine cycle. The idea of using a
compound unit consisting of two (or multiple) wave rotors, one supercharging the other,
is also reported in an early German patent by Muller in 1954 [107], as stated by Azoury

[34].

21

 

X Bypass

 

 

 

 

 

    

/

 

 

 

 

 

 

 

 

' 2
/ High Low \
Compressor Pressure N Pressure
# Turbine Turbine '

Figure 12: Schematic of a double wave rotor cycle, taken from Ref. [98]

Power Jets Ltd (1949-1967)
In parallel with but independent of Seippel’s efforts in 19403, Jendrassik, former chief

engineer of the Gantz Diesel Engine Company of Budapest, was working on the
development of wave rotor machines for gas turbine applications [20, 108—110]. He
developed one of the first concepts for wave rotor applications to aircraft engines,
proposing the wave rotor as a high pressure topping stage for early aircraft engines [111,
112]. His ideas stimulated the govemment-controlled company of Power Jets Ltd in the
UK to become active in the wave rotor ﬁeld in 1949. Even though the initial intent of
Power Jets Ltd was to use wave rotor technology for IC engine supercharging, the
interest was later extended to several other applications including air cycle refrigerators,
gas turbines, pressure equalizers, and dividers [3, 5, 20, 110]. For instance, two prototype
air-cycle refrigerators using wave rotors were designed and employed in gold mines in
India and South Africa for environmental cooling purposes. They performed the same
duty as equivalent vapor-cycle machines, but with lower weight and bulk. After

Jendrassik's death in 1954, theoretical and experimental work continued at Imperial

22

College, University of London, directed by Spalding and Barnes and also by Ricardo
Company in the UK [20, 113]. The experimental divider test rig at Imperial College is
shown in Figure 13. Detailed information related to Power Jets Ltd efforts can be found

in company reports listed in Ref. [5].

 

Figure 13: The experimental divider test rig at Imperial College, taken from Ref. [5]

Most of these efforts were experimental and thus expensive. Computational methods
and digital computer facilities were too poorly developed in that time, so extensive
theoretical methods required to improve the progress were too difﬁcult. Cycle analyses
by hand calculations were tedious and impractical. Spearheading the development of
CFD methods, Spalding of Imperial College formulated a numerical procedure for wave
rotors considering the effects of heat transfer and friction. It utilized novel features to
ensure solutions free from instabilities and physical improbabilities [20]. Based on this

numerical model, a computer program was developed by Jonsson [114] and it was

23

successfully applied to pressure exchangers [1 15—117]. Spalding’s students, Azoury [1 18]
and Kentﬁeld [119], continued their efforts on different theoretical aspects of pressure
exchangers [3, 5, 20, 34, 110, 117, 120-123] despite the dissolution of Power Jets Ltd in
1967 [20].

Ruston-Hornsby Turbine Company: The Pearson Rotor (Mid 19505 - 1960)

In the U.K. of the mid 19505, besides the work at Power Jets Ltd and Imperial
College, the Ruston—Homsby Turbine Company, manufacturer of diesel engines and
industrial gas turbines, supported the construction and testing of a different kind of wave
rotor designed by Pearson [124, 125]. This unique wave rotor, known as the wave turbine
engine or simply the wave engine, has helical channels that change the direction of the
gas ﬂows producing shaft work similar to a conventional turbine blade. With the ﬁnancial
support by the company, Pearson designed and tested his wave rotor in less than a year,

shown in Figure 14.

 

Figure 14: The Pearson rotor (left) and rear and front stator plates (right),
taken from Ref. [124]

The rotor has a 23 cm (9") diameter and a 7.6 cm (3") length. The engine worked
successfully for several hundred hours in a wide range of operating conditions (c.g.,

3000-18000 rpm) without variable porting, and produced up to 26 kW (35 hp) at its

24

design point with a cycle peak temperature of 1070 K and a thermal efﬁciency of around
10%. While the performance results were slightly less than the expected design
performance (mainly due to the combined effect of excessive leakage and incomplete
scavenging), higher performance seemed to be possible with more careful design and
development. The design of the engine was based on many wave diagrams using the
method of characteristics that accounted for all internal wave reﬂections. The engine
utilized extra ports and injection nozzles to control and cancel unwanted reﬂected waves.
The engine had a length of only one third of its diameter despite having only one cycle
per revolution [12]. The sealing and bearings were carefully adapted considering rotor
thermal expansion. Unfortunately, the engine was accidentally wrecked due to over
speeding from an improperly connected fuel line while the company was suffering
ﬁnancial difﬁculties. Despite the success achieved, the wave engine experiments were far
from the norm of ordinary projects at Ruston-Homsby and it was considered a redundant
project. Tragically, the wave rotor project was canceled when success seemed so close.
Further efforts by Pearson to attract additional funding by other sources to commercialize
his engine were unsuccessful.

In the history of wave rotor technology, the Pearson rotor and the Comprex® are
known as the most successful wave rotor machines developed to date [20, 29, 109]. Both
devices have worked efﬁciently over a wide range of operating conditions, demonstrating
good off-design performance although the less-publicized CAL Wave Superheater was an
equal success. Nevertheless, the Pearson rotor is a notable wave rotor for producing a

signiﬁcant power output in addition to being a successful pressure exchanger.

25

General Electric Company (1956-1963)

While Pearson was developing his novel wave engine, General Electric Company
(GE) in the US initiated a wave rotor program in 1956 [126]. The work was motivated by
earlier work at NASA Langley initiated by Kantrowitz and continued by Huber [127]
during the development of a wave engine in the early 19508 and later in 1954-1956
developing pressure gain combustors (constant volume combustion) [126]. GE studied a
new conﬁguration of wave rotor in which combustion took place inside the rotor
channels (internal combustion wave rotors). Such an arrangement eliminates the need for
an external combustion chamber used in the gas turbine cycle, resulting in a signiﬁcantly
lower weight, less ducting, and a compact size. In the period of 1956 to 1959, the
methods used at NASA were analyzed, improved and applied to the design and
fabrication of the ﬁrst internal combustion wave rotor demonstrator. As reported by
Weber [2], the test rig was ﬁrst tested at the California Advanced Propulsion Systems
Operation (CAPSO) of GE. After 20 seconds of operation, due to the heating and
expansion phenomena, the rotor seized between the end plates causing an abrupt stop.
The test demonstrated the difﬁculty of clearance control between the end plates and rotor
during thermal expansion. Rotor expansion is an especially challenging problem in the
design of wave rotors. While the running clearance between the end plates and rotor must
be kept as small as possible, the rotor tends to expand thermally due to hot gases in the
rotor. Henceforth, GE resorted to inferior rubbing type seals, and tested only pressure-
exchange conﬁgurations from 1960 to 1961 [126]. Despite ﬂow leakage, due to the poor
sealing method, respectable wave rotor overall pressure ratios of 1.2 to 1.3 were
achieved. Meanwhile, a feasibility study was initiated for reducing compressor stages of a

T-58 GE-06 engine by using a wave rotor. It showed a considerable reduction in overall

26

engine weight and cost, and a 15% reduction in speciﬁc fuel consumption rate. These
results motivated GE further to come up with a conceptual design layout of such an
advanced engine. However, further rig experiments revealed other mechanical and
aerodynamic shortcomings including start-up, bearing durability, fuel system
complications and control [10].

GE also pursued designing a shaft work output wave rotor. Over the period from
1961 to 1963, Klapproth and his associates at GE in Ohio fabricated and tested a wave
engine using air-gap seals. An ideal wave diagram of this engine is shown in Figure 15.
The engine worked continuously, but it did not produce the anticipated net output power.
It is believed that insufﬁcient attention was given to account for internal wave reﬂections,
thus, the ﬂow ﬁeld calculations were inaccurate [29]. Simpliﬁcations were unavoidable at
that time and generation of wave diagrams by hand required considerable time and effort
and small design changes necessitated a lengthy recalculation. Although the Klapproth
rotor did not produce the expected performance, it clearly demonstrated the possibility of
the complete exchange of energy within the wave rotor. GE development of the wave
rotor was canceled in 1963 due to shifting funds from turbine engine development to
space exploration and rocket propulsion [2], and GE’s commitment to pursue large

engine development exclusively [126, 127].

27

BUFFER .
-F f 4475!?
— 5
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\2
300—. Q
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270—. S
i

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1111
74

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LAYER

 

 

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Figure 15: Ideal wave diagram of the Klapproth rotor, taken from Ref. [126]

28

PORT

DELI VERY

our/14265 Paar

General Power Corporation (Mid 1960s - 1984)

In the mid 19605, General Power Corporation (GPC) started a wave rotor program
originally intended for a road vehicle engine application [29]. Over a period of about 20
years, GPC spent considerable time and money to successfully design and develop wave
rotors. The work was initially supported by Ford Motor Company and later by the
Department of Energy (DOE) and the US Defense Advanced Research Program Projects
Agency (DARPA). Unfortunately, the GPC work is poorly documented making it
difﬁcult to further report about their wave rotor. Some information is brieﬂy reported in
Ref. [128]. As stated by Taussig [29, 109], while the GPC rotor shared some of the
features of the Klapproth and Pearson rotors, it differed in several aspects taking into
account its own unique design and operation. Figure 16 illustrates an ideal wave diagram

of the GPC rotor, intended to produce reactive shaﬁ power utilizing curved blades.

29

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BO 8an :eumancO “:0 82.5 seam—5:50
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30

There have been reported several difﬁculties with the GPC design which resulted in
poor output power. Its performance suffered from excessive blade curvatures, lack of
control of reﬂected waves within the device required to make the wave system periodic
within one revolution, and the absence of any strong impulsive loading of the rotor from
inlet manifolds to produce shaft work. The latter was in contrast with the Pearson rotor
that relied heavily on impulsive loading of the rotor blades to achieve power output.
Furthermore, the GPC rotor had inadequate control on maintaining high performance for
off-design operating conditions. Although GPC developed a computer code to obviate
manual wave pattern design, accurate calculations were still tedious. Unfortunately Ford
ultimately withdrew its support from the wave rotor research before completion of tests,
mainly due to commitments to other engine development programs [129]. As a result,

GPC discontinued development of the wave engine in the early 19805.

Rolls-Royce (1965-1972)
In the mid 19605, Rolls-Royce (R) in the UK began numerical and experimental

wave rotor research [29]. BBC cooperated with R in the development of pressure
exchange wave rotors as topping spools in gas turbine applications [131], with Berchtold
of the ETH and Spalding of Imperial College serving as consultants [3]. Considerable
efforts were made to design a wave rotor as a topping stage for a small helicopter engine
(Allison Model 250) [130]. The BBC-RR engine utilized a reverse-ﬂow wave rotor
incorporated into a single turbine cycle. This was somewhat different from the cycle
suggested by Berchtold and Lutz [63] in BBC gas-turbine-topping investigations, which
employed a through-ﬂow wave rotor integrated with both low-pressure and high-pressure
turbines. BBC’s interests in wave rotors at that time were mostly related to development

of small gas turbines for passenger cars, beset by poor efﬁciencies at sizes of 100 kW and

31

smaller [21]. Similar to previous wave rotor efforts, rotor designs protracted manual
design methods. While the enhanced engine operated nearly as predicated, measured data
revealed low performance mainly due to leakage [29]. Other difﬁculties related to the
start-up and control are reported [10]. The program was abruptly canceled in 1972 amidst
severe company ﬁnancial difﬁculties [131]. As stated by Kentﬁeld [3], contemporaneous
rapid progress in turbomachinery technology may have disfavored high-risk projects,

both at RR and GE.

Mathematical Science Northwest Inc. (1978-1985)
In the late 19705 Mathematical Science Northwest Inc. (MSNW, later Spectra

Technology Inc., and now STI Optronics Inc.) investigated various applications of wave
rotors [21]. Under the sponsorship of DOE and DARPA, they considered a broad range
of stationary power systems such as magnetohydrodynamic cycles (MHD) [29],
combined cycles integrated with gasiﬁcation plants [132], pressurized ﬂuidized bed
(PFB) power systems [133], and also propulsion and transportation applications [109].
Furthermore, signiﬁcant numerical and experimental efforts were performed developing a
laboratory wave rotor known as the MSNW wave rotor [134-137], shown in Figure 17.
With diameter of 45 cm, it consists of 100 channels each with a 40 cm length. It is a four-
port wave rotor with two additional small ports provided to cancel pressure waves at
critical rotor locations providing more uniform port ﬂows and a higher transfer efﬁciency
[137]. The design rotor speed is reported as 1960 rpm and a pressure ratio of
approximately 2.5 was achieved. A measured wave rotor efﬁciency of 74% is reported
[138]. Besides successful tests using several conﬁgurations (clearance variations, port
sizes, etc.) and various operating conditions, experiments were designed to verify the

scaling laws for predicting the performance of larger machines [132].

32

 

Figure 17: The MSNW experimental set up, taken from Ref. [21]

The MSNW wave rotor was initially designed based on the method of characteristics
using only the Euler equations, but later a one-dimensional unsteady computer code (the
FLOW code) was used for optimizations and modifications of the MSNW design [109].
The modiﬁcations led to improvement in obtaining a very good agreement between the
numerical and experimental results in a wide range of operating conditions. Analytic
estimations involving the ideal wave patterns also supported the obtained results. The
FLOW code which was developed speciﬁcally for both pure pressure exchanger wave
rotor and wave engine analyses, uses the ﬂux-corrected transport algorithm solving Euler
equations accounting for heat transfer, viscosity, gradual port opening, and ﬂow leakage.
The sensitivity of wave rotor performance to tip speed, port placement and size, inlet and
outlet ﬂow conditions, channel geometry, number of channels, leakage, and heat transfer

was analyzed for both on-design and off-design conditions. For instance, it was

33

outlet ﬂow conditions, channel geometry, number of channels, leakage, and heat transfer
was analyzed for both on-design and off-design conditions. For instance, it was
concluded that heat transfer losses were negligible and leakage was recognized as a key
problem for efﬁcient wave rotor operation. Numerical work has been also reported for a
nine-port wave rotor concept to resolve the problem of nonuniform port ﬂows and poor
scavenging.

MSNW has used the knowledge obtained through their investigation to establish
preliminary wave rotor designs for a small turbofan engine generating 600 lb thrust at sea
level condition [109, 138]. A conceptual design of such engine integrated with a pressure
exchanger wave rotor is illustrated in Figure 18. Performance calculations for both on-
design and off-design ﬂight conditions using a cycle performance code and the FLOW
code simulation have predicted signiﬁcant performance improvements of such an
enhanced engine. No new material development for such combined engines was required,
proving the possibility of designing such engines by using available technology.

Despite all successful achievements and satisfying results, the wave rotor activity at
MSNW was discontinued in the mid 19803. No speciﬁc reasons for this cancellation are

reported.

34

 

 

 

 

 

 

5* « .- 55,334.; '

 

 

 

Conceptual design of a turbofan engine incorporating a wave rotor, taken from Ref. [138]

Figure 18

Naval Postgraduate School (1981-1986)
In 1981, the Ofﬁce of Naval Research (ONR) agreed to monitor a joint DARPA/ONR

program to evaluate the wave rotor concept and its potential application in propulsion
systems [127]. Following this decision, Turbopropulsion Laboratory (TPL) at Naval
Postgraduate School (NPS), directed by Shreeve, started an extensive numerical and
analytical wave rotor program. To support the accuracy of the computational results, the
wave rotor apparatus formerly used by Klapproth at GE was transferred to TPL and some
preliminary tests were carried out. It is reported that the rotor produced some shaft work
running at approximately 5000 to 6000 rpm [139]. No further experimental details are
reported.

For numerical simulations, two different approaches to the solution of the unsteady
Euler equations were examined in the overall program. First, Eidelman developed a two-
dimensional code based on the Godunov Method to analyze the ﬂow in wave rotor
channels [140-143]. Unlike contemporary one-dimensional approaches [144], the two-
dimensional code showed the effect of gradual opening of the channels. The main
conclusion of these studies is that if the channels are straight, the ﬂow remains nearly
one-dimensional, which in turn leads to minimal mixing losses caused by rotational ﬂow
in the channels [145]. However, when the channel of the wave rotor is curved, even an
instantaneous opening of the channel does not lead to the development of a one-
dimensional ﬂow pattern with small losses. Computation time of such a two-dimensional
code has been reported to be quite long. For faster computations, a one-dimensional, ﬁrst
order time-accurate code was introduced by Mathur based on the Random Choice
Method for solving the Euler equations [146, 147]. The unconditionally stable code,

called WRCOMP (wave rotor component), calculated the unsteady-ﬂow process inside

36

the wave rotor, inlet and outlet opening times and other useful design parameters required
for a preliminary design. The outputs from WRCOMP are used in a second program,
called ENGINE, for turbofan jet engine performance calculations [148-150]. The results
conﬁrmed the signiﬁcant performance improvement expected by integrating a wave rotor
into a turbofan engine. Work was planned toward incorporating the effects of friction and
heat transfer into WRCOMP and also including other engine conﬁgurations in the
ENGINE code. Some improvements to the WRCOMP code were later started [151, 152],
but further development was not pursued after terminating the wave rotor research around
1986.

It is also worth to point out that NPS sponsored the most comprehensive wave rotor
conference in 1985 where worldwide participation in this conference took place [153].

The conference reviewed much of the history to that point.

NASA Glenn Research Center (1990—Present)

Since the late 19805, a sustained research program at NASA Lewis (now Glenn)
Research Center (GRC), collaborating with the US Army Research Laboratory (ARL),
Rolls-Royce Allison has aimed to develop and demonstrate the beneﬁts of wave rotor
technology for future aircraft propulsion systems [10].

In 1993, using a thermodynamic approach to calculate the thermal efﬁciency and
speciﬁc power, Wilson and Paxson [154] published a feasibility study for topping jet
engines with wave rotors. Applied to the case of an aircraft ﬂying at Mach 0.8 , they have
shown that a wave-rotor-topped engine may gain l...2% in efﬁciency and 10...16% in
speciﬁc power compared to a simple jet engine with the same overall pressure ratio and
turbine inlet temperature. Additionally, Paxson developed a quasi-one-dimensional

gasdynamic model to calculate design geometry and off-design wave rotor performance

37

[155, 156]. The code uses an explicit, second order, Lax-Wendroff type TVD scheme
based on the method of Roe to solve the unsteady ﬂow ﬁeld in an axial passage for time-
varying inlet and outlet port conditions. It employs simpliﬁed models to account for
losses due to gradual passage opening and closing, viscous and heat transfer effects,
leakage, ﬂow incidence mismatch, and non-uniform port ﬂow ﬁeld mixing. In order to
verify wave rotor ﬂow predictions and to assess the effects of various loss mechanisms
[157, 158], a three-port wave-divider machine was constructed and tested [159—161] in a
new wave rotor laboratory facility at GRC. Concurrently, the non-ideal behavior and
losses due to multi-dimensional effects were studied by Welch [162-164] and Larosiliere
[165-167]. Welch has also established macroscopic and passage-averaged models to
estimate the performance enhancements of wave rotors [168, 169]. Based on
experimental data, Paxson further improved the one-dimensional model [157, 158, 170,
171] and used it to evaluate dynamic behavior, startup transients, and channel area
variation [172—175]. This model was then used as a preliminary design tool to evaluate
and optimize a four-port wave rotor cycle for gas turbine topping [176]. This through-
ﬂow cycle was chosen based on several perceived merits, including relatively uniform
rotor temperature, and the feasibility of integration with gas turbomachinery. As a result
of these studies, a new four-port wave rotor was designed and built [177] to test the
performance of this concept under scaled laboratory conditions. A photograph of NASA
four-port wave rotor is shown in Fig. 16. However, a study by Rolls-Royce Allison
discussed below indicated that thermal loads on the rotor and ducting predicted for the

NASA wave rotor cycle in real engine conditions may be difﬁcult to manage. In

38

response, Nalim and Paxson [178, 179] proposed an alterative cycle with a combustor

bypass signiﬁcantly lowering thermal loads.

 

Figure 19: F our-port wave rotor of NASA

Additional studies of the performance beneﬁts of wave rotor topped gas turbines have
been reported. In 1995, Welch et al. [180] predicted a 19.21% increase in speciﬁc
power and a 16...17% decrease in speciﬁc fuel consumption compared with the baseline
engines in performance calculations for small (300 to 500 kW) and intermediate (2000 to
3000 kW) wave-rotor-enhanced turboshaft engines. The same calculations for a wave-
rotor-enhanced large turbofan engine, equal in thrust to the baseline engine, have shown a
6...7% reduction in thrust speciﬁc fuel consumption. Welch has also studied the
possibility of curving the channels to create a wave turbine [181, 182].

In 1995, Nalim at NASA published a feasibility assessment of combustion in the
channels of a wave rotor, for use as a pressure-gain combustor [183]. Combustion

prediction capability was added to the wave rotor code by Nalim and Paxson [184],

39

enabling the exploration of wave cycles involving both detonation and deﬂagration
modes of combustion. For uniform mixtures, a single reaction progress variable is
utilized. Multiple species are represented for a variable fuel-air ratio in deﬂagration
modes. Mixing controlled reaction is combined with a simple eddy diffusivity model.
Other notable features that were incorporated are temperature kinetics factors and a
simple total-energy based ﬂammability limit [185]. The performance of detonative and
deﬂagrative cycles was studied by combined CFD and system simulation. It was
determined that deﬂagrative combustion with longitudinal fuel stratiﬁcation could be
accomplished over a reasonable time in wave rotors.

The current NASA wave rotor research has been mostly focused on experimental
tests with special attention to sealing technology [186-188], identiﬁed as a critical

challenge in high-pressure wave rotor design.

Rolls-Royce Allison (l990-Present)
In 1996, Snyder and Fish [189, 190] of Allison Engine Company evaluated the

Allison 250 turboshaﬁ engine as a potential platform for a wave rotor demonstration,
predicting an 18...20% increase in speciﬁc power and a 15.22% decrease in speciﬁc fuel
consumption. They used a detailed map of the wave rotor cycle performance
accomplished by Wilson and Paxson [10, 154, 176]. Allison (by now Rolls-Royce
Allison) has also studied transition duct designs for integration with turbomachiney [191,
192]. This was later followed by investigations of pulse detonation wave rotors in the
newly formed Allison Advanced Development Company (AADC). A novel four-port
device is proposed [193] for supersonic turbofan engines [194], and was investigated in
collaboration with Indiana University Purdue University Indianapolis (lUPUI) as

discussed later.

40

University of Florida (1992-1998)

Motivated by NASA wave rotor successes, Lear at the University of Florida initiated
analytical and numerical methods to investigate different conﬁgurations of wave rotors.
His team developed an unsteady two-dimensional numerical code using a direct boundary
value method for the Euler equations to analyze the ﬂow in wave rotors and their
adjoining ducts, treating the straight or curved channel walls as constraints imposed via a
body force term [195]. The code was later used to simulate the ﬂow ﬁeld of the three-port
NASA wave rotor. They also introduced a preliminary design method for selecting the
wave engine inﬂow and outﬂow blade angles. Furthermore, an analytical thermodynamic
description of wave rotors was developed [196], which predicted potential increase in
speciﬁc power of 69% and a 6.8% increase in thermal efﬁciency over a conventional gas
turbine topped by a wave engine. A parametric study of gradual opening effects on wave

rotor compression processes is reported, too [197].

ONERA in France (1995-1999)
Fatsis and Ribaud at the French National Aerospace Research Establishment

(ONERA) have investigated wave rotor enhancement of gas turbines in auxiliary power
units, turboshaft, turbojet, turbofan engines [198, 199], accounting for compression and
expansion efﬁciency, as well as mixing and pressure losses in the ducting. Their results
show the largest gains and efﬁciency for engines with a low compressor pressure ratio
and high turbine inlet temperature, such as turboshaﬁ engines and auxiliary power units.
These results are consistent with those obtained by NASA GRC [200]. They hve also
developed a one-dimensional numerical code based on an approximate Rieman solver
taking into account viscous, thermal, and leakage losses [198, 201], and applied it to

three-port, through-ﬂow, and reverse-ﬂow conﬁgurations.

41

0 Recent Academic Work
Besides ongoing research mainly at NASA, AADC, and ETH Zurich, a few

universities have been conducting wave rotor research. To the knowledge of the author,

the universities listed below are are active in this ﬁeld.

Purdue School of Engineering and Technology (1997-Present)
Recent research at Indiana University Purdue University Indianapolis (IUPUI) by

Nalim and coworkers has focused on internal combustion wave rotors, following the
initial work at NASA described above. Deﬂagrative combustion with longitudinal ﬁJel
stratiﬁcation has yielded a wave rotor geometry competitive with pressure-exchanger
designs using a separate combustor [185, 202]. Nalim has highlighted the importance of
thermal management of leakage ﬂows of rotor and end—wall temperatures, with
illustration of the impact of the hot ignition gas and the cold buffer zones on the end
walls. This is consistent with the major challenges revealed by the ABB experiment [73].
Radial stratiﬁcation [203] using a pre-combustion partition has been proposed to
introduce a relatively cooler buffer zone close to the leakage gaps, reducing hot gas or
fuel leakage to the rotor cavity. Figure 20 is a contour plot of the temperature contour
from a simulation of deﬂagrative combustion in a stoichiometric partition region
propagating into a leaner mixture in the main chamber. Above and below the partitions,
there is no fuel, and gas may leak out or in without danger of overheating or pre-
ingnition. These thermal management approaches are possible utilizing extensive cycle
design studies and analysis, and seek to alleviate the challenges previously recognized by
ABB and NASA. This technique also helps burn leaner mixtures, resulting in reduced
NOx emissions, similar to other pilot combustion or lean-bum techniques in conventional

engines [204]. For this approach, radial leakage ﬂows [205] and different combustion

42

models [206] have been studied in detail. These ideas have not yet been tested

experimentally.

 

 

   

Figure 20: Temperature distribution of partition exit ﬂow, taken from Ref. [204]

Detonative combustion cycles for propulsion engines have been also studied [207,
208]. Interest in detonative combustion initially focused on pulsed detonation engines
(PDE) has evolved to the consideration of the wave rotor as an effective implementation
of the concept [209], and a means of overcoming challenges to PDE concepts that
involved integration with conventional turbomachinery. In effect the wave rotor provides
automatic high-speed valving, nearly steady inﬂow and outﬂow, and the use of one or
few steady ignition devices for multiple tubes. However, detonative combustion is
fundamentally restricted to highly energetic mixtures and sufﬁciently large passage
widths, and generates strong pressure waves. This results in the outﬂow being highly
non-uniform in pressure, velocity, and possibly temperature. To better utilize the output
of a wave rotor PDE, it has been proposed to add an ejector element to the wave rotor
[210]. The rotary wave ejector admits bypass air after the detonation tubes to transfer
energy and momentum. Numerical simulations using a quasi-one-dimensional code,
modiﬁed to account for radial-type bypass ﬂows, have shown that the speciﬁc impulse at
static thrust conditions can be doubled, after accounting for ﬂow-turning and shock

losses, comparing with an equivalently loss-free PDE cycle. A sample wave diagram and

43

a schematic sketch are given in Figure 21, where the cold ejector gas ﬂow is clearly

distinguishable.

Temperature Pressure Fuel Fraction

    

2.5

 

0 0.5 1 D 0 5 ‘l
Posmon/Length Posrtion/Length Posttion/Lenglh

 

 

Figure 21: Rotary Wave Ejector Pulse Detonation Engine, taken from Ref. [210]

IUPUI has also investigated [211] the four-port detonation wave rotor proposed by
AADC [193], in which a recirculation duct allows air that is compressed by the shock of
a detonation wave to be reinjected with fuel. Air-buffer regions both between the fuel/air-
combusted gas interface and at the exit end plate are inherent in the cycle design,
allowing self-cooling of the walls. The inﬂow and outﬂow of this engine concept is
designed to be nearly uniform and acceptable to modern turbines, compared to

conventional rotary detonation cycles, as shown in Figure 22.

44

Transition duct

    
 
 
 
  
 
  
 

Rotor

Inlet endplate *0 tlll'llllt'le OI’

nozzle

Transition duct
from compressor
or inlet ‘

Exit and plate with
detonation initiation devices

.. High pressure buﬁer air
duets with fuel nozzles

Figure 22: Wave Rotor PDE the ‘CVC’ Engine, taken from Ref. [211]

A computational and experimental program is currently being conducted at IUPUI in
collaboration with AADC to investigate the combustion process and performance of a
wave rotor with detonative and near-detonative internal combustion [212]. A preliminary
design method based on a sequence of computational models has been developed to
design wave processes for testing in an experimental test rig.

University of Tokyo (2000-Present)

Nagashima et al. have developed one-dimensional [213] and two-dimensional [214]
CFD codes to simulate the ﬂow fields inside through-ﬂow four port wave rotors,
including the effects of passage-to-passage leakage. The codes have been validated with
experimental data obtained by a single-channel wave rotor experiment Single-cell
experiments can efﬁciently demonstrate the operation of actual wave rotor engines. The
test rig, shown in Figure 23, consists of a stationary single tube, and two rotating plates

connected to a shaft driven by an electric motor. This group has also explored the idea of

45

using wave rotors for ultra-micro gas turbines manufactured using microfabrication
technology [215].

  
    
 
 
    

Driving Belt

\ H Wﬁullhlll‘l‘w
i

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is ‘ \ ‘ I .II I
. Y 3)
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l l

  
  

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u" vi

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ll

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I
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will I“ .
, ., trill 1“

Figure 23: University of Tokyo single-channel test rig, taken from Ref. [214]

In conclusion, Figure 24 summarizes the history of the wave rotor research reviewed
here. The goal of this review was to report the continued interest in wave rotor

technology and its wide variety of applications. Some of the latest efforts were discussed

in more detail, inspiring further research and development on this topic

46

‘———EUROPE —'—-—>

U.S.

Japan

 

Germany
France

Poland

U.K.

Switzerland

General Electric
Comel Univ.
ll ITE Circuit Breaker

Tokyo Univ.
Sophia Univ.
Univ. of Wuppertal
ONERA
Warsaw Univ.
Cracow Tech. Univ.

Rolls-Royce
Ruston-Homsby

imperial College
Power Jets Ltd

R-R Allison
Univ. of Florida

Swissauto

MSU
IUPUI

NASA
NPS
MSNW
GPC

ETH
BBC

1940

1950

1960

1 970

 

1980 1990 2000

Figure 24: Historical perspective of wave rotor technology. Red: gas turbine application, ( Il'ttfl
1 111g, Blue. refrigeration cycle,

I(

111-11111 51111.1

l

11

x
I . .
9 1 7 .

\i

. ‘5‘“. ”‘2‘”

::ll1‘1.

'2i‘i‘! 3‘: “I“

47

1.,z,‘

1.1. Black: general applications

CHAPTER 3: WAVE ROTOR THEORY

3.1 Energy Exchange with Waves

To understand the principle of wave rotor operation, it is more convenient ﬁrst to
analyze wave processes that occur in a single stationary channel. This is very similar to
analyzing gasdynamic processes in shock tubes. Figure 25 schematically shows a shock
tube and its corresponding wave diagram (time- space diagram). The wave diagram
describes the wave action by tracing the trajectories of the waves and gas interfaces. It is
a plot of the wave motion as a graph of time verses space.

When the diaphragm which has separated two gases at different pressures i5 ruptured,
two types of gasdynamic waves are initiated at the diaphragm location. A shock wave
propagates into the low-pressure gas (driven section) and an expansion wave spreads out
into the high-pressure gas (driver section). The shock wave increases the pressure and
temperature of the driven section rapidly and induces a mass motion behind itself. The
interface between the driver and driven gases, called the contact surface, moves with an
induced velocity in the same direction as the incident shock. The expansion wave
decreases the pressure and temperature of the driver section smoothly. Therefore, across
the contact surface the pressure and velocity are preserved while the temperature and
entropy change discontinuously.

Now, consider a closed tube which is initially sealed on both ends and contains a gas
at rest, as shown in Figure 26. The rapid opening of one end of the tube to a higher or
lower pressure media can initiate waves in the tube similar to the rupturing of the

diaphragm in a shock tube, as it will be described in the following. Two cases are

48

introduced here utilizing the principle of shock tube operation to increase or decrease the

pressure and temperature of the gas inside the tube.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Before
diaphragm High-pressure ﬂow at rest Low-pressure ﬂow at rest
opening
After
diaphragm 4
opening
it
at?
St ,1 4°
'1— s 08“
0° 1 :5
0 ,' {g}
l é§
$0
11 Distance

5’.

:1

3

8

a.

 

 

 

 

 

 

Figure 25: F low in a shock tube after the diaphragm is ruptured

Left end wall Right end wall

 

Gas with zero velocity

 

 

 

Pressure

‘

Distance

 

Figure 26: Closed tube containing stationary gas

49

0 Generation of a Shock Wave

Assume it is aimed to increase the pressure and temperature of the gas trapped in the
tube. This can be achieved by the rapid opening of one tube end to a medium with a
higher pressure. When the tube opens instantaneously to a high—pressure gas, a shock
wave forms and propagates into the channel as shown in Figure 27. This simulates the

compression process of the driven section in the shock tube by suddenly removing the

diaphragm.

Time

 

High-pressure
gas

 

  

After left
end opens

 

 

Gas at rest

Before leﬂ
end opens

‘

 

 

Pressure

 

 

 

 

7

Distance

Figure 27: Left end opening to gas at higher pressure

0 Generation of an Expansion Wave

To decrease the pressure and temperature of the trapped gas, one end of the tube is
opened to a gas with a lower pressure than the pressure in the tube, as depicted in Figure
28. Rapid opening of one end of the tube results in the formation of a centered expansion

fan which propagates into the tube, inducing a gas ﬂow out of the tube. Such a

50

scavenging ﬂow process imitates the expansion process of the driver section in a shock

tube after sudden removal of the diaphragm.

 

 

 

 

 

 

O
.E
i-
Low-pressure After left
835 end opens
G ‘ Before left
as at rest end opens
I Distance
0 1 '
B 1
U)
8
I-
o.

‘
V

Figure 28: Leﬁ end opening to gas at lower pressure

In the operation of wave rotors, shock and expansion waves may also be generated in
other ways. As an example, consider a case where a gas is ﬂowing through a tube at a
steady rate as shown in Figure 29. Suddenly closing one end of the tube can generate
either a shock wave or an expansion wave. Figure 30 illustrated how an expansion wave
can be generated by rapidly closing the leﬁ end of the tube. Because the velocity of the
gas in contact with the closed end must be zero, an expansion wave propagates into the
moving gas, bringing it to rest. Also, the pressure and temperature inside the tube drop
depending on the strength of the expansion wave, reducing the ﬂow velocity to zero.

Using the same concept, suddenly closing the right end of the tube results in the

generation of a shock wave. This is shown in Figure 31. The shock wave propagates in

the tube from right to left increasing the pressure and temperature inside the tube.

 

 

 

 

 

ll
0.)
I-
:3
U)
V)
d)
I-
On
Distance

Figure 29: Opened tube containing ﬂowing gas

6}
Dan .
s10
I]
Iv

E a.,,
[.—

After left

end closes

Before left _> Flowing gas —-—> —>
end closes a

 

1 Distance
1
l

I

 

Pressure

_
V

Figure 30: Left end closing to generate an expansion wave

52

 

 

 

 

 

 

 

ll
.é’ ”0* a/
l—
8600‘. W
ape
\
After left
.’ -—> —"
__'l \I end clOSCS
_ Before left
_, _. Flowmg gas —’ I end closes
ll Distance
2
3
§
9..

 

 

Figure 31: Right end closing to generate a shock wave

3.2 Charging Process in Wave Rotors

The above gasdynamic phenomena can be used in an array of rotating shock tubes
(wave rotors) to increase the pressure and temperature of a ﬂow by expanding another
ﬂow which has a higher pressure and temperature. Several conﬁgurations can be
introduced for this purpose, but only two of them are brieﬂy discussed here.

Figure 32 shows a developed (unwrapped) view of the charging process of a wave
rotor. This ﬁgure shows a sequence of events occurring within the channels moving in
the upward direction. Such a representation is called a wave diagram. The wave process
occurring inside the wave rotor channels is customarily illustrated by the wave diagram,
where the circular motion of the rotor channels is replaced by a straight translatory
motion. It describes the rotor internal operation by tracing the trajectories of the waves
and gas interfaces. The wave diagram is very useful for visualizing the wave process

occurring inside the channels and also for determining wave rotor design parameters, i.e.,

53

port opening and closing times and their locations. The utility of the wave diagram is
analogous to that of a velocity diagram for a conventional turbine or compressor. In
Figure 32, the ducts are set at the correct angle, so that in the rotor reference frame the
flow can enter and leave the rotor aligned with its axis. The shock wave trajectory is

shown by a dashed line. The trajectory of the interface line is indicated by a dotted line.

t

 

Figure 32: Wave diagram of charging process with two primary shock waves

The process begins in the bottom part of Figure 32, where the channel is closed at
both ends and contains low-temperature and low-pressure ﬂow (e.g., air). When the inlet
port opens, the rotor channels are exposed to a high-pressure and high-temperature gas
arrived from a heat source (e.g., the combustion chamber). This hot driver gas penetrates

the channel. Because its pressure is higher than the pressure in the channel, a shock wave

54

is triggered starting from the lower comer of the inlet port. The shock wave runs through
the channel and causes an abrupt rise of local pressure inside the channel. The shock
wave speed is higher than the local speed of sound. Within the relevant design space the
ﬂow speeds are everywhere subsonic. Therefore, the air/gas interface follows the shock
wave with an induced velocity less than the speed of sound. However, behind the shock
wave the compressed air has the same local pressure and speed as the inlet driver gas. As
the shock wave reaches the end of the channel, the outlet port opens through which the
compressed air is then pushed out. At this moment, both the gas and the compressed air
column in the channel have the same local pressure and move uniformly with the same
induced velocity toward the outlet port. By closing the outlet port, a second primary
shock wave originates from the upper outlet edge and propagates from the right to the
left. It reduces the velocity of the compressed air to zero to satisfy the zero-slip boundary
condition at the end wall. The process is ﬁnished when the gas inlet port closes. The
moment is timed with the arrival of the secondary shock wave to the upper comer of the
inlet port.

While the charging process described above is theoretically feasible, it is rarely used
in wave rotor designs. Even though the second primary shock wave compresses the air
further, the doubly compressed air is not delivered to the outlet port. This may lead to
overheating of the channels. Hence, the second primary shock wave has been only
generated to stop the compressed air. Therefore, another type of charging processes
utilizing an expansion wave is often preferred. This conﬁguration is shown in Figure 33
where the ﬂow is stopped by closing the gas inlet port. This way, an expansion wave

originates from the upper comer of the gas inlet port and propagates toward the right end

55

of the channel. Expansion waves (fans) are depicted by thin solid lines. Since this
expansion wave induces a ﬂow velocity equal but opposite to that of the gas flow, the gas
ﬂow behind this expansion wave is stopped and its local pressure is decreased. Both local
and total pressures behind the expansion wave are still considerably higher than the
pressure of the fresh air in the beginning of the process. The speed of the expansion wave
with reference to the channel wall is equal to the gas speed plus the local speed of sound.
As a result, the expansion wave is traveling faster than the air/gas interface, overtaking
the air/gas interface before the latter reaches the right end of the channel. The expansion
wave reaches the end of the channel at the moment when the upper edge of the outlet port
closes the channel. At this moment, the trapped flow within the channel consists of a
large part of the hot gas and a plug of compressed air preventing the hot gas to contact the
end wall. Thus, during the high-pressure process, penetration of the hot gas into the outlet

ﬂow is avoided.

 

Figure 33: Wave diagram of charging process using single shock wave and
expansion wave

56

3.3 Principles of Four-Port Wave Rotor Operation

A variety of wave rotor conﬁgurations have been developed for different applications.
The number and azimuthal location of the wave rotor ports along with heat addition
schemes distinguish them for different purposes. As described in the previous chapter,
four-port conﬁgurations have been mainly used as superchargers for internal combustion
engines. Three- port wave rotors have been employed in pressure dividers and pressure
equalizers in which the pressures of different ﬂuids are increased or reduced. Two-port,
four-port, ﬁve-port, and nine-port wave rotors have been extensively investigated for gas
turbine engine topping applications. As an application of current interest, a four-port
wave rotor integrated into a gas turbine cycle is brieﬂy discussed below to illustrate wave

rotor operation and options.

3.3.1 Through-F low versus Reverse-Flow Wave Rotors

In the described wave rotor illustrated in Figure 6, both gas and air inlet ports are
located on one side of the rotor while the outlet ports are located on the other side of the
rotor. This conﬁguration is known as the through-ﬂow (TF) wave rotor in the literature.
Alternatively, another type of wave rotor has been designed where the fresh air enters and
exits at the same end of the rotor (air casing) while the burned gas enters and exits the
rotor at the other end (gas casing). This conﬁguration is called reverse-ﬂow (RF) wave
rotor as shown in Figure 34. These two conﬁgurations may provide identical topping and

overall performance enhancement, but they differ substantially in their internal processes.

57

Combustion Chamber

Wave Rotor

 

 

 

Compressor Turbine

Figure 34: Schematic of a gas turbine topped by a reverse—ﬂow four-port wave rotor

In a TF four-port wave rotor, both hot gas and relatively cold air traverse the full
length of the rotor, keeping the wall at a relatively uniform intermediate temperature.
This self-cooling feature of TF wave rotors has prompted interest in them for gas turbine
engine topping applications where gas temperatures are high. The RF conﬁguration does
not inherently result in such a self-cooled rotor. The cold air never reaches the other end
of the rotor as seen from Figure 34. As a result, the air side of the rotor is relatively cool
while the gas side of the rotor is relatively hot. To achieve a better self-cooled RF design,
a mirrored reverse—ﬂow cycle design of the RF conﬁguration can be constructed, which
orients the cycle alternately right and left on the rotor [189]. This approach introduces
symmetry and assures that both sides of the rotor are washed by the relatively cold fresh
air. Unfortunately, it also poses severe ducting challenges. However, for small gas
turbines the gas temperatures are oﬁen lower than those of large scale gas turbines.
Therefore, the RF approach seems to be the viable choice for microturbines. Also, it has
been claimed that the RF cycle provides better separation of air and gas in the rotor
channels [2]. Due to this separation of air and gas regions in the channels, the analysis of

the ﬂuid ﬂow inside RF wave rotor channels is easier. Knowing about all these facts, RF

58

conﬁgurations have been mostly used in the relatively low-temperature application of car
engine supercharging although such conﬁguration for gas turbines has been also
investigated [131, 189, 198]. The General Electric Company has obtained experimental

data on a gas turbine engine enhanced by a RF wave rotor [126].

3.3.2 How Does it Work Inside?
Figure 35 represents NASA wave diagrams [168] for through-ﬂow (left) and reverse-

ﬂow (right) four-port wave rotors for one cycle operation of the rotor. The journey of a
channel of the wave rotor is periodic. The top of each wave diagram is considered to be
looped around and joined to the bottom of the diagram. This requirement presents a
fundamental challenge in the simulation and design of wave rotors. A successful
prediction of the wave rotor implies that the state of the working ﬂuid in the channel at
the end of the cycle must be the same as that postulated at the beginning of the cycle.

To show how a four-port wave rotor works, the events occurring in one cycle are now
described. For the RF conﬁguration, the process begins in the bottom part of the right
wave diagram where the ﬂow within the channel consists of a large part of the hot gas
and a buffer layer separated by a contact surface. For the TF wave rotors (leﬁ), the gas
ﬁlls the whole channel. As the right end of the channel opens to the relatively low-
pressure outlet port, an expansion fan originates from the leading edge of the outlet port
and propagates into the channel, discharging only the gas to the turbine. The expansion
fan reﬂects off the left wall and reduces the total pressure and temperature in the channel
further. This draws fresh air provided by the compressor into the channel when the air
inlet port opens. When the reﬂected expansion fan reaches the outlet port, it slows the
outﬂow and reﬂects back as compression waves, while the outlet port closes and halts the

ﬂow inside the channel. The compression waves form a single shock wave as they travel

59

toward the inlet port. As the shock wave reaches the upper corner of the inlet port, it

closes gradually. At this moment, the channel ﬂuid is at rest relative to the rotor.

60

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61

The above sequence of events is called the low-pressure part of the cycle (scavenging
process). Its purpose is to deliver a high-pressure gas into the turbine, partially purge the
rotor channels, and ingest fresh air received from the compressor. In the high-pressure
part of the cycle (charging process) that follows, the rotor channels are exposed to the
burned gas arrived from the combustion chamber. The wave phenomena here are same as
those described in Figure 33. The only difference here is the existence of a reﬂected
shock wave in the NASA wave diagram. In fact, as the shock wave reaches the end of the
channel and the outlet port opens a reﬂected shock wave originates at the lower outlet
edge, propagating back into the channel. The reﬂected shock wave compensates for the
combustor pressure loss. On the other hand, for an ideal combustion chamber (no
pressure loss), the reﬂected shock wave would not appear. This has been shown in Figure
77 which is discussed later. The double-compressed ﬂow behind the reﬂected shock
wave leaves the wave rotor toward the combustion chamber. In the RF conﬁguration the
discharged ﬂow into the burner is pure air, while in the TF conﬁguration usually both air
and once-burned gas are delivered into the burner. Detailed ﬂuid ﬂow investigations have
suggested that approximately 30 to 50% of burned gas is recirculated to the combustion
chamber in the TF conﬁguration [180]. Again, a favorite case is considered when the
closure of the gas inlet port is timed with the anival of the reﬂected shock wave. At this
moment, an expansion fan originates from the upper corner of the inlet port and
propagates toward the other end of the channel which eventually brings the channel ﬂow
to rest. When the expansion fan reaches the end of the channel, the outlet port closes and

the ﬂow in the rotor channels stops. At this point, the ﬂow with zero velocity is at nearly

62

the peak pressure and temperature of the cycle. It is now ready to be discharged into the

turbine during the low-pressure process.

3.4 Gasdynamic Equations

To ﬁnd ﬂow properties inside wave rotor channels, it is required to predict the
gasdynamic processes occurring inside the channels using boundary conditions provided
by a thermodynamic cycle analysis. Proper design of a wave rotor requires a reasonable
solution for the internal ﬂow ﬁeld.

To derive necessary equations for a gasdynamic analysis, several assumptions are
made throughout this work. The aspect ratio of the wave rotor channels is assumed to be
large enough, so the ﬂow can be treated as one-dimensional. The ﬂow within the rotor is
considered frictionless and adiabatic. However, the wave rotor efﬁciency is used to

account for dissipation losses inside the channels. The gases are treated as ideal gases.

3.4.1 Moving N ormal Shock Wave Relations

Consider a moving normal shock wave which propagates with an absolute
velocity ws from right to left into a channel which contains a ﬂow moving with velocity
u; from left to right, as shown in the top part of Figure 36. The ﬂow velocity decreases
from u, to u; due to the mass motion induced by the shock. The term unsteady reference
frame is attributed to this case. To ﬁnd the ﬂow properties in region 2 (compressed ﬂow
after the shock), the governing normal shock equations which are only valid for a
stationary frame of reference can be used.

In a reference frame moving with the shock wave, the shock appears stationary and
the downstream and upstream ﬂow velocities are u, + ws and u; + ws , respectively as

shown in the bottom part of Figure 36.

63

 

Movmg shock wave ——> _.__>

u, + wS E 112+ Ws

 

Stationary shock wave —> —p

 

Figure 36: Transformation from a moving shock wave to a stationary shock wave

Applied to a control volume considered around the stationary shock wave, continuity,

momentum, and energy equations become:

,0,(W5 +ul):p2 (W5 +u2) (2)
pl‘l‘p/(WS‘H’IY =p2+p2(w5+u_,)2 (3)

2 2
h,+(ws+u') :h2+(ws+u2)

4
2 2 ()

Assuming a calorically perfect ﬂuid, solving the above equations leads to the relations
given below, which give the local temperature ratio (T 2 /T 1) and local density ratio (pg/p1)

across the shock wave as a function of the local pressure ratio [75 =p2 /p, :

 

 

T y+i+ 5
-4=Hsy’ (a
T: I y+1
+— s
y-I
“Ii-(17.)
3i: 7’1 (o
pl XII—{+175
7-1

64

As seen, local thermodynamic properties across a moving normal shock are physically
independent of the ﬂow ﬁeld velocities. Furthermore, the induced velocity can be

expressed as:

l _2_7_

+ 1

uimluce—shuck = 5:11. (”S _ I) y 7 '— I
I] _

V S + 7 +1

(7)

 

where 0, represent the speed of sound in the undisturbed region 1. Therefore, the ﬂow

velocity after the moving shock in region 2 becomes:

u 2 = “I — u induce—shock (8)

The moving shock velocity can be obtained as [216]:

 

1
w, = a, J%(HS - 1)+ 1 (9)

However, the velocity of the shock relative to the gas in the region 1 becomes ws - u 1 or:

 

wS =a, %l(175—1)+1—u, (10)

Furthermore, the shock Mach number is introduced as:

 

ML: ﬁl<ﬂs_1)+1 (1])

M =
S a, 2}!

Finally, the entropy change across the shock is obtained by:

A; = CPln? —C,,1np-’

l l

 

 

(12)

It is worthwhile to note that the total pressure increases across a moving shock while

it decreases across a stationary shock wave. Also, the total enthalpy is not constant across

65

a moving shock wave. It increases for a moving shock wave, but it remains unchanged

for a stationary shock wave.

3.4.2 Expansion Wave Relations

Due to the isentropic nature of expansion waves, the method of characteristics can be
used to ﬁnd the ﬂow properties across an expansion wave. This method is a very general
and powerful technique for analyzing compressible ﬂow. For the speciﬁc problem
considered in this study, only one-dimensional version of this method is used. Because
the method of characteristics is a well-know theory, only the application of this method is
used here and details of this theory can be found in the literature [216].

Consider any given point (x, , t,) in the x-t plane as shown in Figure 37. It is possible

to ﬁnd a speciﬁc path through point (x,, 1,) along which the following equation holds:

 

a =c0nstant=J+ (13)
y—l

ll+

where u is the ﬂow velocity and a is the sound speed. Such a path is called a C+
characteristic line (right-running wave) in the x-t plane. This speciﬁc path is chosen so
that its slope becomes dt /dx=1/(u + a).

In addition, a C_ characteristic line can be found through the point (x, , 11) in Figure 37,
where the slope of the C__ characteristic (left-running wave) is dt/dx=1/(u - a) and along

which the following equation obtains:

a =c0nstant=J_ (14)
7—1

 

"—

The constant values of J+ and J_ are known as the Riemann invariants.

66

dx/dt=u—a dx/dt=u+a

 

‘ZJ
§
3»
e
g
.9
C.)
e
(U
a?
‘A

0

 

 

 

‘
V

xl x

Figure 37: Illustration of the characteristic lines in the x-t plane

With the above results, it is possible to solve for the ﬂow ﬁeld in a one-dimensional
expansion wave. As an example, consider again the generation of an expansion wave by
suddenly closing a tube, as previously described in Figure 30. This ﬁgure is modiﬁed in
Figure 38 to include a C_ characteristic line as well. Note that here the C + characteristic
lines are physically the paths of the expansion waves in the x-t plane.

11 15‘,

11.30812)”

  
 
  

6’6 (C

a

11'” es)

 

Figure 38: Left end closing to generate an expansion wave

67

Knowing the ﬂow properties in front of the head of the expansion wave (region 1),
solution for the ﬂow behind the tail of the expansion wave (region 2) can be obtained as
now described.

Because J_ is constant through the C_ characteristic line, Eq. (14) applied to both

regions 1 and 2 gives:

2 2 ,
u,— a, =u — a‘ =J_ (15)

7—1 2 7—1

 

 

However, the wall condition on the leﬁ side implies that u2=0. Therefore, the sound

speed ratio between region 1 and 2 can be written as:

 

 

3.2. -_- 1-2/:1 u_, (16)
a, 2 a,
or,
2
T _
_Z_=I:1_7_Iu_’] (17)
T, 2 0,
Because the expansion ﬂow is isentropic, therefore:
_7_ ~’_7
-1 _ —-l
p? =[—T—~’—Jr =[1——7 I 59—]7 (18)
PI T1 2 a]
and,
_’_ .3.
._l _ -/
pg: 11.], =1__}’_{31_,_7 (19)
pl TI 2 at

The expansion wave induces a mass motion in the direction opposite to its propagation,

but equal in value to the ﬂow in region 1. Therefore, it is also possible to ﬁnd a relation

68

for the induced velocity based on the pressure ratio across the expansion wave. This can

be easily found by using Eq. (18) as:

 

 

2 . 2r
“induce—exp umiun = a, I - (L) (20)
7‘1 F!

It can be shown that this relation obtained for the induced velocity is independent of
ﬂow velocities of the medium in which the expansion wave moves. Here, the velocity of
the ﬂow in region 2 was zero, however, Eq. (20) always calculates the change of ﬂow
velocity due to the wave expansion. This conclusion is also valid for the induced velocity
generated by a moving shock wave described before, as shown in Eq. (7). Both induced
velocities are only functions of the pressure ratio across the waves, the speed of sound,
and the 7 value of the downstream ﬂow.

Finally, it is important to mention that the head of the expansion wave moves with the
velocity u, + a, to the right because it is a C + characteristic line. The same reasoning
shows that the tail of the expansion wave moves at the slower velocity 0 + a; in the same

direction. Hence, the expansion wave spreads out as it propagates down the tube.

69

CHAPTER 4: THERMODYNAMIC ANALYSIS

4.1 Microturbines

A growing market for distributed power generation and propulsion of small vehicles
has motivated a strong interest in design of small gas turbine systems in the range of 30-
300 kW. Known as microturbines, they are now widely used in the US for distributed
power generation, shaving peak loads, and providing backup power for critical needs.
They propel small commercial aircraft, unmanned air vehicles (UAV), and terrestrial
vehicles. Microturbines are often the preferred alternative to IC engines, due to their
higher power density and robustness. They present several advantageous features such as
compact size, simple operability, ease of installation, low maintenance, fuel ﬂexibility,
and low NOx emissions. Furthermore, due to the recent electric-power crises and
environmental concerns, a strong interest in the research, development, and application of
microturbines has been stimulated

Despite their attractive features, compared with larger gas turbines, microturbines
suffer from lower thermal efﬁciency and their relative output power, due to their limited
cycle pressure ratio and peak cycle temperature. For many applications improvement of
their performance is desirable to enhance advantages over competing technologies.

To achieve such improvements, current efforts are mainly focused on utilizing heat
recovery devices and developing new high-strength high-temperature materials for
turbine blades [217]. Geometries of microturbines make blade cooling very difﬁcult.
Hence, their lifetimes using typical materials used for larger gas turbines are shorter
[218]. Therefore, there is signiﬁcant research toward developing advanced metallic alloys

and ceramics for high-therrnal-resistance turbine wheels used in microturbines[219, 220].

70

Currently, recuperators play a key role in performance enhancement of microturbines.
For example, experimental and theoretical research has shown that microturbines with
pressure ratios of 3 to 5 without recuperation systems achieve only about 15 to 20%
efﬁciency [221, 222]. Utilizing conventional recuperators based on the use of existing
materials can improve the thermal efﬁciency of microturbines up to 30% [222-225]. An
excellent example of a commercial microturbine with a recuperator is the Capstone 30
kW unit, with an efﬁciency of 26% using natural gas fuel [226].

Despite the attractive feature of the recuperator concept, a recuperator introduces
pressure losses reducing the output power and it adds about 25 to 30% to the overall
engine manufacturing cost, which is a challenge for commercialization of microturbines
[227-229]. The current trend of the microturbine market is to reduce the investment cost
per kW. Therefore, alternative devices need to be considered to achieve higher
performance at lower component costs. Topping a microturbine with a wave rotor device
is an appropriate solution. Wave rotor investigations [154, 164, 198-200] have shown a
signiﬁcant potential for performance gain in smaller gas turbines, where the compressor
pressure ratios are typically lower than those of larger machines.

The objective of the present chapter is a comprehensive and systematic performance
analysis of two actual microturbines known as the C-30 and C-60 engines made by
Capstone Turbine Corporation which are topped with a four-port wave rotor in various
wave-rotor-topping cycles. The challenges and advantages associated with the different
implementation cases are discussed. While the performance evaluation of several gas
turbine engines has been studied extensively [154, 198-, 200], to the knowledge of the

author, there exits no comprehensive work investigating the potential beneﬁts of various

71

implementation cases of wave rotor topping cycles for small gas turbines. The presented
results have been obtained using basic thermodynamic equations along with the wave-
rotor characteristic equation previously validated using computational tools [154]. The
model can be employed to predict the performance improvement of various wave-rotor-
t0pping cycles without the need for knowing the details of the complex ﬂuid mechanics

within the wave rotor. 2

4.2 Gas Turbine without Recuperation
4.2.1 Implementation Cases

There are several possibilities to top a gas turbine with a wave rotor. Considering
possible design restrictions and preferences, ﬁve different advantageous implementation
cases for a wave rotor into a given baseline engine can be introduced as follows:

Case A: same compressor, same turbine inlet temperature

Case B: same overall pressure ratio, same turbine inlet temperature
Case C: same combustor

Case D: same turbine

Case E: same compressor, same combustion end temperature

These ﬁve cases have been described in detail in the following. According to the state
numbering introduced in Figure 4 (or Figure 34), Figure 39 visualizes all ﬁve cases in
schematic T-s diagrams. Path O-lb-4b-5b represents the baseline cycle and O-li-Zi-3i-4i-5i
(i=A, B, C, D, E) indicates the wave-rotor~topped cycles, where the subscripts indicate
the case. One of the ﬁve cases might be preferable for a practical design. However

intermediate design cases are possible.

 

2 Parts of materials presented in this chapter have been accepted for publication in 2005 ASME Journal of
Engineering for Gas Turbines and Power.

72

o-lb-4b-5b baseline engine
o'lrzr3r4i-5i topped engine
i=A, B, ..., E

,3A

  
    

_ Tiara»- _ _

Temperature

 

 

Entropy

Figure 39 : Schematic T-s diagrams for a baseline cycle and ﬁve wave-rotor-topped
cycles

Case A: In Case A the pressure ratio of the compressor is kept unchanged, so the
physical compressor of the baseline engine can also be used for the wave-rotor-enhanced
engine provided the mass ﬂow is kept approximately the same. The pressure in the
combustion chamber of the enhanced engine is increased by the compression ratio of the
wave rotor. This may require modiﬁcations to the structure of the combustion chamber
and to the fuel injection system. The heat addition in the combustor is the same as for the
baseline engine, but it takes place after the energy exchange in the wave rotor, hence the
heat addition starts at a higher temperature. Thus, the combustion end temperature is even
higher than that of the baseline engine, possibly requiring additionally a thermal
enhancement of the combustor structure. The turbine of the topped engine might need to

be adapted to efﬁciently utilize the higher pressure ratio. The turbine inlet temperature,

73

however, is the same as that of the baseline engine. As will be shown later, this
implementation case provides the highest thermal efﬁciency and speciﬁc work and the
lowest value of SF C .

Case B: In Case B the overall pressure ratio for the wave-rotor-enhanced engine is
kept equal to that of the baseline engine, so that the combustor works under the same
pressure. However, for the wave-rotor-topped engine, the heat addition in the combustor
and the combustion end temperature are greater than those of the baseline engine. This
may require some adaptation of the combustor, especially in the outlet region. The
turbine and compressor work with lower pressure ratios, reducing the design challenges.
Thus, both may be adapted advantageously. This might reduce the cost of the compressor
and turbine due to reduction of stages in multi-stage types (mostly axial), or due to
reduction of the tip diameter in radial types (mostly single-stage). With a smaller tip
diameter the wheels can be manufactured more economically over a shorter time from
cheaper materials with less strength and on smaller machines. Besides an attractive
performance enhancement, this case additionally provides the highest turbine outlet
temperature of all ﬁve cases investigated. The temperature of the leaving exhaust gas is
much higher than that of the baseline engine. Therefore, this case is attractive for an
external heat recovery application or for internal recuperation that can enhance the
performance further.

Case C: Case C assumes that it is desirable for the wave-rotor-enhanced engine to use
the unmodiﬁed combustor of the baseline engine. So the overall pressure ratio and
combustor inlet and outlet temperatures for the wave-rotor—enhanced engine are kept

equal to those of the baseline engine. The heat addition in the combustor is consequently

74

the same.3 The implementation of the wave rotor considerably reduces the pressure ratio
of the turbine and compressor. The compressor pressure ratio is as low as in Case B, and
the turbine pressure ratio and turbine inlet temperature are even lower than those in Case
B. Thus, the turbine and compressor could be made from less thermally resistant material.
Compared to the baseline engine, they also could be smaller and hence less expensive, as
discussed in Case B. This might be the main implementation reason because
unfortunately, but not surprisingly, the unambitious combustor constrains the
performance enhancement. It is nearly negligible for the smaller C-3O engine and even
negative for the C—60 engine.

Case D: Case D employs the same physical turbine as the baseline engine. Due to the
wave-rotor—topping, the compressor needs to produce a lower pressure ratio than that of
the baseline engine. This allows for a smaller and less expensive compressor as discussed
for Cases B and C. The pressure in the combustion chamber and the combustion end
temperature are higher than those of the baseline engine, but lower than those of Case A.
Hence, less effort might be required to adapt the structure and fuel injection of the
combustion chamber. As a result of the lower pressure ratio in the compressor, hence
lower compressor discharge temperature, the heat addition in the combustor has to be
more than that for the baseline engine to utilize the same allowed turbine inlet
temperature. This case gives the second highest performance increase for both baseline
engines.

Case E: Case E is similar to Case A but the combustion end temperature (the cycle

peak temperature) is restricted to the turbine inlet temperature of the baseline engine in

 

3 The wave rotor compression efﬁciency is greater than the compressor efﬁciency. Therefore, the combustor inlet
temperature is in fact negligibly smaller and hence the heat addition is negligibly greater than that in the baseline

engine.

75

order to avoid additional thermal requirements on the combustor design. The overall
pressure ratio is the same as in Case A because this case employs the same physical
compressor as for the baseline engine. Thus, the overall pressure ratio is greater than that
of the baseline engine, by the wave rotor pressure ratio. The heat addition in the
combustor is less than that for the baseline engine because to the wave rotor compression
work is added to the ﬂuid before combustion. The turbine in the topped cycle works with
a slightly greater pressure ratio than the turbine of the baseline engine, but the turbine
inlet temperature is less than that for the baseline engine. In fact, it is the lowest of all
cases investigated. This may give the option to produce the turbine wheel at a lower cost

out of less thermally resistant material.

4.2.2 Thermodynamic Calculations

To evaluate the performance enhancement of topping gas turbines with wave rotors, a
computer program based on a thermodynamic model has been developed to determine the
thermodynamic properties of the gases in different states of the cycles. The results are
used to calculate the theoretical performance (expressed by thermal efﬁciency ,7] , net
speciﬁc work w , and speciﬁc fuel consumption SF C) and the actual T-s diagrams of both
wave-rotor-topped and baseline engines. The methodology is similar to that introduced
by Wilson and Paxson [154] with some modiﬁcations. The component performance
parameters for the baseline engines are based on information provided by the

manufacturer and are listed in Table l.

76

Table 1: Baseline engine data, assuming T 0 =300 K, Cp,,,—,=l.005 kJ/kgK, Cpg..s=l.l48
kJ/kgK, yair=l.4 , yga, =l.33

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Baseline engine C-30 C-60

turbine inlet temperature Tm, (11151566? [1,: $272576? I];
466.5 K 505.4 K

compressor outlet temperature Tm, :3 80° F) (450. F)
compressor pressure ratio p, 1M0 3.6 4.8
compressor isentropic efﬁciency rzc 79.6% 82.6%
turbine isentropic efﬁciency Ur 84% 85%
compressor polytgpic efﬁciency mpg 82.9% 85.9%
turbine polytropic efﬁciem ”N 81.7% 82.3%
combustor pressure ratio [Lo—”d, 0.98 0.98

 

For each engine it is assumed that the compressor inlet condition is known and is the
same for both baseline and wave-rotor-enhanced engines. Considering the same
“aerodynamic quality” of the wheels, the polytropic efﬁciencies are kept the same for the
enhanced and baseline engine, for the compressor and turbine respectively. Incomplete
combustion of the fuel is reﬂected by a combustor efﬁciency of 98% (17g = 0.98). No
pressure losses in the intake air ﬁlter, exhaust silencer and additional piping, or heat
losses or mechanical losses are taken into account. Such losses reduce the predicted
performance. The gases are treated as ideal gases. For both air and burned gas constant
values of speciﬁc heat coefﬁcients (Cpair = 1.005 kJ/kgK, Cpgas = 1.148 kJ/kgK) and
speciﬁc heat ratios (nu-r: 1.4 , ygas= 1.33) are considered. This assumption simpliﬁes the
performance calculations signiﬁcantly without affecting the qualitative nature of the
results. Air enters the compressor at 300 K.

In the following it is assessed how the wave-rotor-topping enhances the performance

of C-30 and C-60 baseline gas turbine engines.

77

0 Path 0-1: Compressor
With the given compressor inlet temperature (T 0=T ,0), the compressor outlet total

temperature and pressure are calculated from the adiabatic relations:

 

T run-I
mam—1p. -1)
77C
:21 = EL = ”C (22)
p10 p0

where the compressor isentropic efﬁciency (tic) relates the compressor pressure ratio

(17 C) to the compressor polytropic efﬁciency (npc) through:

run-1
17 —1
77c = C r (23)

yr"! —1

17C Ymr’IPC‘ _1

 

 

 

For Cases A and E, the compressor pressure ratio is equal to that of the baseline engine
(e.g. for the 030 engine 17C = 3.6 and for the C-60 engine 17C = 4.8). However, for
Cases B and C its value is calculated by dividing the baseline compressor pressure ratio
with the wave rotor compression ratio (PRW = p,2/p,,). The wave rotor compression ratio
plays an important role in the performance analysis of wave-rotor—tapping engines by
signiﬁcantly altering the engine performance. A higher value of a PRW leads to a higher
engine performance. F atsis and Ribaud have varied PRW from 1.4 to 3 in their
calculations [198, 199]. PRW values lower than 1.8 have considered unacceptable due to
the poor compression processes inside the rotor. The authors believe that a PRW value
higher than 3 is unrealistic. This is conﬁrmed by detailed calculations in a four-port wave
rotor by Welch [164]. Wilson and Paxson have used PRW = 1.8 from GE data [154]. They

have used the term “advanced” for their wave rotor working with PRW = 3.6. Therefore, a

78

wave rotor compression ratio of 1.8 appears to be conceivable for the envisioned
application and is chosen for the following discussion. In this work all performance plots
are shown for various wave rotor pressure ratios, indicating its effect on cycle
performance enhancement. For Case E, the value of Hg is calculated in a way that keeps
the pressure at the turbine inlet equal for both baseline and topped engines. This
calculation can be performed by inreversly solving of Eq. (21) through (41). Finally, the

compressor speciﬁc work is obtained as:

C . T! r..,,-1
WC = Cp... (T., —T..) = fume —1) (24)

77C

 

0 Path 1-2: Compression in the Wave Rotor
The ﬂow properties after the wave rotor compression process are obtained from the

adiabatic relations similarly to those of the compressor:

 

T
T,_, = T,, + ——"— PRW - 1 (25)
”WC
ELL-zfiﬂiszw 'Hc (26)
p10 pr] p10

where 27% is the wave rotor compression efﬁciency. Later my; , which is deﬁned as the
wave rotor expansion efﬁciency, will be used. Consistently with previous wave rotor
investigations [154, 164, 198, 199], the wave rotor compression and expansion
efﬁciencies are assumed as mm = my; =0.83 in this study. In fact, the efﬁciencies are not
constant values. However, the range of variations have been almost determined. In
previous wave rotor studies it was not possible to evaluate mm and my}; separately.
Instead, their product was determined. Moritz found mm - :ng = 0.6 in experiments at

Rolls-Royce [130]. Reference [153] has reported a range of 0.77 < mm = 17mg < 0.86.

79

Wilson and Paxson assumed mrc = ”WE = 0.83 based on previous work on wave rotors
[154]. Recently, Fatsis and Ribaud have taken into account the effect of wave rotor
efﬁciencies on the performance of several different types of gas turbine engines topped
by a wave rotor [198, 199]. They have carried out their calculations with efﬁciencies in a
range between 0.8 < mm = my}; < 0.86. For a small size turbojet engine considered in
their study, they have shown that the gains of speciﬁc power and speciﬁc fuel
consumption are sensitive to the values of wave rotor efﬁciencies. These authors have
emphasized that it is important to optimize efﬁciencies for the compression and
expansion processes.

0 Path 2-3: Combustion Chamber

The value of ﬁrel/air ratio (f =mf/mair) can be obtained by applying the energy (ﬁrst law)

equation to the combustion chamber:

”Q f hPR =(1 + f)Cpgas 7:3 _ Cpair T12 (27)
where hpR = 43000 kJ/kg is the heating value used for all calculations here. This equation

gives:

__ Cpgas 7‘13 —Cpair 22
”Q hPR — Cpgas 7:3

 

(28)

Alternatively, f can be expressed based on the turbine total inlet temperature (T ,4) and the
compressor total exit temperature (T ,1). For this purpose, the wave rotor compression and

expansion speciﬁc work (per unit mass of air ﬂow) are deﬁned respectively as follows:
WWC = Cpair (Tu _ TH) (29)

wwg = (1 + f ) C19,... (Tu - 7.4) (30)

80

Here, it is considered that m, = n'z, = m and m, = m, = mg, + #2,. Other cycles exist in

which the mass flow rates m, and m, are not the same, and, correspondingly the mass
ﬂow rates m, and n'z, may not be equal either. These cycles are not considered here.
Using Eqs. (29) and (30) , Eq. (27) can be expressed as:

779 f hm = (1+f) Cpg... T... + WW5 - wwc -Cp...-, TH (31)
Because the net output work of the wave rotor is zero (wwc = WW5), solving for f leads
to:

_ Cpgas 7:4 —Cpair 7;]
”Q hPR -Cpgas 7:4

 

f (32)

For Cases C and E, T ,3 is equal to the baseline inlet turbine temperature, therefore Eq.
(28) is used to calculate f. For Cases A, B, and D, where T,., is a known value and is
equal to the baseline inlet turbine temperature, f can be obtained from Eq. (32). The

relation between T,3 and T ,4 can be found by equating Eqs. (29) and (30):

 

 

 

T mr‘", C .
7:3 :734 +l: tl [PRWyr‘m —1]] pair (33)
77wc (1 + f) C10,...

Finally, the total pressure after the combustion chamber is obtained:

& : 111$ = 17
P10 Pr) on

PR,,, .17, (34)

comb

0 Path 3-4: Expansion in the Wave Rotor
To obtain the turbine inlet total pressure (19,4), it is convenient to express the wave rotor

compression work and the expansion work in terms of pressure ratios:

ch=m Cp..,(T» -T,) = ”” PRW'ZT—z
”we

' C .T run-l
m pair tl( ) (35)

81

-I. 'yl‘l‘

P0
:(muir +m /)Cpgas (773—14): (mair +m /)Cpgus ”WE 7:3 I—£_—_u—] (36)

7mm

17 PR

comb

Where P0=p,4 /p,, is the pressure gain ratio across the wave rotor. Equating the

compression work to the expansion work leads to:

7 gm ‘ I " ng

 

17 PR

comb

C . T nr’
—’-’”—”-—"(PRW —1]= (I+f)Cp... n... T” I— {—Ig—ﬂ <37)
”WC

Substituting 7),; from Eq. (33) into Eq. (37) and some algebra gives:

 

 

 

 

r ‘ r,.a,/rg...-l
A—1———B
P0: 17m, PR 1- '7” "I” > (38)
1+A——B
( ”WC ,
where
C .
: pal, (39)
(1+f)Cpg...
B_Tz___1_ [PRW (r... -l)/r.,,-, _1] (40)
Tu

Equation (38) is a modiﬁed version of the “wave-rotor characteristic” equation
introduced in the literature [154]. This equation represents the performance of the wave
rotor. It predicts the pressure gain ratio across the wave rotor as a function of the
temperature ratio across it (7)., /T,1). For ”WC = my); = 0.83 , f = 0, 1760”,), = 100%, and
PRW= 1.8, Figure 40 clearly indicates monotonically increasing of the pressure gain as a
function of the temperature ratio across the wave rotor. For a ﬁxed turbine inlet
temperature (e.g., 7",.) = constant), using a smaller compressor with a low pressure ratio

leads to a higher pressure gain across the wave rotor. Similarly, for a ﬁxed compressor

82

 

pressure ratio, increasing the turbine inlet temperature results in a greater pressure gain
across the wave rotor. The most signiﬁcant performance gain has been found for engines
with low compressor pressure ratios and high turbine inlet temperatures [154, 164, 198,
199]

Equation (38) can be also used again to investigate the inﬂuence of the wave rotor
compression ratio on the pressure gain ratio across the wave rotor. This is shown in
Figure 41 for ”WC: 17m; = 0.83 ,f= 0, 17mm), = 100%, and 7)., /T,, = 2.2. The plot clearly
indicates that the pressure gain ratio across the wave rotor is an increasing function of the
wave rotor compression ratio, while the highest relative gains are obtained at low
compressor ratios. Therefore, higher values for PRW results in a higher net absolute

performance gain.

 

 

 

 

 

 

 

 

 

 

 

1.2
as: .-
11.: f
g 1.15 I
a
m
s
a r-
o
1.1
g I
3 .
2
E ,.
O
s 1.05
m
a
>
a L-
3
b.5411.8112.11I2.4112.71 T3

Wave Rotor Temperature Ratio (run ,1)

Figure 40: Pressure gain ratio across the wave rotor versus temperature ratio across it

83

 

 

 

 

 

 

 

 

 

 

 

1.2
0::- /
:3 /
‘35 115 //
N
m _
C
a r
o /
2 1.1
3
.. 1 /
m
2 /
a, _
O
*5 1.05
K
0
>
N
3
11 1.5 2 2.5 T3 ‘ 13.5

 

Wave Rotor Compression Ratio (PRW)

Figure 41: Pressure gain ratio across the wave rotor versus wave rotor compression ratio

By using Eq. (3 8), the turbine inlet total pressure is obtained as:

_p_IL=£1_I_)L/_=P0.17C , (41)
P10 Pu pro

For Cases C and E, where T ,4 is an unknown value, the turbine inlet temperature is
obtained by using Eq. (33), and T,3 is equal to the baseline inlet turbine temperature.

0 Path 4-5: Turbine

The turbine speciﬁc work (per unit mass of air ﬂow) can be calculated knowing the

pressure ratio across the turbine:

has -1

w, = (1+ f)Cp.... (TM —T..) = (1+ m. 6p... 7,. I—(ﬂ) <42)

:4

 

assuming the turbine expands the hot gas leaving the wave rotor to atmospheric pressure
(p,5=p0). The isentropic turbine efﬁciency (m) can be obtained from the polytropic

turbine efﬁciency mar:

84

( p0 )(Ym.—l)’7rr ’ll’g... _ I

’77- : pl4 (43)

(&)(y£m_l)/7rm _1

 

p14

Therefore, the total temperature of the gas leaving the turbine can be calculated as:

W
T =T — T 44
’5 ’4 (I+f)Cpgm ( )

 

After calculating the thermodynamic properties of all states in the cycle, it is possible
to calculate the engine performance parameters. The net speciﬁc output work produced
by the engine can be calculated by subtracting the turbine speciﬁc work from the

compressor work:

71:1“ —I

 

1 Cpuir 7; 7017‘]
W=77MWT_WC=77M777CpgusTr4 1_(££) 7W _———_0(17C7‘”" _1) (45)

:4 77C

 

Where pm is the turbine shaft mechanical efﬁciency. In this study an ideal transmission
case (77M =100%) is considered.
With the amount of speciﬁc heat addition through the combustion process being

deﬁned as q=f. hpR , the thermal efﬁciency can be written as:

rgui —l

 

 

 

1 Cpuir 7; 7"” -l
m, m Cp... T... I—(Ei) ’e — ———0(17c — I)
W p14 77 C
’7 = — = ' (46)
q f hPR
Finally, the speciﬁc fuel consumption (SF C) is calculated from:
SF C = —f— (47)

W

85

4.2.3 Predicted Performance Results

Cases A and B are the most common cases discussed in the literature, therefore, they
have been discussed here in more detail.

Case A: Figure 42 shows the actual T-s diagrams for the baseline engines C-30 and
C-60 as well as for the topped engines, simulated with a wave rotor pressure ratio of 1.8
(PRW = 1.8). The overall pressure ratio of the enhanced engines is 1.8x3.6=6.48 and
l.8x4.8=8.64, respectively. The T-s diagrams qualitatively show that the topped engine
has a higher thermal efﬁciency compared to the baseline engine. This is because the
turbine has a higher speciﬁc work output, while the consumption of speciﬁc work by the
compressor and speciﬁc heat addition to the cycle remain the same as for the baseline
engine. Details of calculations for creating such plots are described step by step in

Appendix B.

86

1500 - 045-4195,, baseline engine

 

 

 

 

 

 

0-1 A-Z {3 .1'4 A-S A topped engine
3A
1200 '
Ttnrbine 4A 4
.— _ U _________________ Q h _. _.
_. \
E \
a 900 ' \
‘5 5..
a 5*
3
o.
E 600 '
E-
lb=lA
300 ' .
0 G30 Engine
0 A l L L j
1.5 1.8 2.1 2.4 2.7 3.0
Entropy lid/kg K]
1500 , 0-lb-4b-5b baseline engine
0'1A'2A'3A-4A'5A topped engine 3A
b Tiarbine _______________ _
1200 '
E
i‘.’ 900 '
:1
3
z
o.
g 600 P
l- ]b
300 ' .
0 C-60 Engine
0 l l l l I
1.5 1.8 2.1 2.4 2.7 3.0

Entropy lkJ/kg K]

Figure 42: T-s diagrams for the C-30 and C-60 wave-rotor—topped engines for
Case A implementation

87

Figure 43 illustrates the increase of cycle thermal efﬁciency (dash dot) and speciﬁc
work (dashed), and the decrease of speciﬁc fuel consumption (solid) with increasing
wave rotor pressure ratio PRW for both the C-30 and C-60 topped engines. The plot
visualizes how the effect develops from the baseline engine with PRW =1 until PRW =2
which might be a practical limit for the investigated application. However, if the wave
rotor pressure ratio increases beyond this limit, the trend already shows that the rate of
increase of the effect diminishes while technical problems may increase. With a
conceivable wave rotor pressure ratio of 1.8, the thermal efﬁciency of the baseline cycle
increases from 14.9% to 20.0% for the C-30 engine and from 19.4% to 24.2% for the C-
60 engine. Simultaneously, the speciﬁc work increases from 128 kJ/kg to 171 kJ/kg for
the C-30 engine and from 184 kJ/kg to 231 kJ/kg for the C-60 engine. The speciﬁc fuel
consumption (SF C) of the C-30 engine decreases from 0.156 kg/kN.s to 0.116 kg/kN.s
and it reduces from 0.120 kg/kN.s to 0.095 kg/kN.s for the C-60 engine.

A better picture of the performance improvement is obtained by calculating the
relative increases of thermal efﬁciency, speciﬁc work, and the relative decrease of SF C
as shown in Figure 44. For Case A, the relative increases of thermal efﬁciency and
speciﬁc work (dash dot) are precisely the same as it is obvious ﬁ'om Eq. (46) where the
heat addition q=f. hPR is the same for both topped and baseline engines. For a wave rotor
pressure ratio of 1.8, Figure 44 indicates an attractive relative performance improvement
in thermal efﬁciency and speciﬁc work of about 33.8% and a 25.2% reduction in SF C
(solid) for the C-30 engine. The C-60 engine shows a 25.1% enhancement in thermal

efﬁciency and speciﬁc work and a 20.1% reduction in SF C.

88

Overall Pressure Ratio

   

 

— T“=1116.5 K
7; 0.28 5 0.28 “95033 240
z“ ; =0.82
i‘ 0.24 - 0.24 =0.98 .. 220
3’ ~ A
V ' 3’
I: 0.2 - 0.2 200 \
.9 - 1
ii I 5 :‘5
g 0.16 7 g; 0.16 180 1;
In ‘9 g
g . a:
o 0.12 - m 0.12 160 51;;
3 ' 3
E — a.
u 0.08 — 0.08 140 w
9.: _
g _
n. 0.04 - 0.04 120
‘0 :

L

0 — 0 100
Wave Rotor Compression Ratio (PRw)
Overall Pressure Ratio

3 Tu=1227.6 K
13 0.28 _- 11,5006
2' Z =0.82
1‘ 0.24 — =0.9s
m r A
i . g.
g 0.2 - \
a : e 3

._ E
g 0.16 L 2 .g
in _ 2 3
s . v: ..
o 0.12 - w E
_ h o
0 .
e - a
o 0.08 5 u,
SE .
g -
o. 0.04 - ._
"’ : C-60

o L

 

 

Wave Rotor Compression Ratio (PRw)

Figure 43: Thermal efﬁciency, speciﬁc work, and SF C for the wave-rotor—topped engines
versus the wave rotor pressure ratio and overall pressure ratio, Case A consideration

89

Overall Pressure Ratio

Tu=1116.5 K
npc=0.83
=0.82

=0.98

C-30

 

Relative Increase of Efﬁciency and Specific Work in “/5
Relative Change of Specific Fuel Consumption in “/0

Wave Rotor Compression Ratio (PRw)

Overall Pressure Ratio

1',=1227.5 K
$50.86
=0.82
=0.9e

e96

C-60

 

Relative Increase of Efficiency and Specific Work in %
Relative Change of Specific Fuel Consumption in %

Wave Rotor Compression Ratio (PRw)
Figure 44: Relative values of thermal efﬁciency, speciﬁc work, and SF C for the wave-

rotor-topped engines versus the wave rotor pressure ratio and overall pressure ratio,
Case A consideration

90

Case B: As described before, Case B considers another way to implement a wave
rotor beneﬁcially. While keeping the overall pressure ratio of the topped engine equal to
that of the baseline engine, the compressor pressure ratio is reduced in the wave-rotor-
enhanced engine. Lowering the compressor pressure ratio usually leads to a higher
isentropic compressor efﬁciency (for comparable aerodynamic impeller quality, here
simulated by using the same polytropic compressor efﬁciency), less the mass of the
compressor, and probably lower manufacturing costs.

Figure 45 depicts the actual T-s diagrams of the baseline engines and the wave-rotor-
topped engines for Case B. Now, the overall pressure ratio is ﬁxed at 3.6 and 4.8 for the
C-30 and the C-60 engines, respectively. It is evident that the compressor work of the
topped engine is less than that of the baseline engine. However, the turbine work is less
too, but the heat addition for the topped engine is greater than that of the baseline engine.
So, without calculating the thermal efﬁciency and speciﬁc work it is problematical to
determine whether the topped engine has a higher performance than the baseline engine.
Clearly shown in Figure 45 is that the turbine outlet temperature is considerably higher
than that of the baseline engine (T 53> T 51,), making this case attractive for heat recovery

applications due to the available additional thermal energy from the exhaust gas.

91

1500

1200

900

600

Temperature [K]

300

1500

1200

900

600

Temperature [K]

300

. 0"0'40'5b baseline engine

 

 

 

 

 

 

0-1 A-Z A-3 (4 {5.1 topped engine
E Tllrbjne ________
h
0 G30 Engine
1.5 1.8 2.1 2.4 2.7 3.0
Entropy [Id/kg K]
. 04545-55 baseline engine
0-13-23-33-43-53 topped engine 3
e
L 1.1-dine ___________ . _______ . _
p
In 53
10
o C-60 Engine
1.5 1.8 2.1 2.4 2.7 3.0

Entropy [kJ/kg K]

Figure 45: T-s diagrams for the C-30 and C-60 wave-rotor-topped engines for

Case B implementation

92

Values of the thermal efﬁciency, speciﬁc work, SFC and as well as their relative
increase of the wave-rotor-topped C-30 engine can be obtained from the plots in Figure
46 and Figure 47, respectively. Similar to Case A, both thermal efﬁciency and speciﬁc
work are monotonically increasing functions of the wave rotor pressure ratio PRW while
SF C is decreased. Now, the relative increase of thermal efﬁciency (dash dot in Figure 47)
is considerably less than the relative increase of the speciﬁc work (dashed), because the
heat addition is greater in the wave-rotor-topped cycle. Similar results are obtained for
the C-60 engine, not shown here.

For Case B, the results show that with a wave rotor pressure ratio of 1.8 the thermal
efﬁciency of the C-30 engine increases from 14.9% to 15.8%, and from 24.8% to 27.3%
for the C-60 engine. Similarly, the speciﬁc work increases from 128 kJ/kg to 150 kJ/kg
for the C-30 engine, and from 116 kJ/kg to 137 kJ/kg for the C-60 engine. Finally, SF C
of the C-30 engine decreases from 0.156 kg/kN.s to 0.147 kg/kN.s and from 0.094
kg/kN.s to 0.085 kg/kN.s for the C-60 engine.

For the same PRW = 1.8, wave-rotor-topping of the C-30 engine gives a relative
increase of 5.9% and 17.1% for the thermal efficiency and speciﬁc work, respectively,
and 5.6% reduction in SF C. For the C-60 engine, the performance improvement is less
with 9.7% and 18.0% increase of thermal efﬁciency and speciﬁc work, respectively, and

8.7% reduction in SF C .

93

Compressor Pressure Ratio
3.6 3.0 2.57 2.25 2.0 1.8

   

T"=1116.5K

1; 0.28 :' 0.28 ”#:0‘83
2' : np,=0.82
i 0.24— 0.24 ” =9”
en 2 A
i » 3’
c 0.2 - 0.2 \
.9 ’ '1
a ’ a 5
g 0.16- 130.16 180 g
2 ~ ,3 3
O 0
o 0.12— m 0.12 160 5
a Z 8
if - o.
o 0.03 _- 0.08 140 w
E .
8 _
o. 0.04 - 0.04 120
"’ C C-30

0 '- 0 100

 

Wave Rotor Compression Ratio (PRw)

Figure 46: Thermal efﬁciency, speciﬁc work, and SF C for the wave-rotor-topped C-30
engine versus the wave rotor pressure ratio and compressor pressure ratio,
Case B consideration

Compressor Pressure Ratio

 

 

.\° 3.6 3.0 2.57 2.25 2.0

.s 0-

C _

.9 .\°
3 ' ..\° .5
E -6- E 15
2 - 5 ;°
° 5 u
‘1’. 1 1,; s
.2 '12" m a
a - “6 2
.3 _ 3‘, o
a. ” iii
0, ~18- e a
“5 ~ E 8
0 k a; 5
2’ E o
w -24— 5» 2
.c 0, i3
0 - [z 3
E - n:
N

a -30-

1!

Wave Rotor Compression Ratio (PRW)

Figure 47: Relative values of thermal efﬁciency, speciﬁc work, and SF C for the wave-
rotor-topped C-3O engine versus the wave rotor pressure ratio and
compressor pressure ratio, Case B consideration

94

0 Comparison of Case A with Case B

The performance enhancement of Case A and Case B can be compared visually by
using the plots in Figure 48 and Figure 49 for both C-30 and C-60 engines, respectively.
For Case B the results are shown with dashed lines and the corresponding compressor
ratio is shown in the upper scale in orange. The corresponding overall pressure ratio for
Case A is shown in purple and the results are shown with solid lines. Case A clearly

shows a more beneﬁcial performance enhancement than Case B.

0 Cases C, D, and E

The actual T-s diagrams of the baseline engines and the wave-rotor-topped engines
for Cases C, D, and E are shown in Figure 50 to Figure 52, respectively. More detailed
documentation of these cases are not presented here. The reader is referred to Ref. [230].
Instead, numerical values of the predicted performance enhancement of all ﬁve
investigated cases with a wave rotor pressure ratio of 1.8 are summarized in Table 2. In
this table, 177 represents the turbine pressure ratio. Subscript “gain” indicates the relative
increase of thermal efﬁciency and speciﬁc work and decrease of SF C. Table 2 shows that
Case A gives the highest performance increase for both baseline engines. Aﬁer Case A,
Case D gives the highest overall performance for the C-30 engine as for the C-60 engine.
However, Case B provides the second highest thermal efﬁciency and the lowest SF C for
the C-60 engine.

Figure 53 and Figure 54 show maps of the relevant design space for Cases A, B, and
D for each engine. The only ﬁxed parameters are turbine inlet temperature, the polytropic
efﬁciencies of the compressor and turbine corresponding to the respective baseline
engine, and the combustion chamber pressure loss as indicated in the upper right corner

legend of each map. Performance maps valid for Cases C and E of the C-30 and the C-60

95

engines are shown in Figure 55 and Figure 56 which have lower turbine inlet
temperatures than that indicated in Figure 53 and Figure 54. Instead, Cases C and B have
the same combustion end temperature as the baseline engine, as indicated in the upper

right hand corner of these maps.

96

.___ Panm 3.60 3.00 2.57 2.25 2.00 1.80
Pale 3.60 4.32 5.04 5.76 4.68 7.20

 

   

: T“=1116.5K
A 0.28 7 0.28 ”30.33 240
"3 1 =0.82
Z .
i 0.24 — 0.24 ‘ 220
m _
“ 1 a
E 0.2 — 0.2 200 f
.2 ' ﬂ
‘6. I 3 i‘,
_ E

g 0.16_ 2 0.16 180 g
m . ,2 g
g ' t u
o 0.12— No.12 160 E
a : g
:1 _ D.
'3 0.08- 0.08 140 w
E I
0 -
a 0 04 - 0 04 120
"’ : C-30

0 L 0 100

 

Wave Rotor Compression Ratio (PRW)

Figure 48: Thermal efﬁciency, speciﬁc work, and SF C for the wave-rotor—topped C—30
engine versus the wave rotor pressure ratio and overall pressure ratio for Case A (solid)
and versus wave rotor pressure and compressor pressure ratio for Case B (dashed)

.__— PNIPl0 4.80 4.00 3.43 3.00 2.67 2.40
PaIPm 4.80 5.76 6.72 7.68 8.64 9.60

 

 

I Tu=1227.6K

0.28 — 0.28 ,1 =0“ 240
if : ”=0 32
5 ‘ =0 98
\ 0.24 _- 0.24 ' 220
g : Pl"? ’5
c 0.2 - 0.2 200 f
.2 ' 'i
a : a 5‘—
g 0.16 r E 0.16 180 1;
In ‘2 g
g H t u
o 0.12— LU0.12 160 E
3 - 3
if i a
u 0.08- 0.08 140 «n
1.: .
.5 C
3 0 04 f o 04 120
"’ i C-60

0 ‘- 0 100

 

Wave Rotor Compression Ratio (PRW)

Figure 49: Thermal efﬁciency, speciﬁc work, and SF C for the wave-rotor-topped C-60
engine versus the wave rotor pressure ratio and overall pressure ratio for Case A (solid)
and versus wave rotor pressure and compressor pressure ratio for Case B (dashed)

97

1500 . 0‘lb'45‘58 baseline engine

 

 

 

 

 

 

o‘lc'zc'3c'4c'5c topped engine
1200
Z
3 900
h
5
ti
5:
o.
g 600
5..
300 .
0 C-30 Engine
0 l I J l I
1.5 1.8 2.1 2.4 2.7 3.0
Entropy [lei/kg K]
1500 . 0‘18'45‘58 baseline engine
0--lC-2C-3C-4C-5C topped engine
_ Tim!“ _________________________
1200 '
i
2 900 '
:1
E".
8.
g 600 -
[—
300 ' .
0 C-60 Engine
0 I I J L I
1.5 1.8 2.1 2.4 2.7 3.0

Entropy [kJ/kg K]

Figure 50: T-s diagrams for the C-30 and C-60 wave-rotor-topped engines for
Case C implementation

98

 

 

 

 

 

 

1500 i 0‘10'48'55 baseline engine
0-10-20-30-49-5.) topped engine

1200
2
2 900
5
E
8.
g 600
1-

300 .

0 C-30 Engine
0 I I I I I
1.5 1.8 2.1 2.4 2.7 3.0
Entropy [Id/kg K]
1500 . 0-1,,-4,,-5b baseline engine
0'10’20‘30'40'50 topped engine 3:)
-Tivrlnne ____________________

1200 r
2
2 900 "
3
3
0
o.
,E, 600 -
[-

300 ' .

0 C-60 Engine
0 L I I I I
1.5 1.8 2.1 2.4 2.7 3.0

Entropy [id/kg K]

Figure 51: T-s diagrams for the C-30 and C-60 wave-rotor-topped engines for
Case D implementation

99

 

 

 

 

1500 . 0-1b-4b-5b baseline engine
071E72E73E74E75E topped engine
1200 :Tlnrbine . 3E 4b
7777777777777777.9477/3777
. . E I \
_ 1’
a z: 1
,, 900 - ,’ '- \
I- I ._ 5
g ’z . 5
g V’ ” 5E
:- 2 ”
‘5 600 - :1 ’..
1- . ., a ’
Iii-lip“
I
300 - ’ .
0 C-30 Engine
0 J I I I I
1.5 1.8 2.1 2.4 2.7 3.0
Entropy [Id/kg K]
1500 r0'15'4b'50 baseline engine
mix-2,73,34,75E topped engine
_T1urbine_. _ _ __ , it 2: _
1200 - ' '7 7 7"7'7'7 7 7 7 7 :211‘747; 7 7'7
9. E! \
If. \
E. 1’ i‘\ “
900 ' ’1‘”, I '1‘.
a .r” ’ \
a .1 ,I s,
E 600 ' g ”
[7' lb=lE 6’ I
I
.l
I
300 ' .
0 C-60 Engine
0 I I I I I
1.5 1.8 2.1 2.4 2.7 3.0

Entropy [Id/kg K]

Figure 52: T-s diagrams for the C-30 and C-60 wave-rotor-topped engines for

Case E implementation

100

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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101

Rcomp=3.6

 

    
 
 

 

   

 

1:1 C-30 Engine Tu=1118.5 x 240
' n,c=o.33
:5 Info-32
2' mangoes 220
X
5’ -
E 200 g
‘u‘: . 3
X
180 g
' 180 a?
‘ 8
- 8
, ‘ 14o
' 120
- ' 1oo

 

 

1 2 3 4 5 6 7 8 9101112131415

Compressor Pressure Ratio

Figure 53: Performance map for wave—rotor-topping of the C-30 engine, Cases A, B,
and D

Rcomp=4.8
C-60 Engine
‘ \7 _ ...,

 

7.512275 K

_ 11 "$.36
'1. 11,155032

~ [Imam

   

,. '—$: 4- "

 

220

.l.

  
  

 

src (kg/kN.s)

 

Speciﬁc Work (kJ/kg)

No Gain
Region

 
 

120

 
 

100

 

1 2 3 4 5 6 7 8 9101112131415
Compressor Pressure Ratio

Figure 54: Performance map for wave-rotor—topping of the 060 engine, Cases A, B,
and D

102

Rcomp=3.6

   
   

 

 

 

     

| C-30 En ine Tu=1116'5 K
I g 11,5033 24°
‘5? No Gain Info-32
2' I Region “co-r5033 220
i‘ |
g’ A
3 l 200 g
“a“: 2
180 f
O
3
0
E
O
8
U)

 

Compressor Pressure Ratio

Figure 55: Performance map for wave-rotor—topping of the C-30 engine,
Cases C and E

Rcomp=4.8

C-60 En ine T =1227.6 K
g n:c=o.85 240

 

 

 

 

 

 

 

src (kglkN.s)

Speciﬁc Work (kJ/kg)

 

 

 

12 3 4 5 6 7 8 9101112131415

Compressor Pressure Ratio

Figure 56: Performance map for wave—rotor-topping of the C—60 engine,
Cases C and E

103

The maps allow predicting the performance of the wave-rotor-enhanced engine in
terms of thermal efﬁciency (green), speciﬁc work (blue), and SF C (red) for any
combination of the compressor pressure ratio (abscissa) and the wave rotor pressure ratio
PRW (parameter labeled in black). In these maps, the multiplication of compressor
pressure ratio p,,/po and wave rotor pressure ratio PRW determines the overall cycle
pressure ratio [7,/pg (orange). The locus of optimum compressor pressure ratio points (for
highest thermal efﬁciency and speciﬁc work at each achievable wave rotor pressure ratio)
are connected by black solid lines. The optima for SF C are found at the same
combination of the compressor pressure ratio and PRW as the optima of the thermal
efﬁciency.

Such maps are not only very useful to explore the possible enhancement of already
existing baseline engines, but they also serve well for selecting a design point or region
for designing a new wave-rotor-topped engine. In all plots, the performance points of the
baseline engine (PRW=1) and the wave-rotor-enhanced engines of all cases with a wave
rotor pressure ratio of PRW=1.8 can be found. For instance in Figure 53 and Figure 54,
starting from the performance point of the baseline engine, the performance values for
Case A are found by moving vertically upwards (e.g. along the dashed line for constant
compressor pressure ratio p” /po) until the corresponding performance curve of the
expected wave rotor pressure ratio is crossed. Case B is found by moving along a line of
constant overall pressure ratio pg /po (orange).

The results indicate that for every compressor pressure ratio in each design space
shown here, the performance of the topped engine is always higher than that of the

corresponding baseline engine with the same compressor pressure ratio (Case A

104

 

consideration). The increase of PRW always increases the performance. However, for
higher compressor pressure ratios the beneﬁt of using a wave rotor progressively
diminishes. In fact, for compressor pressure ratios greater than around 11, almost no
beneﬁt can be obtained for the C-30 engine. An identical statement applies to the 060
engine for compressor pressure ratios above around 15. The beneﬁt is clearly the greatest
for lower compressor pressure ratios. This suggests that the wave-rotor-topping for
microturbines with low compressor pressure ratios can produce the greatest relative
beneﬁt. Moreover, as expected and known for baseline engines (PRW =1), it is also true
for wave-rotor—topped engines that the compressor pressure ratio for the maximum
speciﬁc work is always less than that of the maximum thermal cycle efﬁciency.
However, with increasing wave rotor pressure ratio, the optima come closer together,
while moving towards lower compressor pressure ratio. This can be viewed as an
additional advantage for applying wave rotors to small gas turbines with low compressor
pressure ratios. So as the plots show, adding a wave rotor with a 1.8 pressure ratio to C-
30 or C-60 baseline engines with a compressor pressure ratio p,1/po= 3.6 or 1),, /pg= 4.8
respectively, already brings the design point into the optimum range for highest speciﬁc

work and nearly half way closer to the optimum for highest thermal efﬁciency.

4.2.4 Comparison Between Adding a Second Compressor Stage with Wave-Rotor-
Topping
The wave-rotor-topping competes mainly against adding a second compressor stage

to the single stage baseline engine. In this competition one major advantage of the wave-
rotor—topping is that the wave rotor favorably operates mechanically independent from
the high-speed engine shaft. Therefore, adding a retroﬁt wave rotor does not require the

redesign of the challenging dynamic system. Even if the compressor or turbine wheel is

105

 

adapted subsequently to utilize the full potential of the wave-rotor-topping, the dynamic
system may change but not as dramatically as if a second compressor stage was added.
Thus, by default the wave rotor is a system for achieving similar thermodynamic
advantages as by adding a second compressor stage or a high pressure spool, but with
many fewer dynamic challenges.

To justify the wave-rotor—topping approach further, the performance results of both
competing solutions are compared below. For the addition of a second compressor stage,
performance data are calculated for the ﬁve most probably relevant pressure ratios of the
second stage described here:

' 1762 =PRW

A (perhaps) logical way to compare both systems would be to assume the same
compression ratio for the second compressor stage as for the wave rotor. Hence the
compression ratio of the second compressor stage would be 170:1.8, because the
assumed wave rotor compression ratio is PRW =1.8.

' "’0: WC]

More likely, when the effort of adding a second compressor stage is undertaken, the
designer would not limit the pressure ratio of the second stage to 17c2=l.8. It might be
desired to a add second compressor wheel that is similar to the existing ﬁrst stage (or the
same) for reasons such as using existing experience, or producing both wheels cost
effectively as identical wheels, or producing them geometrically similar using the same
or slightly modiﬁed tools. This approach can be modeled by setting the compressor shaft
work of the second stage equal to that of the ﬁrst stage, simulating the same angular

momentum change of the ﬂow in both stages at the same shaft. Because the inlet air

106

 

temperature for the second stage is much higher, the pressure ratio of the second stage is
less than that of the ﬁrst stage 17C2<17c1 .

' 1762: 1761

Alternatively, it could simply be assumed that the pressure ratio of the second stage is
equal to the pressure ratio in the ﬁrst stage. This is a common design approach.

° 17c (w)opt. and ”C(71)”;

It might be desirable to compare the wave-rotor-topped engine with a two-stage
compressor engine that has an overall pressure ratio 1752 '170 corresponding to the
optimum for maximum speciﬁc work 17c (w),,,, or the optimum for maximum efﬁciency
17c (Wopr- The resulting values for speciﬁc work and thermal efﬁciency respectively are
the maximum values actually obtainable by enhancing the pressure ratio of a
conventional compressor. The values of 17(w)op, and INmOP, can be found by using
performance maps in Figure 53-Figure 56 and following the curves for PRW =1 to their
optimum points.

In all performance calculations above, it is assumed that the polytropic compression
efﬁciency of the second stage is equal to that of the single stage baseline compressor. The
performance values of all these two-stage-compressor cases as well as intermediate cases
can also be read off the performance maps in Figure 53 to Figure 56 following the curves
for PRW =1. The compressor pressure ratio at the abscissa then corresponds to the overall
pressure ratio 170170. The performance results are compiled for the C-30 engine in
Table 3 and for the C-60 engine in Table 4 for all ﬁve two-stage-compressor cases
described above. The two-stage-compressor engines are compared with the wave-rotor-

topped engine Cases A and B. These cases are more suitably compared with two-stage-

107

 

compressor engines for a few reasons. Both cases employ the same compressor as the
ﬁrst stage of the baseline engines. Case A has shown the highest performance
improvement and it represents the maximum performance achievable for a wave-rotor-
topping cycle. In Case E, the baseline compressor is the same and the combustion end
temperature is the same as for the baseline engine, not requiring any thermal
enhancement of the combustor. It is understood that this is exactly the case for the two-
stage-compressor engine, where the combustion end temperature is simultaneously the
turbine inlet temperature (which is never the case for wave-rotor-topping). This has been
illustrated in Figure 57 that compares the T-s curves of the baseline cycle, the modiﬁed
cycle with the two—stage compressor, and the wave-rotor-topped cycle Case E with a
wave rotor pressure ratio of 1.8 for the C-30 engine. The two-stage-compressor values
may also be compared with the wave-rotor-topping Cases B, C and D using the supplied
data in Table 2.
Table 3: Performance comparison between adding a conventional second compressor

stage and wave-rotor-topping Case A and E with a wave rotor pressure ratio of 1.8;
baseline engine C-30

 

 

 

 

 

 

 

 

 

 

C-30 Two-Stage Compressor Wave Rotor Topped Baseline
Feature 17cm)... 17C3=PRw17c(rz).p.lvcz=wc, 170:1}, Case A Case E
170 1.52 V 1.8 2.22 2.42 3.6 1.8 1.8
- ﬂuff/ESL ‘_ 13.6 135, , 129 .131 1.35 all]. .331--. ‘28
SFCIkg/st] 0.133 0.128 0.126 0.127 0.141 _ 0.116 0.126 0.156
77 0.175m “0.181 0.184 0.183 0.164 0.200 0.184 0.149 C
(10),... [%] 6.3 . 5.5 0.8 -23 -25.8 33.8 7.7

 

 

 

(81:0,... [%] 14.5 17.9‘l18.9 18.6 9.6 25.2 18.9

 

 

 

 

 

 

 

 

 

 

 

(1])gain[%] 17.5 21.5 23.4 22.8 10.1 33.8 23.4

 

108

Table 4: Performance comparison between adding a conventional second compressor
stage and wave-rotor-topping Case A and E with a wave rotor pressure ratio of 1.8;
baseline engine C-6O

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

C-60 Two-Stage Compressor Wave Rotor Toppedl Baseline
Feature 17C(w)0,,,. f HC2=PRWHCQ])_OE_WC2=WC, ”C2=17Cl Case A Case E
170 1.52 1.8 2.55 2.79 4.8 1.8 1.8
w [kJ/kg] 193 192 179 173 118 231 190 184
SFCIkg/kN.s] 0.105 0.101 0.098 0.098 0.115 0.095 0.102 L 0.120
F.._--_ﬁs._-1u._ﬁ _ , - - ‘ _ ,.1-- ﬂ-.___-1.2 a- A.“ 1_
77 0.222 0.229 0.236 0.236 0.203 0.242 0.227 0.194
(w)ga,-,, [%] 4.9 4.3 -2.7 -6.0 -35.9 25.1 2.9
(SF C)ga,-,, [%] 12.5 15.8 18.3 18.3 4.2 20.1 14.9
(7] )gum [%] 14.4 18.0 21.6 21.6 4.6 25.1 17.5
1500 ' 0-1 1:41:51, baseline engine
0'12'23'35'42'51: topped engine
0'11'41'51 two-stage compressor engine
1200 :.TL'!!|1-1¢._.. ......... , ---,-.- _____ _ ________
x
_. I
E. ’ ,
a 900 - / ’_,/
:5 ’ I ’ I
a ’ ’Jg'ﬁ'! ’ ’
l5 1: ’ "J .J’ ’ I
2' 600 - 2 ‘5’”), I ’
a: E a a ’
111:] 2:11 ’
300 "
0 G30 Engine
0 I I I I I
1.5 1.8 2.1 2.4 2.7 3.0

Entropy [Id/kg Kl

Figure 57: T-s diagrams for baseline cycle, conventional cycle with two-stage compressor
(double compression work) and wave-rotor-topped cycle Case E with
PRW =1.8, C-30 engine

109

Table 3 and Table 4 show that the gain in predicted thermal efﬁciency, speciﬁc shaft
work, and SF C of the wave-rotor-topped engine in Case A is always greater than any
obtainable values for the two-stage-compressor engine. A look at the maps in Figure 53
and Figure 54 clearly veriﬁes this for the relevant design space. In these plots, the
performance points for Case A lie well above any point at the curves for PRW =1 where
all the performance data of the two-stage-compressor engine can be found. For the C-30
engine, Case E in Table 3 still shows a higher performance than any two-stage-
compressor conﬁguration. For the C-60 engine, however, Case E in Table 4 achieves
nearly the same performance as a two-stage-compressor engine with a second-stage
compression ratio in the range between the two optima for maximum speciﬁc work and
maximum thermal efﬁciency (minimum SF C), 17c2=1.52. .255. Finally, the results show
again that the compressor pressure ratio for the maximum speciﬁc work is always less
than that of the maximum thermal cycle efﬁciency (minimum SF C).

Besides the drawbacks of the two-stage-compressor implementation already
mentioned, a second compressor stage adds not only a second compressor wheel, it
always requires an enhanced combustor capable for higher combustion pressure and a
turbine adapted to considerably a higher pressure ratio. Finally it requires an enhanced
engine shaft, transmitting much more compression work. This situation is quite opposite
for the wave-rotor-enhanced engine in which the transmitted compression work for all
considered wave-rotor-topping cases is always either less (Cases B, C, and D) or the
same as for the baseline engine (Cases A and D). The combustion pressure ratio can be
kept the same (Cases B and C). Furthermore, the pressure ratio that has to be

accommodated in the turbine is always less for the wave-rotor-topped engine than for the

110

 

two-stage-compressor engine with the same or greater overall pressure ratio, likely
causing fewer problems when adapting the turbine. For the wave-rotor-topping Cases B
and C, the turbine pressure ratio is even less than that of the single stage baseline engine.
Additionally, in Cases C and E the wave-rotor-topping even lowers the turbine inlet

temperature, which allows the designer to use a turbine made of a cheaper material.

4.2.5 Substituting the Compressor in the C-60 Engine with the C—30 Compressor
plus Wave Rotor

An interesting practical engineering option is to substitute the current compressor of
the C-60 microturbine with the smaller and cheaper compressor designed for the C-30
microturbine by adding a wave rotor to obtain the same overall compression ratio of 4.8
required for the C-60 engine. This case is similar to the Case E shown in Table 2, only
the compressor polytropic efﬁciency is switched from 85.9% for the C-60 compressor to
82.9% for the C-30 compressor. Also different from the Case E, the wave rotor pressure
ratio needs to be adapted to 1.33 (instead of the value 1.8 in Table 2) to obtain the C-60
baseline overall pressure ratio of 4.8 in combination with the C-30 compressor that has a
compression ratio of 3.6. For such a modiﬁed C-6O engine, Figure 58 shows the thermal
efﬁciency, speciﬁc work, and SF C as a function of wave rotor compression ratio (lower
abscissa) and compressor pressure ratio (upper abscissa) if the overall compression ratio
is ﬁxed at 4.8 and the compressor has a polytropic efﬁciency of 0pc =0.829 of the C-30’s
compressor. It is seen that by using a wave rotor with a pressure ratio of 1.33 (orange
points), the modiﬁed C-60 engine would have an thermal efﬁciency of 19.1%, a speciﬁc
work of 190 kJ/kg, and SF C of 0.122 kg/kN.s. Compared to the C-60 baseline engine
(yellow points where PRW =1) with a thermal efﬁciency, speciﬁc work, and SFC of

19.4% and 184 kJ/kg, and 0.120 kg/kN.s, respectively, the topped engine would not

111

enhance the efﬁciency and SF C, but the speciﬁc work would increase by about 3%.
Furthermore, this engine would use a smaller compressor that has a lower manufacturing
cost and already exists. Other, more advantageous combinations with an even smaller but

newly designed compressor and higher wave rotor pressure ratio are also conceivable, as

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

shown in Figure 58.
Compressor Pressure Ratio
4.8 4.0 3.43 3.0 2.67 2.4
F T f r r r r r r r
0.28 l 0.28 ---,--Pa/ 9 3'5 “21:3“ 240
7»? . 1 "P" '
z : n,,=o.82
f 0.24 b 024 _.__.._...,..-_ .. ..... - ””5033 220
a ’— A
i r 11 \N ’4 P"
1: 0.2 _~ 0 2 - _ _ . . (w ~ _ 200 a
a: L. 5‘ —————:f",/ -1 3
I. L. I ""'”""".T_.:
§O.16_ .§°16/ I nan-g
8 i E '7 ' src i E
o 0.12 mm 0.1-2 m7 160 E
- ” ‘ 0
§ 2 | 1 a
0 0.08 - 0.08 ~ ~ 1 ~~~~~~~~ ._ 140 w
E C | I
8 - . .
0.0 04 - 0.04 — ~- 120
01 ~ | .
: palplﬂ=!1.33 :
° ' 01 1.2 1.4 1.6 1.8 2”"

Wave Rotor Compression Ratio (P R“)

Figure 5 8: Thermal efﬁciency, speciﬁc work, and speciﬁc fuel consumption of the wave-
rotor-topped C-6O engine using the C-30 compressor versus wave rotor compression ratio
and compressor pressure ratio for an overall pressure ratio ﬁxed at 4.8 (yellow points for
original C-60 baseline engine, orange points for wave-rotor-topped C-60 engine with C-
30 compressor)

4.2.6 Effect of Compressor Inlet Temperature
It is well known that the performance of gas turbines is affected by varying the

ambient conditions [231]. For instance, both the thermal efﬁciency and output power of a
gas turbine engine with a ﬁxed turbine inlet temperature decrease when the compressor

inlet temperature rises. This is disadvantageous if a stationary gas turbine is installed in

112

 

hot-weather locations. Under these conditions, it is proved in the following that the wave-
rotor-topping is even more advantageous.

For simulation it is assumed that the compressor isentropic efﬁciency stays constant
although a slight decrease might be expected. It is ﬁirthermore assumed that the speciﬁc
compression work stays constant since the compressor geometry does not change. Hence,
the compressor pressure ratio decreases upon increase of the ambient temperature from
basic thermodynamics. For the baseline engine it is obvious that this results in a lower
turbine pressure ratio and lower speciﬁc work produced by the engine. The results for
compressor inlet temperatures of 250 K, 300 K, and 350 K are compiled in Table 5 for
the C-30 engine and in Table 6 for the C-60 engine. The corresponding performance
values of the wave-rotor—enhanced engines of Cases A and B, which are the most
considered cases discussed in the literature, are also shown in these tables.

For the wave-rotor-enhanced engines the general trend is the same. However, while
the absolute performance degrades at higher ambient temperatures for baseline and
topped engines, the performance gains of the enhanced engines relative to the baseline
engines increase, making this technology more desirable for applications under hot-
weather conditions. For a range of wave rotor compression ratios PRW =1...2, this
reversed effect is visualized in Figure 59 and Figure 60, especially in the lower part
showing the relative gains of the topped engines. While Figure 59 shows the absolute
gain and the relative performance enhancement for Case A, Figure 60 illustrates similar

results for Case B.

113

 

Table 5: Ambient temperature effect on performance - comparison between baseline
engines and Cases A and B of wave-rotor—topping with PRW =1.8; baseline engine C-30

 

 

 

 

 

C-30 To: 250 K To: 300 K To: 350 K
Cases Baseline CaseA CaseB Baseline Case A CaseB Baseline CaseA CaseB
w [kJ/kg] 168 217 190 128 171 150 95 132 117
SFC[kg/kN.s] 0.125 0.097 0'112 0.156 0.116 0 147 0.195 0.141 0.179
g A- ,- .. f - ,, .1 2 _ - ,. - ,. w“ .. A- - 1 . m.
77 0.185 0.239 0'219 0.149 0.200 0.158 0.119 0.165 0.130
(w)gm-,, [%] 29.0 13.0 33.8 17.1 38.6 23.1
(SFC)g,,,-,, [%] 22.5 3.6 25.2 5.5 27.8 8.6
( r] )gm-n [%] 29.0 3.8 33.8 5.9 38.6 9.4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 6: Ambient temperature effect on performance - comparison between baseline
engines and Cases A and B of wave-rotor-topping with PRw =l.8; baseline engine C-60

 

 

 

 

 

C-60 To: 250 K To: 300 K To: 350 K
Cases Baseline Case A Case B Baseline Case A Case B Baseline Case A Case B
w [kl/kg] 234 285 258 184 231 208 144 185 168
SFC[kg/kN.s] 0.010 0.082 0'39 0.120 0'29 0.117 0.144 0.113 0 139
PM.-. 2. -2 2. -~_ a. 2*. ._ _7-- ,H2 2. A" #71 gMMum A—- -1. # A .1 ﬂ
,7 0.232 0.283 0'723 0.194 l 0'224 0.199 0.161 0.206 0.167
(w)g,,,-,, [%] L 21.8 10.5 25 l 13 0 28.2 16.4
(SF C)g,,,-,, [%] 17 9 1.8 20.1 2.6 22.0 4.0
by _n_. 1 _ __ .ﬁ ..7.. ..1.___._ﬁ LA ..- .k- ‘_-._A
(77 )8“... [%] 21.8 1.8 25.1 2.7 28.2 4.2

 

 

 

 

 

 

 

114

 

 

 

 

 

 

 

 

Specific Fuel Consumption ( kg / kN.e)

r,=111s.5 K ‘ T_=1227.s K
0-23 ' nwsoes ' ' nwaoas
=0.82 =o.az
0.24 - =o.ea . - =o.9e
0.2 -
>~
2
0.15 - a
‘2
e-
on — w .
0.08 —
0.04 _-
o h.

 

Wave Rotor Compression Ratio (PRW) Wave Rotor Compression Ratio (PRW)

60
T“=1 116.5 K T“=1227.6 K

11,630.86

50 30.82

40

30

20

0
Relative Change of Speciﬁc Fuel Consumption in "I.

 

Relative Increase of Efficiency and Specific Work in "/2.

Wave Rotor Compression Ratio (PRw) Wave Rotor Compression Ratio (PRW)

Figure 59: Effect of ambient temperature: absolute and relative changes of thermal
efﬁciency, speciﬁc work, and SF C versus wave rotor pressure ratio for Case A

115

 

Specific Work (kJ/ kg)

Speciﬁc Fuel Consumption ( kg l kN.s)

 

 

1 r =111s.5 K Tu=1221.6.5 K
0.28 — “ ‘ ’ ' qw=o.es
. . =o.az
0.24 — . . =o.9a
0.2 r
0.16 L
0.12 —
0.08 -
0.04 _-
0 1.
Wave Rotor Compression Ratio (PRw) Wave Rotor Compression Ratio (PRw)
g
c 0 — 0
E ru=111s.5 K ru=1227.s.5 K
.g . . 71,5086
1: ..\°
g -2 .s 25 =o.9a
in _ >.
C U
0 C
9 .4 _ g 20
, .-
.2 f.
g -6 — $15
0 u:
a » a
m 8
‘6 -8 - 510
81 . ,3
C 5
g a
0-10 - g 5
G
.2 *
..
%-12 - 0
I!

 

Wave Rotor Compression Ratio (PRw) Wave Rotor Compression Ratio (PRW)

Figure 60: Effect of ambient temperature: absolute and relative changes of thermal
efﬁciency, speciﬁc work, and SF C versus wave rotor pressure ratio for Case B

116

 

Speciﬁc Work (kJ I kg)

Relative Increase of Specific Work In %

4.3 Gas Turbine with Recuperation

Figure 61 shows a block diagram of a recuperated gas turbine topped with a four-port
wave rotor. The wave rotor is placed after the compressor and before the recuperator.
Figure 62 and Figure 63 show T-s diagrams for the C-30 baseline engine and the
corresponding wave-rotor-topped engine for Cases A and B, respectively. Path O—lb-Zb'~
4b—5b-5b' represents the baseline cycle and path 0-11-21-21.-31-41-51-51. (i=A and B)
indicates the wave-rotor-topped cycles. Only Cases A and B are considered in this study
due to their high potential for performance improvement under recuperated conditions.
All calculations have been presented only for the C-30 engine, while the conclusions are

valid for the C-60 engine, too.

Combustion Chamber

    

Wave Rotor

 

 

 

Compressor Turbine

Figure 61: Schematic of a recuperated gas turbine topped by a four-port wave rotor

Compared to the baseline engine without the recuperator, the recuperated baseline
engine here has a lower turbine inlet pressure due to the pressure loss of the recuperator
air-side. However, it has a higher turbine outlet pressure due to the pressure loss of the

recuperator gas—side.

117

Temperature [K]

1500

1200

900

600

300

 

 

 

F0-1b-2b'4b-5b-5b' baseline engine
0-1 A-ZA-2A33A-4A-SA-5A' topped engine: Case A
b
L Tartan _ -, _
b
- 1A
. o .
C-30 Engine
1.5 1.8 2.1 2.4 2.7 3.0

Entropy [kJ/kg K]

Figure 62: T-s diagrams for recuperated baseline and wave-rotor-topped cycles, Case A

Temperature [K]

1500

1200

900

600

300

'0-1b-2b'4b-5b-5b' baseline engine
0-13-23-23333-43-53-5,’ topped engine: Case B

 

C-30 Engine

 

l I l l I

 

1.5 1.8 2.1 2.4 2.7 3.0
Entropy [Id/kg K]

Figure 63: T-s diagrams for recuperated baseline and wave-rotor-topped cycles, Case B

118

4.3.1 Thermodynamic Calculations

In this section, calculations have been performed for both Cases A and B of the C-30
engine equipped with a recuperator. As before, in the wave-rotor-topping cycle, it is
assumed that the compressor inlet condition, turbine inlet temperature, compressor and
turbine polytropic efﬁciencies remain unchanged and are the same as for the baseline
engines.

For the recuperated cycle, the pressure losses across the air and gas sides of the
recuperator are assumed the same and equal to 2% (”recap—air = 17mm... = 0.98). The
effectiveness of the recuperator is considered to be ”recup=90%- These values have been
selected based on typical existing recuperators. The major challenges in providing a
recuperator with greater effectiveness are size and cost. For stationary applications, size
and weight are not critical, but for mobile applications these limitations produce
recuperator effectiveness commonly less than 90% [224].

Complying with Figure 61, the following steps are used to calculate the
thermodynamic properties of the gases in different states of the topped cycle [232]:

0 Path 0-1: Compressor

Equations (21) to (24) are again used to calculate the air properties at the compressor exit.
0 Path 1-2: Compression in the Wave Rotor

As shown before, Eqs. (25) and (26) are used to calculate the air properties after the wave
rotor compression process.

0 Path 2-2‘: Recuperator Air-Side

The recuperator effectiveness based on the actual and maximum heat transfer from the

turbine exhaust gas to the air can be expressed as:

119

T T,

(2. ’5

T,—T,

I I

77...... = (48)

Using this equation, the temperature at the exit of the recuperator air-side (Tm) becomes:
713' 2 T!) + ”ramp (715 — 7.12) (49)
In this equation, the turbine outlet temperature (7)5) is still an unknown parameter. More

equations are required to ﬁnd the unknown values Ty and T,5 . These additional

equations will be derived later. The corresponding pressure at state 2‘ is:

:17 pr2zn

I
p12
recup~uir recap—air

Po p0

 

PRW 17.: (50)

 

where I],e,.,,p_a,-, = pm /p,2 represents the pressure loss across the air side of the
recuperator.
0 Path 2:3: Combustion Chamber

Similar to Eq. (27), the following equation is used to calculate the ﬁrel/air ratio:

”thPR:(1+f)Cpgas 773—Cpair712' (5])

where T, 3 is an unknown value. However, f can be alternatively expressed as follows:

77g f hPR = (1+ f)Cpgas 7:4 '1' WWE _ WWC _ Cpuir (7:2 " 7:2) _ Cpair Tn (52)

Since the net output work of the wave rotor is zero, f can be expressed as:

_ Cpgus 774 -Cpair (712’ _772)_Cpair711

 

f (53)
77g hPR T Cpgas T14
This expression also will be later used to ﬁnd 732*, T,5 , and f .
Now, the total pressure after the combustion chamber is obtained by:
b. : £1521.— : ”comb ”recap—air PRW 17C (54)

p0 pt). p0

120

 

0 Path 3-4: Expansion in the Wave Rotor

The wave-rotor characteristic equation (38) is used to calculate the turbine inlet total
pressure expressed in Eq. (41).

0 Path 4-5: Turbine

The turbine total outlet pressure is:

EL: 1 p'5' = I (55)

po 17 recoup-gas p0 17 recap—gas

 

where Hrecupgaﬁ p,5¢/p,5 represents the pressure loss across the gas side of the regenerator.
It is justiﬁed to assume that the total pressure of the gas leaving the recuperator (p.54) is
equal to the ambient pressure ([70).

The turbine speciﬁc work according to pressure ratio across the turbine is calculated
by using Eq. (42) where pg must be substituted by p,5.

By obtaining the turbine speciﬁc work, the turbine total exit temperature (7)5 ) is

given as:

r... -l

w p /p
T —T - T =T — T I— —’5—-1 7“" 56

 

 

Now, to ﬁnd values of Ty, T ,5 , and f , it is necessary to use Eqs. (49), (53), and (56),
along with Eqs. (38), (41), and (55). The procedure is now explained. Substituting Eq.
(49) into (53) gives:

_ Cpgus 774 _ I7rer'up Cpair (715 _ Ti} ) — Cpair 77’
”Q hPR _Cpgas 7‘14

 

f (57)

or,

121

 

Cp gas 77 - ( h _ C as 77 )— C air 71
TU=TH+ _ 4 f 779 PR pg 4 p l (58)
I7recup Cpuir

 

Now, equating this equation with Eq. (56) results in:

 

-I
Cp...T. -f(77 h —Cp..T, )—Cp..,T, , /
+ 1, 4 Q PR 1: 4 1:714-777 T74 1_(P5 pa) 7,... (59)

”recap Cpuir pH / p0

 

Using Eqs. (41) and (55) gives:

+ Cpgus 774 - f(,]Q hPR _ Cpgas‘ 7.14 ) _ Cpair Til

 

 

”remap Cpair
I ’_w_’ (60)
=Tr4-777T14 1-( )rg...
recap—gas POHC

where PO is a function of f and is obtained from Eq. (3 8). The above equation computes
f as a function of other known cycle parameters.

0 Path 56‘: Recuperator Gas-Side

The temperature of the gas leaving the recuperator (T ,5.) can be calculated by using the

energy equation across the regenerator as follows:
Cpair (712’ —7;2):(1+f)Cpgas(]-;5—T;5°) (61)
This yields:

C . T . —T
T . = 7:5 _ pair ,2 12 (62)
’5 Cpgw 1+f

 

Now, it is possible to calculate the engine performance parameters as described before.

4.3.2 Predicted Performance Results

Implementing Case A for the C-30 recuperated engine, Figure 64 illustrates the
variations of the cycle thermal efﬁciency (dash dot), speciﬁc work (dashed), and SF C

(solid) with increasing wave rotor pressure ratio PRW. For the baseline engines (without

122

the wave rotor, PRw = 1), whereas the thermal efﬁciency of the recuperated cycle is much
greater than that of the simple cycle (without the recuperator) its speciﬁc work is slightly
less due to the pressure losses across the recuperator. With a conceivable wave rotor
pressure ratio of 1.8, the thermal efﬁciency of the recuperated engine increases slightly
from 24.8% to 26.6%. However, the speciﬁc work increases from 116 kJ/kg to 160 kJ/kg.
The thermal efficiency of the recuperated cycle remains almost unchanged when the
wave rotor is used, because the temperature difference between the air and the gas
entering the recuperator of the topped engine (Tm-Tm) is much less than that of the
baseline engine (Tab-m). This results in reduced recuperation, and hence the greater
heat addition into the topped engine offsets the increased work output. This is in contrast
to the unrecuperated engine where both the baseline and topped cycles have the same
heat addition, as explained before. SF C for the recuperated engine slightly decreases from
0.094 kg/kN.s to 0.087 kg/kN.s. Again the increased heat addition into the recuperated
topped cycle keeps SF C almost constant compared with the simple cycle, which is seen is
Eq. (47).

The relative increases of thermal efﬁciency, speciﬁc work, and relative decreases of
SF C are shown in Figure 65. Even higher than the unrecuperated engine discussed
before, the speciﬁc work of the recuperated cycle shows a signiﬁcant improvement of
about 37.6%, but thermal efﬁciency and SF C are only about 7% greater than those for the

baseline engine.

123

 

Overall Pressure Ratio

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_ 3 6 4.32 5.04 5.76 6.48 7 2
e I T =1116.5K l I 1 i V V V I 1 V l
3 0.36t 0.36 ":50“ :190
- 2 1] =0.82 —
E 0.32 r 0 32 flim=0.98 . _: 180
30.28} 028— - - 11 -—‘170 3
" * ...,..- - ----- ‘ o ’; f
5 0.24 ~ 0.24 :- rrrrr ~ - e, 0 ’ - 160 a
‘15. C 3 o’ i 1‘5
C 1’ _ X
g 0.2: '8 02— «ml—,4 ~ 1150 3
c I E v j 3
8 0.16:- Lu016— .7. 1140 [g
8012: 012—.1’7’1-97 ‘130E
I; ' * r SFC - a?
g 0.08;- 0.08 ’4 -_ 120
O ' 2
D. r _
m 0.04 - 0.04 — ~ - 11o
: C-30 wrth Riecuperator :
'_ 1 1 1 1 1 . 1 1 1 l 1 1 1 1 1 L‘
o o, 1.2 1.4 1.6 1.8 210°

Wave Rotor Compression Ratio (PRW)

Figure 64: Thermal efﬁciency, speciﬁc work, and SF C of the enhanced C—3O recuperated
engine versus the wave rotor pressure ratio and overall pressure ratio for Case A

Overall Pressure Ratio

Relative Increase of Efficiency in °/o
Relative Increase of Speciﬁc Work in %

 

 

Relative Change of Speciﬁc Fuel Consumption in %

Wave Rotor Compression Ratio (PRw)
Figure 65: Relative values of thermal efﬁciency, speciﬁc work, and SF C of the enhanced

C-30 recuperated engine versus the wave rotor pressure ratio and overall pressure ratio
for Case B

124

Similar to Figure 64 and Figure 65, now Figure 66 and Figure 67 show the absolute
and relative increases of the thermal efﬁciency (dash dot) and speciﬁc work (dashed), and
the decrease of SF C (solid) upon implementing Case B to the C-30 recuperated engine.
Similar to Case A, both the thermal efﬁciency and speciﬁc work are increasing functions
of PRW while SF C decreases. For PRW = 1.8, the thermal efﬁciency of the simple cycle is
increased from 24.8% to 27.3%. Similarly, the speciﬁc work is increased from 116 kJ/kg
to 137 lekg. Finally, the SF C is decreased from 0.094 kg/kN.s to 0.085 kg/kN.s for the
recuperated cycle. Therefore, wave-rotor-topping of the recuperated engine gives a
relative increase of 9.7% and 18% for the thermal efﬁciency and speciﬁc work,

respectively, and an 8.7% reduction in SF C .

125

Compressor Pressure Ratio
3.6 3.0 2.57 2.25 2.0 1.8

  

: T =1116.5K

0.36 — 036 u 190
,u? : 11,5083
5 0.32 _— 0.32 180
E. i ......
i‘. 0.28: 0.28 170 :25
E: 0.24:- 0.24 160 :,
3 02?~§°2 Hog
50m; Eme 1403
U . : LU . 5%
e » 8
: 012- 012 130 g
‘1“. i "’
1% 0.08; 0.08 120
w r-
6} 0.04 3 0.04 110

o L o 100

 

Wave Rotor Compression Ratio (PRW)

Figure 66: Thermal efﬁciency, speciﬁc work, and SF C of the enhanced C-30 recuperated
engine versus the wave rotor pressure ratio and compressor pressure ratio for Case B

Compressor Pressure Ratio

         

 

.\° 3.6 3.0 2.57 2.25 2.0

.5 ° 7 0

E -

2 s

2'1 ~ s .E

S -6- -E e

2 3‘ g

8 - E u

a 2 %
_ - it:

u:- 12 E (E-

0 ' o

E u-

8 - 3

g -18 — e 3

s - E E

0 a: E

8’ E c»

19 ~24 - s E

.1: q, I!

U Q: T)

.‘2’ - 1:

N

3 -30 -

1::

Wave Rotor Compression Ratio (PRW)

Figure 67: Relative values of thermal efﬁciency, speciﬁc work, and SF C of the C-30
recuperated engine topped with a wave rotor versus the wave rotor pressure ratio and
compressor pressure ratio for Case B

126

Table 7 summarizes the results showing a comparison between performance
improvements for these for implementations of both Cases A and B. It is seen from the
table that the baseline engine with recuperation has a much higher efﬁciency (about
twice) and lower SF C than those of the unrecuperated engine due to the heat addition
reduction in the combustion process. Implementation of Case A into the baseline engine
results in a signiﬁcant performance improvement of the unrecuperated engine, however,
the thermal efﬁciency and SF C improvements of the recuperated engine are lower than
that of the speciﬁc work. In Case B, the topped cycle without the recuperator still beneﬁts
from the wave rotor even though the performance enhancement is less than that of Case
A. However, the topped recuperated cycle has a higher performance compared to the
unrecuperated engine and its thermal efﬁciency and SF C gains are even more than those
of Case A. The results clearly demonstrate the beneﬁt of implementing Case A for
unrecuperated engines and implementing Case B for recuperated engines. Among all
topping cycles, the best relative performance improvement is obtained with Case A
topping of the unrecuperated engine.

A comparison between the recuperated baseline engine and the topped simple cycle
engine (PRW =1.8) reveals that for the speciﬁc example, the recuperator enhances the
thermal efﬁciency and SF C, while the wave-rotor mostly enhances the speciﬁc work
output. Therefore, substituting a recuperator with a wave rotor may be guided by a
preference for high power output and reduced unit cost, considering that a recuperator
contributes about 25-30% to the unit cost and a wave rotor may be cheaper. Finally,
topping a recuperated gas turbine with a wave rotor can increase the performance

especially if the topping cycle operates with the same turbine inlet temperature and same

127

overall pressure of the baseline engine (Case B implementation), which is preferable for

the combustor and fuel injection design.

Table 7: Performance comparison between baseline engines and two cases of wave-rotor-
topping with a wave rotor pressure ratio of 1.8

 
 
  
 

   

17L,

at

   

*LclkL/kgi

 
  
  

  

W

   
  

   

.552 #99337
0.266

37,-6

. 6:2, ,
7.2

 
 
   

 
 
 
   

  

(SFQML .

 

4.4 Turbojet Engines Topped with Wave Rotors

Figure 68 illustrates how a four-port wave rotor is used to top a simple turbojet
engine (without afterburner). Point “a” here refers to the ambient condition at the inlet
difﬁiser. Thermodynamic analysis of such a topped cycle is different form that of a
stationary topped cycle due to the presence of the inlet (diffuser) and nozzle and the fact
that in a turbojet cycle the net output work is zero, whereas in a grounded gas turbine the

turbine work is greater than the compressor shaft work.

128

Combustion Chamber

Diffuser
1| 0

\ l/

Compressor Turbine

 

Wave Rotor

 

 

 

 

Figure 68: Schematic of a turboj et engine topped by a four-port wave rotor

Figure 69 visualizes all ﬁve wave rotor implementations in a schematic T-s diagrams.

Path a-O-lb-4b-5b—6b represents the baseline cycle and path a-O-li-Zi-3i-4i-5i-6i (i=A, B, C,

D, E) indicates the wave-rotor-topped cycles.

Temperature

    

a--0-1b-4b~5b-6b baseline engine
a-0-1i-2i-3i-4i-5i-6i topped engine
i= A, B, ..., E

  

Tturbine

 

 

Entropy

Figure 69 : Schematic T-s diagrams for a baseline turbojet cycle and ﬁve different wave-

rotor-topped cycles

129

4.4.1 Thermodynamic Calculations

For the thermodynamic calculations, a reasonable case considered here is an aircraft

equipped with a simple turbojet engine ﬂying at an altitude of 10,000 m at Mach 0.8. The

component performance parameters of the C-30 engine are used as the baseline engine.

Other component efﬁciencies are listed in Table 8. Now, a higher pressure drop of 5% in

the combustion chamber (17mm = 0.95) is considered in the calculations. It simulates a

higher pressure drop in the combustor due to a possible size reduction of the combustion

chamber which is attractive for small propulsion systems. At the altitude of 10,000 m,

ambient air enters the inlet diffuser at Ta=223 K and [20:26.5 kPa .

Table 8: Turbojet baseline engine data, assuming T 0 =300 K, Cpair=l .005 kJ/kgK,

Cpgas =1.148 kJ/kgK, yair=l.4 , yga, =1.33

 

Baseline Engine

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Turbine inlet temperature T“ 11165K
Compressor pressure ratio pub/3,0 3.6
Diffuser isentropic efﬁciency 770 93%
Compressor isentropic 0
efﬁciency ”C 80 /°
Compressor polytropic o
efﬁciency ”PC 83 /°
Combustion efﬁciency ”Q 98%
Combustor pressure loss Hm”, 0.95
Mechanical transmission 0
efﬁciency '7” 99 /°
Turbine isentropic efﬁciency m 84%
Turbine polytropic efﬁciency mar 83%
Nozzle isentropic efﬁciency my 95%

 

 

0 Path a-0: Inlet Diffuser

With the given ﬂight Mach number M, the stagnation temperature across the diffuser

(Tm=T,0) is:

130

=Ta[I+ZL”—_—1M2)
2

(63)

Using the deﬁnition of the diffuser isentropic efﬁciency (:70), the diffuser outlet total
pressure is obtained from the equation :

2 7a” / fair '1
p10 u
- = 1+ ——“——— 64
p“ [ 770 2Cp T] ( )

air a

 

- Path 0-1: Compressor

The approach used before is implemented for this stage.

0 Path 1-2: Compression in the Wave Rotor

The approach used before is implemented for this stage.

0 Path 2-3: Combustion Chamber

The approach used before is implemented for this stage.

0 Path 3-4: Expansion in the Wave Rotor

The approach used before is implemented for this stage.

0 Path 4-5: Turbine

In turbojet engines, considering the mechanical transmission efﬁciency, the compressor

shaft work equals the turbine output work:

Cpair(Til—7~10)=17M(1+f)cpgas(7~14—T'15) (65)

Therefore, the total temperature of the gas leaving the turbine can be calculated as:

C . T -T,
715:7,” pair( (I 0) (66)

— m. (I + f) 6p...

 

To ﬁnd the total pressure of the gas leaving the turbine (p,5), the value of the turbine

isentropic efﬁciency (In) is needed. There are two ways to calculate 771 for a turbine:

13]

(&)(7gas‘l)’ll'r”Iran" _]

 

p,
77f : I: .
( f5 )(rgm-l)/7gac _1
p14
and,
T5
_r_ -1
(TM)
777 :

 

(ﬁi)7ga§_’/7gut _1
pm

It is preferred to use the turbine polytropic efﬁciency mar to obtain p,5

equating the above two equations, one ﬁnds:

rgm

[9,5 =(Ir_5_)(7,....-I)npr

T

p14 :4

or,

ygm

 

££:!ﬁ1(£
pa 1).. T

(4

)(rgm —I)’II’T

0 Path 5-6: Nozzle

For a given local pressure ratio at the nozzle exit (pig/pa), it is true that:

19.5 = p... 12.5 g = 10,4 19,5 /p.. 1
pr 12.. p” p. p. I’M/p. p. /p..

 

 

(67)

(68)

. Therefore, by

(69)

(70)

(71)

Parameter p,5/p6 is a useful quality for calculating the local gas temperature leaving the

nozzle (T6) by using the deﬁnition of the nozzle isentropic efﬁciency (771v):

73m"

." gas

p15 /p6

— d

T6 : T15 _77NTr5

 

 

I32

(72)

Since T,5=T,6 , the nozzle exit velocity (u6) can be calculated from the expression:

 

“6 : JZCpgas (7:5 — T6) (73)

Therefore, the nozzle exit Mach number is obtained:

 

 

u
M, = 6 (74)
vygas Rgas T6
Finally, for the given pé/pu , the ratio of p,6/pa is calculated from the expression:
y _ 1 7m“
f1:_p’_6_p_i=(1+ 5““ M62)7w'"1p_6 (75)

 

p. p. p. 2 p.
After calculating the thermodynamic properties of all states in the cycle, it is possible to
evaluate the engine performance parameters. For instance, the thrust produced by the
engine is given by:
T=(mai,+m/)u6-marrua+(pa-Pa)A.v (76)
where A N is the exhaust area of the nozzle. Thus, the speciﬁc thrust (thrust per unit mass

ﬂow of air) can be written as:

 

Av
ST=(1+f)u6-ua+(p6—pa)m‘ (77)

air

The pressure term in the above equation can be presented [233] as:

 

 

RmT/T 1— /
)g 6 a .0. 106] (78)

R' u6/aa yair

("r

A
(p.-p.) .” =a.[(1+f
m

where an is the sound velocity of the entering air. Therefore, the speciﬁc thrust can be

calculated as:

 

\R805 7:5 /Ta 1—pa/p6 (79)
IR' u6 /aa rair

0”

ST=(I+f)u6 —ua+au [(144

133

The thrust pressure term (pé-pa)AN is non-zero only if the exhaust jet is supersonic. In
this study, all calculations have been performed for pé/pu =1 resulting in zero value for
the thrust pressure term.

The speciﬁc fuel consumption is given by:

 

SFC = i- (80)
ST
Finally, the overall efﬁciency is derived as:
n - ST ”“ (81)
0 m.

4.4.2 Predicted Performance Results

A detailed documentation of the results is not presented here. The reader is referred
instead to Ref. [234]. However, numerical values of the relevant cycle data and the
performance enhancement are summarized in Table 9, where indicates an attractive
relative performance improvement in overall efﬁciency and speciﬁc thrust of about
15.4% and a 13.3% reduction in speciﬁc fuel consumption using a wave rotor pressure
ratio of 1.8. Table 9 shows that for Case A perhaps the best combination of high overall
efﬁciency, speciﬁc thrust and low speciﬁc fuel consumption is achieved. However, Case
B provides the lowest value of the speciﬁc fuel consumption and the highest overall

efﬁciency.

134

Table 9: Performance comparison between baseline turbojet engine and ﬁve cases of
wave—rotor-topping with a wave rotor pressure ratio of 1.8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Case A Case B Case C Case D Case E
o compressor ° 3:3” 0 turbine inlet 0 compressor
Same as _ ratio I ocombustor pressure. Baseline
baseline - turbine . turbine 0 turbine inlet - combustor
engine inlet temp. 7 inlet IEIDP- temp. 7end temp.
@[K] 7 1116.5 1116.5 7 1056.2 1116.5 7 1042.75
7 717(7 7 7 3607 7 72.070 7 7 290 7 2.65 7 7 73.670 7
7 177T 7 7 1.73 7 717.307 71.32 1.47 7 1781
wc [kl/kg] 140 68 68 100 140
§UNikgl B .562 z, Eli z 7493' 550,, a.-- $131
§fglgg0<us1_,a9§§§ -z,0-9446_,_M411w ,o-oazﬂ 0-0378
7],, 0.145 0.125 0 126 0.135 0.147
(577")gui,,[%] 715.747 7 777.717 0:6 7 12.17 7757.67
.(SFczgamtm. 13:3 0.1 - .. 341,3 6:9 . 7,1415 “
(770 )guin [%] 15.4 -0.9 0.3 7.4 17.0

 

It is interesting to note that for Case A the turbine pressure ratio is the same for both

the baseline and enhanced engines (17;- =l.73) since the compressor is unchanged.

However, due to the higher turbine inlet pressure, the turbine inlet volume ﬂow is lower,

which may require an adaptation of the turbine. Also, Case D here uses only the same

turbine inlet temperature and inlet pressure of the baseline engine. Note that for the case

of a stationary gas turbine, the same physical turbine as the corresponding baseline

engine can be used because the inlet and outlet turbine conditions remain unchanged,

which is different from the turbojet cycle considered here.

Figure 70 shows a map of the relevant design space for Cases A, B, and D. Again,

Cases C and E can not be shown on the same map because in both cases the turbine inlet

temperature is less than indicated in the upper right comer of the map. Therefore, the

performance map of Cases C and E is shown instead in Figure 71, where both cases have

135

the same combustion end temperature as the baseline engine, as indicated in the upper
right comer of the map.

The same trend as observed for stationary gas turbines is seen here again. For higher
compressor pressure ratios, the beneﬁt of using a wave rotor diminishes. Figure 70 shows
that for compressor pressure ratios greater than about 20, almost no beneﬁt can be
obtained for Cases A, B, and D. The beneﬁt is the greatest for lower compressor pressure
ratios. Implementing Cases C and E even results in performance enhancement for

compressor pressure ratios just lower than about 10, as seen from Figure 71.

 

 

 

 

 

 

 

 

; 650
‘5, Tu=1116.5 K
t '1 I,c=0.832
._ 7191:0330
g 7 ;_ ncm=0.95 550
x ' .. I
3’ . a ' 3’
J - a
. i. . 2
1.0;) .- 450 :
, i_ 7 g
. .E
- q '—
- 350 ,g
No Gain ' 8
. ‘ D.
.7 Region in
A » - 250
i A A A El I A A A A I A A A A A A A A I A A A A A A A A A 1 50
0 5 10 15 20 25 30

Compressor Pressure Ratio

Figure 70: Performance maps for wave-rotor-topping of the C-30 turbojet engine, Cases
A, B, and D

136

Rcomp=3.6

 

Tn=1116.5 K

Optima n Pc=0.832
*1 ”=0.830

 

 

SFC (kg/kN.s)

Speciﬁc Thrust (N.s/kg)

No Gain
Region

 

   

Compressor Pressure Ratio

Figure 71: Performance map for wave-rotor—topping of the C-30 turbojet engine, Cases C
and E

137

CHAPTER 5: PRELIMINARY WAVE ROTOR DESIGN

5.] Analytical Approach

The ﬁrst step in designing a wave rotor is to decide on a TF or RF wave rotor
conﬁguration. One problem with TF and RF wave rotor conﬁgurations is that fully
purged processes are not often achieved due to a high turbine inlet pressure. As an
example, Figure 35 shows ﬂuid ﬂow patterns which clearly indicate that the high
pressure part (charging process) in both the TF and RF conﬁgurations does not start with
channels ﬁlled uniformly with fresh air. In the case of the TF wave rotor, the channels are
already ﬁlled with fresh air received from the compressor and residual burned gas from
the low-pressure part of the cycle, resulting in exhaust gas recirculation back to the
combustor which in turn causes excessive temperatures in the discharge ducts and parts
of the end wall. In the RF conﬁgurations, the channels are ﬁlled with the fresh air and
some residual air and gas from the low-pressure part of the cycle known as the buffer
layer [180]. This residuum can remain permanently in the channels. Repeated
compression waves increase the temperature of this buffer layer. This leads to high wall
temperatures in the center of the channels [179]. It is possible to achieve a full
scavenging process for both the TF and RF conﬁgurations using bypassing or bleeding of
certain mass ﬂows [9, 178, 179]. However, all of this leads to more complex
conﬁgurations.

Knowing the above challenges, one-dimensional analytical gasdynamic models are
used here to analyze the high-pressure phase of four-port wave rotors. Two approaches
can be considered. The simplest way is to assume that the high-pressure cycle begins

with channels fully ﬁlled with air at the ambient state for a Comprex or the state of the

138

compressor discharge in the case of a gas turbine application. This approach gives
reasonable results for the Comprex analysis where the charging process begins when the
channels are fully ﬁlled with fresh ambient air. This approach has been discussed in
detail in our previous publications [235, 236]. A similar approach, which is a more
realistic model for most gas turbine applications, calculates the channel temperature and
pressure at the beginning of the high pressure part, which are mostly different from those
of the compressor discharge air (as will be described in this Chapter). The overall
procedure allows a fast estimate of the wave rotor dimensions which is useful for
preliminary design considerations. The accuracy of this approach can be enhanced by
solving the ﬂow ﬁeld in the low-pressure cycle and using the results as the initial state at
the beginning of the high-pressure part. The difference with the initially obtained values
is then an indicator of the accuracy and convergence of the calculations.

Even though all these approaches can be applied to the TF and RF conﬁgurations,
they may be more suitable for the RF conﬁgurations where the channels are ﬁlled mostly
with air at the beginning of the charging process. When applied to the TF conﬁgurations,
however, the models are still consistent with the thermodynamic performance
calculations. Performance calculations for the TF conﬁgurations sometimes do not
account for the pre-burned gas contained in the combustor inlet ﬂow and the ﬂow is
considered only to be air (similar to the previous chapter). If the gas recirculation is
considered, higher accuracy could be obtained for the thermodynamic performance
calculation and the wave rotor design.

The present work follows a methodology similar to that used by Keller [44] with

several modiﬁcations. Keller does not consider tuning of the reﬂected shock wave as

139

-.1

 

 

does the current work. As a result, Keller’s approach obtains only a relative small gas
penetration length as shown later. Two main cases are considered: a charging process (1)
with a single shock wave (2) with two shock waves. The ﬁrst is simpler, but gives less
favorable results. Furthermore, two different wave patterns are considered in the more
complex second case. For low wave rotor compression ratios, the idea of using weak
shock waves (isentropic) relations instead of existing normal shock wave equations has
been implemented by Selerowicz and Piechna to simplify the gasdynamic calculations
[85]. For beneﬁcial high values of wave rotor compression ratios, however, the
assumption of weak shock waves inside channels is not accurate and jump relations
across the shock waves are necessary for the calculations [6, 85, 91, 93]. Gyarmathy
emphasizes the importance of using jump relations across the shock waves when
describing the operation of a conventional Comprex [6], but he performs no calculations.
Taking into account the jump conditions, Weber et a1. [91, 93] describe a design
procedure for a type of Comprex which is different from the conventional Comprex as
Keller has considered [44]. The procedure described here calculates wave speeds, the
location of air/gas interface, and port widths in the high-pressure phase. Other useful
design parameters including channel height and rotor size can be subsequently estimated.
Besides the assumptions described in Chapter 3, it is assumed that pressure equalization
occurs faster than mixing. Thus, the thickness of the mixing layer between the burned gas
and the fresh air inside the channels is negligibly small and mixing between the gases is

therefore ignored.

5.1.1 Charging Process with a Single Shock Wave

As a ﬁrst step, the charging process is considered in the absence of the secondary

shock wave. This situation may occur when the combustor pressure loss is relatively

140

small and hence the total pressure difference between the incoming hot gas and outgoing
compressed air is small. For a particular level of combustor pressure drop, this may lead
to equal static pressures at the high-pressure gas inlet and high-pressure air outlet,
avoiding the primary shock reﬂection off the outlet port. The wave diagram of the
charging process is shown in Figure 72 which is the same as Figure 33 with some

additional information.

1 j

-/i:

U
-
‘1... ...ir: ., "7.1"
’
.7 7 7‘ . 7 77 41
53E.J:_f---43~J: - {i-f; '_
.
.
O
V .- :3
h 7,. ‘1... .
O
0

    
 
 

as

 
    

'

 

   

O
O
O

0::
z: ::

  

:33

.V'

 

 

" 7|

Figure 72: Wave diagram of high-pressure part with only one shock wave

As explained in Chapter 3, the burned gas coming from the combustor enters the rotor
through the inlet port 3. The air, compressed by the hot gas, leaves the channels through
the outlet port 2. Region 1' represents the channel closed at both ends, containing only
air. By further rotation, the hot burned gas (region 3) penetrates the channel triggering a
shock wave (dashed line). The air/ gas interface (dotted line) follows the shock wave with
an induced velocity uz. Behind the shock wave (region 2), the compressed air has the

same local pressure and speed as the inlet exhaust gas (p2: p 3 and u 2: u 3). As the shock

141

walnut-1“]; "A...
Ii . ‘ _'

 

wave reaches the right end of the channel, the compressed air ﬂows out into the
combustion chamber. This is usually called the tuning condition for the primary shock
wave [6, 40, 44, 79]. In the following it is referred to as the tuning condition 1 (TC 1).
The tuning condition 2 (TC 2) dealing with the generation of a reﬂected shock wave will
be introduced in the next section.

When the air/gas interface approaches a timed location, typically around the middle
of the channel, the ﬂow is stopped by closing the gas inlet port. An expansion wave
originates from the upper comer of the inlet port and propagates toward the right end of
the channel. In Figure 72, Lg“, denotes the gas penetration length indicating how far the
burned gas can penetrate the rotor channels. It is nearly the location where the ﬁrst
disturbance of the expansion wave catches the air/gas interface. W3 and W; are the inlet
and outlet port widths, respectively. Tuning condition 3 (TC 3) is fulﬁlled if the
expansion wave reaches the right end of the channel at the moment when the upper edge
of the outlet port closes the channel [6, 40, 44, 79].

To calculate an example of the ﬂow properties in the channels shown in Figure 72,
the inlet and outlet stagnation pressures and temperatures listed in Table 10 are used as
port boundary conditions, assuming air enters the compressor at atmospheric pressure.
These data are obtained from the results of the thermodynamic calculations performed in

Chapter 4 for the wave rotor implementation of Case A into the C-30 engine.

142

 

Table 10: Enhanced engine data, assuming PRW = p,2/p,; =1.8 for Case

 

 

 

 

 

 

 

 

 

 

 

 

Port Thermodynamic States Enhanced C-30
Inlet air pressure p,, 3.60 XI01 kPa
Outlet air pressure pm 6.48 XI01 kPa
Inlet gas pressure p,3 6.35 XIOI kPa
Outlet gas pressure 1)“ 4.38 XIOl kPa
Inlet total air temperature T, , 466.6 K
Outlet total air temperature Ta 569.3 K
Inlet total gas temﬁrature T,3 1204.7 K
Outlet gas total temperature TM 1116.5 K

 

 

The following ﬁve equations are used to ﬁnd the ﬁve unknown parameters p 3, T 3 , uz , pr,

Tr:

 

u:

70w

 

 

 

 

27w?
=‘iyur‘rRair 7; 7P_3771 yﬂr+l
p], &+ 7.... -1
\pl/ yair +1
u 2
:2 =T2 + 2
2Cpair

 

 

 

£9. .. [Z1]"‘""
p3 T3

(82)

(83)

(84)

(85)

(86)

Note that p; =p3 , u; = 113 , and T 2 is obtained by using Eq. (5) derived in Chapter 3.

Also, the velocity of the ﬂow in the inlet duct is (uzz + V2) 1 2 where V stands for the

tangentially averaged rotor speed. However, it has been justiﬁed [44] that the rotor speed

143

 

 

is small compared with the ﬂow sound speeds, i.e. V/a2<< 143/a}. Therefore, V has been
omitted on the right sides of Eqs. (83) and (85).

For Case A described before, the above ﬁve unknowns are calculated as:
p3=6.24><101kPa , T3=1199.3 K , u2=111.8 m/s ,p,r= 4.45X101kPa , and T]'=510.8 K.

The results can be further used to calculate the gas penetration length (nearly exactly
L,,,.). It is an important design parameter which will be used to determine the required
rotor port widths. The maximum possible value of the compressed air mass ﬂow is
reached when the gas penetration length becomes equal to the rotor channel (L,,., /L= l).

The two following equalities are used to ﬁnd the dimensionless penetration length L,.,,/L:

 

 

L
w —E/i+ "“5 (87)
u, V u2+a3
W L L—L ,
7_3+ gas gas 77:-*7 W (88)

Equation (87) results by equating the time required for the air/gas interface originating
from the lower comer of the inlet port to be overtaken by the expansion wave (LN/uz)
with the opening time of the inlet port (W3 /V) plus the time required for the expansion
wave to catch the air/gas interface (L,,../u2+a3). Tuning conditions TC 1 and TC 3 result
in Eq. (88). Note that when the expansion wave meets the air/gas interface, its velocity
reduces from u2+ a3 to u2+ 02. Multiplying both sides of Eqs. (87) and (88) by er and

dividing them by length L and simplifying gives:

144

 

 

 

 

 

 

 

 

M5 M+a’
Lgus 0] 89
L — 1 W, ( )
W, 1 1
+ _
M(1+M,) M+$ a; M+ 0.»
a, a], a],

where M: 112/01' and M3 = u; /a3 are the ﬂow Mach number and the Mach number of the
burned gas in region 3, respectively. M5: w5 /a,r is the shock Mach number where ws
represents the shock wave velocity. The only unknown term here is the port width ratio
W 2 /W3 . It is obtained by writing the continuity equation across the combustion chamber
which gives:
m, = m, + ”'2! (90)
or,
p321, A, =p2 u, A2+n'1f (91)
Because u2= m and A3 = W3 . z , A2: W2. 2 where z is the height of the channel, the

above equation can be simpliﬁed to:

£2.52

= 1 + f (92)
[’2 W2

Here, it is assumed that m, = mm, = #2,. Other cycles can exist in which the mass ﬂow
rates "'2, and n’z, are not the same, but they are not considered here. By using the ideal gas

equation, Eq. (92) transforms to:

—l
W7 - 7 R’05

.. : Ralr I; I = A £(I'l‘f) (93)
W3 Rgus T3 (1+f) Rair T2

 

 

 

145

Thus, substituting this equation into Eq. (89) estimates the gas penetration length. L,..,./L
as a function of the initial cycle thermodynamics and the port boundary conditions. For
Case A and corresponding thermodynamic parameters listed in Table 10, a very low
value of Lm/L =0.05 is obtained. Similar to Keller’s results [44], the present result
indicates that for typical wave rotor pressure ratios, e.g. PRW < 3, the gas penetration
length Lga,./L is very small, e. g. less than 0.2 for such tuned wave rotors. Hence a relative
small amount of air compared to the wave rotor channel volume is delivered to the outlet
port. Furthermore, it leads to very small and unrealistic port width W3 and W2 . For

instance, inlet gas port width can be found by using Eq. (87) as:

 

W3=Lg,, —1—-+ I V (94)
uz u2+a3

Also, Eq. (93) gives the port width ratio Wz/W 3 from which the absolute value of W; can
be obtained, by using W; from Eq. (94). In this study, W3 = 0.45 cm and W2= 0.21 cm are
obtained for a rotor with a channel length of L = 20 cm and a rotational speed of V=50
rer. A huge (and unrealistic) channel height of H...” = 35 cm would be then necessary for
the desired volume ﬂow, resulting in an undesirably bulky wave rotor. An alternative
solution should be sought to increase the gas penetration length. This is the topic of the

next section.

5.1.2 Charging Process with Two Shock Waves

Several wave diagrams can be introduced that employ two shock waves. Two of these
wave patterns were introduced in Chapter 3 and they are studied in more detail in this
section. As it will be shown in the following, considering a secondary wave inside the
channels results in reasonable geometry parameters with a greater relative gas penetration

length.

146

 

 

0 Charging Process with Two Primary Shock Waves

Figure 73 depicts a wave diagram which employs two shock waves. It is similar to the
previous wave diagram sketched in Figure 72, but both inlet and outlet port widths have
been proportionally extended and a shorter rotor height is used, matching the same mass
ﬂow rate through the rotor as considered before. Also, the gas inlet port becomes much
greater than the air outlet port keeping the same port width ratio as before to allow more

gas to penetrate the rotor channels.

 

 

 

" 'I

Figure 73: Wave diagram of high-pressure part with two primary shock waves

By closing the outlet port, a second primary shock wave originates at the upper outlet
edge and propagates from the right to the left with velocity w5.a;, into the part of the
channel that contains air ﬂowing with velocity u; from left to right. It reduces the velocity

of the air to zero to satisfy the zero-slip boundary condition at the end wall. After

147

 

 

crossing the air/gas interface, the velocity of the second primary shock wave increases
from W50), to Magus, where the latter is the velocity of the shock wave in gas region 3.
The increase in the velocity is mainly due to the fact that the burned gas has a higher
temperature than the compressed air. The double-compressed air region and the

compressed gas region after the second primary shock wave are introduced as regions 2'

and 3', respectively. Here, a favorite case is considered where the closure of the gas inlet
port is timed with the arrival of the second primary shock wave. This is called the tuning
condition 2 (TC 2) [6, 40, 79]. Hence, expansion waves are not generated after closing
the gas inlet port.

To ﬁnd the useful design parameters for such a wave diagram, all previous derived
equations are used again, but new expressions must be derived for W3 , W2 , and L,” /L.
Hence, the two following equalities are used to ﬁnd the new value of dimensionless

penetration length L... /L:

 

 

 

“w = — + + g” (95)
u 2 “’5 V WS-arr
L L
E1 7 gas + gas (96)
V ”2 WS-gas

The ﬁrst equation is found by equating the time required for the air/gas interface
originating from the lower corner of the inlet port to reach the second primary shock
wave (Lga, /u2) with the time required for the ﬁrst primary shock wave to run through the
channel (L/ws) plus the outlet port opening time (W 2/ V) plus the time required for the
second primary shock wave to meet the air/gas interface ((L-Lgas ) /w5-a,-, ). The second
equation is obtained by equating the inlet port opening time (W3 / V) with the time

required for the air/gas interface originating from the lower comer of the inlet port to

148

 

 

reach the second primary shock wave (Lga, /u2) plus the time required for the second
primary shock wave to travel through the gas region and to meet the upper corner of the
inlet port (Lgas/wsgm). Multiplying both sides of Eqs. (95) and (96) by all and dividing

them by length L and simplifying gives:

1 4,
L803 _ MS WS—gjr

L 1 a, W, 1 a,
—— +———— —-—— —- +
M W3- W; M wS

air —gas

 

 

 

(97)

 

In this expression, WSW and ws-gas are unknown and must be found. To ﬁnd WSW-r , the
strength of the second primary shock wave propagating in the air region 2 (175.0”: pzl/pz)

is ﬁrst obtained by using Eq. (7):

 

 

 

 

 

2 yair
a / + I
”2 = —2_(US—air ‘" I) 7 _1 (98)
yet? \ ”S-air + 2,0;—
yair + 1
Then Ws.a,'r is found by using Eq. (9):
yair + 1
WS-air = 02/ J—(HS-air —1)+1 (99)
2701?
Finally, the corresponding Mach number can be calculated by using Eq. (1 1):
. I
MS—air :\/72mr + —(HS—air ‘1)+1 (100)
yair

A similar approach is used to ﬁnd 175,33“,= p 3 ’/p 3 (strength of the second primary shock
wave propagating in the gas region 3), WSW, , and M51305 using the gas properties.
Therefore, Eq. (97) calculates the dimensionless penetration length for the wave diagram

shown in Figure 73. For the case considered in this study L,.../L = 0.66 is calculated which

149

.‘\

 

 

 

is a considerably greater value than that calculated for the wave diagram with the single
shock shown in Figure 72. The higher value of this parameter is due to the extension of
the gas inlet port width, allowing more burned gas to enter the rotor channels. By
obtaining L,,../L it is now possible to estimate the rotor geometry, as described before.

By obtaining W 2 and W 3 using Eqs. (95) and (96), respectively, the inlet opening time

(t3) and outlet opening time ((2) become:

W3
’3 :7 (101)
and,
t —— W3 (102)
" V

Additionally, the time required for the primary shock wave to run through the channel (II)
is simply equal to L /w5.

To ﬁnd the other geometry characteristics, the following procedure is described. For a
given baseline air volume ﬂow rate, Q2 , the cross-sections of the outlet port 2 and inlet

port 3 can be obtained from:

A: = - (103)

 

and,

23:21

A, = (104)

where Q; is the volume ﬂow rate of the burned gas, which is different from the air
volume ﬂow rate Q; . Considering the mass added from the fuel, the continuity equation

relates Q2 and Q3 as follows:

150

 

 

 

 

p3Q2(1+f) (105)

p3

Q3:

The cross-section of the outlet port A2 is typically 10% of the frontal area of the rotor [6].

Therefore, the diameter of the rotor represented by 0,0,”, can be found from:

2
A2 E19. 75.91"“:— (106)
100 4

Assuming a rectangular cross-sectional channel, the channel height is given by:

 

Han = ‘ (107)

which is identical to H(.,.u=A3/W3 due to the continuity equation used in the derivations
above.
The rotational speed of the rotor (nmm), which is a variable to be chosen by the

designer, is related to the rotor angular speed (co) by:

n = — (108)
where a) = V/Dmm/ 2 . Finally, the channel width can be found from the expression:

” D romr
Wu” = -—N—— (109)

where N is the number of channels for a one-row rotor, e. g., in the order of 30.
Implementing the procedure explained in this study for Case A, the discussed channel
and port parameters were calculated and are listed in Table 11 and Table 12. Results
clearly show that the scenario employing two primary shock waves provides realistic
results. Figure 74 shows the location and size of inlet and outlet ports of the charging

process based on the results obtained in the present study.

151

 

Table 11: Flow properties inside rotor channels for L=20 cm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Flow properties Two shocks
Channel pressure in region 1' (p,) 4.45 XIOI kPa
Channel temperature in region 1' (T2) 511 K
Inlet and outlet ﬂow velocity (:13 = 112) 112 m/s
Outlet air Mach number (M2) 0.24
Inlet gas Mach number (M3) 0.17
Static pressure ratio across the primary shock ([75) 1.40
Velocity of the ﬁrst primary shock (ws) 525 m/s
Mach number of the ﬁrst primary shock (Ms) 1.16
Static pressure ratio across the second primary shock in the air region (H5.a_,-,) 1.38
Static pressure ratio across the second primary shock in the gas reﬁnﬂgw) 1.24
Velocity of the second primary shock in the air region (w5.,,_,~,) 436 m/s
Velocity of the second primary shock in the gas region (wag) 630 m/s
Mach number of the second primary shock in the air region (M5_,,_,~,) 1.15
Mach number of the second primary shock in the gas region (Mew) 1.10
Penetration length (L823) 13 cm

 

 

Table 12: Wave rotor design parameters for L=20 cm, N=30, nmmr=10000 rpm, and
m, =0.318 kg/s,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Geometry and port characteristics Two shock

Rotor tangential speed (V) 50 m/s
Rotor diameter (Dung) 9.63 cm
Cell width (Ween) 1.01 cm
Cell height (Hail) 2.23 cm
Circumferential length from 0 to opening of air outlet port (W 2) 1.92 cm
Width of high pressure air outlet port (W2) 3.26 cm
Width of high pressure gas inlet port (W 3) 7.02 cm
Port width ratio (W 2/ W 3) 0.46

Angle from 0° until opening high pressure air outlet (61,) 22.85 °
Opening angle of high pressure air outlet (a2) 38.76 °
Opening angle of high pressure gas inlet (a3) 83.52 °
Time from 0 until opening of high pressure air outlet (’1) 0.38 ms
Outlet air opening time (t2) 0.65 ms
Inlet gas opening time (t3) 1.39 ms

 

 

152

 

 

   

(11:22.8 °
‘7 t1 =0.38 ms

    

 

 

073 :835 o W1=1.92 Cm
t3 =1.39 ms (12 =38,8 '
7 W3 =7.02 cm 12 =o,55 ms
'- W, =3.26 cm
Gas Inlet Port Air Outlet Port

Figure 74: Preliminary port designs of the charging process, corresponding to the wave
diagram sketched in Figure 73

0 Charging Process with Two Shock Waves and an Expansion Wave

Figure 75 shows a second different wave diagram that also utilizes a secondary shock
wave, but now as a reﬂected shock wave. Compared with the wave diagram shown in
Figure 72, this wave diagram has a shorter (more compact) wave rotor design as seen in
the ﬁgure and a relatively great gas penetration length. Creating such a reﬂected shock
wave always implies that the static pressure in the high-pressure air outlet port is greater
than that of the high-pressure gas inlet port. This can easily be the case when using a
combustion chamber with a higher pressure loss as discussed further below in Figure 77.
Here, for the same combustor pressure loss as considered before, employing a reﬂected
shock wave results in a shorter wave rotor and a reduced velocity of the leaving high-
pressure air. Because of mass continuity, the high-pressure air outlet port is now wider.

Figure 75 shows that with a reﬂected shock wave and a shorter rotor, the expansion
wave arrives aﬁer the upper outlet edge, if the tuning condition for the shock wave at
lower outlet edge (TC 1) is kept and the inlet width remains unchanged. This also has

been mathematically proved by Keller [44]. In this ﬁgure W2 is still the outlet port width

153

 

 

 

before shortening the rotor (in the absence of a reﬂected shock wave as discussed before).
Here, W0", is deﬁned as the outlet port width and WT is introduced as the outlet port width
at which the tuning condition for the expansion wave at the upper outlet edge is fulﬁlled
(TC 3). Thus, the actual outlet port width Wm), for the shortened rotor is wider. In fact, it
is in the domain of W2 g W0", 3 WT. For a perfectly tuned design, Wm,r should equal W7,
however it is highly possible that for a given PRW, TC 3 might not be achieved. Then,
another shock wave would originate from the upper comer of the outlet port in order to
satisfy the zero-slip boundary condition at the end wall. This is undesirable and not

shown in Figure 75, although it describes a general case where TC 3 is not fulﬁlled. For

such an untuned situation, the restriction W2 5 WW, 5 W T must still be satisﬁed.

 

Figure 75: Outlet port width comparison before and alter rotor shortening

In the shown case, to reduce the outlet velocity, the reﬂected shock wave originates at

the lower outlet edge and propagates ﬁ'om the right to the leﬁ. The double-compressed air

154

 

after the reﬂected shock wave leaving the wave rotor toward the combustion chamber is
region 2. In Figure 75, L, represents the location where the reﬂected shock wave
intersects the air/ gas interface. The compressed gas region after the reﬂected shock wave
is named region 3'. The reﬂected shock wave reduces the speed of the ﬂow leaving the
channel from u2r to W and causes a further local pressure rise within the channel from p2'
to p2. By fulﬁlling TC 2, the entire ﬂuid column in the channel consisting of two
different ﬂuids moves to the right with the pressure and velocity of the outlet air. In this
way, the entire ﬂuid column is compressed by the reﬂected shock allowing more gas to
enter the channel.

Similar to the same approach described for the charging process without a secondary

shock wave, again Eqs. (82)-(86) are used but with some modiﬁcations:

 

 

 

 

 

 

 

 

Zyair
,/ . R. T, —._T
“27 7 air rm , £377] 7,", +1 (110)
ya” pl” _p_3+&'_’;1
pl/ rair +1
u ’ —u - ir
T,,=T,+(2 “)2 (111)
2Cpair
70"
...,—l
££=[_]i_2]7 (112)
[’2 T2
2
_T + ”2’ (113)
I3 3 ZCPgas
7pm
m-I
&=[_Tg_]’ (114)
P3 T3

where T2 is now equal to:

155

 

 

73:77:;
.. [T 7.2

l

 

T.
T3 (115)

T 2./ T ,4 and T2 /T2 , and functions of the local pressure ratio across the primary
(175: p 3/p 1 .) and reﬂected (HIM-r: p2/p3) shock waves, respectively, and can be obtained
by using Eq. (5). Furthermore, um), is the induced velocity caused by the reﬂected shock
wave and can be calculated from Eq. (7) using 172-02,. Thus, 6 unknown parameters p 3, T 3
, u2' , pp, Ty, and 17mm exist for the above ﬁve Eqs. (110)—(114). This shows that
introducing the reﬂected shock gives an additional degree of freedom. This allows
selecting arbitrary values of 175 and [722-02, in a conﬁned range. Thus, several solutions can
be obtained for given engine data. However, if it is assumed that pp: p, and T1,: T 2 the
number of unknowns is reduced and unique solutions can be obtained for 4 unknown
variables p3, T 3 , u 2, , and 1722.“... This has been shown in a simple ﬁve-step procedure for
a triple tuned wave rotor examined in our previous work [235, 236]. For example, Figure
76 shows the variation of 175 and 1722...), versus PRW for a given 170mb . It is clearly seen
that, for the given [760”, higher PRW values can be only achieved by simultaneously
increasing 175 and decreasing 1712-02,. By assuming pp: p, and T1,: T2 , the results also
show that for a PRW greater than around 1.8, 172-0), becomes less than unity which is
physically impossible for a shock wave. However, a higher pressure drop allows for a

higher 1722-02, as it is shown in Figure 77 described below.

156

1.5

 

mm: 0.83
T“= 1116.5 K

 

 

1.8 1.4

 

1.6 / 1.3

1.4/

 

115
1“IR

 

 

 

 

 

 

 

 

 

 

 

 

11m,,&"‘0';’9r 1-2
1.2 / 1.1
11.5 l 1.6 1.7 1.8 I 1.9 l 21

Wave Rotor Compression Ratio (PRw)

Figure 76: Variation of 175 and HR versus PR W for a constant value of 1760,"),

Assuming pp: p; and T 2': T, , Figure 77 provides a general map relating the shock
strengths 175 and 1722-02, to various combinations of PR W and 110mb. The right boundary of
the map is the 1712-02, =1 line, where the reﬂected shock wave vanishes. The lower and leﬁ
boundary is the u2=0 line, simulating the closed wall condition where no ﬂow leaves the
channel anymore. This condition occurs when 1723.“), approaches the value of 175 . Beside
zero exit velocity line, several lines of ﬁnite exit velocity are shown between the leﬁ and
right boundaries. They show that for a given PRW, increasing 17mm causes a reduction in

u2 due to the induction of mass motion behind the reﬂected shock wave.

157

A‘ 4mm."

 

 

 

 

 

 

 

3'0 (13.-51111111; I-i 71-497041-3 1-2 1.1 HR=I
. .— " ‘ ‘15.? 200
7 l a I. " ‘ 0.03 I ‘ 1
275 _ I 100 T 7 \
- . - ‘ r ~ . [0.02
I , T ~ . 1.9
3 1.7\ .
M 2.0 * ~
:4 opt' - - - - '. '
As/R=0.01 l ‘ ~ I g
* ' - . - . .2 1.5‘
1.5 — ' ~ . ._ ‘
0.0025 ‘~ ~
” "1:0 HS=L1
1.0 L 1 41 L l 1 J_
0.6 0.7 0.8 0.9 1
mental)

Figure 77: Shock strengths 175 , 17R , entropy production As/R , and exit velocity u2 versus
P R W and [lamb

The most interesting feature in the map may be the lines of constant entropy
production in the channels due to the shock waves. The entropy production As is
calculated by summing the entropy generations across the primary and reﬂected shock
waves. They have been normalized by gas constants of the air and gas, respectively. For a
desired ﬁxed PRw , coming from the HM.) =1 line with only a primary shock, As/R
reduces as the reﬂected shock 17R...” becomes stronger. This shows the potential for
higher efﬁciency, if the wave rotor is designed with a reﬂected shock wave. However,
this continues only until the optimum line (solid black) is reached. The optimum line
indicates the combinations of PRw and [lamb at which the least entropy production

occurs in the shock wave compression for a given PRw. Left of it, the entropy production

158

 

increases again for a given PRW. Simultaneously in Figure 77 from the right to the left,
[760mb decreases. This allows for a combustor with a higher pressure drop that may be
smaller and more compact, which may be especially desirable for aerospace applications.
In fact, Figure 77 shows that for the desired PRW=1.8, the value of [Zomb=0.98
considered in the thermodynamic analysis gives a value of 17R-a),<1. Thus, a higher
pressure drop (meb<0.98) should be selected for higher values of 1722-02,, which may
preferably reduce the size of combustion chamber. Finally, the optimum line can be
achieved at lower exit velocities, here lower than 100 m/s.

After calculating the above unknowns, it is now possible to obtain an equation for the
ratio of the outlet port to the inlet port width (Wow /W3) using the continuity equation at

the inlet and outlet ports:

W
1££L_L=1+f (no
'02qu

To ﬁnd the gas penetration length for the charging process with the reﬂected shock
wave, the following equation is introduced by equating the time required for the air/gas

interface to be overtaken by the expansion wave with the opening time of the inlet port

plus the time required for the expansion wave to catch the air/gas interface:

21:, Lg... 1; .L._«‘_Ls_
V u2+a3. u}. u,

 

(up

where 03' is the speed of sound in the hot gas region 3' after the reﬂected shock wave has
passed through it and L, is the location where the air/gas interface meets the reﬂected
shock wave. The temperature in the region 3' is found by using Eq. (5). The local pressure

ratio across the reﬂected shock wave propagating in region 3 (1722-80,: p y/p 3) is calculated

159

1.7-

 

 

similar to the method described for the charging process with the secondary shock wave.
The value of L, can be found by equating the time required for the air/gas interface
originating from the lower comer of the inlet port to reach the reﬂected shock wave
(Ls/212') with the time required for the primary shock wave to run through the channel
(L /wS) plus the time for the reﬂected shock wave to meet the air/gas interface
((L'Ls) /WR-air )3

L, L L - L5
- = — + (118)
“2" W5 WRoair

 

 

Therefore, LS/L can be obtained through this equation as:

 

 

I l
M +w /a
L arr ,-
L 15 R 1 ’ (119)
—+
M WR_W / a 1’

Multiplying both sides of Eq. (117) with a], and dividing it by the length L leads to:

 

AF" 1 ]_ W:

L L M u,/a,, MRL

gas: I

L 1 1 (120)

(u, + a3.)/a,, - u_, /a,,.

 

 

where MR is the rotor Mach number deﬁed as MR: Way. The value of W 3 / MR /L can be

obtained from TC 2:

 

 

W3 L + L—L‘ + L‘ (121)

MR MS wkw, /a,, wk“ /al,

which leads to:

W" = I + I + I — I L‘ (122)
MRL Ms wR_m.r/al,. w /a, wR_a,,/al, L

R—gas l

 

 

 

 

160

Therefore, Lg“. /L can be obtained based on Eq. (120) for a given PRW and 172.0,“. Figure
78 shows the variation of Lg“, /L versus PRW for different values of 172mb. It clearly
shows that the newly obtained value of Lem/L is much greater than that calculated before.

Therefore, employing a shock wave increases the gas penetration length signiﬁcantly.

 

 

 

 

 

 

 

 

 

 

 

 

1
nwc=o.03 ’//
TM=1116.5K 7‘3/ ‘
08 / ’A/Q’ISA”
, , v.0,
//‘ x09 a"
_ /,/ "A “’6’?
06 1 -/ ," ’1’?“
' ' ‘ . 0
i // ’I’d'I’ 00‘“
a ’ " .r"
—J a’ ,e’
0.4 a
,/
0.2
Q1.5 1.6 I 1.7 I 1.8 l 1.9 I :2

 

Wave Rotor Compression Ratio (PRW)

Figure 78: Variation of L,gm /L versus PRW and 17comb

Finally, the maximum outlet port width WT for which TC 3 is satisﬁed can be
obtained by equating the time required for the primary shock wave to run through the
channel (L /ws) plus the opening time of the outlet port (W;- /V) with the opening time of

the inlet port (W3 / V) and the time required for the expansion wave to reach to the end of

 

thechannel:
L L—L
i+ﬂ=ﬂ+—gm_+ gas (123)
ws V V u7,+a, u2+a2

161

 

 

By multiplying both sides of this equation by a), and dividing it by length L and

simplifying:

 

 

1 Lg“, 1 1 I
_ + ._ .—
M5 L (u2+a2)/a, (u2+a3,)/al, (u2+a2)/a

I

W, Wj/MRL

I"

 

(124)

As mentioned above, for any given PRW, W 7- Wow 20 should be a restriction. However,
a designer may prefer to obtain a value of W2— Way, as small as possible to fulﬁll TC 3.
Figure 79 shows the variation of dimensionless values of (WT— WWJ/ WT versus PRW for

various values of n.0,...

100
um: 0.83
9° Tu=1116.5K
so
°\° 70
l.-
; 60
”g 50
3
- 40
.-
5 30
20
10

 

qI.5 1.6 1.7 1.8 1.9 2
Wave Rotor Compression Ratio (PRw)

Figure 79: Variation of dimensionless values of (W2- W0.,,)/ WT versus PRW and 17mm

The plot shows that the expansion wave reaches the right channel end at different
times for different combinations of PRW and [Zomb- Thus, for a desired ﬁxed PRW there

is only a unique 17mm), that satisﬁes TC3 and vise versa. Furthermore, with increasing

162

 

 

 

PRW the divergence from TC 3 increases, which is not preferred for an efﬁcient design. In
a practical design, the rotor may incorporate various cavities (pockets) in the end plates to
reduce the sensitivity to a mismatch of the various wave arrival times [44]. These

mismatches can be caused by other events such as variation in the engine load.

5.1.3 Validation

To validate the accuracy of the analytical prediction model described above,
comparisons with numerical results of a test case have been made. Very little
experimental data are published in the literature concerning successful operation of wave
rotors. Therefore, it was decided to validate the above analytical model with existing
numerical predictions which have been validated with some previous wave rotor
investigations.

The data of the wave rotor numerically designed at University of Tokyo are selected
[215]. Table 13 shows the pressure and temperature of the inlet and outlet ports of this
wave rotor as they were imported into a computer. code written in MathCAD software
according to the above analytical approach. Additional design parameters of the Tokyo.
wave rotor are shown in Figure 80 in which HP stands for high pressure and LP is used

for low pressure.

Table 13: Inlet and outlet port data of University of Tokyo wave rotor,
taken from Ref. [215]

 

 

 

 

 

 

 

 

 

 

 

 

Inlet air pressure p,, 3.0 XI01 kPa
Outlet air pressure p,» 10.1 XI01 kPa
Inlet gas pressure pd 9.2 XI01 kPa
Outlet gas pressure 712,4 3.7 XIOI kPa
Inlet total air temperature 7",, 440 K
Outlet total air temperature T,2 907 K
Inlet total gas temperature 733 12048 K
Outlet gas total temperature TM 973 K

 

163

 

 

 

   

 

 

7, 7.77 \\\\ 7 ,7’;// \“\‘7\ 7
\ 1 :1
I I’ \\ o " 1 " ‘1
— ----- 4——/ “° ----- 1 ex W
l 7 1’ . ‘ /' \ 77 I
15920° - r \\ s 1‘ ’ ‘ .’ °
/ 27' Gas-HP \\ I,” ii \ « \21-5
Air—LP X72: 7:. / n43° 128‘” .%77 / A'I'HP
11563” Gaﬂpisf 616
Charge Side (Left Side) Discharge Side (Right Side)

 

Figure 80: University of Tokyo wave rotor, taken from Ref. [215]

Using some of the data provided in Table 13 (cell number, cell length, mean rotor
radius, and rotational speed) and port locations shown in Figure 80, the analytical model
predicts reasonable wave rotor geometry parameters as listed in Table 14 in comparison
with the published data of the Tokyo wave rotor The listed data of the model are obtained

for initial channel pressure and temperature of p,» = 4.34 XI01 kPa and T]: = 707 K.

 

Good agreement between the predicted results and those obtained by numerical

164

 

simulations is observed. The existing errors can be related to the assumptions made in the
model, e.g., assuming pure air at the beginning of the high-pressure part of the cycle and
using a constant speciﬁc heat at different states of air and gas. In the ﬁrst step, the
accuracy of the present model can be improved by solving the ﬂow ﬁeld in the low-

pressure cycle to determine how much the temperature and pressure of the air ﬁlling the

entire channel at the beginning of the high-pressure part model should be changed.

Table 14: Comparison between the analytical model and the numerical data of

the Tokyo wave rotor

 

 

 

a ‘V

 

 

 

 

 

 

 

 

Geometry and port characteristics Model Tokyo WR

Cell width (If/egg) 0.49 cm 0.4 cm
Cell height (H2221) 0.3 cm 0.3 cm
Circumferential length from 0 to opening of air outlet port (W 1) 0.76 cm 0.88 cm
Width of high pressure air outlet port (Wow) 1.65 cm 1.62 cm
Width of high pressure gas inlet port (W3) 2.1 cm 1.87
Port width ratio Lng/ W 3) 0.78 0.87
Angle from 0° until opening high pressure air outlet (61,) 18.7 ° 21.6 °
Opening angle of high pressure air outlet (a2) 40.6 ° 40.0 °
Opening angle of high pressure gas inlet (a3) 51.7 ° 46.0 °

 

 

 

 

 

5.2 Numerical Simulation

Even though the above analytical procedure can provide some useful design
parameters for a preliminary design, for a complete design, a whole-cycle analysis
including the low-pressure part of wave rotor operation is required. In analytical
approaches, the method of characteristics is often used to solve the ﬂow ﬁeld especially
in the scavenging part of the cycle [98-105]. An alternative technique is to use CFD
methods, allowing more accurate simulation of the ﬂow ﬁeld. It has been common
practice to create and use specialized codes. As reviewed in Chapter 2, several precise
numerical codes have been developed so far. However, most of these codes are not

commercially available, hence a few groups of researchers who have developed these in-

165

 

 

house codes are using them, e.g., the NASA wave rotor code. A broader accessibility of
appropriate CFD tools could facilitate a wide range of wave rotor analysis. The
development of modern multi-purpose commercial software packages has reached a level
that allows the successﬁil modeling and analysis of the operation of many technical
devices, including wave rotors. Several commercial codes are now available that can be
applied to investigate many problems related to wave rotor design and operation. Such
codes are particularly interesting for 2D and 3D modeling of full devices and some
special problems. They offer tools that allow for relatively easy geometric preparation, a
range of typical boundary conditions, relatively fast and robust solving, and a wide range
of post-processing which is valuable for engineers and scientists. Yet, the use of such
codes is not as fast as the use of common ofﬁce soﬁware. Although the geometry and
boundary conditions can be modeled relatively fast, using such complex commercial
software packages, the computational effort is still enormous, so that ﬂow ﬁeld is often
only available after lengthy, time—consuming computations. Therefore, such simulations
are not suitable for an initial geometry search or a geometry optimization but can be
performed as a last stage of investigation, verifying solutions of particular problems or
the full operation of a complete wave rotor. For preliminary investigations, initial design,
and optimization they are not necessarily as efﬁcient as specialized codes.

Some attempts at simulating pressure wave superchargers with the help of
commercial codes already have been undertaken. One such code is GT-POWER, in
which pipe elements have been divided into a series of objects for which the conservation

equations have been solved. An interesting description of techniques for optimization of

166

. -..- '_.. .....-.

n

 

 

timing, shaping, and control of pressure wave changers using GT-POWER has been
described by Podhorelsky et al. [237].

The present work shows an attempt at developing 2D models of wave rotors using the
commonly available CFD software FLUENT. This enables the realization of detailed
gasdynamic phenomena inside the rotor channels. For most of the results investigated
here, very little experimental data exist for possible veriﬁcation. Therefore, the results are
compared with some previous numerical data which are available for conventional wave
rotors. The presented results shall not be interpreted as a proposition of a particular
geometry for practical applications. While they can be a base for further investigations,
they rather present some illustrations of physical phenomena generated in different

conﬁgurations and phases of operation and showing the capability of the code.

5.2.1 Approach

The numerically designed Tokyo wave rotor (see Table 13 and Figure 80) was
selected as a model and reference for the numerical simulation presented here. The main
reason for selection of this wave rotor is the availability of detailed geometry data and
operating conditions. Figure 81 shows the temperature contour in this wave rotor
predicted at the University of Tokyo. While for this port arrangement the pressure
contour is not reported, it is available for another operating condition as shown in Figure
82. The goal here is to produce similar results using the commercial software FLUENT.
The numerical procedure described here can be performed for almost all other wave rotor

conﬁgurations.

167

"m 11".?“ mﬁizkummww

 

 

 

 

 

 

 

,. “1111 kiiu
W‘WIWM ’ . 1111'“ 1313.111113- m.

ann1mlllllm1llllwllm “1m

"‘an 7rmlItmumiuuunhlllii

    

rotation

Figure 81: Non-dimensional total temperature contour of the Tokyo wave rotor
taken from Ref. [215]

168

 

IPrimr h kal

 

 

    
  

 

 

Secondaﬂ
Shock Wave

 

 

 

Air-LP

  

rotation

Figure 82: N on-dimensional total pressure contour of the Tokyo wave rotor,
taken from Ref. [215]

The ﬁrst step in the ﬂow simulation with FLUENT is the mesh generation for a
particular geometry. GAMBIT is the default grid generator for FLUENT. Even though
GAMBIT has the ability to draw the geometry, a more advanced drawing software like
AutoCAD was used to sketch the geometry. Using AutoCAD signiﬁcantly reduces the
sketching work. After creating the geometry in AutoCAD, the geometry was imported

into GAMBIT for the grid generation. The mesh size in the channels and inlet and outlet

169

 

ports was carefully selected to provide accurate and fast solutions. Tests showed that
ﬁner meshes would not provide more accurate results, only causing additional
computational time. Figure 83 shows the unstructured mesh used for this study at the
beginning of the wave rotor operation. In the computational model, 30 channels (not all
shown) move in a block upward perpendicular to the ports. A ﬁner mesh has been used in
the interface of the channels and all four ports, as shown in Figure 84. This enhances the
simulation of gradual port opening and closing.

The most challenging step in generating the mesh is the creation of a stationary
interface between the moving channels and the stationary ports. This was carried out by
creating a relatively small gap (0.1 mm) between the wall ends of the channels and the
end plates with the ports where an interface connects them. The yellowish line in Figure
85 indicates the interface where the entire right side connected grid lines slip along it
during the wave rotor operation. An additional advantage of generating such a gap is the
actual simulation of the leakage between the stationary end plates and the rotating
channels.

Once the grid was generated in GAMBIT, the rotor geometry was imported into
FLUENT. Boundary conditions, the solving method, the working ﬂuids, the convergence
criterion, and several other functions (parameters) must be deﬁned prior to the
simulation. Air as an ideal gas was chosen as the working ﬂuid. Such an assumption
reduces the accuracy of the results because the effects of gas at the inlet and outlet ports
and the burned gas in the rotor channels are neglected. However, the trends of the
gasdynamic processes occurring in the channels are consistent with more accurate models

as shown later.

170

‘I

 

 

 

 

 

Jul 19. 2004

 

 

 

 

 

 

 

 

 

Grid
FLUENT 6.1 (2d. dp. segregated. lam]
Figure 83: Generated mesh
1
1
Grid Jul 19, 2004
FLUENT 6.1 (2d. dp. segregated, lam]

 

Figure 84: Finer mesh at port interface and channels, inlet high-pressure port

171

 

 

Interface

/

 

 

 

 

Grid Jul 19. 2004
FLUENT 6.1 (2d. dp. segregated. lam]

 

 

 

Figure 85: Interface and gap between stationary ports and moving channels

In this study, the governing equations of 2D unsteady laminar ﬂow were selected and
solved explicitly with a ﬁrst order upwind difference scheme. The boundary conditions at
the inlet and outlet ports play important roles in the ﬂow ﬁled simulations. These
boundary conditions consist of ﬂow pressure and temperature values at the inlet and
outlet ports. The compressor exit pressure and temperature were assigned as initial values

for the upward moving channels.

5.2.2 Numerical Results

Figure 86 shows contours of total pressure at two succeeding time steps when the ﬁrst
moving channel is exposed to the high—pressure gas inlet port. As expected, the high-
pressure ﬂow starts compressing the low-pressure ﬂuid in the channel by generating
compression waves. The pressure stratiﬁcation along the channel length indicates the

action of compression waves. These compression waves form a stronger single shock

172

wave running through the channel. The bottom part of Figure 87 can be interpreted in this
way. Figure 87 also shows how the second channel is inﬂuenced by the high-pressure gas
inlet port as the computational time increases.

Figure 88 depicts the moment when the head of the compression waves nearly meets
the right end wall. The top ﬁgure indicates the moment before the wall contact, and the
bottom ﬁgure presents the contact moment. The pressure contours clearly indicate the
compression of the air by the compression waves.

Once the compression waves reaches the end wall, a reﬂected shock wave is
generated as shown in the two following time steps in Figure 89. The reﬂected shock
wave compresses the air further. The doubled-compressed air leaves the channel by the
opening of the high-pressure air outlet port. A pressure peak well above the inlet pressure
is seen at the moment the high-pressure air outlet port opens.

Figure 90 shows how the ﬂow is scavenged from the rotor channels by opening the
gas outlet port. This process is supported by the generation of expansion fans traveling
through the channels toward the air inlet port.

Finally, the fresh air is ingested into the rotor channels by opening the air inlet port as
shown in Figure 91. It is interesting to note that a region of lowest pressure even less than
the air inlet pressure (here shown numerically as negative pressure) is created just before
opening the air inlet port. This can be interpreted as the reﬂection of the expansion wave
off the left end plate shortly before the air inlet port opens. This low- pressure region
signiﬁcantly assists the ingestion process. Closing the air inlet port terminates the ﬁrst
operating cycle. The pressure contour at this moment is shown in Figure 92. All of the

above described phenomena can be seen in this ﬁgure.

173

W. " ”at" ’ﬁaaiiiﬁfﬁnmelmw

 

Figure 93 shows the enhancement of ﬂow circulation in the beginning parts of the
channels, which becomes dominant resulting in some outﬂow from of the channel shortly
before the high-pressure gas inlet port closes. In the lower corner of the gas inlet port, the
effect of gradual channel opening is magniﬁed as shown in the top section of Figure 94,
also showing some leakage effects. The bottom part of Figure 94 represents a focused
zone of the upper comer of the high-pressure gas inlet port, indicating less leakage
because of ﬂow blockage caused by intense ﬂow recirculation in the beginning of the
channels. This ﬂow blockage suggests that a narrower gas inlet port may be substituted
for the current wider port.

Figure 95, Figure 96, and Figure 97 show by colors the axial component of relative
velocity vectors at air outlet port, gas outlet port, and air inlet port, respectively. For all
three ports, recirculated ﬂow regions exist in the upper port corner which again suggests
that the port widths for all three ports can be reduced. Whereas the air inlet port shows
the strongest recirculation, the air outlet port shows the least.

For the above described ﬁgures only the ﬁrst operating cycle is simulated. To reach a
periodic solution, the code needs to be run with more cycles. For instance, convergence
to a periodic unsteady state has been achieved after 20 rotations in a study by Fatsis and
Ribaud [198, 201]. This can be performed in FLUENT in a 2D simulation by extending
the current port arrangements, as shown in Figure 98 for a 3 cycle port arrangement
(coarser mesh is used for a better visualization of the reader). Another approach would be
a quasi 2D simulation with a 3D model Unfortunately, the generated data requires a huge

memory, which was not available for this research. Such a periodic solution, however,

174

can be simply obtained in the 2D simulation of a radial-ﬂow wave rotor. This is the topic

of the next chapter.

 

 

175

lDle'US

9.75eou5
9.3913‘05
9.02e*05
, seams

 

3.84e‘05
3.47e‘05
3.10e'05
2.73e’05

 

 

Contours of Total Pressure (pascal) (Time= l. Bﬂﬂﬂe- 06] l08. 2004
FLUEN T6.l (2d. dp. coupled imp. lam.U unsteady)

 

1.026’06
9.8413415
9.48e‘05
9.12e‘05
8.7813415
8.40eo05
"Ti 8.05e'05
7.69e‘05
7.33e'05
8.976'05
; saunas
__, 6.256’05
5.90e*05
5.546‘05
5.18e+05
4.826'05
4.46e‘05
4.106‘05
3.75erﬂ5
3.3991‘05
3.036'05

 

 

 

 

Contoms of Total Pressure (pascal) (Time= 6. SDUUe-l] (1.8 2004
FL EN T8. 1 (2d. ldp. coupled imp. lam.U unsteady)

 

Figure 86: Total pressure contours, the effect of high—pressure gas inlet port on
the lst channel

176

 

 

 

l.l4e~06
l.lI]e*1115 ‘
1.056416
1.016‘06
9.7Ue~05
9.256’05
8.86e'05
8.44e*|]5
8.036’05
7.619%15
7196415 750:1";
6.776’115 '

6.361345
5.941e‘05
1;: 5.526’05
5.111e*05
4.68e‘05
4.278‘05
3.85:.”05
3.43et05
3.1116'05

 

 

 

 

 

   

 

 

Contours of Total Pressure (pascal) (Time= 15111106 Jul [18. 211114
FLU ETN 6.1 (2d5. dp. coupled imp, lam. unsteady)

 

1.156’06
1.11er06
1.08e*06
1.0213416
9.78e005
9.32.3415
8.88e*05
8.43e*05
7.9913415
7.55e+[15
7.119'05

 

 

 

 

3.125415
2.686415

   

 

Centaurs of Total Pressure (pascall (Time=2.8110[1e Jul 118. 211114
FLU NT61(205.d]dp. coupled imp. larn. unsteady]

 

Figure 87: Total pressure contours, the effect of high-pressure gas inlet port on
the lst and 2nd channels

177

 

1,1 1.3-416
1.017e416
1.[|3e*06
9.878415
9.46er05
9.0E1er05
8.65e4)5
8.245415
7.846415
7436*05
7.03ev05
6.626415
6.21e‘05
5.816415
5.40er05
; 4.99erﬂ5

4.59e‘05

4.18e‘05

3.77:415

3.37e415

2.96e'05

1111‘-

 

 

 

 

 

Contours of Total Pressure (pascal) (Time= 3 3000e-0 5) Jul 08. 2004
FL LUENT 6.1 (2d. dp. coupled imp. lam. unsteady)

1.056'06
1.026'06
9.7923415
9.416‘05
9.03e'05
8658*05
8.27e415
7.89e*05
7.51e*05
7.1363905
6.7513415
6.37e415
5.9913415
5.611905
5.246415
4.86e415
4.48e415
4.106415
3.72e~05
3.34et05
2.96e*05

2%:

 

 

 

 

Contours of Total Pressure (pascal) (Time= 3 9000e-0 Jul 08. 2004
LUNEN16.1 (2d. dp. coupled 1mp. lam. unsteady)

Figure 88: Total pressure contours, traveling of compression waves toward end wall

178

7.655415
7.195.115
': 61312.05

5; 6.2712415

‘2 5.3151115

5.35.9415
“ 4.8913415
4.43e415
3.9713105
3.51e‘05
3.0413415
2.5812415

 

 

 

 

 

 

Contours of Total Pressure (pascal) (T1Fme= 4. 9000e- 05 1 Jul 08. 2004
FLUENT 6.1 (2d. dp. coupled imp. lam. unsteady)

 

1.18er06
1.1 le*06
1.0713416
1.0312416
.1 9.84e'05
' sums
8.9813415
85515.05
8.12e‘05
reams
7.2319415
6.8315415
6.40e+05
5.97e*05
5.54e+05
5.1115415
4.6812415
4.25e+05
3.82e405 ,
3.3812415 " "i=r
2.95e+05

   

    

 

11

 

1

 

 

 

 

 

Contours of Total Pressure (pascal) (Time=5.70006-051 Jul 08. 21104
FLUENT 6.1 (2d. dp. coupled imp. lam. unsteady)

 

Figure 89: Total pressure contours, generation of reﬂected shock wave at two different
time steps

179

 

 

 

.28e‘06
.23e416
.166416
.136436
.06e’06
1.036'08
9.84er05
9.35e415
8.85e‘05
6.366r05
7.87e*05
7.37e*05
6.86e415
6.39e‘05
5899*05
5.40e415
4.90e‘05
4.4le*05
3.92e*05
3.42e415
2.93e415

._._-._.._..,4

an":

 

 

 

 

 

 

 

 

 

      
 

(Time= 2. 0300e-0 Jul 08. 2004

Contours of Total Pressme (pascal)
FTLUEN 6.1 (2d. ldp. coupled imp. lam. unsteady)

 

 

1.34e*06
1.298416
l.24e*06
1.19e‘06
1.13e*06
l.08e+06
1.03e'06
9.76e‘05
9.24e‘05
8.7113905
8.19e*05
7.666+05
7.14e+05
6.6]e‘05
6,096+05
5.576415
5.04e415
4.52e415
3.99e'05
3.47e'05
2.9449415

 

 

 

 

 

 

 

 

 

Contours of Total Pressure (pascal) (Time= 2. 2900e-0 41 U] 08. 21104
LUEN NT 6 1 (2d. dp. coupled 1mp lam. unsteady)

 

Figure 90: Total pressure contours, air scavenging process at two different time steps

180

 

1.449415
1.36e'06
1.2913‘06
1.21906
1.146415

9 1.07e+06
5‘2: 9.93905
9.19e‘05
— 8.4551115
7.7169115
6.978905
8.236415
5.491905
4.756415
4.003905 ‘
3.26e905
2.52e‘05
l.78c*05
1.04e'05
3.02904
-4.38e*04

 

 

 

 

 

Contours of Total Pressure (pascal) (Time-2790060412400
FLUENT 6.1 (2d. dp. coupled imp. lam. unsteady)

 

 

1.346415
1.26e'06
1.1813006
l.lle*06
1.0313006
9.53e+05
8.77e'05
8.00e'05
7.23e*05
6.473905
5.70905
4.93905
4.16905
3.40e+05
2.833905
1.86er05
1.09e005
3.27e+04
-4.40e'04
-1.21e~05
'l.97e*05

 

 

 

3.5100 08. 2004
NT 6.1 6(204d )dp. coupled imp. lam. unsteady)

Contours of Total Pressure (pascal) (T1me=

 

Figure 91: Total pressure contours, ingestion of fresh air at two different time steps

 

181

 

L38e+06
l.31e+06
1.236416 1
1.15e+06
1.08e+06
1.01e+06 g _ 1 1
9.3le+05 1________ 1' f" f t; ' '
7.80e+05
7.04e+05
6.29e415
5.5319415
4.78e‘05 . ,
4.02e+05 7" "
3.2se+05 ;
2.51e+05 }

1.75e+053 ~ 7% 9 1"" r :
9.975414 1 -- .- .. " "
l

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2.41e+04 '
-5.15e+04
4.2712415

 

 

 

 

 

 

 

Contours of Total Pressure (pascal) (Time=3.9400e-041 Jul 08. 2004
FLUENT 6.1 (2d. dp. coupled imp. lam. unsteady)

 

 

 

Figure 92: Total pressure contour, end of the ﬁrst operating cycle

6.23e+02
5.66e412
5.09e+02
‘-: 4.51e*02
E 7.1 3.945412
.1 3.37e+02
2.79e412
2.22e412
l.65e+02
l.089+02
5.03e+01 ‘
4‘056’00 Fit-5-" -:77. . .
- 6.4 4 e + 0 1 “f: ;;;:}::::‘;1 12:71.. 322'; '7 1 .
-l.22e+02 .' :; ;:..
~1.799r02
“2.36e+02 '%.1“.j..__:;5';‘:_..._... .
-2.94e+02 ' 13326:6312}???'5"
-3.5]e+02 ' F
~4.08e+02 . ‘-~_;“’N
-4.66e*02 ~75 5__e;~
*5.236*02 ' ‘

 

  

 

 

 

'

...
.0
i

I

 

 

 

 

 

 

relative-velocity Colored 6y X Velocity (m/sl (Time=4.4700e-041 Jul 16. 2004
FLUENT 6.1 (2d. dp. coupled imp. lam. unsteady)

 

 

 

Figure 93: Axial component of relative velocity vector, gas inlet port and channels

182

6.23e+02
5.66e+02
5.09e‘02
, _ 4.5113402
is 3.94e‘02
3.37e+02
2.79e'02
2.22e‘02
1.65e412
1.066v02

 

 

 

5.03: 0%
-7.05e*00
-6.44e411
-l.22e‘02
'1.796’02
-2.36e'02
-2.94e+02
-3.51e412
'4.06e*02
”4.66er02
-5.23e'02

 

 

 

 

 

 

relative- velocity Colored By X Velocity [WE] (Time-4.4700e-01.6 2004

 

NT 6.1 (2d. dp. coupled imp. lamlJ unsteady)

 

6.23e*02
5.6613412
5.09e412
4.51e412

 

 

3.94am:
3.3712412
2.7913412
2.22e~02
1.85e412
1.086412
5.03e+01
-7.05e+00
-6.44e*01
4.2212702
'1.796’02
-2.36e~02
-2.94e+02
'3.516‘02
-4.08e412
“1.6812412
-5.23e+02

 

 

 

relative- velocity Colored By X Velocity FIm/s) [Time= 4. 470 0e 16 2004

UT
FL UENT 6.1 (.2d dp. coﬂu4pled imp. 1am. unsteady)

 

Figure 94: Axial component of relative velocity, lower and upper comers of gas inlet port

 

183

6.23e+02

 

5.66e*02
5.09e412
16:1 4.516‘3?
i:- ’.i 3.94et02
3.37e+02
2796*02
2.22e*02
l.65e*03
l.066+02
5.03e*Ul
-7.05e+00
-6.44e*01
-l.22e41?
-l.7Qer02
-2.36e+02
-2.94e*02
-3.51e*02
-4.08e+02
-4.66e*02
‘5.23e412

  

 

 

1'. 7‘.
‘1' l
.

 

 

 

 

 

relative-velocity Colored By X Velocity (m/s) [Time=4.4700e-04) Jul 16. 2004
FLUENT 6.1 (2d. dp. coupled imp. lam. unsteady)

 

Figure 95: Axial component of relative velocity, air outlet port

 

 

 

 

6.23e+02 IQ; .
5.66e+02
5.09e412
. 4.51e+02
3.9412402
3.37e412
2.79et02
2.22e412
1.65-$02
1,089+02
5.03e*01
-7.05e410
-6.44e+01
-l.22e+02
-1.79e+02
-2.36e*02
-2.94e+02
-3.51e+02
-4.08e+02
”1.6613417 ~14?“

I. "E‘:'

-5.23e+02 5"“.

 

  

1";

 
 

 

 

 

 

 

 

 

T

 

 

relative-velocity Colored By X Velocity lm/s) (Time=4.4700e-041 Jul 18. 2004
FLUENT 6.1 (2d. dp. coupled imp. lam. unsteady)

 

Figure 96: Axial component of relative velocity, gas outlet port

184

 

 

6.2313412l . J
5.66e+02
6.0.9902
4.5163412
3.94e+02
3.37e+02
2.7919412
2.2263412
1.65e4l2
1.06e412
5.03e+01
-7.05e*00
-6.44e‘01
-l.22e+02
-1.7Qe+02
-2.366+02
-2.94e+02
“3.5let02
-4.08e*02
'4.669+02
'5.23e+02

 

 

 

 

 

...,
‘.

 

 

 

 

 

relative-velocity Colored By X Velocity [m/s) (T1me=4.4700e-04) Jul 18. 2004
FLUENT 6.1 (2d. dp. coupled imp. larn. unsteady)

 

 

 

Figure 97: Axial component of relative velocity, air inlet port

l5:“)

 

l
1.-..FLJ

 

 

 

 

“‘1'" .- .

 

 

 

Grid Jul 19. 2004
FLUENT 6.1 (2d. dp. segregated. lam)

 

 

Figure 98: Mesh for multi cycles

185

o ":. 'n

 

 

CHAPTER 6: INNOVATIVE WAVE ROTOR DESIGNS AND APPLICATIONS

The MSU wave rotor group has initiated studies to evaluate the possible beneﬁts of
utilizing wave rotor technology in several thermal cycle applications. Besides the efforts
described so far, the team has also investigated and developed several innovative

conceptual designs, which will be brieﬂy discussed in this chapter.

6.1 Radial Wave Rotor Concept

As described in the wave rotor history of Chapter 2, for various wave rotor
applications several different conﬁgurations of mainly axial-ﬂow wave rotors have been
studied so far. The four-port version with straight channels has been used most widely.
However, pure scavenging is a challenging task in axial-ﬂow conﬁgurations. In gas
turbine applications, neither the TF conﬁgurations nor the RF conﬁgurations can often
achieve a full scavenging process, as discussed in Chapters 3 and 5. An innovative design
taking advantage of centrifugal forces can improve the scavenging process, as described

in the following.

6.1.1 Radial-F low Wave Rotor with Straight Channels

Here, the radial-ﬂow wave rotor concept (wave disc) is introduced employing a ﬂow
in the radial and circumferential directions. This can substantially improve the
scavenging process by using centrifugal forces. Figure 99 shows schematically a simple
radial-ﬂow wave rotor with straight channels and a constant rectangular cross-sectional
area. Similar to axial-ﬂow wave rotors, if the driven ﬂow enters and leaves at the inner
radius and the driver ﬂow enters and leaves at the outer radius, such a conﬁguration can

be referred to as a reverse-ﬂow (RF) design. For a through-ﬂow (TF) conﬁguration, most

186

.t)". PM,
o

w '..I-' ".r .' .0 - ‘.
e I“:

 

of the driven ﬂow travels through the full length of the channel and leaves at the outer

end.

 

Figure 99: Radial-ﬂow wave rotor with straight channels

To verify the general idea of wave disc operation, where our knowledge is still very
limited, the commonly available software package FLUENT has been used to simulate
the ﬂow ﬁeld inside a RF wave disk. Here, using FLUENT it has been shown that the
novel concept of wave discs and their particular problems can be investigated by 2D
models. The model considered here has 60 straight channels with an inner radius of 0.15
m and outer radius of 0.3 m, rotating clockwise at 8330 rpm as shown in Figure 100. The
results presented below have been provided by a collaboration between the MSU wave
rotor group and the wave rotor team at Warsaw University in Poland.

Two sets of four ports, as in a conventional four-port wave rotor, are used. It is
assumed that the fresh air enters the channels with temperature a of 300 K at a pressure of
105 Pa. It is compressed by a high-pressure, high-temperature gas entering at 1000 K and

2.105 Pa. The gas expands and leaves at the lower value of 105 Pa.

187

 

 

 

pressure
air inﬂow Compressed

 

Figure 100: Reverse-ﬂow wave disk with straight channels

From top to bottom, Figure 101 shows contours of local pressure, local temperature,
and velocity. Due to the symmetry, only half of the solution is shown. Relatively uniform
regions shown before and after the four ports indicate that the combination of diameters,
speed, and port arrangement is not optimized, because nearly no changes are seen in these
regions. Only the pressure contours show a certain radial stratiﬁcation in these regions,
indicating the action of centrifugal forces. They also clearly show the effects of
compression and expansion waves and how the low-pressure region is created by
supporting the ingestion of fresh air. The temperature contours clearly show the ﬂow
casing especially for the burned gas which is a typical feature of RF conﬁgurations.
Furthermore, the contours of pressure and velocity show effects of gap leakage,

especially on the left side before the high pressure gas port.

188

 

 

 

 

 

Figure 10]: Wave disc with straight channels: contour plots oflocal pressure (top), local
temperature (middle), and velocity (bottom) at timc:8.6c-04 s

189

 

 

6.1.2 Radial-F low Wave Rotor with Curved Channels

As Figure 102 suggests, the channels alternatively may be curved and varied in cross-
sectional area. Compared to straight wall channels, curved channels provide a greater
length for the same disc diameter, which can be important to obtain certain wave travel
times for tuning. With curved channels also the angle against the radius can be changed
freely. Furthermore, curved channels may be more effective for self-propelling and work
extraction in the case of a wave turbine or work input for additional compression,

analogous to the principle of turbomachines.

 

Figure 102: Radial-ﬂow wave rotor with curved channels

Figure 103 is the conﬁguration used in the numerical simulations presented here. The
disc radii and number of channels are the same as for the wave disc with straight
channels, as explained before. The rotational speed now is 8300 rpm. While the
temperature boundary conditions for the ports also are the same as before, now two
different high pressure levels are used. In the lower right comer 3.105 Pa is set for both
high-pressure gas inlet and high-pressure air outlet ports while 2.10S Pa for these ports in

the upper left comer, both parts being independent.

190

 

Figure 103: Reverse-ﬂow wave disk with curved channels

The results are presented in Figure 104, showing that compression and expansion are
generally working. The static pressure contours show a radial pressure stratiﬁcation in the
regions where both ends of the channels are closed as discussed in Figure 101. The
complete temperature contours now show a similar casing like in Figure 101, but with
deeper gas penetration and an unexpected carry on of expanding hot gas stretches after
the exhaust gas port opens. The penetration is obviously deeper at the right side where the
high-pressure level is 3.105 Pa.

More results and observations in the simulation of rather complex time-dependent
ﬂow phenomena that occur in radial wave rotors are not presented. The reader is referred

to Ref. [238] for a complete discussion.

191

 

 

 

 

Figure 104: Wave disc with curved channels: contour plots of local pressure (top), local
temperature (middle), and velocity (bottom), at timc=1.73e-3 s

192

6.1.3 Radial-F low Wave Rotor Concept

The wave disc described above can be also stacked together as shown in Figure 105.
This way a modular construction is possible that can be adapted for designs with different
mass ﬂow rates. Furthermore, similar to the known two-row axial Comprex, the channels
are subdivided, which can allow for acoustic noise reduction. Such a wave disc stack can
be used in the same way as a single disc wave rotor for different applications like

refrigeration, gas turbine topping and supercharging of IC engines.

 

Figure 105: Stack of wave discs with straight channels

Stacked wave discs provide the unique opportunity to place a radial ﬂow
turbomachinery at the periphery of the axis of the wave disc stack with an angle such that
the turbomachine impeller periphery interfaces all active discs of the stack. In this way, a
peripheral continuous outﬂow from turbomachine impeller and inﬂow to the disks is
possible without any additional ducting, collector, volute, diffuser or nozzle between the
turbomachine impeller and the wave rotor. Thus, ducting losses are eliminated, resulting
in a higher efﬁciency of the assembly. Figure 106 shows a conﬁguration in which a radial
compressor is placed inside a wave disc stack. Only the inner plate is placed between the

impeller and the inner wave rotor. Due to the angle between the axes of the impeller and

193

 

 

wave rotor, the end plate between both can be spherical for minimum thickness (ducting
length). This also allows switching on and off outer discs by varying the angle between
impeller and wave rotor axis. The port opening can be a continuous oblique slot that
interfaces with the impeller periphery. Since the end plates are stationary they can form
one part with the housing of the turbo impeller as shown in Figure 106 for the outer
impeller shroud and axial duct. The shape at the outer diameter of the wave rotor stack is
generated by the shape at the inner diameter, the channel length, inclination and timing of
each disc. Still if the outer shape is similar to the inner shape of the wave disc stack, the
timing on each disc is different and is determined by the circumferential distance from

one port to the other at the inner diameter , as shown schematically in Figure 106.

 

Figure 106: Stacked radial wave discs and radial compressor

In gas turbine applications preferably the turbo-compressor impeller is placed inside
the wave rotor. Such a design eliminates the need for a diffuser which has been replaced
by a more effective shock deceleration process [1, 2] in the wave disc channels. Using an
outward-ﬂow turbine, the turbine could be placed at the outer diameter with its axis also
set at an angle to the wave rotor axis but rotating around the wave rotor axis with respect
to the compressor axis, allowing a certain time between opening the channels at the inner

and outer diameters. Such a conﬁguration might be too challenging and would require

194

separate shafts for compressor and turbine. To avoid a gear box, their coupling could be
achieved electrically via generator and motor. Figure 107 shows a simpler conﬁguration
with a direct shaft coupling compressor and turbine as in a gas turbine. This requires a
ﬂow collector from the wave rotor outer end plate and certain ducting that directs the
ﬂow to the turbine as shown schematically in Figure 108. The conﬁgurations shown here
uses an internal combustion wave rotor that allows for outward ﬂow only in the wave
rotor. If a conventional external combustor is used than an additional port opening is
necessary for the burned gases leaving the combustor and the high pressure air entering
the combustor. External ducting may then be eliminated by having combustion in the

pressure exchange channels.

 

Figure 107: Cut~view of a radial wave rotor topping a gas turbine

Figure 109 shows an exploded View of a radial wave rotor topping a gas turbine. The
outer end plate is shown with an oblique slot as it would also be suitable for a
peripherical outer radial outﬂow turbine. However, for an external turbine as shown here,

the slot of the outer end plate can have any form that will adapt most conveniently to the

195

outlet opening time. In the ﬁgures sketched here, a turbine volute is used to distribute the
ﬂow around the turbine. The exhaust gas leaves the turbine axially. The reader is referred

to Ref. [239] for additional discussions about the radial wave rotor concept.

Fresh air

Fresh air intake
Intake wave rotor
port/plate

Internal combustion
radial wave rotor

    
 
 
   
    

Compressor

Driving shaft

Turbine volute

Turbine

Expanded gases
exhaust

Expanded burned
gases

Figure 108: Flow through an internal combustion radial wave rotor topping a gas turbine

196

9'

I \ 5. compressor inlet port

4. radial compressor

/ 3. Internal end plate
2. external end plate
1.3tacked radial wave

/ rotors
6. turbine volute
/ 7. turbine

 

  

turbine exit port

   

 

Figure 109: Component parts of a radial wave rotor topping a gas turbine

6.2 Condensing Wave Rotor

Wave rotor technology has the potential to enhance performance and reduce the size
and cost of refrigeration cycles using water (R718) as a refrigerant. While R718
refrigeration systems have shown attractive features compared to commonly used
refrigeration system, these units often suffer from the expensive and bulky multi-stage
turbo-compressors which are crucial units for the R718 chiller technology. Utilizing the
wave rotor in such water cycles appears to be a potentially promising solution.

While possibility of integrating four-port wave rotors in R718 cycles has been

discussed in a previous study [240], the attention is on utilizing a three-port wave rotor in

197

R718 units. Utilizing a three-port wave rotor, know as the condensing wave rotor,
appears more promising because it combines the function of a compressor stage and the
condenser in one compact unit. Both pressure rise and condensation occur inside the

wave rotor channels as described below.4

6.2.1 How Does a Condensing Wave Rotor Work?

Figure 110 depicts schematically a R718 cycle with a direct condenser and a direct
evaporator. Figure 111 shows a schematic of a R718 cycle using a three-port condensing
wave rotor. A comparison between these two cycles indicates that the wave rotor has
substituted three subsystems: the condenser, one compressor stage, and the intercooler
(now shown). A schematic design“ of a condensing wave rotor is depicted in Figure 112.
In this innovative design, condensation of vapor occurs inside the wave rotor channels as
depicted in Figure 113 and Figure 114. Figure 113 schematically shows a model for the
condensation process inside a channel of a three-port condensing wave rotor. Figure 114

shows a corresponding schematic wave and phase-change diagram.

 

4 Materials presented in this section have been accepted for publication in 2005 ASME Journal of
Engineering for Gas Turbines and Power.

198

 

 

 

 

Wm. 1

Condenser

 

—

   

 

Wcr+ Wcz

 

 

Condensing
wave rotor

 

 

 

Figure ll 1: Schematic of a R718 cycle enhanced by a three-port condensing wave rotor
substituting for the condenser and for one compressor stage

199

MP water
(3)

'0

End plate

     

Channel I
End plate
Vapor collector ’ . V
‘ Q3 . Condensing Wave rotor
' .,- ‘ (drum)
I HP water
(6)

 

Figure 112: Schematic of a three-port condensing wave rotor

 

 

Contact surface Normal shock wave

Figure 113: Regions modeled for each compression and condensation

i

axial length
I

Rotation

i

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( uninti surface

luﬂdlILIIJ-ill'lllﬂmﬂllllﬂl‘m
111m11111m11111uw

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Shack mm.-

 

 

200

Following the points introduced in Figure 111, coming from the turbo-compressor
(2), the superheated vapor ﬂows continuously through a vapor collector (shown in Figure
112) to the inlet port of the wave rotor located at one of the two stationary end plates. By
rotating the wave rotor between the two end plates, the wave rotor channels are opened to
the port and ﬁlled with the incoming superheated vapor. The region (a) in Figure 113 and
Figure 114 is the state after the ﬁlling process is completed. After further rotation, the
channels meet the second-inlet port (6) through which the high-pressure low-temperature
water (e) comes in and is exposed to the low-pressure high-temperature superheated
vapor in region (a). Due to the sudden pressure drop (from p6 to p;), all the heat cannot be
contained in the incoming water as sensible heat and the heat surplus is transformed into
latent heat of vaporization. This is the so called ﬂash evaporation or ﬂashing
phenomenon [241, 242]. Therefore, one portion of the incoming water suddenly
vaporizes (c) and the remaining part cools down (d). The frontal area of the saturated
vapor (c) generated by the ﬂash evaporation is called the contact interface and acts like a
fast moving piston. It causes a shock wave triggered from the leading edge of the inlet
port traveling through the superheated low-pressure vapor which exists inside the channel
(a). The shock wave travels with supersonic speed (Vshm) faster than the contact
interface (14mm...) Therefore, the trajectory of the shock wave (solid line in Fig. 6) has a
smaller slope than the incoming water and the contact interface of the generated vapor
(dashed line). Behind the moving shock wave (b) the temperature is increased from T 2 to
T 2' and the pressure is increased from p; to p2'=p3 due to the shock compression. The
latter is a design decision similar to a tuning condition. With it, the pressure at the inlet

port (p6) is set to an appropriate value that generates the pressure ratio p6 /pz required to

201

trigger the desired shock wave. The superheated vapor will be condensed at pressure p3 .
This shows that the ﬂuid in its liquid state serves as a “work capacitor” storing pump
work to release it during its expansion in the wave rotor channels for the simultaneous
vapor compression. Therefore, in the enhanced system the pump in the cooling water
cycle not only has to provide the work necessary to overcome the pressure loss in the heat
rejecter cycle (wPL) but also the work necessary for the shock wave compression in the
wave rotor channels (war). The pressure behind the shock wave (b) is imposed on the
vapor generated by the ﬂash evaporation (c). It is the pressure at the water surface and the
equilibrium pressure at which the evaporation decays p(c)=p(b)=p3. Hence, both
generated vapor and the cooled water obtain the saturation temperature T 3=Tm(p 3).

Due to the contact of the superheated compressed vapor (b) with the cold incoming
water, the superheated vapor is desuperheated and its heat is transferer (I) to the
incoming water. This continues until the equilibrium temperature T3 is achieved in region
(b) and the superheated vapor is changed to saturated vapor. Subsequently, the incoming
water compresses the saturated vapor further and condenses it, while the latent heat is
transferred to the incoming water (g). The water, which is nearly a fully condensed two-
phase vapor with a typical quality of 0.005, is scavenged through the only outlet of the
wave rotor (3). The scavenging process may be supported by gravity and pump power.

The schematic pressure-enthalpy (p-h) and temperature-entropy (T-s) diagrams of
both the baseline and the wave—rotor-enhanced cycle are depicted in Figure 115 and

Figure 116, respectively.

202

------ l-2b-3b-5b-6b-3b-4b baseline cycle
1-2-2' -3-5-6-3-8 wave rotor enhanced cycle

 

/

Pressure

 

 

 

 

 

Enthalpy

Figure 115: Schematic p-h diagram of a R718 baseline cycle and a
wave rotor enhanced cycle

------ 1-2b-3b-4b baseline cycle
l-2-2’ -3-5-6-3-8 wave rotor enhanced cycle

 

 

 

 

 

 

 

i:
3
N
6
D-
6
3
[— 6 ...Y.
i 3
1135/ '1
“'33" 4:8

 

Entropy

Figure 116: Schematic T-s diagram of a R718 baseline cycle (cooling water cycle not
shown) and a wave rotor enhanced cycle

Both cycles start at the outlet of the evaporator l, where the vapor is saturated. State
21, represents the compressor outlet of the baseline cycle whereas state 2 is the

compressor outlet of the wave—rotor-enhanced cycle that allows using a compressor with

a lower pressure ratio. State 2' is an intermediate state inside the wave rotor channels that
corresponds to the ﬂow properties in region (b) right after the shock wave. The slope
between points 2 and 2' is greater than that between points 1 and 21, because the shock
compression typically occurs with a higher efﬁciency. Still inside the wave rotor channel,
the superheated vapor is desuperheated to the equilibrium temperature T 3 (2'—>3). State 3
is actually much closer to the liquid region than shown in Figure 115 and Figure 116
because the mass ﬂow rate of the cooling water cycle is much greater than that of the
refrigerant cycle. Knowing this, it becomes clear that the distances between points 3, 5,
and 6 are exaggerated in both diagrams. The expansion process (6—>3) releases the
energy consumed by the compression process of the vapor (2—>2’) all within the wave
rotor channels. Coming from the only outlet port of the wave rotor (3), the ﬂow diverges.
The small fraction used as refrigerant is directed to the expansion valve and is expanded
in a constant enthalpy process (3—> 4), while most of the ﬂow out of the wave rotor goes
into the heat rejecter (cooling tower or similar) where it cools (3—> 5). Afterwards the
pressure is again increased (5——>6) by the pump, providing the energy for the vapor
compression in the wave rotor (wwc) and compensating for the pressure loss in the heat

rejecter and associated piping (wPL).

6.2.2 Performance Evaluation

Using a thermodynamic model described in Ref. [243, 244], a performance map of
the enhanced cycle is obtained as shown in Figure 117. For each point on this plot, the
mass ﬂow ratio between the cooling water cycle and the chilled water cycle is considered
as a ﬁxed value, e.g. here, K =m6 /m2=125. Reference [245] describes how increasing the
mass ﬂow ratio above 200 appears ineffective for the COPgam (COP increase of enhanced

cycle relative to the base cycle divided by the COP of the baseline cycle) and the gradient

204

of the COPgam increase above K =125 is not signiﬁcant. According to Figure 117, each
point on this plot shows the maximum COPgam that can be obtained by the optimum
choice of PRW for a given evaporator temperature and a temperature-lift meaning that for
each evaporation temperature and temperature-lift combination, PRW is varied within a
certain range (1 <PRW < 4.5). Then among all COPgams recorded, the maximum COPgam
is the optimum value. On the other hand, the constant PRW lines indicate the optimum
PRW that yields the highest possible COPgam indicated by the max. COPgain lines. This
graph reveals a theoretical improvement of the COP up to 22% using the three-port

condensing wave rotor.

45

 

/ _,. .
.— .. __ _, .— —- "' max. COPgain=6°lo

4o-

 

 

 

 

 

45

 

Figure l 17: Performance map representing maximum performance increase and optimum
wave rotor pressure ratios

205

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I-l.’ J's
5.

 

J_¥——‘.

6.3 Wave Rotors for Ultra-Micro Gas Turbines

Ultra-micro gas turbines (UuGT) are seen as appropriate solutions for propelling mini
unmanned air-vehicles (UAV) and powering miniaturized wireless sensors and
equipment on board. This vision has been inspired by the improvements in Micro
Electrical Mechanical Systems (MEMS) technology. Furthermore, these microfabricated
gas turbines are suitable for high-density, distributed and redundant power generation on

aircrafts, and other vehicles. A typical size of an UuGT is 20mm x 20mm x 4mm. An

UpGT can achieve higher power densities compared to larger gas turbines as indicated by
the so called Cube-Square Law. Because the output power is proportional to the mass
ﬂow rate of working ﬂuid through the engine and the mass or volume of the engine is
proportional to the cube of the characteristic length, the power density (Power/Mass=L")
is increased as the device is miniaturized.

However, investigations of such UuGT have shown difﬁculties in obtaining high
thermal efﬁciencies and output power levels. The reasons are additional losses due to
miniaturization and the dominant 2D geometries typical for MEMS-fabrication
technology. The ﬂow paths often encounter 90-degree bends, which are known to
decrease efﬁciency and mass ﬂow rate. The compressor isentropic efﬁciency typically
drops to _<_ 60% [246]. Here, innovations are sought to attractively enhance the low
performance of the UuGT. Utilizing wave rotor technology, the idea of integrating a
wave rotor into an UpGT is proposed by the MSU wave rotor team. This section brieﬂy
review some of the challenges and advantages of conceptual designs integrating wave

rotor devices into such small engines [247].

206

6.3.1 Performance Enhancement of UuGT Using Wave Rotors

The way in which the wave-rotor topping enhances the cycle at the micro scale often
differs from that at larger scales. In the large scale, for a ﬁxed turbine inlet temperature,
the goal is generally either to increase the cycle overall pressure ratio (Case A) or to
substitute for costly high pressure turbomachinery stages with the wave rotor (Case B).
At the ultra-micro scale, the optimum cycle pressure ratio is very small, e.g. around 2,
due to the low component efﬁciencies. Thus, a single-stage centrifugal compressor can
easily generate the low optimum overall pressure ratio and a further increase with the
same efﬁciency actually decreases the desired performance. Therefore, the wave rotor
integration is most effective if its compression and expansion efﬁciencies are greater than
those of the turbomachinery components of the baseline engine. This enhances not only
the overall compression and expansion efﬁciencies, but it also increases the optimum
cycle pressure ratio to a greater value that allows for additional performance
improvements. In such a case in which the wave rotor compression efﬁciency is higher
than that of the spool compressor, a wave rotor can enhance the performance of an UpGT
that was already designed for an optimum pressure ratio. While the optimum overall
compression ratio increases with the wave rotor integration, usually the pressure ratio of
the spool compressor decreases. This can additionally enhance the isentropic efﬁciency
of the spool compressor, provided its polytropic efﬁciency (aerodynamic quality) stays
the same [246].

As described before, the efﬁciencies of both compression and expansion processes in
a wave rotor have been found to be greater than 70%. This may be considered as
matching the efﬁciency of large scale compressors or turbines and as almost double of

that achieved with UuGT compressors. Exploring the application of even smaller wave

207

rotors suitable for UuGT, the question arises if such wave rotor efﬁciencies can be
maintained at the ultra-micro scale. No experimental and theoretical values are available
at this scale. However, recent results show no or only a small decrease in efﬁciency with
reduced size, which encourages investigations at the ultra-micro scale. Recent research
has shown only a small decrease in efﬁciency of small wave rotors. For example, for a
small-scale wave rotor with a channel length of 69 mm, the wave-rotor characteristic
equation introduced in Eq. (38) calculates a compression efﬁciency of 79% using the
parameters provided by Okamoto and Nagashima [213, 215].

Using the available wave rotor efﬁciencies above versus the corresponding wave
rotor channel length, a trend can be generated as shown by the solid green line in Figure
118. The simple linear extrapolation predicts a wave rotor efﬁciency at the ultra—micro
scale (about 1mm channel length) that is greater than 70%. Such a compression
efﬁciency of a microfabricated wave rotor is much better than the obtained efﬁciencies of
around 50% for microfabricated compressors [246]. Furthermore, efﬁciency values of
compressors suitable for or corresponding to the reported wave rotor topped cycle are
shown in red in Figure 118. This allows showing the red broken trend line for the
compressor efﬁciency corresponding to the wave rotor length. Both trends for the wave
rotor efﬁciency and for the compressor efﬁciency coincide at the larger scale. However,
towards the smaller wave rotor size for UpGT, the compressor efﬁciency trend falls far

below the wave rotor efﬁciency trend. This clearly suggests an advantage of using a wave

rotor for the UpGT.

208

 

100 '

90
U Tokyo
80 . ..._. 3.7“ w-7 E ———-»7 4_ i
' ’ “7“” u Tokyo

      
   
 
    

GE lﬂSNV‘L. 7 _.

 

 

 

 

 

 

 

 

 

 

 

 

 

5 6° wamra‘
5 50 \
g Compressor
3° H“"_ Potential @ ___#¥. *C—_
20 -77#*— u-scale
0 l I l I
0 100 200 300 400 500

Wave Rotor Channel Length (mm)

Figure 118: Efﬁciency trend of compression process. Green: wave rotor efﬁciency,
Red: compressor efﬁciency

6.3.2 Conceptual Designs of Wave Rotor Implementation into UuGT

Based on possible design restrictions and preferences, three different advantageous
conceptual designs for a four-port (or multiple of that) wave rotor integrated into a
baseline UuGT are suggested. Figure 119 shows a wave rotor added at outer diameter of
the disk of the “classical” MIT baseline design [248]. This innovative design has several
advantages. The wave rotor rotates with the compressor/turbine disk, i.e., there is still
only one rotating disk in the engine. Additionally, because the wave rotor is a self-cooled
device, it isolates the compressor disk from the combustion chamber while in the wave
rotor the heat is given to the compressed air adding a recuperative effect. The end plates
with the ports at either side of the wave rotor can be etched in the same wafer as the

stationary guide vanes.

209

Ultra-Micro Turbine Design (UuGT) - Design 1

Classical Design

Comprssor

  
 
   

 

 

 

 

 

 

 

 

 

 

 

Wave Rotor
Rotating Disk
\ Air intake

! 1'1

1 1 / 1

1 5 1:3; 1 l 1 ' i
1 - . ... 3
l
.47 ’2 \
i 5
Exhaust T Turbine Fuel
E d Plat
n as Combustor

4 cycle] revolution
(16 port wave rotor)

Figure 1 19: First conceptual design of an UuGT equipped with a four-port wave rotor

The second possible design implies using additional wafers allowing a multi layer
rotor, as shown in Figure 120. However, the diameter of the rotor is smaller than in the
ﬁrst design. This results in a smaller frontal area, which is favorable for the propulsion of
air vehicles and may generate less stress in the disks. Further, this design allows for better
separation of the combustion chamber from the compressor, reducing the heat introduced
to the compressor. The major challenge with this design is the perfect axial alignment of
the compressor/turbine unit with the wave rotor, which may be achieved with the
common laser aligning method. The ﬂow connection from the compressor to the wave
rotor may be viewed as a challenge with respect to keeping the pressure loss small.
However, the equivalent diameter of this connection may be designed sufﬁciently large.
Further, this may aid in isolating the compressor case from the combustor heat, especially

with the counter ﬂow of the compressed air, where a regenerative effect is seen again.

210

Ultra-Micro Turbine Design (UuGT) - Design 2
Two-Layer Design

Air Intake

Combustor
Turbine Compressor
\ .

. .
it;

 

.11 ll.

/ <: Y\ m'

Wave Rotor End Plates
E) haust

 

 

 

 

 

 

 

 

 

Figure 120: Second conceptual design of an UuGT equipped with a four-port wave rotor

The third design concept introduces a new idea having multiple wave rotors arranged
circumferentially around the compressor/turbine unit, as shown in Figure 121. The
advantage of this design is that no additional stresses occur in the main rotor, which is the
compressor/turbine disk. The stresses in the separate small diameter wave rotors are
negligible since they can rotate at a relatively low speed. Similar to the ﬁrst classic
design, this design requires fewer layers then the second design, which translates into a
lower fabrication cost. The challenge associated with this design is driving all wave
rotors at appropriate speed. This may be achieved by arranging the wave rotor ports at

proper and suitable angles, so that the impulse of the ﬂuid streams can be utilized.

211

Ultra-Micro Turbine Design (UuGT) - Design 3

Exterior Wave Rotor Design

“mung ”'5“ Air Intake Comprssor

 

Exhaust

Fuel
Turbine
End Plats

Comer

Figure 121: Third conceptual design of an UuGT equipped with a four-port wave rotor

212

CONCLUSION

Wave rotor technology has shown unique capabilities to enhance the performance and
operating characteristics of a variety of engines and machinery utilizing thermodynamic
cycles. Although there have been numerous efforts in the past dealing with this novel
concept, this technology is not yet widely used and barely known to engineers. Here, an
attempt was made to investigate this novel technology further and to show its signiﬁcant
potential for performance gain in small gas turbines. This work has extensively discussed

the following topics:

0 Comprehensive Review of Wave Rotor Technology

The presented research summarizes both the previously reported work in the literature
and ongoing efforts around the world. It covers a wide range of wave rotor applications
including the early attempts to use wave rotors, its successful commercialization as
supercharges for car engines, research and development for gas turbine topping, and other
developments. The review also pays close attention to more recent efforts, e.g. utilization
of such devices in pressure-gain combustors, highlighting possible further efforts on this
topic. Observations and lessons learnt from experimental studies, numerical simulations,
analytical approaches, and other design and analysis tools were presented.

0 Performance Prediction of Gas Turbine Topping Wave Rotors

Signiﬁcant performance enhancement of two microturbines (30 and 60 kW) was
predicted by implementing various wave rotor topping cycles. Five different
advantageous cases were considered for implementation of a four-port wave rotor into the
given baseline engines. The compressor and turbine pressure ratios and the turbine inlet

temperatures vary in the thermodynamic calculations, according to the anticipated design

213

 

 

objectives of the cases. Advantages and disadvantages were discussed. To evaluate the
performance enhancement of topping small gas turbines with wave rotors, a computer
program based on a thermodynamic approach was created to determine the
thermodynamic properties of the gases in different states of the cycles. The results were
used to calculate the theoretical performance and the actual T-s diagrams of both wave-
rotor-topped and baseline engines. The unrecuperated engines were predicted to have
thermal efﬁciency and speciﬁc work enhancement up to 34% for the smaller engine and
25% for the larger engine, using a four-port wave rotor with a compression ratio of 1.8.
Similar approach has predicted an improvement up to 15% of overall efﬁciency and
speciﬁc thrust in a turbojet engine using the wave-rotor—topping cycle of 30 kW
microturbine ﬂying at an altitude of 10,000 m at Mach 0.8. The 30 kW recuperated
engine was predicted to achieve thermal efﬁciency and speciﬁc work enhancement up to
7% and 38%, respectively, by implementing Case A where the baseline compressor and
turbine remain unchanged. In addition, the impact of ambient temperature on
performance of both baseline and topped engines was investigated and it was shown that
the wave-rotor-topped' engines are less prone to performance degradation under hot-
weather conditions than the baseline engines
0 Development of an Analytical Preliminary Design Procedure

A one-dimensional analytical gasdynamic model of the charging process was
employed to calculate ﬂow characteristics inside the wave rotor channels. Three different
wave patterns were considered. The ﬁrst scenario uses a single shock wave and an
expansion wave. This situation may occur when the combustor pressure loss is relatively
small and hence the total pressure difference between the incoming hot gas and outgoing

compressed air is small. It was shown that for a tuned condition, this particular scenario

214

requires a huge channel height, resulting in an undesirably bulky wave rotor. The second
wave pattern employs two primary shock waves to perform the charging operation. It was
discussed that while the charging process is theoretically feasible, this case is rarely seen
in wave rotor designs. Therefore, the attention was given to last studied case which
employs a primary shock wave, a reﬂected shock wave, and an expansion wave. This is
the wave diagram often exists in the charging process of four-port wave rotors. For all
three cases considered here, useful design parameters such as port widths and rotor size
are determined by computing transit times of the waves traveling inside the channels. To
verify the accuracy of the model, the results were compared with some previous
numerical data which are available for conventional wave rotors. Reasonable agreement
has been found comparing the predicted results of the model with the numerical data.
0 Numerical Simulations with FLUENT

A successful attempt at developing 2D models of wave rotors using the commonly
available CFD software FLUENT was performed. It was shown that such an available
software package can be effectively used for simulation of rather complex time-
dependent ﬂow phenomena that typically occur in wave rotors, resulting in a wide range
of information about the unsteady processes.
0 Innovative Wave Rotor Designs and Applications

Several innovative and cutting-edge application of wave rotors were suggested and
discussed. These concepts include radial-ﬂow wave rotors, integrating wave rotors in
ultra—micro gas turbines (UuGT), and water refrigeration systems working with wave

rotors.

215

As the ﬁnal note, while new knowledge and technology innovations have provided a
new opportunity to consider wave rotor concept as innovative technology, still sealing
and thermal expansion appear to be dominant problems despite continued research.
Special technology developments and additional research on wave rotors overcoming

some of present challenges and limitations are needed.

216

 

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217

 

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230

 

 

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232

 

 

APPENDICES

233

 

 

Appendix A: Shock Waves and Diffusers

In order to obtain the plots shown in Figure l and Figure 2, it is first necessary to ﬁnd
relations between pressure gain and Mach number changes of upstream and downstream

ﬂow across a moving shock wave and a diffuser.

A.l: Moving Shock Wave
According to the top part of Figure 36, for a moving shock wave propagating in a
frictionless channel, the flow upstream and downstream Mach numbers are respectively

defined as:
M,=—,M,=— (125)

The goal here is to find an equation relating the pressure gain across the shock wave (175)
to the Mach numbers introduced in Eq. (125). To do so, it is necessary to transfer the
non-steady frame of reference to a stationary coordinate as. shown in the bottom part of
Figure 36. In such a stationary frame cf reference, the flow upstream and downstream

Mach numbers now are defined respectively as:

W +u W +u
=__5 ’,M2 =__5 2 (126)

a, a2

M1

"3
The governing normal-shock equations for a stationary shock wave results in:

2: (y—I)M,_52+2

MZ-s 2
27M,.. -(r-1)

 

(127)

Now the task is to find relations for M 1-, and M2-,. relating them to 175 , M1 , and M2 and
substitute all of them in Eq. (127). As shown in Eq. (11), the definition of the shock

Mach number (M5) becomes the same as M 1-, , therefore:

234

 

 

MH =M$ =\/—(175 —1)+1 (128)

“2
+ ——- (129)

 

Ws +52: Ws +“z "“2 +12: Ws “‘1 114,22. (w, “‘1 .__liLJEL

01',

IT
M2-.=(Ms-M,) }’-+M2 (130)
2

Substituting Eq. (128) and (130) into (127) yields:

y+1

 

(17,—1)+1]+2

1 T 2 (7‘1)[—2—
H -”2;(175—1)+1—M,]\/;+M,] = y (131)
y 2 2r[Y+1(175-1)+1]-(r—1)

 

'37
where T2 [T1 is a function of the local pressure ratio as well, calculated by Eq. (5).
Therefore, this equation calculates 175 as a function of M1 and M2 as illustrated in Figure
l for y =l.4.
To calculate the shock wave efﬁciency (7mm) as a function of the local pressure
ratio, ﬁrst the deﬁnition of the shock wave isentropic efﬁciency is considered. It is

defined as the ratio of the isentropic enthalpy change to the actual enthalpy change:

( A h )isentropir
u" = (132
”SIN L (Ah )0 )

 

rrual

Treating the flow as a perfect gas, for which h = Cp T and CI) is constant:

(AT).

isentropir

_(T,s/T, )-—1
_ (TZ/T, )—1

. _ 133
”Shark (AT)0 ( )

 

 

dual

235

 

 

Using the isentropic relation for T2, ,ﬂ‘, and substituting Eq. (5) for T2 /T, assuming a

frictionless channel yields:

 

 

17 "”7 —1
”Shook : 7:1 (134)
""" s
5 y_1+1 —1
1 L175
y-I

Figure 2 illustrates the variation of 775;“), as a function of the local pressure ratio for y

=l.4.

A.2: Diffuser
The diffuser isentropic efficiency known as the diffuser effectiveness is deﬁned

similar to that of a shock wave:

_ (A h)isentropir _ (T2.s /T1)— 1

( A T l,
77 1' User - :
DI, ( A h )arrual ( AT )artual (T2 / Tl )- I

sentmpr'r

 

(135)

Because the total temperature remains constant across the diffuser (T,1=T,2), the diffuser
isentropic efﬁciency can be expressed based on the static pressure ratio across the

diffuser (p2/p1) and the ratio of the total pressure (p,2 /p,,) that represents the pressure loss

r-l/r
1” (136)

”Diffuser : r-I/r
[£122.] _ 1
p12 pl

The variation of now," as a function of the static pressure ratio across the diffuser, p2/p1,

across the diffusers:

 

for certain losses of total pressure across the diffuser expressed by the ratio, p,2 /p,,, is

shown in Figure 2 and compared with the shock wave isentropic efficiency 775mm.

236

 

 

Now, to obtain a relation between the pressure gain and changes of inlet and outlet
Mach number, the following procedure is pursued:
The following equations relate the ratio of the total pressure to static pressure of the flow

in the inlet and outlet, respectively:

y/y—l
_19,_,:[1+__721M12) (137)
P1
and,
r/y—l
&=[1+__721M22] (138)
P2

Dividing these two equations gives:

 

 

 

 

y/r—l
1+);le
31$: 2 (139)
Prz P) 14.51)":
By substituting this equation in Eq. (136):
r-l/r
”Dijfuser pl_} (140)
1+L-M,2
2 -1
I+———}/‘;1MZ2
or,
.. nr/Y-l
1.+.Z::;£UA[IZ
p3 — 1+ 2 141
_ _ ”Diffuser _I ( )
p, 1+1—M 2
l. 2 2 _

 

 

237

Using this equation and Eq. (131), Figure 1 therefore compares the variation of pressure
gain between a moving shock wave (solid) and a 100% efficient (7701mm, =1) diffuser
(dotted) as a function of the incoming flow Mach number, M 1 , for two different values of

M2 assuming y 21.4.

238

 

 

Appendix B: Temperature-Entropy Diagram Construction

The temperature-entropy (T-S) diagrams serve as valuable tools for visualizing the
second law aspects of processes and cycles. They are frequently used in performance
demonstration of gas turbine cycles. The performance enhancement of a gas turbine
topped by a four-port wave rotor can be easily observed by comparing the T-S diagrams
of the baseline and topped engines. Here is shown the calculations necessary to draw T-S

diagrams for a four-port wave rotor topped cycle and the corresponding baseline engine.

B1: Compressor and Turbine Isentropic Efﬁciencies
For a compressor, the isentropic efﬁciency relates actual work per unit mass flow to that

of an ideal machine with equivalent pressure change as below:

Ideal work
Actual work

 

(142)

romp

By neglecting the changes in kinetic and potential energies associated with the ﬂuid
stream ﬂowing through the compressor and assuming an adiabatic process, the above

relation can be expressed as:

( A h )isentropir
mm a ‘ 143)
77 p ( Ah )0 (

 

dual

Treating the flow as a perfect gas, for which h = Cp T:

 

( A T )isentropir
com 5 (144
77 p ( AT )0 )

dual

Assuming a process between points j and j+1 :

,7 : Tj+l.s —Tj : (Tj+l.s /Tj )_1 (145)
“""P T -T. (T). /T}.)—1

#1 J

 

+1

239

By substituting Tj+,d./ T]. .__( ij/pj) 2.1/2
77 —(pj+I/pj)7—l/y_1
rump (T. /Tj )_1

1+!

 

(146)

This equation can also written as:

Tj+l :(ij/pj )7'1/7 -1

+ I ( 147)
Tj ”romp

 

 

For a turbine, the isentropic efﬁciency relates actual work per unit mass flow to that of an

ideal machine with equivalent pressure change as below:

77 _ Actual work
Mb Ideal work

 

(148)

Similarly, the turbine isentropic efficiency for a process begins from a point j to a point

j+1 can be expressed as:

 

(T., /T.)—1
”lurb : j I 17‘“? (149)
(pj+l /pj) -1
01',
T. .
'j—+l- = 771mb (pJH )7‘1/7 —1 +1 (150)
T, j

B2: Compressor and Turbine Stage Efficiencies

For a multistage compressor, each stage has an isentropic efficiency. For a N-stage

l/N
)

compressor with equal stage pressure ratios (JE,-c=( p,” /p,- ) and equal stage

efficiencies ( n“. ), it can be shown that [233]:

,7 —7 ”erlr-ll/r _] (151)
...,... [1+(1/77... )(7:_,,.""”’ —1)]” —1

 

240

This equation can give us the compressor stage efficiency as:

77 = (152)
N( y—l V7
—1
41 + 7t“ — 1

 

 

”(amp
For a multistage turbine, when all stages have the same stage pressure ratios

(IQ, =( pj+7/pj )1 “V ) and equal stage efﬁciencies ( 77,, ), the turbine isentropic efficiency can

be written as:

_ 1_[1_"“(1_7[“(r-1777)]N

 

 

 

turb ._— 1 NIr-lr/y (153)
_ ”st
So, the turbine stage efficiency is then becomes:
_1-W-nturb(1-ﬂst‘ ) 154
’7“_ 1_” (r-n/r ( )
5!

B3: Compressor and Turbine Polytropic Efficiencies
In general, for a polytropic process beginning from a point j to a point j+1 the following

relation can be written:

 

. T.
pj+l :( 7+1 )n/(n-ll (155)
Pj T}

where n is called polytropic component. The values of n for some familiar processes are
given below:

Isobaric process 77 = 0

Isothermal process 72 = I

Isentropic process n = y

241

The polytropic efficiency is a useful concept for a non-isentropic process. For a non-

isentropic process:

 

 

For compression: 2 W7”
71 —1 y — I
F ~ . _ r
or expansron. —
n - 1 (7 _ 1 )77P

where 77p is the polytropic efficiency.
Therefore, for a non-isentropic process beginning from a point j to a point j+1 we can
write:

For a compression process:

 

pj+l =(_7;7_+L )rnpﬂ r—l) (156)
177' T1
For an expansion process:
pl“ : (111)7/(Ur(7‘1)) (157)
P7 T1

For a compressor, assuming that the polytropic efficiency is constant over the pressure
ratio, the relation between the isentropic efﬁciency and the polytropic efﬁciency is given

by using Eq. (146) and (156) as:

)(r-Wr —l

(pj+l lpj
(pj+| lpj)(7‘1)/(7’lrr) _l

 

(158)

comp _

Similarly, the relation between the turbine isentropic efficiency and the polytropic

efficiency can be shown as:
(pm/p,- )""”’""” -I

tur - (159)
b (pj+l/pj)

 

lr-ll/r _1

242

.-'. ‘ .29.“!

 

 

B4: Entropy Calculation

The relation which relates the entropy generation between points j and j+1 is:

 

T. .12
31+, =sj.+Cpln( ’*’ )—R1n(-I3i) (160)
T7 p].

The above all equations can be used to sketch T-s diagrams of the gas turbine cycles. For
a case when a four-port wave rotor is added to a baseline gas turbine engine, similar
methodology described above is adopted as will be explained in the following. Referring
to state numbering introduced in Figure 6 (TF wave rotor) or Figure 34 (RF wave rotor),
the below procedure is performed to sketch the T-s diagrams for such enhanced cycle.

0 Path 0-1: Compressor

The stage pressure ratio can be simply obtained by:

7:. = N BLI— (161)

Po

where N can be considered any arbitrary integer number greater than unity. N =12 is
chosen for this study. By knowing the compressor isentropic efﬁciency, Eq. (152) gives
the compressor stage efﬁciency. Then, Eq. (147) calculates the temperature at each
pressure stage j+1 . Finally, the entropy at each stage can be obtained by using Eq.
(160).

0 Path 1-2: Compression in the Wave Rotor

A similar approach described for the compressor path can be used for the wave rotor

compression process by simulating a wave rotor stage pressure ratio obtained by:

7r =N — (162)

0 Path 2-3: Combustion Chamber

243

 

The treatment of this path is different from the earlier paths because there is no
information about isentropic paths across the combustion process. Indeed, Eq. (155)
should be used to ﬁnd the polytropic component along path 2-3. To do so, for the path 2-

3, Eq. (155) can be expressed as:

 

 

n: (163)
1_ln( 713/712 )
pr3/p72
Of,
1
n: (164)
1-171(713/712)

comb

By calculating 72, Eq. (155) will be used to calculate the pressure (or temperature) in each
temperature (or pressure) step.

In calculating the entropy at each step, the values of Cp (speciﬁc heat) and R (gas
constant) need to be determined. For the ﬁrst approximation, the following estimation

have been made:

Cp air + Cp gas

 

C — 165
pave 2 ( )
Rair + R gas
Rm = —-———‘- (166)
2
0 Path 3-4: Expansion in the Wave Rotor
The stage pressure ratio is obtained by:
225 = N 132. (167)
p 74

244

 

Therefore, the wave rotor expansion stage efficiency can be obtained by using Eq. (154).

Then, using Eq. (150) leads to the temperature at each pressure stage j+1 .

0 Path 4-5: Turbine

A similar approach described for the wave rotor expansion process now is used for the

flow expansion in the turbine using the turbine stage pressure ratio obtained by:

7: = N & (168)
p15

245

«r. .o' ‘ haw)". an: Ju- uﬂ'
. b - I I.

 

 

   
  
 

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