the design of a micro-turbogenerator
TRANSCRIPT
The Design of a Micro-turbogenerator
by
Andrew Phillip Camacho
Department of Mechanical Engineering and Materials ScienceDuke University
Date:
Approved:
Jonathan Protz, Supervisor
Devendra Garg
Rhett George
Thesis submitted in partial fulfillment of the requirements for the degree ofMaster of Science in the Department of Mechanical Engineering and Materials
Sciencein the Graduate School of Duke University
2011
Abstract(micro-turbogenerator design)
The Design of a Micro-turbogenerator
by
Andrew Phillip Camacho
Department of Mechanical Engineering and Materials ScienceDuke University
Date:
Approved:
Jonathan Protz, Supervisor
Devendra Garg
Rhett George
An abstract of a thesis submitted in partial fulfillment of the requirements forthe degree of Master of Science in the Department of Mechanical Engineering and
Materials Sciencein the Graduate School of Duke University
2011
Copyright c© 2011 by Andrew Phillip CamachoAll rights reserved except the rights granted by the
Creative Commons Attribution-Noncommercial Licence
Abstract
The basic scaling laws that govern both turbomachinery and permanent magnet gen-
erator power density are presented. It is shown for turbomachinery, that the power
density scales indirectly proportional with the characteristic length of the system.
For permanent magnet generators, power density is shown to be scale independent
at a constant current density, but to scale favorably in actuality as a result of the
scaling laws of heat dissipation.
The challenges that have affected micro-turbogenerators in the past are presented.
Two of the most important challenges are the efficiency of micro-turbomachinery and
the power transfer capabilities of micro-generators.
The basic operating principles of turbomachinery are developed with emphasis on
the different mechanisms of energy transfer and how the ratio of these mechanisms in
a turbine design relates to efficiency. Loss models are developed to quantify entropy
creation from tip leakage, trailing edge mixing, and viscous boundary layers over
the surface of the blades. The total entropy creation is related to lost work and
turbine efficiency. An analysis is done to show turbine efficiency and power density
as a function of system parameters such as stage count, RPM, reaction, and size.
The practice of multi-staging is shown to not be as beneficial at small scales as
it is for large scales. Single stage reaction turbines display the best efficiency and
power density, but require much higher angular velocities. It is also shown that for
any configuration, there exists a peak power density as a result of competing effects
iv
between the scaling laws and viscous losses at small sizes.
The operating principles of generators and power electronics are presented as
are the scaling laws for both permanent magnet generators and electro-magnetic
induction generators. This analysis shows that permanent magnet generators should
have higher power densities at small sizes. The basic concepts of permanent magnet
operation and magnetic circuits are explained, allowing the estimation of system
voltage as a function of design parameters. The relationship between generator
voltage, internal resistance, and load power is determined.
Models are presented for planar micro-generators to determine output voltage,
internal resistance, electrical losses, and electromagnetic losses as a function of geom-
etry and key design parameters. A 3 phase multi-layer permanent magnet generator
operating at 175,000 RPM with an outer diameter of 1 cm is then designed. The
device is shown to convert 10 W of input shaft power into DC electric power at an
efficiency of 64%. A second device is designed using improved geometries and system
parameters and operates at an efficiency of 93%.
Lastly, an ejector driven turbogenerator is designed, built, and tested. A thermo-
dynamic cycle for the system is presented in order to estimate system efficiency as
a function of design parameters. The turbo-generator was run at 27,360 RPM and
demonstrated a DC power output of 7.5 mW.
v
Contents
Abstract iv
List of Tables ix
List of Figures x
Acknowledgements xv
1 Introduction 1
1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.3 Review of Previous Work . . . . . . . . . . . . . . . . . . . . . 22
1.2 Research Objectives, Goals, and Design Overview . . . . . . . . . . . 31
2 Micro-turbine Design and Performance 33
2.1 Concepts of Turbomachinery . . . . . . . . . . . . . . . . . . . . . . . 33
2.1.1 Operating Concepts . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1.2 Conservation Laws and Governing Equations . . . . . . . . . . 37
2.1.3 Turbine Reaction, Flow Type, and Stage Count . . . . . . . . 39
2.2 Micro-turbine Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.1 Turbine Efficiency and Loss Accounting . . . . . . . . . . . . . 42
2.2.2 Characterization of the Flow . . . . . . . . . . . . . . . . . . . 47
2.2.3 Boundary Layer Losses . . . . . . . . . . . . . . . . . . . . . . 48
vi
2.2.4 Trailing Edge Losses . . . . . . . . . . . . . . . . . . . . . . . 51
2.2.5 Tip Clearance Losses . . . . . . . . . . . . . . . . . . . . . . . 54
2.2.6 Scaling Effects on the Efficiency of Gas Turbines . . . . . . . . 57
2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3 Permanent Magnet Micro-generator Design and Performance 68
3.1 Concepts of Permanent Magnet Generators . . . . . . . . . . . . . . . 68
3.1.1 Operating Concepts, Scaling Laws, and Generator Selection . 68
3.1.2 Magnetic Circuits and Modeling . . . . . . . . . . . . . . . . . 76
3.1.3 Electric Circuits and Modeling . . . . . . . . . . . . . . . . . . 86
3.1.4 Magnetic Materials and Configuration . . . . . . . . . . . . . 89
3.2 Device Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.2.1 Device Geometry and Parameters . . . . . . . . . . . . . . . . 94
3.2.2 Flux and Induced Voltage . . . . . . . . . . . . . . . . . . . . 99
3.2.3 Generator Coil Resistance . . . . . . . . . . . . . . . . . . . . 102
3.2.4 Maximum Power Transfer Capabilities . . . . . . . . . . . . . 104
3.2.5 Loss Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.2.6 Power Output, Efficiency, and Layer Count . . . . . . . . . . . 110
3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4 Experimental Results of an Ejector Driven Micro-turbogenerator 113
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.2 Thermodynamic Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.3 Experiment and Results . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5 Conclusion and Future Work 121
5.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 121
vii
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Bibliography 125
viii
List of Tables
3.1 Voltage results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.2 Maximum power dissipation . . . . . . . . . . . . . . . . . . . . . . . 105
3.3 Eddy current losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.4 Ohmic losses in the stator windings and power electronics . . . . . . . 109
3.5 Load power and efficiency as a function of layer count . . . . . . . . . 110
4.1 Efficiency approximations . . . . . . . . . . . . . . . . . . . . . . . . 117
4.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
ix
List of Figures
1.1 A micro-turbine engine operating a Brayton cycle . . . . . . . . . . . 3
1.2 Scaling laws for heat dissipation in a conductor . . . . . . . . . . . . 6
1.3 Energy density of micro-turbogenerators relative to Li-ion batteries . 7
1.4 TS diagrams for a Brayton cycle . . . . . . . . . . . . . . . . . . . . . 8
1.5 Gas flow path for a micro-turbine . . . . . . . . . . . . . . . . . . . . 10
1.6 Velocity triangles of a turbine stage . . . . . . . . . . . . . . . . . . . 11
1.7 Thermal resistance structures . . . . . . . . . . . . . . . . . . . . . . 12
1.8 A magnetizing head . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.9 Leakage flux permeating incorrect sectors during magnetization . . . 14
1.10 Trapezoidal magnetization profile of an annular permanent magnet . 14
1.11 Discrete magnet pieces arranged peripherally around the generator rotor 15
1.12 A silicon frame for discrete peripherally arranged magnet pieces . . . 15
1.13 Coil winding patterns for 2D micro-generators . . . . . . . . . . . . . 17
1.14 Silicon laminations within the stator back iron to reduce eddy currents 18
1.15 Temperature effects on magnetization . . . . . . . . . . . . . . . . . . 18
1.16 A single-zone combustor schematic . . . . . . . . . . . . . . . . . . . 19
1.17 Schematic of a dual zone combustor . . . . . . . . . . . . . . . . . . . 21
1.18 A catalytic micro-combustor . . . . . . . . . . . . . . . . . . . . . . . 21
1.19 A 10 mm axial impulse turbine . . . . . . . . . . . . . . . . . . . . . 22
1.20 SEM photograph of the MEMS turbocharger . . . . . . . . . . . . . . 23
x
1.21 Cross sectional view of the turbomachinery . . . . . . . . . . . . . . . 23
1.22 A bird’s eye SEM photograph of the MEMS turbine . . . . . . . . . . 23
1.23 A split SEM view showing the concentric turbine stages . . . . . . . . 24
1.24 A fully packaged micro-combustor . . . . . . . . . . . . . . . . . . . . 25
1.25 Schematic of a dual zone combustor . . . . . . . . . . . . . . . . . . . 25
1.26 A catalytic combustor schematic . . . . . . . . . . . . . . . . . . . . . 26
1.27 Schematic of a ball bearing supported turbo-generator . . . . . . . . 26
1.28 Original and optimized winding patterns . . . . . . . . . . . . . . . . 27
1.29 A comparison between SmCo and NdFeB generators as a function oftemperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.30 Power density and ejector efficiency as a function of the nozzle throatdiameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.31 Suction draft performance versus entrainment ratio . . . . . . . . . . 29
1.32 A schematic of a MEMS power plant utilizing a Rankine cycle . . . . 29
1.33 Cross sectional schematic of the fully integrated permanent magnetturbogenerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.34 The original turbogenerator and power electronics . . . . . . . . . . . 31
2.1 A control volume for a general turbo-machine . . . . . . . . . . . . . 33
2.2 Velocity triangles for the exit flow from a rotor blade . . . . . . . . . 34
2.3 A fluid element in a centrifugal field . . . . . . . . . . . . . . . . . . . 36
2.4 Velocity triangles for an impulse turbine rotor blade row . . . . . . . 39
2.5 Velocity triangles for a reaction turbine rotor blade row . . . . . . . . 40
2.6 Multistage turbine arrangement . . . . . . . . . . . . . . . . . . . . . 41
2.7 A radial micro-turbine . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.8 Effectiveness of impulse turbines and a reaction turbine as a functionof the velocity ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.9 An irreversible flow process through a throttle . . . . . . . . . . . . . 46
xi
2.10 Simplified velocity triangles for idealized impulse and reaction blades 48
2.11 Dissipation coefficients for various boundary layers . . . . . . . . . . . 50
2.12 Wake mixing behind the trailing edge of two turbine blades . . . . . . 51
2.13 A dump diffuser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.14 Total pressure drop in a diffuser . . . . . . . . . . . . . . . . . . . . . 53
2.15 Thick trailing edges of a micro-turbine . . . . . . . . . . . . . . . . . 55
2.16 A schematic of tip leakage over a turbine blade . . . . . . . . . . . . 56
2.17 Turbine effectiveness for a single stage reaction turbine . . . . . . . . 58
2.18 Turbine effectiveness for a single stage impulse turbine . . . . . . . . 59
2.19 Turbine effectiveness for a two stage impulse turbine . . . . . . . . . 59
2.20 Turbine effectiveness for a three stage impulse turbine . . . . . . . . . 60
2.21 Turbine efficiency for a single stage reaction turbine . . . . . . . . . . 60
2.22 Turbine efficiency for a single stage impulse turbine . . . . . . . . . . 61
2.23 Turbine efficiency for a two stage impulse turbine . . . . . . . . . . . 61
2.24 Turbine efficiency for a three stage impulse turbine . . . . . . . . . . 62
2.25 Power density for a single stage reaction turbine . . . . . . . . . . . . 64
2.26 Power density for a single stage impulse turbine . . . . . . . . . . . . 64
2.27 Power density for a two stage impulse turbine . . . . . . . . . . . . . 65
2.28 Power density for a three stage impulse turbine . . . . . . . . . . . . 65
3.1 A current induced magnetic field . . . . . . . . . . . . . . . . . . . . 69
3.2 A permanent magnet used to create a magnetic field . . . . . . . . . 70
3.3 The induced voltage being used to create a current and produce work 70
3.4 A three phase power system . . . . . . . . . . . . . . . . . . . . . . . 71
3.5 A phasor diagram for a wye connected generator . . . . . . . . . . . . 72
3.6 A diode bridge rectifier . . . . . . . . . . . . . . . . . . . . . . . . . . 72
xii
3.7 The original AC voltage forms and the rectified waveform . . . . . . . 73
3.8 Rectifier current pathways . . . . . . . . . . . . . . . . . . . . . . . . 73
3.9 Scale effects on basic magnet interactions . . . . . . . . . . . . . . . . 75
3.10 Scale effects on magnetic interactions with increased current density . 76
3.11 A B-H graph for a hard magnetic material . . . . . . . . . . . . . . . 77
3.12 A simple permanent magnet circuit . . . . . . . . . . . . . . . . . . . 77
3.13 A simple permanent magnet circuit with an air gap . . . . . . . . . . 78
3.14 The B-H operating conditions for a permanent magnet system . . . . 80
3.15 A toroidal ferromagnetic core wrapped within a current conducting wire 80
3.16 A close up of the B-H curve for a hard magnetic material . . . . . . . 81
3.17 A circuit model for a permanent magnet . . . . . . . . . . . . . . . . 83
3.18 A circuit model for a permanent magnet enclosure with an air gap . . 83
3.19 A circuit model for a planar permanent magnet generator . . . . . . . 84
3.20 A simplified circuit model for a planar permanent magnet generator . 84
3.21 A B-H curve for a permanent magnet material undergoing magnetization 85
3.22 A magnetizing head used to create permanent magnets . . . . . . . . 86
3.23 A single phase AC circuit . . . . . . . . . . . . . . . . . . . . . . . . 86
3.24 Load power as a function of efficiency . . . . . . . . . . . . . . . . . . 89
3.25 Demagnetization curves for various magnetic materials . . . . . . . . 90
3.26 A comparison between SmCo and NdFeB generators as a function oftemperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.27 An annular magnet and mounting adaptor . . . . . . . . . . . . . . . 92
3.28 Discrete magnet pieces arranged peripherally around the generator rotor 93
3.29 A B-H curve for a magnetic material . . . . . . . . . . . . . . . . . . 94
3.30 The arrangement of an uncoupled micro-turbogenerator . . . . . . . . 94
3.31 A permanent magnet generator with three coil layers . . . . . . . . . 96
xiii
3.32 A representational circuit diagram for a generator with a variable layercount, n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.33 Effect of layer count on system parameters . . . . . . . . . . . . . . . 97
3.34 A single phase of the 4 pole-pair, 3 turns per pole stator . . . . . . . 98
3.35 Planar continuum layers for a planar PM generator . . . . . . . . . . 99
3.36 A magnetization profile for an annular permanent magnet . . . . . . 100
3.37 Back iron thickness schematic . . . . . . . . . . . . . . . . . . . . . . 101
3.38 Laminations embedded within a magnetic conducting stator . . . . . 106
3.39 Eddy currents within a radial conducting segment with and withoutlaminations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.40 A representational circuit for power balance . . . . . . . . . . . . . . 109
3.41 Energy density of designed micro-turbogenerators relative to Li-ionbatteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.1 Ejector control volume . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.2 After-burning thermodynamic cycle . . . . . . . . . . . . . . . . . . . 115
4.3 The original micro-turbogenerator . . . . . . . . . . . . . . . . . . . . 117
4.4 3D printed turbo-generator . . . . . . . . . . . . . . . . . . . . . . . . 118
4.5 LEDs powered by the micro-turbine generator . . . . . . . . . . . . . 119
4.6 Turbine shaft power and load power as a function of rotor speed . . . 120
xiv
Acknowledgements
I would first like to thank my advisor, Dr. Jonathan Protz, for all of his help and
guidance over these past few years. With the exception of my parents, no one has
ever done as much for me. I would also like to thank my parents and sisters for their
support over this time. Most of this work is a direct result of the work of others
before me at MIT and GaTech, and I wish to strongly acknowledge that and thank
all of them for contributing to the scientific literature on this subject. I want to
thank Will Gardner and Ivan Wang for not only greatly assisting me academically,
but for also being good friends throughout this period. I also wish to thank Yuxuan
Hu for her emotional support over the years. You’ve always been there for me, both
during the highs and the lows. And while I will forever deny it, everyone claims that
you motivated me to become a hard worker and a better person. Things did not work
out between us, but I wish you my best. I know you will have a happy, fulfilling,
and successful life. I also want to thank Gwen Gettliffe for being by my side during
much of this time and giving me a reason to keep on going. You are a wonderful
person in so many ways, and I wish things had worked out differently between us,
but I take comfort in knowing that at least we will continue to be good friends. A
strong degree of gratitude goes to Kathy Parrish for simplifying all of the graduate
school requirements, I would have been lost without you. I also want to thank all of
my professors over the years for being patient and allowing me to build a stronger
fundamental understanding of some very abstract concepts, in particular Dr. George
xv
for his E.E. insights and Dr. Shaughnessy for his compressible flow insights. I also
want to thank Dr. Garg for allowing me to pursue my interest in robotics during this
time. Kip Coonley and Justin Miles were especially kind in allowing me to use their
electronic equipment, sensors, and SMT soldering equipment. I would not have been
able to do my experiments without that support, thank you. Lastly, I would like to
thank Logos Technologies, DARPA, and Duke University for financial support over
this period.
xvi
1
Introduction
1.1 Background and Motivation
1.1.1 Concept
In the mid 1990’s, the idea of using micro-sized devices to produce electrical power
from combustible fuels was presented by Dr. Epstein et al. (1997). The intent was
to use micro-sized heat engines in lieu of chemical batteries or fuel cells in specific
applications due to certain advantages that micro-heat engines would posses. To
demonstrate this, a quick synopsis of the strengths and weaknesses of these different
power sources must be given.
Chemical batteries are typically associated with high power densities but low
energy densities. As an example, conventional lithium-ion batteries are capable of
high power densities in the 1000 W/kg range, but also typically exhibit low energy
densities in the 150 W-hr/kg range [Panasonic Corporation (2009)]. In contrast,
fuel cells typically demonstrate high energy density due to the large specific energy
of their fuels and a good conversion efficiency, but also low power densities. A
compilation of commercial fuel cells by Narayan and Valdez (2008) showed energy
densities as high as 805 W-hr/kg, but with an accompanying specific power of only
1
2 W/kg. Other fuel cells were shown that demonstrated power densities as high as
18 W/kg, but this occurred at the expense of a much lower conversion efficiency as
demonstrated by a specific energy of 121 W-hr/kg, less than a lithium ion battery.
Thus fuel cells and batteries exist at opposite ends of a trade-off spectrum, with
batteries demonstrating low energy density but high power density, and fuel cells
demonstrating high energy density but low power density. In between these two
extremes are hydrocarbon combustion based heat engines. As an example, a 1000 W
Honda generator using 2 gallons of gasoline has a specific energy of 506 W-hr/kg and
a specific power of 40 W/kg [Honda Corporation (2001)]. Reasonably high energy
densities can be achieved despite a low conversion efficiency due to the large specific
energy of hydrocarbon fuels (12,500 W-hr/kg for gasoline).
As a result of these trade-offs, these device classes are each well suited to different
applications. In particular, in cases where reasonably high power and energy densities
must be obtained such as in automobiles and aircraft, heat engines are most often
used. This trend continues down to the size where relatively cheap and conventional
machining practices can be utilized. As an example, small sized piston and gas
turbine engines are used for long endurance RC hobby aircraft. Below such sizes
however, relatively cheap manufacturing processes with the necessary precision no
longer exist. For this reason, batteries and fuel cells are currently used exclusively
at these smaller sizes for applications that would be better suited for a heat engine.
This work discusses the design of a heat engine, a micro-turbogenerator, that
could be manufactured to fill this void. Its operation is analogous to a convention-
ally sized natural gas power plant, where a gas turbine utilizing a Brayton cycle is
driven by the combustion of fuel, and the resulting shaft power is outputted to an
electric generator to create electricity. A figure of the engine core without the elec-
trical components can be seen in figure 1.1. At the macro-size level, gas turbines are
excellent devices for converting heat from combustion into shaft power due to their
2
Figure 1.1: A micro-turbine engine operating a Brayton cycle, Spadaccini et al.(2003)
simple operation, high power density, and ability to use a wide range of fuels. And at
the micro-size level, a turbogenerator is an ideal choice also due to its theoretically
simple operation and low complexity; there are minimal moving parts, the compres-
sion, combustion, and expansion processes are steady state, and all the components
can be designed with planar geometry, allowing the utilization of micro-fabrication
techniques. In addition, when compared to its conventionally sized counterpart a
micro-heat engine has the advantage of benefiting from the cube-squared law, where
as the characteristic length of the engine decreases, its power density increases. This
can be shown as follows.
For a control volume, the output shaft power of a turbine is
9Wt ηt 9mcppTt4 Tt5q (1.1)
where ηt is the turbine efficiency, 9m is the mass flow, cp is the specific heat capacity at
constant pressure, Tt4 is the total temperature pre-turbine, and Tt5 is the total tem-
perature post-turbine. Factoring out Tt4 and using an isentropic expansion process,
this can be shown to be
9Wt 9mcpTt4p1 rPt5Pt4sγ1γ q (1.2)
where Pt represents the total pressure of the gas. As can be seen, for fixed conditions
such as efficiency, pressure, and temperature, the power output will scale linearly
3
with the mass flow. And again for fixed conditions, the mass flow will scale linearly
with the area, or with the square of a characteristic length, L2. As a result, the
power output will scale with the square of the characteristic length.
9W 9 9m 9 A 9 L2 (1.3)
The volume of the device however will simply be proportional to characteristic length
of the system cubed, L3. Therefore for fixed conditions, the ratio of power to volume,
or the power density of the device will be
9W
V9 L2
L39 1
L(1.4)
Such a simple analysis shows the advantage of micro-sized gas turbines. If one were
to simply scale a gas turbine with an outer diameter of 1 meter to the size of a micro
gas turbine with an outer diameter of 1 cm and maintain its operating conditions
and efficiency, the power density of the device would increase 100 fold.
In order to extract electrical energy from a gas turbine however, an electrical
generator must be attached to its output shaft. It follows then that a similar analysis
must be presented to show the scaling laws that govern electrical generators. For
this work, a permanent magnet micro-generator was selected for reasons to be later
explained.
As shown by Faraday’s law of induction, the induced voltage in a loop of coil is
equal to the time rate of change of the magnetic flux in the area circumscribed by
the coil.
ε dΦ
dt(1.5)
where the flux is determined by
Φ » »
~B d ~A (1.6)
4
As such, the induced voltage should be proportional to the area, or a characteristic
length squared, and the rotational speed of the magnetic field over the generator
coils.
ε 9 A Ω 9 L2 Ω (1.7)
As shown in the maximum power transfer theorem, for any given voltage, the maxi-
mum output power is related to the resistance of the generator coils by
9Wmax ε2
4 Rcoil
(1.8)
with the resistance of the coils being
R ρ LA
9 1
L(1.9)
Therefore, plugging in our voltage and resistance relationships, the power of the
generator system should scale as
9Wmax 9 L5 Ω2 (1.10)
However, with regard to turbomachinery, the tip speed of the turbine blades must
always be commensurate with the speed of the gases, which do not scale with the
size of the device. As such
Ω 9 1
L(1.11)
Inserting this conclusion into the power relationship shows that
9W 9 L3 (1.12)
As such, the power density of the magnetic generator should scale as
9W
V9 L3
L39 1 (1.13)
5
and therefore no benefit would be seen with miniaturization. However, such a benefit
is observed in real world scenarios as a result of scaling laws that affect heat dissipa-
tion. As shown in figure 1.2, for a given current density, the heat creation will scale as
L3. However, conductive and convective heat dissipation will scale with the surface
area or L2, making the ratio of heat dissipation to heat creation scale as 1L
. As a
Figure 1.2: Scaling laws for heat dissipation in a conductor, Cugat et al. (2003)
result, larger current densities are acceptable in micro-sized generators than in their
conventionally sized counterparts. This allows generator power density to an extent
also scale with the inverse of the characteristic length of the system at the expense
of efficiency. This has been confirmed in permanent magnet micro-generator tests
by Arnold et al. (2006b), where a power density of 59MW m3 was reported, whereas
typical macro-sized generators typically achieve around 20MW m3. Alternatively,
with size reduction, volume that was previously dedicated to cooling elements can
instead be dedicated to power producing elements, allowing the power density to
increase while maintaining efficiency.
As demonstrated, given equal non-scaling operating conditions such as tip speed,
efficiency, pressure ratio, etc., the power density of both the gas turbine and perma-
nent magnet generator scale indirectly proportional to the characteristic length of
the turbogenerator system. However, as shown in this thesis, the efficiencies of these
components typically do not scale favorably with size, and thus the power density
does not scale quite as well as this simple analysis would suggest. Regardless, the
power density does scale well enough for micro-turbogenerators to provide sufficient
amounts of power for many applications. Likewise, given sufficient components effi-
6
ciencies, the utilization of hydrocarbon fuels enables these devices to achieve energy
densities many times greater than conventional batteries. As shown in figure 1.3,
using ethanol as fuel, a thermal efficiency of roughly 13% and a generator efficiency
of 80% results in a device whose energy density can reach five times that of current
lithium-ion batteries. Thus there are many potential advantages associated with
Figure 1.3: Maximum energy density of a micro-turbogenerator relative to lithium-ion batteries as a function of thermal efficiency, generator efficiency, and fuel source
micro heat engines, particularly turbogenerators. They exhibit high energy density
relative to batteries due to their fuel source. They also possess sufficiently high
power densities due to scaling laws to power mobile devices and should theoretically
exhibit higher power densities than commercial fuel cells. For these reasons, micro-
turbogenerators are ideal power sources to run certain classes of devices, in particular
mobile devices that require endurant power sources and have relatively large power
requirements such as robotics, uav’s, and high power portable electronics. There
exists one other potential advantage associated with micro-turbogenerators. Due
to the power density scaling laws, if the devices are fabricated cheaply using large
volume manufacturing techniques as in the semiconductor industry, multiple micro-
7
turbogenerators could be used in parallel to provide a large power output at a fraction
of the size of an equivalent macro sized machine.
1.1.2 Challenges
Despite all the advantages that could be attributed to micro-turbogenerators, a self
sustaining engine has never been designed and run due to the many difficulties and
challenges associated with their design, fabrication, and operation.
Gas turbines often utilize a Brayton cycle, where ideally gases are isentropi-
cally compressed, then heated reversibly, and then isentropically expanded through
a turbine. As can be seen in figure 1.4, due to heat addition, the change in total
temperature across a compressor and turbine with the same pressure ratio diverges.
This allows a gas turbine to not only create its own pressurized gas source, but also
deliver excess shaft power to a generator to create electric power. In real systems
however, all of these processes are done in an irreversible entropic manner. The
differences between the reversible and non-reversible cycle paths can also be seen in
figure 1.4. For both the compressor and turbine portions of the cycle, the amount of
Figure 1.4: Reversible and non-reversible TS diagrams for a Brayton cycle
8
energy transfer will be
9W 9m cp ∆Tt (1.14)
As can be seen in figure 1.4 however, for any given pressure ratio, ∆Tt is larger for
the compressor and smaller for the turbine in the non-isentropic case relative to the
isentropic case. The result of this is that the compressor requires more input power
to achieve its pressure ratio, and the turbine delivers less power output for the same
pressure ratio. And because the compressor receives its shaft power from the turbine,
there then exists some efficiency below which the Brayton cycle will no longer close
as a result of the compressor requiring more power than the turbine can provide at
a given pressure ratio. This can be shown as follows. As stated before, the power
output for a turbine is
9Wt ηt 9mcpTt4p1 rPt5Pt4sγ1γ q (1.15)
Likewise, the power input for a compressor is
9Wc ηc 9mcpTt2prPt3Pt2sγ1γ 1q (1.16)
Equating these two until the power requirements are equal results in
ηt ηc ¥ Tt2prPt3Pt2sγ1γ 1q
Tt4p1 rPt5Pt4sγ1γ q
(1.17)
and therefore, for any given pressure and temperature ratios, there are minimum
component efficiencies that are required for the turbine to supply sufficient shaft
power to the compressor. If the efficiencies are below this threshold, the turbine
cannot power the compressor, and therefore no high pressure gas will be available to
power the turbine, and the cycle will not close.
For micro-turbomachinery unfortunately, the typical sources of entropic losses are
much greater than in their conventional counterparts due to relatively higher viscous
9
effects and manufacturing constraints. Some of these turbomachinery loss sources
are tip clearance losses, trailing edge mixing losses, end wall losses, and viscous losses
in the boundary layer.
In addition, there are many forms of losses that are practically unique to micro-
turbine engines. For example, due to the planar geometry dictated by MEMS fabri-
cation constraints, there exist many right angle turns in the flow path of the working
fluid as shown in figure 1.5, which lower the total pressure of the fluid. Another issue
Figure 1.5: Gas flow path for a micro-turbine, Frechette et al. (2005)
that affects micro-turbomachinery especially is residual swirl leftover in the wake of
the turbine. As can be seen by analyzing the velocity triangles in figure 1.6, there ex-
ists some rotational speed (or tip speed) for the turbine blades where the tangential
velocity of the exit flow is zero, with the remaining flow having either only a radial
or axial component. This speed corresponds to the optimum speed of the turbine,
because all of the tangential kinetic energy of the gases has been transmitted to the
turbine shaft. However, as described in section 1.1.1, the turbine tip speed is inde-
pendent of size. As a result, extremely high values of RPM (often above 300,000)
are required for micro-turbomachinery, and this necessitates the use of high-speed
air bearings for the engine. Such bearing systems are extremely complex however
and have only recently shown sufficient progress to be reliably utilized as demon-
strated by Frechette et al. (2005). They also have yet to be correctly integrated
into a complete turbogenerator system. The highest observed rotational rate for a
fully integrated turbogenerator system utilizing gas bearings is currently only 40,000
10
Figure 1.6: Velocity triangles of a turbine stage
RPM as demonstrated by Yen et al. (2008b).
Micro-turbine engines also typically suffer from issues associated with heat trans-
fer due to the small characteristic length of the devices. As an example, it is unfavor-
able for heat to transfer from the hot exhaust gases of the turbine to the gases within
or traveling to the compressor. This can be explained conceptually as follows. For a
turbomachine with an incompressible working fluid, the power transfer between the
fluid and the blades will be
9W 9m∆Ptρ
(1.18)
Conceptually then, one can think of a Brayton cycle as working across a fixed pres-
sure ratio, with the combustor existing solely to heat the temperature of the gas
and lower the density so that the turbine can extract more power during expansion
11
than the compressor requires for compression. For this reason, if the gas within the
compressor or the gas in route to the compressor is heated, then for a given shaft
RPM determined by material constraints, a lower pressure ratio will be achieved and
result in lower system efficiency. In conventional sized machines, the heat conduc-
tion from the turbine section to the compressor is often negligible relative to the
mass flow. However, for micro-sized machines with small characteristic lengths, the
thermal resistance is sufficiently low that strong measures must be taken to limit
heat conduction. As can be seen in figure 1.7, structural changes such as adding
Figure 1.7: Comparison of a first generation design relative to a more recent designshowing the thermal resistance structures between the turbine and compressor, Lang(2009)
thin and long thermal barriers between the turbine and compressor sections must be
made. This figure contrasts an early generation design versus a more contemporary
design. However, this small scale heat transfer effect can be used advantageously
as well. As an example, a recuperator with a high effectiveness transferring heat
12
from the exhaust gases to the pre-combustion gases can be made with a very small
form factor due to these scaling laws. For conventionally sized machines, the size
and weight penalties of these heat exchangers often outweigh their thermodynamic
benefits. Micro-sized heat engines however can use heat exchangers very effectively
to theoretically improve performance because heat exchangers are relatively much
smaller in micro-systems due to the scaling laws of heat transfer.
In addition to the efficiency problems associated with micro-turbomachinery,
there are many problems encountered by micro-generators due to their small size
as well. However, these problems are not as severe as those that affect micro-
turbomachinery components. The issues effecting permanent magnet micro-generators
will be discussed, because this type of generator is the one that is ultimately chosen.
One of the primary difficulties with micro-sized permanent magnet generators
is magnetizing the hard magnetic material. Ideally, there would be a square wave
magnetization pattern on the magnets alternating between north and south polarity
as shown in figure 1.10 (cm 0). However, due to leakage flux (figure 1.9) in the
magnetizing heads (figure 1.8) used to magnetize the material, trapezoidal magnetic
Figure 1.8: A magnetizing head used to permanently magnetize magnet mate-rial, Gilles et al. (2002)
patterns are typically encountered. This results in a decreased time rate of flux
13
Figure 1.9: Leakage flux permeating incorrect sectors during magnetization, Gilleset al. (2002)
Figure 1.10: Trapezoidal magnetization profile of an annular permanent mag-net, Das et al. (2006)
transversing the generator coils and therefore smaller voltages. This in turn results
in a reduced efficiency for a fixed power output requirement and a fixed internal coil
resistance. A solution to this problem is to use discrete magnet pieces that can be
easily magnetized and to place them around the periphery of the generator as shown
in figure 1.11. This also helps solve the issue of balancing the generator rotor because
the magnets can be weighed and appropriately arranged around the periphery in
order to minimize imbalance. Unfortunately, this technique also necessitates the
14
Figure 1.11: Discrete magnet pieces arranged peripherally around the generatorrotor, Herrault (2009)
use of a robust silicon frame to keep the magnet pieces situated. This is because
the magnetic material is no longer annular and therefore cannot resist radial forces
associated with centripetal motion. An example of such a silicon frame can be
seen in figure 1.12. Despite the frame however, using discrete magnet pieces will
Figure 1.12: A silicon frame for discrete peripherally arranged magnet pieces, Her-rault (2009)
always lower the maximum achievable RPM relative to an annular magnet because
15
an annular magnet is able to support some ring strain by itself. The result of lowering
the maximum RPM is the lowering of the generator output voltage. This negatively
affects power output and efficiency.
There are also many fabrication constraints that negatively affect performance.
For example, due to the limited number of materials which can be electrically de-
posited using current MEMS techniques, materials that don’t possess optimal char-
acteristics are often chosen. A good demonstration can be seen with Yen et al.
(2008a), where a nickel-iron alloy with a low saturation density of only 0.8T (versus
the standard Hiperco with a saturation value of 2.4T) was chosen as the back iron
material for their generator rotor. In addition, due to a MEMS depositing thickness
limitation, not only was the choice of material compromised, but so was the geome-
try, with the actual back iron being only 20% of the ideal thickness. This decreased
the effective permeability of the back iron and therefore lowered system voltage and
efficiency.
Permanent magnet generator performance is also negatively affected by rotor
stability constraints imposed by the gas bearings. The journal bearings typically have
specific damping and stabilizing performance values, and higher rotor weights and
angular velocity values cause the rotors to cross this threshold. As a result, generator
rotors are often constructed with low aspect ratio (low thickness to diameter ratios)
magnets and magnetic back irons. As described in chapter 3, this decreases the flux
output of the generator and therefore lowers system voltage, power, and efficiency.
Fabrication constraints imposed by the 2D planar geometry also negatively af-
fect the internal resistance of the generator coils. As can be seen in figure 1.13,
complicated multi-plane coil patterns utilizing many electrical vias are required to
achieve multiple coil turns per pole. This has the effect of greatly increasing the in-
ternal resistance beyond what one would expect from standard area, resistivity, and
length calculations, and therefore lowers the efficiency of the system for any given
16
Figure 1.13: Coil winding patterns for 2D micro-generators, Arnold et al. (2006b)
power output.
Micro-generators are also more negatively affected by eddy current and hysteresis
losses than conventionally sized machines. Hysteresis losses typically scale with the
frequency of the device, f , and eddy current losses scale with the frequency squared,
f 2. And as previously explained, the rotational speeds of micro-turbines must be
extremely high. This results in high electrical frequencies for the attached gener-
ator and therefore relatively high eddy current and hysteresis losses. In order to
reduce these losses, a few possible options can be taken. The first is to have a non-
conducting, non-magnetic stator iron so that eddy currents and hysteresis cannot
exist. This will typically have the effect however of increasing the reluctance of the
magnetic circuit and therefore lowering the generator output voltage and efficiency.
Another option is to laminate the stator back iron as shown in figure 1.14. This
will reduce the severity of the eddy current losses without greatly increasing the
reluctance of the magnetic circuit.
Another issue that has a great impact upon generator performance is the effect
that temperature has on the magnetic remanence of permanent magnets. As can be
17
Figure 1.14: Non-conducting silicon laminations within the stator back iron toreduce eddy current losses, Herrault et al. (2010)
seen in figure 1.15, the remanence of magnetic materials typically degrades with tem-
Figure 1.15: Relationship between magnetization and temperature for rare earthcobalt materials, Campbell (1996)
perature. This is especially true in the case of neodymium boron material, which is
typically the highest performing material at room temperature. This thermal degra-
dation not only reduces the magnetic performance of the engine, but also completely
eliminates certain magnetic materials from consideration and forces designers to use
18
alternative materials that are typically considered as inferior.
Lastly, of immense importance for micro-sized permanent magnet generators is
their power output capabilities. For any generator design operating at a fixed RPM,
there will be a maximum amount of mechanical shaft power that the generator
can convert to electrical power and dissipate into the surroundings as heat through
loss mechanisms. Should the generator be unable to utilize or dissipate its input
shaft power at its design RPM, it will accelerate beyond this speed and most likely
mechanically fail. For this reason, assuming a high value of efficiency is sought, it is
required that the maximum power transfer capabilities of the generator be high and
sufficient enough to convert the input shaft power into electric power. This presents
a challenge however because the power density of the turbomachinery scales much
more favorably with size than does the power density of micro-generators, making it
difficult for the generator to utilize the relatively large shaft power input at a high
efficiency.
Another key component of a heat-engine is the combustor as shown in figure 1.16.
This component also sees significant challenges due to the scaling laws that affect its
Figure 1.16: Schematic of a silicon wafer single-zone combustor, Spadaccini et al.(2003)
performance. As described by Spadaccini et al. (2003), there is a length of time that
is required for most of the injected fuel to combust, τreaction, which is related to the
19
chemical reaction rate by an Arrhenius equation.
τreaction rfuelsA rfuelsa rO2sb eKaRT (1.19)
There is also a length of time known as the residence time, τresidence, which is the
length of time that the gas mixture requires to travel through the combustor volume.
τresidence volume
volumetric flow rate V ρ
9m V P
9m R T(1.20)
The ratio of these two values is known as the Damkohler number.
Dah τresidenceτreaction
(1.21)
In order to combust all of the injected fuel and achieve a high combustion efficiency,
the Damkohler number must be greater than one. This is difficult to achieve with
micro-combustors because, due to the size of device, τresidence is quite low. For
fixed operating conditions such as pressure, temperature, and combustor volume,
the residence time can be increased by reducing the mass flow rate through the
device. This however will reduce the power density of the device, making it a non-
viable option. The other option is to dramatically increase the reaction rate which
can be done by one of two methods: increasing the combustion temperature or using
catalytic surface material. Increasing the combustion temperature can be achieved
by burning closer to stoichmetric conditions (in typical gas turbines, combustion
takes place far from this point). The gases must still be cooled before entering the
turbine however, and therefore this setup requires the use of a dual-zone combustor,
as shown in figure 1.17. The first zone is utilized to burn the fuel quickly, at high
temperature, and closer to stoichiometric conditions. The second zone is used to mix
and cool the gases. The combustion reaction rate can also be increased by utilizing a
catalytic surface as shown in figure 1.18. This insert works by lowering the activation
20
Figure 1.17: Schematic of a dual zone combustor, Spadaccini et al. (2003)
Figure 1.18: A silicon micro-combustor with a platinum plated catalytic in-sert, Spadaccini et al. (2002)
energy of the reaction and thereby increasing the reaction rate.
As shown, micro-turbogenerators are extremely complex and there are many chal-
lenges associated with their operation. Each of the components, the turbomachinery,
combustor, and electric generator all face severe hurdles due to manufacturing lim-
itations and scaling laws that govern their performance. However, significant work
has been done towards developing these components and integrating them into a
working engine.
21
1.1.3 Review of Previous Work
A substantial amount of work has been done towards developing all the components
of a micro-turbogenerator as shown by Lang (2009). Most of this work was done
independently however, with the components being meant to function with external
support as opposed to working in an fully integrated engine.
Significant hurdles have already been overcome with gas bearings and micro-
turbomachinery. Peirs et al. (2003) demonstrated an axial flow impulse turbine
with an outer diameter of 10 mm generating 28 W of shaft power at 160,000 RPM
(figure 1.19). Savoulides et al. (2008) fabricated and tested a silicon MEMS tur-
Figure 1.19: The 10 mm axial impulse turbine tested by Peirs et al. (2003)
bocharger up to an RPM of 480,000 (figure 1.20). The turbocharger was powered by
an integrated micro-turbine and created a pressure ratio of 1.21. All of the turboma-
chinery components were supported by hydrostatic thrust and journal gas bearings.
Similarly, Frechette et al. (2005) fabricated and tested a single stage 4.2 mm diameter
radial inflow MEMS turbine (figures 1.21 and 1.22). The turbine was supported by
hydrostatic thrust and journal bearings and reached speeds greater than 1,000,000
RPM with a corresponding shaft power of 5 W. Lastly, Lee et al. (2008) demonstrated
22
Figure 1.20: SEM photograph of the MEMS turbocharger, Savoulides et al. (2008)
Figure 1.21: Cross sectional view of the turbomachinery, Frechette et al. (2005)
Figure 1.22: A bird’s eye SEM photograph of the MEMS turbine, Frechette et al.(2005)
23
a multistage radial inflow turbine operating at 330,000 RPM and with a mechanical
shaft power output of 0.38 W.
Figure 1.23: A split SEM view showing the concentric turbine stages, Lee et al.(2008)
A great deal of effort has also been successfully dedicated to micro-combustor
research. Mehra et al. (2002) designed and tested a hydrogen gas fueled micro-
combustor (figure 1.24). The device demonstrated stable operation in a silicon
structure with a flame temperature in excess of 1600 K. Subsequently, Spadaccini
et al. (2003) designed a dual zone combustor with the intent of increasing fuel burn
efficiency and allowing hydrocarbon combustion (figure 1.25) . Exit gas temper-
atures were as high as 1800 K and the power density of the device corresponded
to 1100MW m3. In order to increase combustion efficiency of hydrocarbon fuels,
platinum plated catalyst inserts were tested in micro-combustor experiments (fig-
ure 1.26). The insert allowed the combustion of hydrocarbon fuels in this device
which was not possible without the catalytic inserts. Combustion efficiencies in ex-
cess of 40% were demonstrated with the insert.
Electrical generator technology has also extensively progressed over this time pe-
24
Figure 1.24: A fully packaged micro-combustor, Mehra et al. (2002)
riod. Holmes et al. (2005) showed a low pressure ratio integrated turbo-generator
(figure 1.27) achieving a power output of 1.1 mW at 30,000 RPM. The device oper-
ated with ball bearings and had discrete magnetic components embedded within the
turbine structure. Raisigel et al. (2006) created a generator that operated at 380,000
RPM with external support and created 5 W of power. The same device was able to
function without external support at 58,000 RPM and produce 14.6 mW of power.
Figure 1.25: Schematic of a dual zone combustor, Spadaccini et al. (2003)
25
Figure 1.26: Schematic of a combustor with a catalyst insert, Spadaccini et al.(2002)
Figure 1.27: Schematic of a ball bearing supported turbo-generator, Holmes et al.(2005)
Das et al. (2006) and Arnold et al. (2006b) modeled, fabricated, and tested perma-
nent magnet generators up to 120,000 RPM with DC power outputs of 1.1 W. That
generator design was then further optimized by Arnold et al. (2006a) by improving
the winding pattern (figure 1.28) and operating at a higher RPM. The optimized
windings showed significantly reduced resistance values for the generator coils. This
resulted in a net electrical power output of 8 W at 305,000 RPM. Herrault et al.
(2008) then tested and characterized micro-generator performance at high tempera-
tures, confirming as expected that generator performance would decrease at elevated
temperatures. He demonstrated however that the generator performance can remain
adequate for micro-heat engine electrical power generation with careful selection of
materials (see figure 1.29). Herrault et al. (2010) then took the first steps to inte-
grate micro-generator technology into a silicon structure capable of supporting gas
26
Figure 1.28: A comparison of the original and optimized winding patterns, Arnoldet al. (2006a)
Figure 1.29: A comparison between SmCo and NdFeB generators as a function oftemperature, Herrault et al. (2008)
27
bearings and micro-turbomachinery, a key requirement for creating a self-sustaining
micro-turbogenerator. His device also contained a laminated stator-iron to reduce
eddy current losses and increase electrical efficiency.
At Duke, research has been conducted into ejectors and injectors. These are static
devices which utilize vaporized gases from boilers to pump secondary fluids. Initial
work was done to show that an ethanol vapor powered ejector could create a sufficient
suction draft to replace a compressor for a micro-turbine or to power hydrostatic gas
bearings during the startup transient periods of operation, Gardner et al. (2010b).
Subsequent work created detailed loss models for ejector performance, Gardner et al.
(2010a). In particular, the relationship between power density and size as a result
of scaling effects were shown (figure 1.30) as was the relationship between suction
draft and mass flow (figure 1.31). This work is significant, as will be explained
Figure 1.30: Power density and ejector efficiency as a function of the nozzle throatdiameter, Gardner et al. (2010b)
in detail later, because it allows a micro-turbine to operate without a compressor
and the cycle closing disadvantages of a Brayton cycle and without the condensation
28
Figure 1.31: Suction draft performance versus entrainment ratio, Gardner et al.(2010a)
disadvantages associated with a Rankine cycle.
As shown, many of the components which make up micro-turbogenerator systems
have been heavily studied and researched. However, few studies have been conducted
towards integrating the various components into a working system. Frechette et al.
(2004) conducted high level design work for a micro-heat engine utilizing a Rankine
cycle (figure 1.32). His analysis showed an electrical power output of 1-12 W with
Figure 1.32: A schematic of a MEMS power plant utilizing a Rankine cy-cle, Frechette et al. (2004)
29
an overall conversion efficiency of 1-10%. However, his device was never fabricated.
Yen et al. (2008a) designed an integrated turbogenerator system that utilized a
pressurized gas source. His system was fabricated on a seven wafer thick silicon
stack. The permanent magnets were discrete pieces and were imbedded within the
silicon structure of the turbomachinery, which was supported by hydrostatic journal
and thrust bearings. For this device, the generator windings and their connectors
were integrated into the silicon structure. This was the first integrated system to
have an electric generator powered by micro-turbomachinery and to have all the
components supported by gas bearings. A schematic of the device can be seen in
figure 1.33. It was designed for a rotational speed of 360,000 RPM and a power
Figure 1.33: Cross sectional schematic of the fully integrated permanent magnetturbogenerator, Yen et al. (2008b)
output of 1.5 W. However, due to a structural oversight, it was only able to accept
sufficient pressure to reach 40,000 RPM, resulting in a power output of 19 mW.
Regardless, this is arguably the most advanced silicon micro-turbogenerator to date.
As shown in more detail in chapter 4, Camacho et al. (2010) integrated two micro
turbines with a permanent magnet generator, power electronics, and an ejector. The
system was unique in that it lacked a compressor, but rather used an ejector to create
the pressure gradient to drive the turbine. The ejector was powered by combustion
30
from an ethanol boiler, making this the first micro-engine to convert the heat from
combustion into electricity. The first of the two turbogenerators can be seen in
figure 1.34.
Figure 1.34: The original turbogenerator with the protruding generator leads con-nected to the power electronics board
As shown, a great deal of work has gone into developing the subsystems and
components for a micro-turbogenerator. The knowledge from this research is only
just now starting to be integrated together to form functioning engines. As such,
should there be a strong economical argument for micro-turbogenerators, a great deal
of useful information exists in the scientific literature from which one could begin to
design a working system.
1.2 Research Objectives, Goals, and Design Overview
The goal of this thesis is to achieve a few specific tasks that are required to design a
micro-turbogenerator and accurately model its performance. The first task is to show
the relation between efficiency and various turbine parameters such as reaction, stage
count, RPM, and size. This is of critical importance because it allows the designer
31
to predict the actual mechanical shaft power output for given operating conditions
such as inlet pressure and temperature. It is also especially important in the case of
any Brayton cycle, because as described in 1.1.2, this efficiency value is necessary to
determine if the cycle can close for given operating conditions. The second task is to
accurately model micro-generator performance. This is important because it allows
the designer to predict the total system efficiency, from fuel burn to electric power
output, of a micro-turbogenerator. The total system efficiency is a crucial value in
determining whether a micro-engine would be superior to a fuel cell or battery for any
given application. It is also important to properly model the generator performance
so that turbomachinery and generator components can be correctly matched. If they
are designed independently and not correctly matched, then either the generator,
the turbomachinery or both will be working off design, with detrimental effects on
performance. Lastly, it will be demonstrated that, with no moving parts, an ejector
can be utilized to not only start-up a micro-turbogenerator through the transient
startup phase, but to also drive it in steady state and create electrical power, thus
presenting a relatively easy means to convert the heat from combustion into electric
power.
32
2
Micro-turbine Design and Performance
2.1 Concepts of Turbomachinery
2.1.1 Operating Concepts
A turbomachine functions by absorbing or imparting energy, more specifically total
enthalpy, to the fluid traveling through it. A diagram of a general turbomachine can
be seen in figure 2.1. As it is a rotating machine, the power transfer is simply the
Figure 2.1: A control volume for a general turbo-machine, Mattingly et al. (2006)
33
torque times the rotational velocity.
9Wt τ ω (2.1)
The torque on the blades of the turbomachine can be determined from the change
in angular velocity between the fluid leaving and exiting the device.
τ 9mpr1Vu1 r2 Vu2q (2.2)
The equation for power can then be re-written as
9Wt 9mωpr1Vu1 r2Vu2q (2.3)
This equation is known as the Euler turbine equation. It is useful however to expand
this equation in a way that sheds light on the various components of energy transfer.
Defining the product of ω and r as tip speed, U , we see that
9Wt 9mpU1Vu1 U2Vu2q (2.4)
Observing the velocity triangle in figure 2.2, we see that the flow leaving a turbine
Figure 2.2: Velocity triangles for the exit flow from a rotor blade
34
blade will have some absolute exit velocity, V2, that is made up of its axial and
tangential components, Vm2 and Vu2. Note that the blade velocity U2, relates the
absolute velocity V2 to its relative velocity Vr2. Using geometry, we see that
V 2m2 V 2
2 V 2u2 (2.5)
V 2m2 V 2
r2 pU2 Vu2q2 (2.6)
Equating these two expressions results in
V 22 V 2
u2 V 2r2 pU2 Vu2q2 (2.7)
This can be further expanded and then simplified
V 22 V 2
u2 V 2r2 U2
2 2U2Vu2 V 2u2 (2.8)
U2Vu2 1
2pV 2
2 U22 V 2
r2q (2.9)
A similar process can be shown for the blade inlet and results in
U1Vu1 1
2pV 2
1 U21 V 2
r1q (2.10)
Inserting these values into the Euler turbine equation, eq. 2.3, results in
9Wt 9m1
2
pV 21 V 2
2 q pU21 U2
2 q pV 2r2 V 2
r1q
(2.11)
While the Euler equation is most often used in calculations, this form of the equation
is superior at showing the underlying mechanisms of energy transfer between the
rotor and the fluid. It is worthwhile to discuss which energy transfer mechanisms
the different terms correspond to. The first term, V 21 V 2
2 , is the easiest term to
understand, as it directly corresponds to the absorption of kinetic energy by the
rotor blades. The second term, U21 U2
2 requires more explanation. As described
by Shepherd (1956), if one were to observe a fluid element in a rotating environment
35
Figure 2.3: A fluid element in a centrifugal field
as shown in figure 2.3, then clearly some type of force is being applied to the element
towards the center. In this case, the only force available is pressure. In order to
maintain a fluid element in rotation about the center of rotation, a centrifugal force
of dmω2r must be present, where dm drdAρ. This centrifugal force is a result of
a pressure gradient across the fluid element in the radial direction, dPdA. Equating
the two forces results in
dPdA drdAρω2r (2.12)
Rearrangement gives
dP
ρ ω2rdr (2.13)
Integrating both sides of the equation gives
»dP
ρ»vdP 1
2ω2pr2
2 r21q
1
2pU2
2 U21 q (2.14)
From the combination of the first and second laws of thermodynamics and for an
adiabatic process, the left hand side of the above equation is equal to the change
in enthalpy, and thus shaft work done to the fluid. As shown, this is equal to the
term 12pU2
2 U21 q, and therefore this term corresponds to the work done to the fluid
for traveling from one radius to another in a centrifugal pressure field. The last
36
term, 12pV 2
r2V 2r1q, corresponds to changes in the relative fluid velocity, which simply
corresponds to changes in static pressure and therefore enthalpy.
As shown, a turbomachine works by changing the kinetic energy of the working
fluid, changing its static pressure through centrifugal pressure fields, and changing
the static pressure by accelerating the fluid flow in the relative frame. The sign of
these value changes determines whether or not enthalpy is being added to the flow
or being extracted, that is whether or not the machine is working as a turbine or
a compressor. Turbomachines can be further classified by the extent to which they
utilize these different energy transfer mechanisms. In particular, the ratio of energy
transfer via static pressure changes to the overall energy transfer is an important
feature called reaction, which as will be shown, is a key parameter when classifying
turbomachine operation.
2.1.2 Conservation Laws and Governing Equations
For turbomachine design, the basic equations which govern fluid flow must be used,
that is the conservation of mass, energy, and momentum, a state equation, the isen-
tropic relations, and the equation for the speed of sound in a gas. These will be
quickly reviewed.
The conservation of mass simply states that for any control volume, which typ-
ically coincides with a blade row, the mass into the control volume must equal the
mass exiting the control volume
9m ρV A const. (2.15)
The conservation of energy in a control volume simply states that for a control volume
in steady state, the rate of energy transfer out of the control volume must equal the
rate of energy transfer into the control volume plus the rate of heat addition and the
37
rate of work done to the control volume.
»ρpe u2
2q~V ~dA dQ
dt dW
dt(2.16)
However, because work can be done at the system boundaries by the pressure forces
of the fluid itself,³p~V ~dA, it is often useful to consider this separately in the energy
equation so that shaft work and pressure work at the boundaries can be separated.
Doing this and combining terms results in
»ρpe p
ρ u2
2q~V ~dA dQ
dt dWshaft
dt(2.17)
The left hand term is seen to be the total enthalpy of the fluid. Using this variable
instead, the scalar form of this equation on a per unit mass basis becomes.
ht2 ht1 9q 9wshaft cppTt2 Tt1q (2.18)
The equation of state is the perfect gas law.
P ρRT (2.19)
The isentropic equations show the relation between changes in temperature and
pressure
Tt2Tt1
pt2
pt1
γ1γ
(2.20)
Using the speed of sound, these isentropic relations can show useful relationships
between total and static values and Mach number.
TtT 1 γ 1
2M2 pt
p
1 γ 1
2M2
γγ1
(2.21)
With the use of all these equations, a 1-D mean line analysis can be performed to
design isentropic turbomachinery.
38
2.1.3 Turbine Reaction, Flow Type, and Stage Count
One of the primary ways to characterize a turbine is to define its reaction, which is
the ratio of energy transfer due to static pressure changes to overall energy transfer.
R rpU21 U2
2 q pV 2r2 V 2
r1qsrpV 2
1 V 22 q pU2
1 U22 q pV 2
r2 V 2r1qs
(2.22)
This ratio is very important because many turbomachinery properties such as design
RPM and efficiency are strongly correlated with reaction. The relationship between
efficiency and reaction will be shown in section 2.2.1, and the RPM relationship will
be shown here.
Imagine some jet stream flowing into a row of turbine blades at an angle of alpha
as shown in figure 2.4. Because the relative velocity Vr1 is equal to Vr2, the fluid was
Figure 2.4: Velocity triangles for an impulse turbine rotor blade row
neither accelerated nor decelerated in the blade row and therefore the static pressure
did not change overall. The absolute kinetic energy did change however, as noted by
the fact that magnitude of V2 is less than V1. Therefore, because none of the energy
transfer was associated with a static pressure change, the reaction of this turbine
row is zero. Now imagine that we wanted to drop the pressure within the blade row,
this would involve increasing the absolute value of Vr2 so that it was greater than
39
Vr1. If we wanted to absorb the same exit kinetic energy however, such that V2 was
still the same as before, we see that the rotational speed U would have to increase
as shown in figure 2.5. This demonstrates then, that for absorbing the same amount
Figure 2.5: Velocity triangles for a reaction turbine rotor blade row
of kinetic energy from the incoming flow, a reaction turbine needs to operate at a
higher RPM than an impulse turbine. If it was not able to, this would result in
sufficient non-utilized kinetic energy remaining in the exit flow.
Often, the case is that material constraints or the bearing system limit the max-
imum allowable RPM of the turbine blades. As a result, significant kinetic energy
could remain in the flow in the form of tangential velocity or swirl. The solution to
this problem is to use multiple blade rows as shown in figure 2.6. The term for this
is multi-staging, and it allows you to absorb all tangential kinetic energy in the flow.
As will be discussed though, this is not without a cost, as it increases the production
of entropy within the turbine blades, with corresponding losses in efficiency.
Turbines also have different types of flow configurations and are named radial,
mixed, or axial. For a radial turbine as shown in figure 2.7, the fluid flows in the
direction perpendicular to the axis of rotation. Likewise, for an axial turbine, the fluid
flows in the direction parallel to the axis of rotation. In a mixed flow machine, the
flow enters axially and then leaves radially, or vice versa. Due to the manufacturing
process constraints for micro-turbines, their configuration is often limited to a purely
40
Figure 2.6: Multistage turbine arrangement
Figure 2.7: A radial micro-turbine, Lang (2009)
41
radial design when constructed out of silicon.
2.2 Micro-turbine Design
2.2.1 Turbine Efficiency and Loss Accounting
As described in section 1.1.2, an important issue with turbomachinery with regard
to closing thermodynamic cycles is efficiency. For a turbine dedicated to converting
enthalpy into shaft work, the efficiency can be define as
η 9Wt,realized
9Wt,potential
(2.23)
where the potential work extraction rate has already been shown to be
9Wt,potential 9m1
2
pV 21 V 2
2 q pU21 U2
2 q pV 2r2 V 2
r1q
(2.24)
As mentioned in previous sections, a simple way to lower the net work from a turbine
is to leave swirl velocity in the exit flow due to insufficient angular velocity. This
swirl velocity represents wasted kinetic energy that could have been extracted by the
turbine had it been operating at its optimal speed. Assuming completely isentropic
flow, the term for the ratio of actual energy absorption to ideal absorption is known
as effectiveness.
ε rpV 21 V 2
2 q pU21 U2
2 q pV 2r2 V 2
r1qsrV 2
1 pU21 U2
2 q pV 2r2 V 2
r1qs(2.25)
The effectiveness can be derived analytically for reaction values of R=0 and R=1/2
as follows. For an impulse turbine, because it derives all of its energy from kinetic
energy, its effectiveness ratio is simply
ε U∆Vu12V 2
1
(2.26)
42
Referring again to the velocity triangles for the impulse case in figure 2.4, we see
that
Vu1 V1cospα1q and Vu2 V2cospα2q
So
∆Vu V1cospα1q pV2cospα2qq
We know from relative velocities however that
V1cospα1q Vr1cospβ2q U and V2cospα2q V2rcospβ2q U
Plugging these values into the above equation results in
∆Vu Vr1cospβ1q U pV2rcospβ2 Uq Vr1cospβ1q pV2rcospβ2q
Because this is an impulse turbine by definition, the relative inlet and exit velocities
must be equal and therefore
Vr1 Vr2 and β1 β2
Plugging these values into the above equation results in
∆Vu 2Vr1cospβ1q 2 rV1cospα1q U s (2.27)
Plugging this into the impulse effectiveness equation, eq. 2.26, results in
ε 2U rV1cospα1q U s12V 2
1
4U
V1
cospα1q 4
U
V1
2
(2.28)
As can be seen by this equation, a key parameter in determining the effectiveness of
a turbine is the ratio of UV1 which is known as the velocity ratio.
The same procedure can be done for the case of a reaction turbine assuming that
V 20 V 2
1 and using the defining relations of a 50% reaction turbine.
V1 Vr2 V2 Vr1 α1 β2 α2 β1
43
The end result is
ε 2pUV1
qcospα1q U
V1
2
(2.29)
These plots, as well as those for a two stage and three stage impulse turbine, can be
seen in figure 2.8, and demonstrate two key features. The first is, as was said before,
Figure 2.8: Effectiveness of impulse turbines and a reaction turbine as a functionof the velocity ratio for α1 20
that in order to achieve the same effectiveness for a fixed value of V1, a reaction
turbine must operate at a higher RPM, and single stage turbines must operate at a
higher RPM that multi-stage turbines. The second conclusion is that there is a value
of UV1
which maximizes the effectiveness, and therefore for fixed operating conditions,
there is a specific value of RPM which maximizes effectiveness. It should be noted
that while reaction turbines require higher values of RPM to operate effectively, their
advantage lies in increased efficiencies. This is a result of two principle features. The
first is that, in contrast to an impulse turbine, there is a favorable pressure gradient
throughout the entire device (NGV’s, rotors, and stators). As will be shown, this
44
reduces viscous losses in the boundary layers. The second is that for the same
pressure ratio, the average cubed velocity value of the gas stream (proportional to
the rate of energy loss) in a reaction turbine is less than in a impulse turbine. This
is a result of the manner in which reaction turbines and impulse turbines expand
their gases. For an impulse turbine, all of the pressure is converted to dynamic head
within the NGV’s. This results in very high gas speeds at the NGV exit and within
the first few rotors and stators before this kinetic energy is absorbed. For a reaction
turbine, the pressure is converted into dynamic head at a gradual rate over the length
of the device, and this kinetic energy is constantly being absorbed by the rotors. The
result is that, for the same overall pressure ratio, extremely high gas speeds are not
observed in reaction turbines as they are in impulse turbines. Therefore, the average
cubed value of velocity is less in reaction turbines, resulting in less energy dissipation.
As shown, the losses associated with effectiveness are unrelated to irreversibility and
entropy creation, they are simply a result of leaving kinetic energy in the flow in the
form of swirl. Other loss mechanisms in the flow that are irreversible and lower the
power output of the turbine will now be discussed.
For internal flow through turbomachines, it is not common to use drag as a metric
of loss as it is in aircraft. This is because it is difficult to define the direction in which
the drag vector acts in a turbomachine, Denton (1993). A more a common metric
of loss in turbomachines is entropy creation, which in a stationary adiabatic blade
row is a measure of the total pressure loss.
∆s R lnpPo2Po1q (2.30)
The advantage of using entropy generation as the metric for loss is that entropy
does not depend on the reference frame from which it is measured. The same values
of entropy will be measured whether the flow is viewed from stationary or rotating
blade rows.
45
As stated by Greitzer et al. (2004), in a non-ideal flow with no heat exchange or
shaft work, the amount of work required to bring an irreversible flow process back
to its initial state is related to entropy creation by
wrev Tt∆s (2.31)
This equation can be reasoned out quite simply. Imagine an irreversible and adiabatic
flow process through a throttle device as shown in figure 2.9. Because the throttle is
Figure 2.9: An irreversible flow process through a throttle
adiabatic, there is no heat addition to the flow. Likewise, because there are no moving
parts within the throttle, no shaft work is being performed on the fluid. Referring
back to the energy equation, eq. 2.18, this means that the stagnation enthalpy must
be constant as well as the internal energy at stagnation for the inlet and outlet flows
(which is only a function of temperature for a perfect gas). From the first law of
thermodynamics however, if the stagnation internal energy has remained constant,
then
q w (2.32)
It is also known that for reversible heat addition
dqrev Tds
In the abstract then, one can think of an irreversible entropy creating process as one
that is adding heat to itself. From equation 2.32 however, this can only be done
if work was performed by the fluid. Combining these two thought processes, an
46
irreversible process can be thought of as a fluid taking its ability to perform work
and converting it into heat in a manner which cannot be undone, reducing the fluids
ability to perform useful work on its surroundings. And just as heat transfer can be
determined by measuring the entropy change, so too can this internal conversion of
energy.
9wlost T 9ds (2.33)
In order to determine the efficiency of a fluid system then, one must keep track of
all the sources of entropy creation in a flow process, relate this entropy creation to
lost work, and compare this value to the fluids initial ability to perform work. This
requires the ability to quantify entropy creation for all the irreversible processes in
turbomachine flow such as boundary layers profile losses, fluid mixing losses, and tip
clearance losses. Before these losses can be calculated though, many parameters of
the flow must be characterized.
2.2.2 Characterization of the Flow
For all three loss mechanisms modeled in this work, various parameters of the flow
over the blades must be known. For example, the velocity distribution must be
estimated over each blade in order to estimate both the surface pressures and the
boundary layer parameters such as displacement thickness and momentum thickness.
Following Denton (1993), the difference in velocities on the blade surfaces can be
roughly estimated by assuming that the blade loading is constant. By then using
the definition of blade circulation and setting it equal to zero for the non-lifting flow
channel, it can be seen that
Vs Vp p
CVxptanα2 tanα1q (2.34)
Using Vr as a reference value, idealized velocity distributions over the blades can then
be assumed as shown in figure 2.10 for the different turbine types. With a knowledge
47
Figure 2.10: Simplified velocity triangles for idealized impulse and reaction blades
of the velocities over the blades, the boundary layers can be analyzed to determine
useful parameters such as the momentum and displacement thicknesses.
In the case of laminar flow, the Falkner-Skan velocity profiles were solved for
each pressure gradient encountered as shown by Cebeci and Bradshaw (1977). For
turbulent flow, the following equations from Greitzer et al. (2004) were used for
displacement thickness and momentum thickness.
δpxq 0.37x
Re15x
(2.35)
θpxq 0.036x
Re15x
(2.36)
The same equations were used for both favorable and zero-pressure gradient flows,
as the state of the boundary layer was not determined to differ much between the
two, see White (1991)
2.2.3 Boundary Layer Losses
As described by Denton (1993), the entropic change due to profile loss in the bound-
ary layer is conveniently expressed non-dimensionally as the viscous dissipation co-
48
efficient.
Cd T 9Saρ V 3 (2.37)
The value Cd is a function of the Reynolds number based off of momentum thickness.
This function changes for turbulent, laminar, favorable, and non-favorable boundary
layers. From Denton (1993), the dissipation coefficient for zero pressure gradient
turbulent flow was modeled as
Cd 0.0056 Re16θ (2.38)
However, simulations by Cebeci and Carr (1978) show that the dissipation coefficient
for accelerating turbulent boundary layers can be significantly less. This value was
modeled as shown below.
Cd 0.0205 Re0.417θ (2.39)
The Pohlhausen family of velocity profiles may be integrated to attain dissipation
coefficients for laminar boundary layers in accelerating, decelerating, or separated
flows in terms of the Pohlhausen parameter, λ, as shown by Schlichting (1968).
These results can be simplified into a laminar dissipation coefficient.
Cd β Re1θ (2.40)
The value of β can range from 0.220 for highly favorable pressure gradients to 0.173
for boundary layers with no pressure gradient. The graphs depicting the various
values of Cd for turbulent, laminar, favorable, and zero pressure gradients are plot-
ted in figure 2.11. As can be seen, the entropy generation in a laminar boundary
layer is significantly less than its turbulent value near the transition point. For this
reason, it is extremely important to accurately determine the state of the boundary
layer for real systems and to attempt to retain laminar flow for as long as possible.
For this analysis, a value of Reθ 200 was chosen as the transition point(Greitzer
49
Figure 2.11: Dissipation coefficients for favorable,constant pressure, laminar, andturbulent boundary layers
et al. (2004)). Also, it can be seen that for turbulent flows, the dissipation coefficient
in an accelerating boundary layer is significantly less at higher Reynold’s numbers,
highlighting the advantage of reaction turbines. Lastly, it can be observed that for
very low Reynolds, such as those that characterize micro-turbine flows, the dissipa-
tion coefficient is very high, and thus one should expect large losses as a result of
this mechanism. After determining the value of Cd across a blade, the total entropy
generation per unit height within the boundary layer up to the trailing edge can
then be computed for both the suction and pressure surfaces as described by Denton
(1993).
9S ¸
Cs
» 1
0
Cd ρ V3
TdpxCsq (2.41)
This provides a mechanism to quantify the creation of entropy within the boundary
layers for the entire turbine and relate this to lost work potential. A similar analysis
can be done for lower and upper walls of the rotor and stator rows. And as can be
50
seen in figure 2.7, micro-turbines typically have very large chord to height ratios as
a result of high centripetal stresses. This decreases the ratio of mass flow to wetted
area and therefore decreases efficiency.
2.2.4 Trailing Edge Losses
Entropy is also created in the wake of the blade trailing edges as shown in figure 2.12.
Three sources contribute to this entropy creation: the mixing out of the boundary
Figure 2.12: Wake mixing behind the trailing edge of two turbine blades
layers at the trailing edge, a lower base pressure on the blade tip than would exist in
an inviscid case( Roberts and Denton (1996)), and the combined blockage from the
boundary layers and the trailing edge itself. The last mechanism stated is indicative
of the other two and is very analogous to the losses in a dump diffuser. For this
reason, a quick analysis of the losses in a dump diffuser is warranted because its
sheds light on the actual loss mechanisms experienced at the trailing edge of the
turbine blades.
A dump diffuser, as shown in figure 2.13, is a device that expands the flow
abruptly as opposed to using slowly varying areas ratios as dictated by the isentropic
flow equations. This sudden expansion is an entropy creating process and therefore
lowers the total pressure of the fluid.
Creating a control volume within the dump diffuser as shown in the figure, a mass
and momentum balance can be done. From the conservation of mass and assuming
51
Figure 2.13: A dump diffuser
an incompressible fluid we have
u1A1 u2A2 (2.42)
For the conservation of momentum, we need to know the pressure of the diffuser
back wall. It can be reasoned out, but also shown experimentally, that this pressure
is simply equal to the jet pressure. Noting that A2 A1 Awall, this results in
ρu1A1u1 p1A2 p2A2 ρu2A2u2 (2.43)
This can be rearranged to give
p1 p2 ρu22
A1
A2
ρu21
A1
A2
We can add dynamic head terms to both sides of the equation to determine the
total pressure loss and utilize mass flow to substitute out u2. Doing so and dividing
through by the inlet dynamic head results in
pt1 pt212ρu2
1
A21
A22
2A1
A2
1
52
Defining A2
A1as the aspect ratio AR, this can be represented as
pt1 pt212ρu2
1
1 1
AR
2
The graph for this normalized total pressure drop can be seen in figure 2.14. Recall
Figure 2.14: Normalized total pressure drop in a dump diffuser as a function ofaspect ratio
from equation 2.30, that a drop in total pressure corresponds with an increase in
entropy, confirming that dump diffusion is an entropy creating process. As a side
note, notice that for infinitesimally small aspect ratios, the total pressure drop is
zero. Consequently, one can think of an isentropic diffuser as an infinite amount of
dump diffusers placed in series feeding into each other.
Looking back at the trailing edge of a turbine blade then, one can see how the
sudden expansion of the flow behind the trailing edge is identical to dump diffusion.
Therefore, the same exact method can be used to determine the losses as a result
of the trailing edge thickness and as a result of the displacement thickness due to
the boundary layer. A similar method can be performed to demonstrate why mixing
53
out of the momentum boundary layer results in total pressure losses. This analysis
also clearly explains why a lower pressure on the trailing edge would result in total
pressure losses, because it would directly lower the static pressure at the mixed out
location.
Denton (1993) combines all three of these sources into a loss coefficient ,
ζ Cpb tw
2θ
wδ t
w
2
(2.44)
and from this we have an easy method by which to quantify the creation of entropy
from trailing edge mixing.
For micro-turbomachinery, these losses can be significant on account of the large
trailing edge thickness relative to the chord as shown in figure 2.15 due to feature res-
olution and manufacturing constraints (Lee et al. (2008)), high material stress levels
(Osipov (2008)), and the rate at which the displacement and momentum thickness
of the boundary layers grow over such small chord lengths.
2.2.5 Tip Clearance Losses
Entropy is also generated as the lower velocity, higher pressure gas from the pressure
side of the turbine blade spills over and mixes with the higher velocity gas stream of
the suction surface as shown in figure 2.16. As shown, this mechanism is analogous
to a small high velocity jet stream being injected into a larger and slower body of
fluid. As described in detail by Shapiro (1953), the entropy creation from such a
process is
ds
cp 2
VpVs
pk 1qM2
2
dm
m pk 1qM2dm
m(2.45)
By substituting these thermodynamic relationships
k cpcv
M2 V 2
kRTcp cv R
54
Figure 2.15: Thick trailing edges of a micro-turbine as a result of feature resolutionand stress levels, Lee et al. (2008)
the equation can be rearranged to give
Tds 1
mV 2s p1
VpVsqdm (2.46)
We only need a few more pieces of information to utilize this equation. The first is
the velocity profiles that were shown previously. The profiles can be determined by
assuming constant blade loading and using the continuity equation. This results in
the following equations
Vs Vp p
CVxptanα2 tanα1q (2.47)
Vs Vp 2Vxcosα
(2.48)
The second piece of information required is the mass flow of the jet from the higher
pressure fluid traveling over the blade tip and into the suction flow. Using incom-
55
Figure 2.16: A schematic of tip leakage over a turbine blade
pressible flow assumptions, this can be approximated as
dm Cdisg
d2ρ
1
2pV 2
s V 2p q
(2.49)
Cdis is the discharge coefficient that often must be determined experimentally. How-
ever, it was chosen as 0.6 for this analysis as that was seen as a typical observed
value in the literature. Combining all of these equations into a loss equation for tip
56
clearance results in
T∆s CdisgC
V2hp cosα2
» 1
0
VsV2
21 Vp
Vs
gffe1
Vp
Vs
2dz
C(2.50)
Due to manufacturing tolerances, the ratio of tip gap to blade heights is often larger
for micro-turbines than it is for conventionally sized machines. For large scale gas
turbines, the values of tip clearance can be less than 1% of the blade height while
for micro-turbomachinery this value is often much greater (Jovanovic (2008)). As
a result, the losses associated with this mechanism in this analysis are also more
detrimental to micro-turbomachinery than they are to conventially sized machines.
2.2.6 Scaling Effects on the Efficiency of Gas Turbines
Using these equations, the entropy generation for each mechanism was calculated
and the associated power loss was numerically computed for one stage, two stage,
and three stage impulse turbines as well a single stage reaction turbine over a large
range of outer diameter and RPM. The turbines were modeled as axial turbines that
are scaled geometrically based on the outer diameter while retaining the same mass
flux, a total-to-static pressure ratio of Pt1Pe 1.85, and turbine inlet total temper-
ature of Tt1 1400 K. All impulse turbines were designed as velocity-compounded
Curtis turbines where all pressure drop occurs strictly in the first nozzle guide vane,
and subsequent stator rows only redirect the flow. The pitch to chord ratio was
numerically altered in order to minimize the combined losses for the rotors, stators,
and NGV’s. The results for this analysis are shown below.
Figures 2.17 - 2.20 show the effectiveness of the different turbines as a function of
their velocity ratio and diameter. Recall that effectiveness simply measures the ratio
of actual energy absorbed to the potential energy that could have been absorbed at
the optimum RPM. The flow is assumed to be isentropic and therefore lossless. For
57
reference, lines of constant RPM have been drawn into the figures. A few expected
Figure 2.17: Turbine effectiveness as a function of diameter and the velocity ratiofor a single stage reaction turbine
features can be noted from these graphs. The first is that the optimum velocity
ratio required for the single stage reaction turbine is greater than for the single stage
impulse as expected. Likewise, the velocity ratios for the multistage turbines are
lower than their single stage brethren. The other observed feature is that for small
turbine diameters, extremely high values of RPM are required to reach the design
speed of the turbine. Note that even with isentropic flow, the effectiveness does not
reach 100% even at the optimal RPM. This is because there remains unutilized kinetic
energy in the purely axial exit flow. In order to achieve close to 100% effectiveness,
a diffuser would be required downstream of the turbine, allowing it to expand the
gases to sub-atmospheric conditions.
These graphs do not show the effect of the loss mechanisms however, which can
have large effects on overall efficiency. When loss mechanisms are included, the plots
change as shown in figures 2.21 - 2.24. Note that the effect of viscous drag in the
58
Figure 2.18: Turbine effectiveness as a function of diameter and the velocity ratiofor a single stage impulse turbine
Figure 2.19: Turbine effectiveness as a function of diameter and the velocity ratiofor a two stage impulse turbine
59
Figure 2.20: Turbine effectiveness as a function of diameter and the velocity ratiofor a three stage impulse turbine
Figure 2.21: Turbine efficiency as a function of diameter and the velocity ratio fora single stage reaction turbine
60
Figure 2.22: Turbine efficiency as a function of diameter and the velocity ratio fora single stage impulse turbine
Figure 2.23: Turbine efficiency as a function of diameter and the velocity ratio fora two stage impulse turbine
61
Figure 2.24: Turbine efficiency as a function of diameter and the velocity ratio fora three stage impulse turbine
direction of rotation is not accounted for. Therefore, at extremely small diameters,
the performance will be higher then depicted here, as the device begins acting more
as a Tesla turbine than a conventional bladed turbine.
From these figures, multiple observations can be made. The first is that, at
design speeds, reaction turbines show much greater efficiencies as small scales than
their impulse counterparts. This is for a few key reasons. The first is that for reaction
turbines, in the nozzle guide vanes, rotors, and stators, there is a continual favorable
pressure gradient. This favorable pressure gradient lowers viscous dissipation losses
in the boundary layer and reduces the growth rate of the boundary layer. Impulse
turbines have no pressure gradient after the NGV, and thus see large boundary
layer related losses. Another reason is that, assuming the same power output, since
a reaction turbine is rotating at a higher rate, it must be delivering less torque.
The corresponds to the blades being less loaded which implies that the pressure
62
difference between the suction side and pressure side is lower in the reaction case
than in the impulse case. This will reduce losses due to tip clearance. The last
main reason that reaction turbines display higher efficiency is that the average value
of V 3 over the blade span, which is indicative of flow losses, is much lower. For
reaction turbines, static pressure is continually being converted into kinetic energy
as the blade passages converge. Thus the fluid is only traveling at a high relative
velocity towards the end of the blade passages. For an impulse turbine, all of the
available pressure is converted into dynamic head in the NGV’s, and therefore the
maximum velocity is seen throughout the rotor blade rows and stators. Because of
this, the average value of V 3 over the blade spans is higher in impulse turbines than
in reaction turbines, and therefore reaction turbines operate with lower losses.
Another interesting observation that can be made is that multi-staging is much
more effective for large diameters than it is for smaller diameters. This can be
reasoned out quite simply. For any turbine, it is extremely important to operate
at the design RPM in order to maximize effectiveness. If this cannot be done with
a single stage, more stages must be added. Adding stages however increases the
wetted area of the turbine blades and therefore increases viscous boundary layer
losses. Additional stages also increase the instances of trailing edge mixing losses and
tip clearance losses. For large turbomachines, these losses are not very significant
to begin with, and thus the benefit of adding multiple stages which allow on design
operation far outweigh the entropy creation penalties associated with the additional
stages. For smaller turbines however, these losses become much for significant and
a result, the benefits of multi-staging are attenuated by the increased viscous losses.
For this reason, it is of critical importance for micro-turbomachinery that the bearing
system allow the turbomachinery to have as few stages as possible and operate at
the design RPM.
Lastly, graphs can be shown depicting the power density of the various devices,
63
operating at their design RPM, as a function of diameter. The results can be seen
in figures 2.25-2.28. These graphs display two key pieces of information. The first
Figure 2.25: Power density as a function of diameter for a single stage reactionturbine
Figure 2.26: Power density as a function of diameter for a single stage impulseturbine
is that as expected, the power density shows a linear relationship with characteristic
length over a large range of diameters. However, at very small length scales, the
64
Figure 2.27: Power density as a function of diameter for a two stage impulseturbine
Figure 2.28: Power density as a function of diameter for a three stage impulseturbine
65
increased entropic losses negate this effect and begin to level out the power density
and then even reduce it. The other conclusion, not surprisingly, is that the reac-
tion turbine maintains its linear power density relationship over a larger range of
turbine diameters. In addition, due to its overall higher efficiency and single stage
compactness, it possesses the highest power density of all the turbines analyzed.
A final observation can be made by looking at the alternate form of the Euler
turbine equation, eq. 2.11. As can be seen, there exists a mechanism to transfer
energy from the fluid to the rotor via the centrifugal pressure field. However, none of
the loss mechanisms analyzed here, which are the main forms of loss in a turboma-
chine, are related to this centrifugal pressure field. Therefore, energy transfer that
utilizes this mechanism is lossless. More on this is described by Cumpsty (1989). As
a result, for micro-turbomachinery, which is typically associated with high losses, as
large a fraction as possible of the total energy transfer should come from centrifugal
effects in order to increase efficiency. And because centrifugal effects are related to
static pressure changes and require high RPM, this means that the reaction should
be further increased, further highlighting the critical importance of the gas bearing
system.
2.3 Conclusion
In this chapter, loss models were presented for insufficient RPM exit losses, boundary
layer profile losses, tip clearance losses, and trailing edge mixing losses. In particular,
an analysis was presented for how these losses scaled with turbine size, type, and
configuration. The result was a series of graphs that showed efficiency values over
a large range of velocity ratios and diameters. A few key observations were made.
The first was that reaction turbines should be pursued as a result of their increased
efficiency and power density, which is due to their favorable pressure gradients. It
was also shown that the practice of multi-staging, while highly useful and effective
66
at large values of diameter, is not as beneficial for smaller sized machines. It was
also confirmed that over a large range of diameter, the power density followed a
linear relationship as predicted by the scaling laws. At very small sizes, increased
viscous losses negated this effect however, resulting in a peak power density for each
configuration. Lastly, it was mentioned that radial turbomachines that use centrifu-
gal pressure fields for a significant portion of energy transfer should be utilized at
very small scales because an entire component of energy transfer is lossless. These
large centrifugal pressure fields require extremely high values of RPM however and
increase the reaction of the design. This further shows the importance of design-
ing effective gas bearings that can maintain the high RPM values required by high
reaction turbomachinery.
67
3
Permanent Magnet Micro-generator Design andPerformance
3.1 Concepts of Permanent Magnet Generators
3.1.1 Operating Concepts, Scaling Laws, and Generator Selection
As described in Chapter 1.1.1, when a magnetic field varies within a contour in space,
by Faraday’s law of induction, a voltage will be induced around the path encircling
that contour proportional to time rate of change of the magnetic flux within the
contour.
ε dΦ
dt(3.1)
where the flux is determined by
Φ » »
~B d ~A (3.2)
The source of flux can come from different sources however. One solution is to use
electric currents in a wire such that a directed magnetic field is created as shown
in figure 3.1. The other solution is to use previously magnetized material, or a
permanent magnet, to create the magnetic field as shown in figure 3.2. In both
68
Figure 3.1: A current conducting wire used to create a magnetic field, Nave (2011)
cases, the flux source is typically placed on a rotating structure such as a rotor so
that the magnetic field within an area is constantly changing, thereby creating an
alternating voltage source. If an electrical conducting material is then placed around
that contour path and then connected to a load, a current will flow and produce
electric power as shown in figure 3.3. This is the mechanism by which an electrical
generator works.
Typically however, a generator will have three sets of conducting coils arranged in
such a way that their voltages are out of phase by 120 as shown in figure 3.4. This is
termed three phase power. Three phase power is essential for an electrical generator
that receives its shaft power from a turbine because three phase power results in a
constant, non-sinusoidal, power draw from the generator, and therefore a constant
69
Figure 3.2: A permanent magnet used to create a magnetic field, Nave (2011)
Figure 3.3: The induced voltage being used to create a current and produce work
power draw from the turbomachinery. This can be proven as follows. Imagine that
each phase of the generator created some sinusoidal voltage.
v1ptq Vpcospωtq v2ptq Vpcospωt 120q v3ptq Vpcospωt 240q
The resulting system power would then be
P ptq v21ptqR
v22ptqR
v23ptqR
Using the trigonometric identity cos2pαq r1 cosp2αqs2, this can be arranged to
70
Figure 3.4: A three phase power system
give
P ptq 3V 2P
2R V 2
P
2Rrcosp2ωtq cosp2ωt 240q cosp2ωt 480qs
The term on the right always adds to zero for any value of t, and therefore the power
draw is constant at
Pt 3V 2P
2R
This is an essential property for a a device that is connected to turbomachinery, be-
cause turbomachines are designed to operate in steady state conditions. If a turbine
was connected to a single phase generator, then the turbine would see a sinusoidal
load and would therefore constantly be accelerating and decelerating and working
off-design, lowering its performance.
The other advantage with three phase systems is that it lets you arrange the
voltage coils in series as shown in figure 3.4. Notice that across leads A-C, the voltage
would be Vcc1 Va1a. The advantage of this is can most easily be seen in a phasor
diagram as shown in figure 3.5. As can be seen by trigonometry, the magnitude
of the combined voltage from both voltage sources is?
3 greater than the original.
This not only provides an easy manner through which the output voltage can be
increased, but it also increases the maximum power capabilities of the generator
and the efficiency relative to three independent single phase outputs. In many cases
however, an 3 phase alternating power source is not directly useful because many
71
Figure 3.5: A phasor diagram for a wye connected generator
devices utilize DC power. In order to convert from AC to DC, a diode bridge rectifier
as shown in figure 3.6 must be used. A diode is a device that only allows current
Figure 3.6: A diode bridge rectifier
to flow in one direction. Taking this into account and observing the AC voltages in
figure 3.7, one can trace out the emitting and returning pathways for the current as
shown in figure 3.8. By noting that analogous pathways will be taken when each
one of the waveforms is near its AC peak, the observed rectified waveform across the
72
Figure 3.7: The original AC voltage forms and the rectified waveform
Figure 3.8: The emitting and returning current pathways for the rectifier
load as shown in figure 3.7 becomes readily apparent.
However, we have yet to explain why a permanent magnet has been chosen as the
flux source as opposed to an electro-magnet. The reason again concerns the scaling
laws that affect these devices ( Cugat et al. (2003)). For a permanent magnet with
volume v and polarization J , the scalar potential of its magnetic field will be
V pP q v
4πµ0
~J ~rr3
(3.3)
As can be seen then, the vector potential is proportional to length
V pP q 9 L
and therefore scales directly with a scaling factor, k. The magnetic field created by
73
the magnet, ~H, however is equal to the gradient of the scalar potential.
~H ~grad V (3.4)
A gradient is indirectly proportional to length, so the gradient of the potential func-
tion is independent of length and therefore the magnetic field strength is constant.
This shows that as a permanent magnet system is reduced in size by a factor of
k, the same magnetic field will be observed at geometrically similar locations. The
induced emf, ε, however is proportional to dΦdt
, where Φ is proportional to the area
and field strength. But because the magnetic field strength will be constant as just
shown, the flux will be proportional to the scaling factor squared.
dΦ 9 k2
The induced emf is also proportional to the time rate of change of this magnetic
field. As stated previously, the rotational rate of the micro-generator is indirectly
proportional to size in order to maintain constant tip speed. Combining these two
observations results in,
dΦ
dt9 k
as shown in figure 3.9. Note that this analysis also results in power density remaining
independent of k for a constant current density, because power is proportional to ε2Rand the resistance is indirectly proportional to 1.
P
V9 pdΦ
dtq2RL3
9pkq2
k1
k39 1
For a electro-magnetic current induction machine, we can determine the scaling
effects by looking at the Biot-Savart law and assuming a constant current density.
The Biot-Savart law shows that
~dB µ0I ~dL r
4πr2(3.5)
74
Figure 3.9: Overall effect of scale reduction 1/k on basic magnet interactions forconstant current density, Cugat et al. (2003)
The magnetic field density then scales as,
~B 9 I
L
which results in the magnetic flux scaling as
Φ 9 BA 9 I
LL2 9 IL
For a constant current density
I
A constant I 9 L2
This results in the flux being proportional to the characteristic length cubed.
Φ 9 L3
As before, the rotational rate is indirectly proportional to the scaling factor k. Com-
bining these observations results in
dΦ
dt9 k3
k9 k2 (3.6)
75
as shown in figure 3.9. The resulting power density is then proportional to k2.
P
V9 pdΦ
dtq2RL3
9pk2q2
k1
k39 k2
For this reason, it is not advisable to use a micro-electro-magnetic generator in con-
junction with micro-turbomachinery, and this is why a permanent magnet generator
was selected for this design.
As stated previously however, the heat dissipation capabilities of electric windings
also scale favorably with k, which explains why permanent magnet generators show
power density improvements as they scale down in size (Arnold et al. (2006a)). The
effect of this added property is shown in figure 3.10.
Figure 3.10: Effect of scale reduction 1/k on magnetic interactions, taking intoaccount increased admissible current density, Cugat et al. (2003)
3.1.2 Magnetic Circuits and Modeling
Now that a permanent magnet generator has been selected, their operation and flux
creating abilities must be described in more detail. As described by Hanselman
(1994), hard magnetic materials exhibit significant hysteresis in their response to
magnetic fields as shown in figure 3.11 and as a result, display a remanence magnet
field strength, Br, without the presence of an external magnet field. By knowing
76
Figure 3.11: A B-H graph for a hard magnetic material
the B-H characteristics of a hard magnetic material, Ampere’s law can be used to
determine a magnets point of operation. Ampere’s law states that
¾~H ~dl µ0i
Applying this law then to the a permanent magnet encased in a ferromagnetic ma-
terial as shown in figure 3.12 , and noting that no current is present in the system,
Figure 3.12: A simple permanent magnet circuit
results in
Hmlm Hklk 0
77
Hence
Hm lklmHk
Because there is a positive magnetic B field in the encasing material, there must also
be a positive H-field as per ~B µ0~H. From Ampere’s law then, the magnetic H
field in the permanent magnet must be negative and the magnet is operating in the
second quadrant of its B-H curve.
Consider a similar magnetic system with an air gap as shown in figure 3.13.
Ampere’s law will result in
Figure 3.13: A simple permanent magnet circuit with an air gap
Hm lklmHk lg
lmHg (3.7)
Assuming that there is no leakage flux, neglecting fringing of the magnetic field at
the air gap, and assuming some representational area for the encasing material Ak
along its length, by Gauss’s law we will have
Φ BmAm BgAg BkAk
Recall the relationship between the magnetic B field and the magnetic H field
~B µrµ0~H
78
The magnetic H fields then take on the following values
Hg BmAmµr,gµ0Ag
Hk BmAmµr,kµ0Ak
Plugging these results into 3.7 gives
Hm lgBmAmµr,gµ0Aglm
lkBmAmµr,kµ0Aglm
(3.8)
The relative permeability of free space of air, µr,g is approximately 1, whereas for high
permeability materials this value, µr,k, can exceed 10,000. Looking at equation 3.8,
we see that the second term can be mostly neglected. Making this simplification
results in
Hm lgBmAmµr,gµ0Aglm
(3.9)
which can be rearranged to give
Bm
µ0Hm
µr,gAglmAmlg
(3.10)
This is the slope of the operating line for magnetic system, which in conjunction with
the magnetic properties of the permanent magnet, will determine its operating point
and flux density emission as shown in figure 3.14. The basic operating principles
of permanent magnets have been presented. With this information, their operation
will now be presented in a slightly different manner that is highly conductive towards
magnetic modeling.
Magnetic circuits are similar to electric circuits and can be modeled using similar
concepts, this basic concept will be demonstrated here. Applying Ampere’s law to
a geometrically symmetrical object such as a coil wound toroid in figure 3.15 we see
that
H l Ni
79
Figure 3.14: The B-H operating conditions for a permanent magnet system
Figure 3.15: A toroidal ferromagnetic core wrapped within a current conductingwire, Nave (2011)
80
Combining this with the relationship between magnetic fields, B µH, where µ µrµ0, results in
B µNi
l
We can multiply this term by the cross sectional area to get
BA Φ µA
lNi (3.11)
Thus there is a linear relationship between some constant parameters of the system,
µAl
, some forcing function, Ni, and the flux, Φ. This relationship is analogous to
Ohm’s relationship between voltage, current, and conductance, and it is very advan-
tageous then to model magnetic circuits in a similar manner. The parameters of the
system, µAl
, will be termed permeance, P, and the forcing function Ni will be referred
to as the magneto motive force, MMF. We require a method then for determining
the MMF of a permanent magnet.
Looking at figure 3.16, a close-up of the B-H curve, and knowing the relation-
Figure 3.16: A close up of the B-H curve for a hard magnetic material
ship between the magnetic H field, the magnetic B field, magnetization M, and the
81
permeabilities,
B µrµ0H M
we can see that the slope is µr as expected. Through geometry then, we can write a
new expression for Bm.
Bm Br µrµ0Hm
If we multiple both sides by the area, the result is.
φm BrAm µrµ0HmAm
Thus the actual flux exiting the magnet is equal to the highest capable flux that
magnet could provide in a zero reluctance environment (φr BrAm) minus some
term that is related to magnetic H field imposed by its environment as per Ampere’s
law. By multiplying the above equation by ll, we get
φm BrAm µrµ0Amlm
Hmlm
Substituting in the values for magnetic permeability and the forcing function from
Ampere’s law, this becomes
φm φr PmF
In an analogous fashion to electric circuit theory then, a permanent magnet can
be modeled as a constant current source in parallel with a resistance as shown in
figure 3.17. Thus the same rules that model current sources in electric circuits can
be used to model permanent magnets in magnetic circuits. As example, looking back
at our simple magnetic circuit with the air gap (fig. 3.13), this could be magnetically
modeled as shown in figure 3.18.
With magnetic circuit knowledge, and an ability to model permanent magnets,
we can now look at a magnetic circuit for a micro-generator. Such a a circuit is
shown in figure 3.19. If we neglect leakage flux, and assume the permeabilities of the
82
Figure 3.17: A circuit model for a permanent magnet
Figure 3.18: A circuit model for a permanent magnet enclosure with an air gap
back iron and stator iron are infinite, this circuit can be simplified and redrawn as
shown in figure 3.20. From this, two magnetic circuit equations can be written
φm Pg2F φr
Pm2 Pg
2
F (3.12)
Equating for the magneto motive force and setting these equations equal to each
other results in
2φmPg
2φr
Pm Pg(3.13)
Assuming that the magnet area and area gap area are the same, solving for the
83
Figure 3.19: A circuit model for a planar permanent magnet generator,Herrault(2009)
system flux results in
φm φrPgPm Pg
φrµ0Alg
µ0Alg
µ0µrAlm
φrlg
1lg µr
lm
φr1
1 µrlglm
(3.14)
This provides a simple equation by which we can approximately estimate the flux
through a pole-pair coil for initial calculations. From this parameter as well as the
Figure 3.20: A simplified circuit model for a planar permanent magnet generator
84
design RPM, we can begin to estimate and determine system performance.
As shown, the observed flux through the coils depends on the remanence flux
of the magnet, which is equal to the flux through the magnet in a zero reluctance
environment. Permanent magnet material such as SmCo is magnetized by placing
the material in a strong magnetic H-field, and then removing this H field, allowing
the material to recoil along its hysteresis path as shown in figure 3.21. The magnetic
Figure 3.21: A B-H curve for a permanent magnet material undergoing magneti-zation
material is typically subjected to strong fields by placing it in patterned permeable
material wrapped in loops of an electric conductor as shown in figure 3.22 and then
running large currents through the coils.
The mechanism by which a permanent magnet generator creates magnetic flux
has been presented as has as a method to quantify the flux leaving the permanent
magnet in a simplified system. Knowing this, and other system parameters such
as RPM, the open-circuit voltage can be determined using Faraday’s law. With
this, many properties of the generator such as maximum power output and electrical
85
Figure 3.22: A magnetizing head used to create permanent magnets, Gilles et al.(2002)
efficiency can be determined.
3.1.3 Electric Circuits and Modeling
A single phase of a permanent magnet generator can be modeled, as shown in fig-
ure 3.23, as an AC voltage source with an internal resistance Rs, corresponding the
resistance of the generator coils in the stator, in series with a load, RL. The inductive
Figure 3.23: A single phase AC circuit
effects can typically be ignored because the stator coils in micro-generators have so
86
few turns, and the inductance of a coil is proportional to N2.
With a fixed voltage source and an internal resistance, the power properties of
the device take on interesting characteristics. The first of these is that there is a
maximum amount of power that can be transferred to a load. By looking again at
figure 3.23, we see that the power transfered to a load is
PL i2RL (3.15)
where
i V
RS RL
(3.16)
Combining these two equations results in
PL V 2RL
pRS RLq2 (3.17)
Taking the derivative of this equation with respect to RL, setting it equal to zero,
and solving for RS gives RL RS. Therefore, the load resistance should be set to
the phase resistance in order to maximize power transfer. Plugging this condition
into equation 3.17 results in
PL,max V 2
4Rs
(3.18)
Therefore, there is a load resistance value above or below which power transfered to
the load will be reduced. This can be easily deduced by looking at extreme cases.
If the load resistance was close to infinite, then no current would be flowing and
therefore there would be no power. If the load resistance was close to zero, then the
current would be set by the phase resistance, and no power would be utilized by the
load.
The efficiency can also be determined from this information. Taking
PG V 2
RS RL
(3.19)
87
and combining this with with equation 3.17, we get
η PLPG
V 2RL
pRS RLq2pRS RLq
V 2 RL
RS RL
1
1 RSRL
(3.20)
Therefore, the higher the load resistance in comparison to the phase resistance, the
higher the efficiency. Because of this, there seems to be a trade off between power
and efficiency, because as the load resistance is increased, the efficiency increases
while power output drops.
This relationship can be quantified. Noting from equation 3.20 that
pRS RLq RL
η(3.21)
and plugging this into equation 3.17, we get
PL V 2RLRLη
2 V 2η2
RL
(3.22)
We wish to have the load power be only a function of the non-varying parameters V
and RS and the efficiency. So using equation 3.21 again, we get
RL ηpRS RLq RL ηRS
1 η(3.23)
Plugging this result into equation 3.22 gives
PL V 2η2 1 η
ηRS
V 2
RS
ηp1 ηq (3.24)
The general graph for this relation takes on the shape shown in figure 3.24, although
the values for the vertical axis will be determined by the system voltage and internal
resistance. As can be seen from the equation, for a given efficiency, higher generator
voltages will increase the power output across the load. Conversely, for a given power
88
Figure 3.24: Load power as a function of efficiency
output, higher voltages will increase electric efficiency. And as with any engineering
system there is a trade off in design. In this case, selecting the load resistance will
be a trade off between power density and energy density as a result of the relation
between power output and efficiency.
3.1.4 Magnetic Materials and Configuration
Thus far, most of the terms in our equations have been variables and they will change
between different systems. However, permanent magnet materials have physical and
magnetic properties that will stay constant in various designs, and these properties
will be discussed here.
As shown from previous sections, a key parameter in the determination of power
output and efficiency is voltage. This in turn is dependent on dφdt
. As shown in
equation 3.14, dφ is dependent on the remanence flux, dφr (which is equal to BrA),
and µr. The maximum remanence flux density, Br, is a magnetic property of ma-
terial and µr is the recoil line for magnetic materials in the second quadrant as
shown in figure 3.16. These values vary between the different permanent magnet
materials such as AlNiCo, NdBFe(Neodymium Boron), SmCo(Samarium Cobalt),
89
etc. Demagnetization curves for these materials are shown in figure 3.25. As can
Figure 3.25: Demagnetization curves for various magnetic materials, Gieras et al.(2004)
be seen from the figure, ferrites are not typically used due to their low remanence.
AlNiCo materials on the other hand have a high remanence, but their value of µr
is extremely high, resulting in low flux values in typical magnetic circuits. Of the
two remaining choices, NdFeB and SmCo, NdFeB has the higher remanence. How-
ever, as shown by Herrault et al. (2008) in figure 3.26, the performance of NdFeB
degrades much more quickly than SmCo in elevated temperature environments, and
thus SmCo magnets are better suited for working in a micro-engine environment.
And due to this temperature dependence, the generator is forced onto the cold side
or the compressor side of micro-engines, and therefore the magnets do not experience
the high temperatures of the combustion gases.
Another important material property for permanent magnet material is the yield
strength. This is a result of the high stress environment due to rotation. Recalling
90
Figure 3.26: A comparison between SmCo and NdFeB generators as a function oftemperature, Herrault et al. (2008)
from centripetal motion that
F mrω2
and noting that rω 9 1 for a turbomachine system, we can see that the forces
experienced by the material scale indirectly proportional to the characteristic length.
F 9 1
r
Therefore, the materials in micro-turbogenerators experience higher forces than their
macro-size counterparts.
Arnold et al. (2005) studied this effect to determine the maximum RPM values
that these devices were capable of sustaining. For an annular magnet the maximum
radial stress and hoop stress are
σr,max 3 ν
8ρω2 rR2 R1s2 (3.25)
σθ,max 1
4ρω2
p3 νqR22 p1 νqR2
1
(3.26)
91
Given that the ultimate tensile strength of SmCo is 35 MPa (Arnold et al. (2005)),
this sets the maximum achievable RPM before failure. For an outer diameter of
approximately 10 mm, the maximum achievable RPM was reported to be 140,000
RPM. In order to increase the maximum allowable RPM, an circumscribing metal
adaptor was placed around the rotor. A similar example can be seen in figure 3.27.
This adaptor can not only pre-strain the magnet, but also provides additional stiff-
Figure 3.27: An annular magnet with a mounting adaptor, Herrault et al. (2008)
ness. With an adaptor, the speed increased to 325,000 RPM without failure, where
the maximum speed was limited by the RPM of their driving turbine, not the stress
levels of the magnetic rotor.
Another configuration option is to use discrete magnetic pieces that are arranged
around the periphery as shown in figure 3.28. There is a maximum stress level
penalty associated with this configuration as there is no ring stiffness in the magnet.
However, this configuration presents a method to mitigate rotor imbalance through
proper selection and placement of the magnetic pieces around the rotor. In the case
of an annular ring, balancing the rotor presents a major challenge as it is difficult to
address imbalances that are present in the material. The main advantage of discrete
pieces however is that they can be much more easily magnetized.
Another key material property in magnetic systems is permeability, µ. As shown
92
Figure 3.28: Discrete magnet pieces arranged peripherally around the generatorrotor, Herrault (2009)
in the magnetic modeling section, through any material
φ MMF
P MMF
µAl
and therefore permeability in magnetic material is equivalent to conductivity in
electrical circuits, and should be maximized in order to increase the magnetic flux
through the system. An additional important parameter is the saturation flux den-
sity. Looking at a typical B-H curve as shown in figure 3.29, the permeability is
constant, that is there is a linear relationship between H and B, below certain values
of magnetic flux. After this point however, the curve begins to shallow, signifying that
the permeability has decreased substantially. This transition point is termed the sat-
uration density. Typical materials used as back irons are NiFe and Hiperco(FeCoV).
NiFe has a saturation flux density of 0.8T (Laughton and Warne (2003)) and Hiperco
a saturation flux density of 2.4T (Arnold et al. (2006b)). However, NiFe is often used
in micro-systems, because techniques for its sputter deposition are mature.
93
Figure 3.29: A B-H curve for a magnetic material
3.2 Device Performance
3.2.1 Device Geometry and Parameters
Now that the concepts of permanent magnet generators have been presented, a micro-
generator will be designed so that its performance can be estimated. The engine will
have a general arrangement as shown in figure 3.30. The advantage of this design is
Figure 3.30: The arrangement of an uncoupled micro-turbogenerator, Pelekieset al. (2010)
94
that the generator is on a different shaft than the core turbomachinery. This allows
the turbomachinery to operate at its high design RPM, while allowing the generator
to operate at a lower RPM within the material limits of its permanent magnets.
Following the geometry of Herrault et al. (2010), as shown in figure 3.27, the
generator magnets will be designed to have an outer radius of 5 mm and an inner
radius of 2.5 mm. Experimental evidence showed an RPM of 200,000 can be reached
before mechanical failure. Thus, for a factor of safety, a design RPM of 175,000 is
chosen. A micro-turbine was designed to give 10W of shaft power at this speed. The
magnet thickness will be 0.5 mm, the air gap between the magnets and the coils will
be 100 µm thick, and the coils will be be 200 µm thick. The thickness of the back
iron will be determined at a later time in order to avoid saturation. This design will
also have a different key feature than previous designs. As opposed to having one
coil layer, multiple layers will be stacked in series as shown in figure 3.31 in order to
increase system voltage and efficiency.
The effect of this change can easily be seen intuitively by looking at its imple-
mentation in figure 3.32. For every increased layer (n), the output voltage of the
generator will increase at an exponentially decaying rate, and the resistance will in-
crease linearly. Therefore at first, and for a constant generator power output, when
the incremental voltage increase is high per additional layer, this higher voltage ad-
vantage will outweigh the increased resistance. For very high layer counts however,
additional coil layers will be far away from the magnets, and therefore will not in-
crease voltage significantly. The resistance however will still be increasing linearly,
and therefore the performance will be hampered by further increasing the coil layer
count. This was numerically simulated in Matlab for this generator design and the
results can be seen in figure 3.33. The efficiency was shown to begin decreasing at a
layer count of approximately 30. Regardless, the performance gains were not signifi-
cant beyond 5-10 layers and cannot justify the added system complexity. In addition,
95
Figure 3.31: A permanent magnet generator with three coil layers, modifiedfrom Herrault (2009)
Figure 3.32: A representational circuit diagram for a generator with a variablelayer count, n
96
Figure 3.33: Effect of layer count on system parameters
this simple model is not taking into account losses from other mechanisms such as
eddy currents that would further erode performance with higher layers counts. The
thickness of the inter-conducting gap between the coil layers was set to 40µm.
Another change for this design involves the removal of the stator back iron. This
is done in order to avoid significant eddy current and hysteresis losses, at the expense
of less output voltage and therefore power density. An empirical analysis done with
the generator designed and built by Yen et al. (2008a), shows that the effective path
length for the flux in the stator is 736 µm.
The remanence flux density for the magnets was assumed to be 1T as a result of
slightly elevated temperatures. Following the optimization analysis done by Arnold
et al. (2006a), the magnets were patterned with 4 pole pairs, P, and the windings
in the generator have 3 three turns per pole, N. As stated earlier, the generator is
designed with three electrical phases, F. The coils of a single phase can be seen in
figure 3.34.
97
Figure 3.34: A single phase of the 4 pole-pair, 3 turns per pole stator, Herrault(2009)
The widths of the conductors must be approximately estimated so that the phase
resistance can be determined. From geometry, the mean radial width can be deter-
mined from
wr,mean 2πRmean
p2NqFP
which results in a value of 300µm. The widths of the outer and inner segments will
be set to 40% of the mean radial width. The thickness of the radial segments is by
definition equal to the specified thickness of the coils. The inner and outer segments
however must be equal to half of this value in order to accomomdate physical cross-
overs from the different phases. The length of the radial segments is set equal to
the radial length of the magnets, 250 µm. Lastly, the length of the inner and outer
segments can again be determined from geometry
li,o 2πri,o2P
and are equal to 196µm and 393µm respectively. For the power electronics, the use
of active MOSFETs will be assumed with an effective resistance of 0.10 ohms.
With a defined geometry and set system parameters, the performance of the
device can be estimated.
98
3.2.2 Flux and Induced Voltage
In order to more accurately determine system voltage and the generated flux through
the magnets, analytical results from Das et al. (2006) will be used. Das analytically
modeled three phase permanent magnet micro-generators that were very similar to
our design. His model worked by solving 2D Maxwell’s equations through planar
continuum layers as a function of radius (figure 3.35). His results compared very
Figure 3.35: The planar continuum layers used in Das’s analysis, Das et al. (2006)
well with 3D FEA simulations.
Assuming a square wave magnetization profile (cm 0 in figure 3.36), infinite
permeabilities for the back iron and stator iron, and B and H fields independent of
radius, Das’s equation for output voltage reduced to
V0 R2o R2
i
TalTal Tcl Tag
Br
NPω (3.27)
This equation was slightly modified for our analysis because the stator back iron
was removed and an inter-conductor gap was introduced between coil layers. In the
99
Figure 3.36: A magnetization profile for an annular permanent magnet, Das et al.(2006)
denominator, a term for the length of the flux path in the tangential direction was
added, Tfp, as was a term for the inter-conductor gap, Tg. The length of the average
flux path at the mean radius can be determined by geometry.
c1
2 1
2
1
2p2πRo 2πRiqp2P q
(3.28)
An additional 12 is present to be consistent with Das’s analysis, where only half of
the magnetic circuit is analyzed (the other half is symmetrical). For the value of c,
an empirical analysis was done with a similar generator constructed by Yen et al.
(2008a) which also lacked a stator back iron. The value of c came out to 1.5.
The equation for voltage was also modified by another constant, k, as a result
of non-uniform trapezoidal magnetization. Comparing the value predicted by using
this model with the experimentally tested generator by Arnold et al. (2006b) resulted
in a value of 0.72 for k.
The resulting single phase voltage values are shown in table 3.1. Note that this
flux was determined assuming an infinite permeability in the rotor back iron. In order
for this assumption to be approximately true, the back iron must not be saturated.
This is the criteria which sets the back iron thickness. Looking at figure 3.37, and
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Table 3.1: Voltage results
n V°V
°Vwye
1 0.77 0.77 1.33
2 0.68 1.45 2.50
3 0.61 2.06 3.56
4 0.56 2.61 4.52
5 0.51 3.13 5.41
6 0.47 3.59 6.22
7 0.44 4.03 6.98
8 0.41 4.43 7.68
9 0.38 4.82 8.34
10 0.36 5.17 8.96
assuming that Ro ¡¡ Ri such that the tangential flux density is independent of
radius we can determine the following relations. The area of the flux leaving the
permanent magnet is
Am 1
2
πpR2o R2
i q2P
(3.29)
Note that only half of the pole area is used, because each half sees flux traveling in
Figure 3.37: A schematic for approximating the back iron thickness
101
a different tangential direction. Therefore, the total flux entering the back iron is
φ AmBm (3.30)
The area of the back iron normal to the tangential flux is
Abi pRo Riqtbi (3.31)
where tbi is the thickness of the back iron. The flux density through the back iron
with our approximations is therefore
Bbi φ
Abi(3.32)
Combing all of these equations results in
Bbi φ
tbipRo Riq (3.33)
The maximum flux density that the back iron should receive is its saturation flux
density, the value of Bbi therefore should not exceed this. Plugging in this saturation
value into the above equation and re-arranging for the back iron thickness gives an
approximation for how thick the back iron should be to avoid saturation.
tbi φ
BsatpRo Riq (3.34)
Assuming a Hiperco50 (FeCoV) back iron which has a flux saturation density of
2.4T, the thickness of our back iron becomes 161µm.
3.2.3 Generator Coil Resistance
In order to determine power outputs and efficiency, a knowledge of the coil resistance
is required. Looking at figure 3.34 we can determine from geometry the areas and
path lengths required to determine resistance.
102
From the figure, we can see that there are 2 radial conductors per pole pair, 1
outer conductor per pole pair, and 1 inner conductor per pole pair. Recalling that
the equation for resistance is,
R ρl
A(3.35)
and assuming copper conductors (ρ 1.68 108Ohm m), we can determine the
resistance of each radial, inner, and outer segment, and then multiply this by the
number of segments for each type and sum the values to get the resistance per phase.
Doing this results in the following values for resistance per phase per layer.
Rr 15.4 mΩ
Ri 30.2 mΩ
Ro 60.5 mΩ
Rtotal 106 mΩ
Two additional adjustments must be made. The first is to multiple the coil resistance
by a correction factor, k, to account for the difference between the model predictions
and reported values. These differences stem largely from manufacturing defects, vias,
and other geometries that are difficult to model. Applying this same technique to the
generator reported by Herrault et al. (2010), resulted in a correction factor k of 3.
With this adjustment, the phase-phase winding resistance for a single layer becomes
Rphph 0.64 Ω
The second adjustment is to add an effective resistance as a result of the MOSFET’s
in the power electronics. This was assumed to be 100 mΩ.
103
3.2.4 Maximum Power Transfer Capabilities
With knowledge of the resistance and voltage values, the maximum power dissipation
abilities of the generator can be be determined as a function of layer count. This
information has two practical importances. The first is that, as shown in figure 3.24,
the ratio between maximum power to actual load power is strongly related to electric
efficiency. The other reason is that it is necessary to determine the minimum number
of generator layers for the generator to operate at its design RPM. As example,
assuming only electrical losses, for a fixed value of voltage and coil resistance, there
will be a maximum amount of power that the generator can dissipate (by setting
RL = 0). If this amount is less than the input shaft power, then by definition the
torque being applied to the generator from the turbine will be greater than the torque
applied to the generator from the electrical current. As a result, the generator will
accelerate above its design RPM and potentially fail. Thus, there is a minimum layer
count required such that it is within the ability of the generator to utilize or dissipate
the input shaft power and maintain its design RPM.
Recall that for this generator, the phase to phase resistance and power electronics
resistance is
Rphph 0.64Ω Rpe 0.1Ω
Therefore, the total phase-phase resistance will be
Rt nRphph Rpe (3.36)
The maximum power that can be delivered per phase is observed when the load
resistance is set to 0. Therefore, assuming a wye connected 3-phase generator system,
the maximum power that the generator can electrically dissipate is
Pmax 3V 2
wye
Rt
(3.37)
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Table 3.2: Maximum power dissipation
n Vwye pV q Rt pΩq Pmax pW q1 1.33 0.74 7.16
2 2.50 1.37 13.69
3 3.56 2.01 18.94
4 4.52 2.65 23.21
5 5.41 3.28 26.71
6 6.22 3.92 29.61
7 6.98 4.56 32.03
8 7.68 5.19 34.07
9 8.34 5.83 35.79
10 8.96 6.47 37.25
The results as a function of layer count can be seen in table 3.2.
As can be seen, with only a single coil layer, the generator does not have the
ability to dissipate 10W of shaft power. Thus, ignoring non-electrical losses for the
time being, more than 1 coil layer would be required to maintain the generator at
its design RPM.
3.2.5 Loss Mechanisms
Eddy Currents and Hysteresis Losses
In addition to the electrical losses in the resistive elements, there exist other losses
associated with the magnetic character of the generator. One of these are hysteresis
losses. As shown in figure 3.11, magnetic material will follow different paths as it
continually observes a changing H-field. The area encompassed by the hysteresis
curve is proportional to the energy loss, and therefore power loss is proportional to
this area times the frequency. However for our generator, we have decided not to use
a soft-magnetic material as our back iron, and therefore these effects are essentially
zero in the stator. The cost of this decision is higher magnetic reluctance, reduced
flux, lower voltage, and therefore lower electrical efficiency for the same load power.
105
As for the rotor, the magnetic material sees a non-time varying flux, and therefore
does not experience hysteresis.
The other type of magnetic losses present are eddy current losses. The mechanism
behind eddy currents are the same as those responsible for creating a voltage in the
generator coils. Due to Faraday’s law, when a small area in a material is experiencing
a changing magnetic field, voltages will be induced around the material. For non-
conducting material, current cannot flow so no power will be dissipated, but in the
case of conducting material, currents will be present and dissipate power. These
losses can be reduced through the use of laminations as shown in figure 3.38, which
are non-conducting materials that act as barriers to current flow. Again however, we
Figure 3.38: Laminations embedded within a magnetic conducting stator
have chosen to use a non-magnetic non-conducting stator core, and therefore these
losses are avoided. However, these losses are present within the radial segments of the
conductors themselves as shown in figure 3.39 Again, these losses can be minimized
by introducing laminations within the radial conductor segments. An analytical
analyses by Das et al. (2006) showed that these losses can be approximated as
Pprox p2FNClamqσc96TclpRo Riq
πRm
3NClam
3
ω2B2 (3.38)
where Clam refers to the number of laminations in the copper conductors. Our design
has only one lamination (zero non-conducting barriers). The total eddy current losses
within the conductors as a function of layer count are shown in table 3.3.
106
Figure 3.39: Eddy currents within a radial conducting segment with and withoutlaminations, Das et al. (2006)
Table 3.3: Eddy current losses
n Pproxpnq pW q °Pprox pW q
1 0.14 0.14
2 0.12 0.26
3 0.092 0.35
4 0.076 0.43
5 0.064 0.49
6 0.054 0.55
7 0.047 0.59
8 0.041 0.63
9 0.036 0.67
10 0.032 0.70
107
Ohmic Losses and Power Matching
The remaining loss mechanism associated with electromagnetic properties of the
generator is ohmic losses as a result of current running through the coils of the
generator stator and the power electronics. This loss is simply the result of Ohm’s
law and is expressed as
Ps,p.e. Fi2phphRt (3.39)
where F again refers to the number of phases. However, the current through the
coils depends on the load resistance, which has yet to be determined.
In order to determine the load resistance, a power balance is required. That is,
the shaft power input into the generator must equal the load power plus all of power
losses due to the electromagnetic loss mechanisms.
Pshaft Pload Pprox Ps,p.e. (3.40)
This is absolutely essential because it ensures that the magnetic rotor operates at the
design RPM and neither accelerates or decelerates. Every variable in equation 3.40
has been defined with the exception of the load power. Therefore, the resistance of the
load must be set such that this equation holds true. We can re-arrange equation 3.40
and define a new variable, Pcircuit.
Pshaft Pprox Pload Ps,p.e. Pcircuit (3.41)
Pcircuit then represents the amount of power that must be dissipated electric circuit
components of the system as shown in figure 3.40, whether that be in the stator
windings or the load, in order to balance the system. Therefore the required load
resistance can be determined from
Pcircuit FV 2
wye
RL Rs,p.e.
108
Figure 3.40: A representational circuit for power balance
RL FV 2
wye
PcircuitRs,p.e.
and the required current to balance the system will become
Iphph VwyeRs,p.e. Rload
With this, we can now solve equation 3.39 for power dissipation in the coil windings.
The results are shown as a function of layer count in table 3.4. Note that the
Table 3.4: Ohmic losses in the stator windings and power electronics
n RLpnq pΩq ipnq pAq Ps,p.e.pnq pW q1 -0.20 2.47 13.51
2 0.56 1.29 6.90
3 1.95 0.90 4.89
4 3.78 0.70 3.93
5 5.96 0.58 3.37
6 8.38 0.51 3.0
7 10.99 0.45 2.75
8 13.74 0.41 2.56
9 16.59 0.37 2.42
10 19.50 0.35 2.31
resistance value for a single layer is negative, or alternatively that the ohmic losses
109
in the stator and power electronics are greater than the shaft power. This indicates
that generator system has no means, with only a single coil layer, with which to
dissipate or utilize sufficient shaft power to maintain its RPM.
3.2.6 Power Output, Efficiency, and Layer Count
With all of the loss mechanisms having been determined, we can now look at load
power and system efficiency as a function of layer count. With this information,
we can determine which generator design should be physically constructed based on
trade-offs between system performance, complexity, and cost as a result of increased
layer counts.
With generator efficiency defined as,
ηgen PLPshaft
(3.42)
the results are as shown in table 3.5. As before, notice that this generator cannot
Table 3.5: Load power and efficiency as a function of layer count
n PL pW q ηgen %
1 - -
2 2.82 28
3 4.73 47
4 5.62 56
5 6.12 61
6 6.43 64
7 6.63 66
8 6.78 68
9 6.89 69
10 6.97 70
function at this RPM with only a single coil layer.
In addition, as expected from figure 3.33, there are diminishing returns to in-
creasing the layer count. As such, a generator with 6 layers, a power output of 6.4
110
W, and an efficiency of 64% seems to be the best choice with regard to the system
trade offs.
This generator was designed within the physical parameter limits of micro-generators
in the literature. The performance can be improved by venturing outside of this de-
sign space (which may require fabrication innovations). As an example, assume
superior coil fabrication reduced the coil correction factor from 3 to 2, the magnet
thickness was increased from 0.5 mm to 1mm, the magnetization profile was improved
such that its correction factor went from 0.72 to 0.95, 3 laminations were placed in
the radial segments, and non-conductive low-hysteresis high-frequency ferrites were
used as the stator back iron in order to reduce the magnetic reluctance of the flux
path. With such a generator, the optimal efficiency would be observed with only two
coil layers and the generator would have an efficiency of 93%.
3.3 Conclusion
The concepts that govern electric generators were presented. Scaling laws were also
shown that demonstrated why permanent magnet generators offer superior perfor-
mance at small size when compared to electromagnetic induction machines.
The principles of permanent magnet performance were explained as were the
concepts of magnetic circuits. A 1 cm diameter permanent magnet generator was
designed and the performance values were determined. The device demonstrated an
output power of 6.4 W with an efficiency of 64% and a coil layer count of 6. With
specific improvements, the device demonstrated 9.3 W at an effeciency of 93% and
a coil layer count of only 2.
The performance of micro-turbogenerators using these two generators can be
compared against batteries in terms of energy density. Mattingly et al. (2006) gives
111
the performance of an ideally recuperated Brayton cycle as
ηtherm 1 PRk1k
T4T2
Assuming a compressor efficiency of 50%, a core turbine efficiency of 50%, and a
power turbine efficiency of 50% (all obtained using high speed radial turbomachin-
ery), a pressure ratio of 1.5, and a turbine inlet temperature of 1000K, results in
an thermal efficiency of 8.28%. As shown in figure 3.41, with generator efficiencies
of 64% and 93% from our low-end and high-end designs, the energy density of these
engines would be 3x and 4.5x greater respectively than lithium-ion batteries
Figure 3.41: Energy density of our designed micro-turbogenerators relative to Li-ion batteries
112
4
Experimental Results of an Ejector DrivenMicro-turbogenerator
4.1 Introduction
This section demonstrates an alternative thermodynamic cycle that was used to
overcome the challenges of the Brayton cycle at the micro scale. In this cycle, the
engine is designed around static pumping devices, an injector and an ejector [Gardner
et al. (2010a)]. Both are based on the same fundamental principle: a high momentum
motive fluid is mixed with a low momentum suction fluid, resulting in a discharge
fluid with less overall momentum and thus a higher pressure. The ejector is used
primarily to create a pressure gradient across the turbine, while the injector pumps
liquid into a high pressure boiler. The benefit of these components lies in their static
and turbo-machine independent operation. Unlike a turbo-compressor, the ejector is
uncoupled from the turbine and thus provides a pressure gradient across the turbine
regardless of the turbine’s rotor speed so long as heat is applied to the boiler.
This eliminates many problems associated with the startup phase and guarantees
that the cycle will close at any operating efficiency. In addition, because there are no
113
Figure 4.1: Control volume of the ejector mixing region
moving parts associated with these components, they can be more easily fabricated
as their manufacturing tolerances are not as critical as those associated with rotating
micro-turbomachinery components. Lastly, the suction provided by the ejector can
be utilized to prime the hydrostatic gas bearings, removing the need for an external
pressure source.
This section will discuss both the thermodynamic model concerning this type
of cycle and the preliminary experimental results of an ejector-driven micro-turbo-
generator.
4.2 Thermodynamic Cycle
The proposed cycle is derived from a steam locomotive cycle and an after-burning
Brayton cycle (Fig. 2). Combustion takes place downstream of the turbine, and
the generated heat is split between preheating the turbine inlet air with the use of
a recuperator and vaporizing ethanol in the boiler. Note that for this cycle ethanol
114
Figure 4.2: Schematic of an after-burning thermodynamic cycle driven by an in-jector
is initially used both as the fuel and as the motive vapor for both the injector and
ejector. The power from the turbine can be estimated from the incompressible flow
assumption, the ideal gas law, and by recalling that the total pressure at the turbine
inlet is roughly equal to ambient pressure.
9Wt
9ms
ηt∆Ptρ
ηtPamb Pt,s
PambRairTt4 (4.1)
The pressure difference between the ambient and total suction pressures can be
determined by the conservation of mass and momentum for the control volume shown
in Fig. 1, and assuming isentropic expansion in the diffuser such that the exit
dynamic head is approximately zero. The result is
Pamb Pt,s12ρu2
m
σ
2 σ α2σ3
p1 σq2 2ασ2
p1 σq 2ασ
p1 σq
(4.2)
115
where σ is the area ratio and α is the ejector entrainment ratio.
σ AmAd
α 9ms
9mm
The system rejects heat in the ejector discharge flow and by the cooling and conden-
sation of the ethanol vapor. However, the heat rejected in condensation typically far
exceeds the heat rejected from the cooling of the exhaust gases. The overall thermal
cycle efficiency can be approximated as
η 9Wt
9Wt 9mmhfg(4.3)
In addition, by further assuming the turbine power is significantly less than the
rejected heat we can re-approximate the cycle efficiency as
η 9Wt
9mmhfg(4.4)
Combining equations (4.1) - (4.4) we obtain
η ηtRairTt4ασ
2 σ α2σ3
p1σq2 2ασ2
p1σq 2ασ
p1σq
12ρu2
m
hfgPamb(4.5)
where σ, um, and Tt4 are design parameters (um is a function of boiler pressure).
Picking an a reasonable area ratio of σ = 0.5, the highest cycle efficiencies will
occur for an entrainment ratio of 0.53. Using these values and assuming a turbine
efficiency of 50%, we can approximate the overall efficiencies for different parameters
as shown in Table 4.1.
4.3 Experiment and Results
An experiment was conducted to demonstrate that an ejector powered by ethanol
vapor could create a pressure gradient across a turbine and drive its attached micro-
116
Table 4.1: Efficiency approximations
Case 1 Case 2
PboilerPs
3 30
Tt4 500 K 1600 K
η 1.13% 5.36%
generator to deliver electrical power. The turbine was originally designed and micro-
machined as a radial flow impulse turbine with an NGV outer-diameter of 10 mm
and blade heights of 250 µm (Fig. 4.3). However, due to the configuration and choice
Figure 4.3: The original micro-turbine design bonded to the rotor of a permanentmagnet generator with protruding leads
of materials, we believe eddy current losses in the surrounding material prevented
the turbine from operating on design. A new turbine with a rotor outer-diameter
of 11 mm and blade heights of 750 µm was 3D printed out of ABS plastic for the
final experiments. This allowed the turbine to reach a speed closer to its design
RPM. The ejector was micro-machined with a throat diameter of 719 µm, an area
ratio of 1 : 8, and was driven by ethanol vapor from a conventionally-sized boiler.
The turbine rotor was bonded to a 3-phase Faulhaber 1202-H-006-BH permanent
117
magnet DC motor with the outputs being rectified to DC with the use of 1N5817
Schottky diodes (Fig. 4.4 ). The rectified DC source was then connected across a
Figure 4.4: 3D printed turbo-generator connected to power electronics (boiler andejector not shown)
variable resistor that was adjusted until the maximum power output was obtained.
The conditions at the optimal operating point are shown in Table 4.2. Power from
Table 4.2: Experimental results
Property Units Value
Rotor Speed RPM 27,360
Turbine Pressure Ratio - 1.05
Boiler Pressure atm 15
VDC V 1.49
Turbine Inlet Temp. K 293
Power mW 7.5
the engine was also used to light a row of LEDs (Fig. 4.5)
118
Figure 4.5: LEDs powered by the micro-turbine generator
Note that maximum power does not take place by setting the load resistance
to the equivalent stator resistance as prescribed in the maximum power transfer
theorem. The reason for this is that the maximum power transfer theorem assumes
a fixed voltage source, where as for a turbo-generator the voltage is coupled to the
rotor speed. Therefore, the load resistance must be set such that the turbine can
reach its design speed (Fig. 4.6).
4.4 Conclusion
This section has demonstrated that, with no moving parts, an ejector can produce
a pressure gradient to drive a micro-turbine and generate power. An advantage of
this operating mode is that, unlike a standard Brayton cycle, there is no minimum
required efficiency for the cycle to close and the engine to function. In addition, the
ejector provides a means of creating the pressure gradient required by hydrostatic
gas journal bearings. This will allow the bearings to operate hydrostatically at low
speeds until the RPM increases, thereby allowing hydrodynamic bearing operation.
The manufacturing tolerances of these static pumping devices can also be much less
stringent than those of micro-turbomachinery.
119
Figure 4.6: Turbine shaft power and load power as a function of rotor speed
The thermodynamic cycle was analyzed with both compressible and incompress-
ible flow assumptions, and a basic method of estimating the cycle thermal efficiency
was presented. Experiments were conducted to demonstrate the viability of a power
cycle designed around an ejector-driven micro-turbine.
120
5
Conclusion and Future Work
5.1 Summary and Conclusions
The basic operating principles of micro heat engines were presented. They were
shown to represent a good balance between power density and energy density, and
this characteristic, in comparison to fuel cells and batteries, was shown to make them
good candidates for mobile applications that require both decent power densities and
energy densities.
The basic scaling laws that govern both turbomachinery and permanent magnet
generator power density were presented. It was shown that for turbomachinery,
the power density scales indirectly proportional with the characteristic length of the
system. For this reason, their power densities are observed to increase as they are
reduced in size. For permanent magnet generators, their power density was at first
determined to be scale independent. However, the heat dissipation capability of
the generator windings was shown to increase with reduced size as a result of the
increased ratio of surface area to heat generation. This should allow for more current
density and therefore power density at small sizes at the cost of efficiency or allow for
121
the replacement of cooling elements with power producing elements at small sizes,
thereby increasing power density with a constant efficiency.
Multiple challenges that affect micro-turbogenerators were presented. Of prime
importance, were the efficiency of micro-turbomachinery and the power transfer ca-
pabilities of the generator. The efficiency of micro-turbomachinery was shown to be
important not only for reasons related to energy and power density, but also because
for a Brayton cycle, component efficiencies must meet a specific threshold in order
for the cycle to close and create net power. The power transfer capabilities of the
generator were determined to be important because for a generator converting its
input shaft power to electric power, if the power transfer capabilities do not far ex-
ceed the input shaft power, either the conversion efficiency will be low or the device
will accelerate beyond its design RPM and potentially fail.
The basic operating principles of turbomachinery were developed, with the dif-
ferent components of energy transfer being specifically delineated. The ratio of these
energy transfer mechanisms was shown to relate to turbine reaction and device ef-
ficiency. Loss models were developed to quantify entropy creation from tip leakage,
trailing edge mixing, and viscous boundary layers over the surface of the blades.
The total entropy creation was then related to lost work and turbine efficiency. The
results showed the efficiency and power density of various turbine configurations over
a large range of sizes. For the configurations analyzed, the high speed single stage
reaction turbine showed the best performance. The power density was also shown
to scale linearly as expected over a large range of diameters. However, at very small
scales, the effects of viscous losses superseded the benefits from the scaling laws,
resulting in a peak power density. The single stage reaction turbine was shown to
possess the highest peak power density. The practice of multi-staging was shown to
not be as beneficial at small scales as it is at large scales, because the gains associ-
ated with increased kinetic energy absorption are largely offset by the combination
122
of high viscous losses associated with small scale turbomachinery and the increased
wetted area. The conclusion was that for micro-turbomachinery, high speed, high
reaction, single stage radial designs are most effective due to favorable pressure gra-
dients, low wetted area, and large portion of energy transfer taking place through
lossless centrifugal pressure fields.
The operating principles of generators and power electronics were then presented.
The scaling laws for both permanent magnet generators and electro-magnetic induc-
tion machines were developed and showed that permanent magnet generators should
scale down more favorably than electro-magnetic machines. The basic concepts of
permanent magnet operation were explained, and this was tied together with mag-
netic circuit theory to show the flux generating capabilities of permanent magnets
in planar micro-generators. The flux generation of the magnets was related to volt-
age creation through Faraday’s law, which allowed us to model the generator as
an alternating current source with a fixed internal resistance in an electric circuit.
An analysis was done to show the relationship between efficiency, generator voltage,
internal resistance, and load power.
Models were presented for planar micro-generators to determine output voltage,
internal resistance, electrical losses, and electromagnetic losses based on geometry
and key design parameters. A 3 phase multi-layer permanent magnet generator
operating at 175,000 RPM with an outer diameter of 1 cm was designed and an
efficiency of 65% was shown. A similar device was designed with improved features
that would require fabrication innovations and demonstrated an efficiency of 93%.
Lastly, an ejector driven turbogenerator was designed, built, and tested. A basic
thermodynamic cycle was presented in order to estimate system efficiency as a func-
tion of design parameters. Experiments were conducted showing a power output of
7.5 mW at 27,360 RPM.
123
5.2 Future Work
For future work, geometrically similar turbines should be tested in thermodynami-
cally identical operating conditions to determine the validity of the models presented
here. Alternatively, high fidelity CFD computational studies could be conducted. A
similar loss study model should be done for micro-compressors taking into account
different behavior and flow separation as a result of their unfavorable pressure gra-
dients. The basic loss models however should be similar.
In addition, a multi-layer generator with the dimensions and parameters given
here should be fabricated and tested in order to validate or repudiate the performance
and loss models. Alternatively, 3D FEA studies could be conducted.
After these tasks are completed, an externally supported turbogenerator system
with an integrated compressor, combustion chamber, turbine, generator, and gas
bearing system should be designed, fabricated, and tested. With the lessons learned
from this process, a stand-alone self sustaining device that would not require external
support for bearings, fuel injection, etc., should be designed, fabricated, and run as
the worlds first fully functional micro-heat engine.
124
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